by Kiyoshi Wesley Masui - University of Toronto T …...intensity mapping as a sensitive and e cient...
Transcript of by Kiyoshi Wesley Masui - University of Toronto T …...intensity mapping as a sensitive and e cient...
Advancing precision cosmology with 21 cm intensity mapping
by
Kiyoshi Wesley Masui
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of PhysicsUniversity of Toronto
c© Copyright 2013 by Kiyoshi Wesley Masui
Abstract
Advancing precision cosmology with 21 cm intensity mapping
Kiyoshi Wesley Masui
Doctor of Philosophy
Graduate Department of Physics
University of Toronto
2013
In this thesis we make progress toward establishing the observational method of 21 cm
intensity mapping as a sensitive and efficient method for mapping the large-scale struc-
ture of the Universe. In Part I we undertake theoretical studies to better understand the
potential of intensity mapping. This includes forecasting the ability of intensity mapping
experiments to constrain alternative explanations to dark energy for the Universe’s accel-
erated expansion. We also consider how 21 cm observations of the neutral gas in the early
Universe (after recombination but before reionization) could be used to detect primordial
gravity waves, thus providing a window into cosmological inflation. Finally we show that
scientifically interesting measurements could in principle be performed using intensity
mapping in the near term, using existing telescopes in pilot surveys or prototypes for
larger dedicated surveys.
Part II describes observational efforts to perform some of the first measurements
using 21 cm intensity mapping. We develop a general data analysis pipeline for analyzing
intensity mapping data from single dish radio telescopes. We then apply the pipeline to
observations using the Green Bank Telescope. By cross-correlating the intensity mapping
survey with a traditional galaxy redshift survey we put a lower bound on the amplitude of
the 21 cm signal. The auto-correlation provides an upper bound on the signal amplitude
and we thus constrain the signal from both above and below. This pilot survey represents
a pioneering effort in establishing 21 cm intensity mapping as a probe of the Universe.
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Dedication
For Kaito’s generation,
that you may reach a better understanding of nature.
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Acknowledgements
First and foremost, I want to thank my thesis advisor Ue-Li Pen. Early on in the program
I was told that no single decision I made in my career would be as important as the
match between student and advisor. I think we managed to hit the sweet spot between
me having the freedom to pursue creative research and you pushing me to accomplish as
much as possible. I have learnt so much from you.
I would also like to acknowledge the efforts of all my collaborators, without whom this
thesis would not have been possible. A special thank you to the two post-docs that did
all the work: Eric Switzer and Pat McDonald. I am also indebted to the many faculty,
post-docs, staff, and grad students at CITA for the countless bits of help, tidbits of
advice, and allowing me to bounce ideas off of you relentlessly. This is especially true of
Richard Shaw, as well as my office mates who have contributed enumerable snippets of
code.
Beyond the professional, I would like to thank the many friends and family who are
responsible for me growing to love Toronto. Life here has been wonderful because of you.
Thank you to my parents and brother for making me who I am.
Thank you Maggie, for being my partner through all of this.
I can’t wait for what adventures may come.
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Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Cosmology and large-scale structure . . . . . . . . . . . . . . . . . 1
1.1.2 Redshift surveys using the 21 cm line . . . . . . . . . . . . . . . . 4
1.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The background expansion . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . 15
I The potential of 21 cm cosmology 17
2 Constraining modified gravity 18
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Modified Gravity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 f(R) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 DGP Braneworld . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Observational Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Baryonic acoustic oscillation expansion history test . . . . . . . . 24
2.4.2 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.3 External Priors from Planck . . . . . . . . . . . . . . . . . . . . . 29
2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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3 Detecting primordial gravity waves 39
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Tests of inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Statistical detection in LSS . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Forecasts for near term experiments 49
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Redshift Space Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Baryon Acoustic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
II Pioneering 21 cm cosmology 62
5 Data analysis pipeline 63
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Time ordered data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.1 Pipeline design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 Radio frequency interference . . . . . . . . . . . . . . . . . . . . . 67
5.2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Map-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.2 Noise model and estimation . . . . . . . . . . . . . . . . . . . . . 76
5.3.3 An efficient time domain map-maker . . . . . . . . . . . . . . . . 81
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 21 cm cross-correlation with an optical galaxy survey 87
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.1 From data to maps . . . . . . . . . . . . . . . . . . . . . . . . . . 90
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6.4.2 From maps to power spectra . . . . . . . . . . . . . . . . . . . . . 92
6.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 21 cm auto-correlation 98
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Observations and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.1 Foreground Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3.2 Instrumental Systematics . . . . . . . . . . . . . . . . . . . . . . . 102
7.3.3 Power Spectrum Estimation . . . . . . . . . . . . . . . . . . . . . 103
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 107
8 Conclusions and outlook 111
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography 115
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List of Tables
2.1 Projected constraints on f(R) models for various combinations of obser-
vational techniques, for a 200 m telescope. Constraints are the 95% confi-
dence level upper limits and include forecasts for Planck. The non linear
results (column marked NL WL) are for the HS model with n = 1. Results
that make use of weak lensing with constraints above 10−3 are only order
of magnitude accurate. The linear regime is taken to be ` < 140, with the
nonlinear constraints extending up to ` = 600. . . . . . . . . . . . . . . 31
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List of Figures
1.1 NASA/WMAP Science Team depiction of the evolution of the Universe
in the Λ-CDM model. The creation of the cosmic microwave background
is shown as well as the era of structure formation. Inflation and the ac-
celerated expansion is depicted on the left and right sides of the figure
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 From Springel et al. [2005]. Large-scale structure at redshift z = 0, as
seen in the Millennium Simulation. The colour map represents density
with the brightest colours representing the densest regions. The bright
spot near the middle of the image is a galaxy super cluster, containing of
order 10 000 galaxies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Large-scale structure in the VIPERS survey [Guzzo et al., 2013]. This
figure contains roughly half of the 55 000 galaxies in the total survey. The
3D position of each galaxy is represented by a black dot. The figure is then
collapsed along one of the angular dimensions (which has a thickness of
≈ 1). Large-scale structure is clearly visible, especially at z ≈ 0.7 where
the mean galaxy density is highest. . . . . . . . . . . . . . . . . . . . . . 4
1.4 Proper time t and conformal time η as a function of redshift z. The
magnitude of the proper time can be interpreted as the distance trav-
elled by a photon observed today and emitted by a source at redshift
z, while the magnitude of η is the current/comoving distance to that
source. These differ because the source recedes with the Hubble flow as
the photon is in transit. t and η are given in terms of the Hubble time
1/H0 = 14.6 Gyr or alternately the Hubble distance c/H0 = 3.00 Gpc/h
where h = H0/(100 km/s/Mpc) = 0.671. . . . . . . . . . . . . . . . . . . 9
2.1 The Weak lensing convergence power spectra for ΛCDM and the HS f(R)
model with n = 1 and fR0 = 10−4. Galaxy distribution function is flat
between z = 1 and z = 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . 29
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2.2 Projected constraints on the HS f(R) model with n = 1 using several
combinations of observational techniques, for a 200 m telescope. All curves
include forecasts for Planck. Allowed parameter values are shown in the
fR0 − h plane at the 68.3%, and 95.4% confidence level. Results are not
shown for “WL” which were calculated much less accurately (see text). . 32
2.3 Same as Figure 2.2 but for a 100 m cylindrical telescope. . . . . . . . . . 33
2.4 Ratio of the coordinate dA(z) (top) and the Hubble parameter H(z) (bot-
tom) as predicted by the best fit DGP model to the fiducial model. Error
bars are from 21 cm BAO predictions. Fit includes BAO data available
from the 200 m telescope and CMB priors on θs and ωm. . . . . . . . . . 34
2.5 Weak lensing spectra in for DGP and a smooth dark energy model with
the same expansion history. DGP parameters are h = 0.665, ωm = 0.116,
ωk = 0 and ωrc = 0.06. Errorbars represent expected accuracy of the
200 m telescope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Primordial tensor power spectrum obeying the consistency relation for r =
0.1. The solid line is the tensor power spectrum. Error bars represent the
reconstruction uncertainty on the binned power spectrum for a noiseless
experiment, surveying 200 (Gpc/h)3 and resolving scalar modes down to
kmax = 168h/Mpc. The dashed, nearly vertical, line is the reconstruction
noise power. The non-zero slope of the solid line is the deviation from
scale-free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Baryon acoustic oscillations averaged over all directions. To show the
BAO we plot the ratio of the full matter power spectrum to the wiggle-
free power spectrum of Eisenstein and Hu [1998]. The error bars represent
projections of the sensitivity possible with 4000 hours observing time on
GBT at 0.54 < z < 1.09. . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Ability of GBT to measure the BAO and redshift space distortions as
a function of survey area at fixed observing time. Presented survey is
between z = 0.54 and z = 1.09 and observing time is 1440 hours. A factor
of 10 has been removed from the Aw curve. . . . . . . . . . . . . . . . . . 56
4.3 Roughly optimized survey area as a function of telescope time on GBT.
Redshift range is between z = 0.54 and z = 1.09. . . . . . . . . . . . . . . 56
x
4.4 Forecasts for fractional error on redshift space distortion and baryon acous-
tic oscillation parameters for intensity mapping surveys on the Green Bank
Telescope (GBT). Frequency bins are approximately 200 MHz wide and
correspond to available GBT receivers. Uncertainties on D should not be
trusted unless the uncertainty on Aw is less than 50% (see text). . . . . . 58
4.5 Forecasts for fractional error on redshift space distortion and baryon acous-
tic oscillation parameters for intensity mapping surveys on a prototype
cylindrical telescope. Frequency bins are 200 MHz wide corresponding to
the capacity of the correlators which will likely be available. These result
also apply to the aperture telescope but with the observing time reduced
by a factor of 14. Uncertainties on D should not be trusted unless the
uncertainty on Aw is less than 50% (see text). Observing time does not
account for lost time due to foreground obstruction. . . . . . . . . . . . . 59
5.1 Data before (left) and after (right) RFI flagging. Colour scale represents
perturbations in the power, P/〈P 〉t−1. Frequency axis has been rebinned
from 4096 bins to 256 bins after flagging, which fills in many of the gaps
in the data left by the flagging. Any remaining gaps are assigned a value
of 0 for plotting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Noise power spectrum, averaged over all spectral channels (δνν′Pνν′ω/cν),
as measured in the GBT 800 MHz receiver. Units of the vertical axis are
normalized such that pure thermal noise would be a horizontal line at
unity. In individual time samples are 0.131 s long and spectral bins are
3.12 MHz wide. The telescope is pointing at the north celestial pole to
minimize changes in the sky temperature. Descending the various coloured
lines corresponds to removing additional noise eigenmodes, Vνq, from the
noise power spectrum. It is seen that after removing 7 of the 64 possible
modes the noise is significantly reduced and is approaching the thermal
value on all time scales. The modes removed from each subsequent line
are shown in Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 The modes Vνq removed from the noise power spectra to produce the curves
in Figure 5.2. Each mode is offset vertically for clarity, with mode number
increasing from bottom to top. The nth mode in this figure is the dominant
remaining mode in the nth curve in Figure 5.2. . . . . . . . . . . . . . . 80
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6.1 Maps of the GBT 15 hr field at approximately the band-center. The purple
circle is the FWHM of the GBT beam, and the color range saturates in
some places in each map. Top: The raw map as produced by the map-
maker. It is dominated by synchrotron emission from both extragalactic
point sources and smoother emission from the galaxy. Bottom: The raw
map with 20 foreground modes removed per line of sight relative to 256
spectral bins, as described in Sec. 6.4.2. The map edges have visibly
higher noise or missing data due to the sparsity of scanning coverage. The
cleaned map is dominated by thermal noise, and we have convolved by
GBT’s beam shape to bring out the noise on relevant scales. . . . . . . . 96
6.2 Cross-power between the 15 hr and 1 hr GBT fields and WiggleZ. Negative
points are shown with reversed sign and a thin line. The solid line is the
mean of simulations based on the empirical-NL model of Blake et al. [2011]
processed by the same pipeline. . . . . . . . . . . . . . . . . . . . . . . . 97
7.1 Temperature scales in our 21 cm intensity mapping survey. The top curve
is the power spectrum of the input 15 hr field with no cleaning applied (the
1 hr field is similar). Throughout, the 15 hr field results are green and the
1 hr field results are blue. The dotted and dash-dotted lines show thermal
noise in the maps. The power spectra avoid noise bias by crossing two
maps made with separate datasets. Nevertheless, thermal noise limits the
fidelity with which the foreground modes can be estimated and removed.
The points below show the power spectrum of the 15 hr and 1 hr fields
after the foreground cleaning described in Sec. 7.3.1. Negative values are
shown with thin lines and hollow markers. Any residual foregrounds will
additively bias the auto-power. The red dashed line shows the 21 cm signal
expected from the amplitude of the cross-power with the WiggleZ survey
(for r = 1) and based on simulations processed by the same pipeline. . . 106
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7.2 Comparison with the thermal noise limit. The dark and light shaded re-
gions are the 68% and 95% confidence intervals of the measured 21 cm fluc-
tuation power. The dashed line shows the expected 21 cm signal implied
by the WiggleZ cross-correlation if r = 1. The solid line represents the
best upper 95% confidence level we could achieve given our error bars, in
the absence of foreground contamination. Note that the auto-correlation
measurements, which constrain the signal from above, are uncorrelated
between k bins, while a single global fit to the cross-power (in Masui et al.
[2013]) is used to constrain the signal from below. Confidence intervals do
not include the systematic calibration uncertainty, which is 18% in this
space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3 The posterior distribution for the parameter ΩHIbHI coming from the Wig-
gleZ cross-power spectrum, 15 hr field and 1 hr field auto-powers, as well
as the joint likelihood from all three datasets. The individual distribu-
tions from the cross-power and auto-powers are dependent on the prior
on ΩHIbHI while the combined distribution is essentially insensitive. The
distributions do not include the systematic calibration uncertainty of 9%. 109
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Chapter 1
Introduction
The question of the origin of the Universe is arguably as ancient as all of human civiliza-
tion. It is only within the last hundred years that we have begun to form an accurate
understanding of the Universe on large scales and only within the last twenty years that
we have been able to make precise statements about the Universe.
It is thought that using the 21 cm line from neutral hydrogen to map the large-
scale structure of the Universe is a promising technique for making precise cosmological
measurements [Chang et al., 2008]. Such measurements would allow for the detection of
subtle effects, ultimately leading to a better understanding of the Universe and continuing
humanity’s push to answer fundamental questions about its origin. This thesis represents
a significant contribution to the pioneering of 21 cm cosmology, both theoretically and
observationally.
1.1 Background
Here we give a brief overview of the field of cosmology, how the field has evolved, and why
the 21 cm line is anticipated to be a powerful probe of the Universe and an important
part of cosmology’s future.
Some of the text in this section has been adapted from research proposals and other
similar unpublished documents.
1.1.1 Cosmology and large-scale structure
Observationally, physical cosmology has seen tremendous progress in the past two decades.
It has matured from an imprecise field, in which a dearth of observations left the Universe
poorly understood, to one in which a richness of data is allowing for precise measurements.
1
Chapter 1. Introduction 2
Figure 1.1: NASA/WMAP Science Team depiction of the evolution of the Universe inthe Λ-CDM model. The creation of the cosmic microwave background is shown as wellas the era of structure formation. Inflation and the accelerated expansion is depicted onthe left and right sides of the figure respectively.
We now have a standard cosmological model, the Λ-cold dark matter model (Λ-CDM),
depicted in Figure 1.1, which explains all observations of the cosmic microwave back-
ground (CMB), large-scale structure (LSS), and Universe expansion rate. Nevertheless,
several fundamental mysteries remain unexplained within Λ-CDM. One of the most com-
pelling is the observed accelerated expansion of the Universe, the so called dark energy
problem, initially discovered by observations of distant type-1a super-novae [Riess et al.,
1998, Perlmutter et al., 1999]. Within Λ-CDM, this is parameterized by a cosmological
constant, however, its exact nature remains a mystery. Equally intriguing is the question
of what set the initial conditions for the Universe’s evolution. The leading theory is cos-
mological inflation [Guth, 1981], however again the physical nature of inflation remains
a mystery.
The rapid advancement of cosmology has been driven by observations of the CMB,
lead by the space observatories: the Cosmic Background Explorer (COBE) [Mather
et al., 1990, Smoot et al., 1992], followed by the Wilkinson Microwave Anisotropy Probe
(WMAP) [Bennett et al., 2012, Hinshaw et al., 2012], and finally the ongoing Planck
mission [Planck Collaboration et al., 2013a]. While this program has been tremendously
Chapter 1. Introduction 3
Figure 1.2: From Springel et al. [2005]. Large-scale structure at redshift z = 0, as seenin the Millennium Simulation. The colour map represents density with the brightestcolours representing the densest regions. The bright spot near the middle of the imageis a galaxy super cluster, containing of order 10 000 galaxies.
successful, the bulk of the information that can be extracted from the CMB has already
been retrieved. While there remains information to be extracted from the CMB polar-
ization signal, as well as various secondary effects such as weak lensing, new probes of
the Universe will become increasingly important.
The field of large-scale structure studies how the Universe’s matter is distributed in
three dimensional space. Figure 1.2 shows a map of the LSS as simulated in a large n-
body simulation. LSS is a potentially powerful probe of the Universe because it is a three
dimensional field—in contrast to the CMB which is two dimensional—yielding far more
independent observables. There are of order 1018 observable LSS modes in the Universe
[Pen, 2004], while the primordial CMB has at most of order 108 observable modes.
The large-scale structure is sensitive to a large variety of physical processes through-
out the evolution of the Universe, and can thus be used to learn about these processes.
The LSS grew, through gravitational collapse instability, from tiny perturbations that
existed in the first instants of the Universe’s evolution. The structure in the late Uni-
verse maintains information about the precise statistics of these initial perturbations. If
inflation is responsible for generating these perturbations, then the LSS can be used to
Chapter 1. Introduction 4
Figure 1.3: Large-scale structure in the VIPERS survey [Guzzo et al., 2013]. Thisfigure contains roughly half of the 55 000 galaxies in the total survey. The 3D positionof each galaxy is represented by a black dot. The figure is then collapsed along one ofthe angular dimensions (which has a thickness of ≈ 1). Large-scale structure is clearlyvisible, especially at z ≈ 0.7 where the mean galaxy density is highest.
study it.
To study the expansion history of the Universe, and thus gain insight into the anoma-
lous acceleration, the baryon acoustic oscillations (BAO) can be used. The BAO result
from sound waves that propagated in the early Universe. They imprint a characteristic
scale in the statistics of the LSS and thus act as a standard ruler on the sky, which
expands with the expansion of the Universe. Thus a precise measurement of this scale
as a function of redshift can be used to measure the rate of expansion.
1.1.2 Redshift surveys using the 21 cm line
Traditionally, large-scale structure has been measured using galaxy surveys. These in-
volve painstakingly measuring the 3D location of many galaxies. Redshift, measured by
performing spectroscopy on each galaxy to identify the shift in spectral lines, is used as
a proxy for radial distance, giving these surveys their colloquial names of redshift sur-
veys. The galaxies are then catalogued and accumulated into three dimensional maps
that reveal large-scale structure. This is illustrated in Figure 1.3. Because the individual
galaxies are much smaller than the structures being studied, they are hard to detect and
a very sensitive telescope must be used. Over two million galaxies have been surveyed in
this way [Eisenstein et al., 2011], but despite this massive effort, only a small fraction of
the observable Universe has been mapped to date.
Recently it has been proposed that the large-scale structure could be mapped much
more efficiently using the 21 cm line from the spin flip transition in neutral hydrogen
Chapter 1. Introduction 5
[Barnes et al., 2001, Loeb and Zaldarriaga, 2004], which lies in the radio part of the
electromagnetic spectrum. This has several advantages over optical redshift surveys. The
21 cm line is by far the brightest line in this part of the spectrum, leaving little chance
for line confusion. As such, each observing frequency corresponds unambiguously with
a redshift. This eliminates the need to detect individual galaxies at high significance.
The 21 cm brightness from each region of space is taken to be a proxy for the total
amount of hydrogen in that region, which in turn is assumed to be a biased tracer for
the total density. This procedure is referred to as 21 cm intensity mapping [Chang et al.,
2008, Loeb and Wyithe, 2008, Ansari et al., 2012a, Mao et al., 2008, Seo et al., 2010,
Mao, 2012]. The 21 cm signal is, in principle, measurable at all redshifts up to z ∼ 50,
potentially providing a probe of the early Universe. This is in contrast to optical surveys
which are difficult in certain redshift ranges and are impossible above a redshift of z ∼ 6
due to a lack of sources.
The use of the 21 cm line for cosmology is not without disadvantages. The most
concerning is the presence of bright foregrounds. In particular, synchrotron emission
from both galactic sources and extra-galactic sources is of order 104 times brighter than
the signal from neutral hydrogen. However, all known foreground contaminants are
expected to be spectrally smooth. The 21 cm signal on the other hand is a spectral line
and even when combining the line emission over all redshifts, the signal from hydrogen
is modulated by the large-scale structure. It is thus expected that the foregrounds can
be separated from the signal, or at the very least that foregrounds will contaminate only
a small number of the signal modes along the line of sight. In practise, instrumental
effects; such as imperfect spectral calibration, frequency dependence of the instrumental
beam, and contamination of the unpolarized channel by polarized emission; all make
foreground subtraction more difficult than might be initially expected. As such 21 cm
redshift surveys are very challenging.
The first detection of the 21 cm signal from large-scale structure above z = 0.1 was
presented in Chang et al. [2010]. There, data from the Green Bank Telescope (GBT) in
West Virginia was cross-correlated against a traditional optical galaxy survey at a redshift
of z ≈ 0.8. This confirmed the existence of the 21 cm signal from large-scale structure,
but did not require the perfect removal of foregrounds, since residual contamination does
not correlate with the optical survey.
Intensity mapping is both competitive with, and complementary to, traditional galaxy
surveys. Technological advances in radio frequency communications and digital signal
processing now make it possible to design intensity mapping systems that are capable of
surveying large volumes of the Universe at relatively modest cost. This makes intensity
Chapter 1. Introduction 6
mapping especially attractive for measuring large-scale features in the Universe, such as
the BAO. In cases where intensity mapping surveys overlap with galaxy surveys, there are
several synergies that would benefit both surveys. The most basic of these is a cross-check
of results, where experiments with independent systematic errors validate one another.
Going beyond this, the fact that galaxies and neutral hydrogen are different tracers of
the same cosmic structure, allows for the disentanglement of key uncertainties in both
surveys, greatly improving the precision of some measurements [McDonald and Seljak,
2009].
1.2 Formalism
Here we review some of the basic cosmological theory and concepts that will be required
for understanding the following chapters. This formalism is covered in more detail in
Hartle [2003], Dodelson [2003] and Liddle and Lyth [2000].
1.2.1 The background expansion
To zeroth order, the Universe is assumed to be homogeneous and isotropic, which on
very large scales agrees well with observations. The metric for a homogeneous and
isotropic Universe is the Friedmann-Lemaıtre-Robinson-Walker (FLRW) metric, which
can be written as
ds2 = − dt2 + a(t)2[
dr2 + Sk(r)2( dθ2 + sin2 θ dφ2)
], (1.1)
with
Sk(r) =
sin (√kr)√k
if k > 0,
r if k = 0,
sinh (√−kr)√−k if k < 0.
(1.2)
Here, we use units where the speed of light, c, is unity. By assumption of homogeneity
and isotropy, all matter-energy components must be on average mutually at rest in these
coordinates. The spatial coordinates in the above metric are thus dubbed comoving
coordinates (with distances in these coordinates referred to as comoving distance), and
t is the proper time of comoving observers. k > 0, k = 0, and k < 0 correspond to an
open, flat, and closed Universe respectively. We are free to rescale the coordinates such
that a = 1 at the present epoch. Likewise we choose t = 0 at the present epoch and r = 0
at approximately earth’s location. With these definitions, k may be interpreted as the
Chapter 1. Introduction 7
Gaussian curvature of the Universe at t = 0. All observations are currently consistent
the Universe being flat and as such we will henceforth take k = 0.
It is common to use the alternate time coordinate η, related to t by
a dη = dt, (1.3)
and hence
η =
∫ t
0
dt
a(t). (1.4)
η is referred to as the conformal or comoving time. It has an especially simple inter-
pretation, in that a light ray arriving at earth today, and emitted at time η, originated
at a distance r = −cη, where the factor of c has been included for clarity. Since all
observations of the Universe involve light arriving at the earth at the present epoch, this
correspondence between time and distance is especially convenient, and η is often used
to represent either in an observational context (with factors of c omitted).
In our Universe, the scale factor, a(t), has been increasing monotonically with time,
with a = 0 corresponding to the Big Bang and a = 1 corresponding to the present day.
This gives yet another way to refer to a time in the Universe’s evolution. More commonly
used is the redshift, z ≡ 1/a − 1. This is an observationally convenient quantity since,
for light emitted by a distant object receding with the Hubble flow, the wavelength shift
is ∆λ = zλ0 where λ0 is the rest frame wavelength of the light. For z 1 the recessional
velocity is v = cz.
An important quantity is the Hubble parameter defined as H ≡ a′/a = a/a2, where
the prime represents the derivative with respect to proper time and the over-dot repre-
sents the derivative with respect to the conformal time (a′ ≡ da/ dt, a ≡ da/ dη). A
time scale and a length scale can then be defined as 1/H and c/H respectively, corre-
sponding to the Hubble time (also called the expansion time) and the Hubble distance.
The Hubble distance is the proper separation beyond which two points recede at a speed
greater than the speed of light. Points separated by more than this distance at a given
time are said to not be causally connected at that time1. A more relevant scale might be
the comoving equivalent: c/aH = c(a/a)−1.
Relating these time measures requires the Einstein equations, which for the FLRW
1This is not intended to be a precise statement, it simply sets a length scale for causality.
Chapter 1. Introduction 8
metric yield the Friedmann equations:(a′
a
)2
=8πG
3ρtot, (1.5)
a′′
a= −4πG
3(ρtot + 3ptot), (1.6)
where ρtot is the total density and ptot is the total pressure. The over-bar, ¯, indicates that
the quantity is spatially averaged. The total density and pressure get contributions from
matter (m), radiation (r), and dark energy (Λ, assumed to be a cosmological constant).
Differentiating the first equation and substituting it into the second gives
ρ′tot = −3H(ρtot + ptot). (1.7)
This is the energy conservation equation for an FLRW Universe. If we assume that
energy is not interchangeable between the different constituents, then this equation can
be used to solve for the a dependence of the densities. We also require the equation of
state for each constituent. The equations of state for each component, along with the
inferred evolutions of the densities are
pr = ρr/3, ρr = ρr0/a4; (1.8)
pm = 0, ρm = ρm0/a3; (1.9)
pΛ = −ρΛ, ρΛ = ρΛ0. (1.10)
To make this dependence on a explicit, we define the dimensionless present day density
constants as
ΩX ≡8πG
3H20
ρX0. (1.11)
where we have defined the Hubble constant, H0 ≡ H(t = 0). Evaluating the first
Friedmann equation at t = 0 leads to the constraint that Ωr + Ωm + ΩΛ = 1, resulting in
the interpretation that ΩX is the present day energy fraction of component X.
The first Friedmann equation can then be written as(a′
a
)2
= H20 (Ωr/a
4 + Ωm/a3 + ΩΛ), (1.12)
or
a′/a = H0
√Ωr/a4 + Ωm/a3 + ΩΛ). (1.13)
Given the parameters H0, Ωr, Ωm, and ΩΛ, this equation can be integrated numerically
Chapter 1. Introduction 9
10-2 10-1 100 101 102 103
redshift, z
10-2
10-1
100time (Hubble units, 1/H
0)
−t−η
Figure 1.4: Proper time t and conformal time η as a function of redshift z. Themagnitude of the proper time can be interpreted as the distance travelled by a photonobserved today and emitted by a source at redshift z, while the magnitude of η is thecurrent/comoving distance to that source. These differ because the source recedes withthe Hubble flow as the photon is in transit. t and η are given in terms of the Hubbletime 1/H0 = 14.6 Gyr or alternately the Hubble distance c/H0 = 3.00 Gpc/h whereh = H0/(100 km/s/Mpc) = 0.671.
to obtain the full expansion history. Current best fit values for the parameters are H0 =
67.1 km/s/Mpc, Ωr = 8.24 × 10−5, Ωm = 0.318, and ΩΛ = 0.682 [Planck Collaboration
et al., 2013d], with uncertainties at the percent level. Figure 1.4 shows the expansion
history calculated from the above equation and these parameters.
1.2.2 Perturbations
While on large scales, above ∼ 100 Mpc, the Universe is homogeneous and isotropic to a
good approximation, on smaller scales there are perturbations to the mean background
expansion. The evolution of these perturbations is normally calculated using linear per-
turbation theory, although below ∼ 10 Mpc, where the perturbations approach order
unity, simulations are required to treat the full non-linear evolution. While the assump-
tions of homogeneity and isotropy are broken by the perturbations, the Universe is still
Chapter 1. Introduction 10
assumed to be statistically homogeneous and isotropic. That is, each location in the Uni-
verse is assumed to be statistically equivalent, even if the exact values of any fields differ
from place to place. This assumption is often referred to as the cosmological principle.
Basics
In the field of large-scale structure, the primary observable is the matter density field,
or some proxy such as the galaxy number density or 21 cm brightness. We define the
density perturbations as
δ(~x, t) =ρ(~x, t)− ρ(t)
ρ(t). (1.14)
Cosmology does not make predictions about the precise density at a given location. It
instead predicts the statistical correlations between densities. The most important quan-
tity predicted by theory is the 2-point function, which can be written as the correlation
function:
〈δ(~x, t)δ(~x+ ~r)〉 = ξ(r), (1.15)
which is independent of ~x by assumption of statistical homogeneity and independent of
r (the direction of ~r) by assumption of statistical isotropy.
The perturbations are much more easily described in the spatial Fourier domain,
defined by
δ(~k) = F [δ(~x)] =
∫δ(~x)e−i
~k·~x d3~x. (1.16)
This has the advantage that all Fourier modes evolve independently. This results from
our assumption of statistical homogeneity and is true only at linear order in perturbation
theory.
In the Fourier domain, the two-point statistic is
〈δ(~k)δ∗(~k′)〉 = (2π)3δ(3)(~k − ~k′)P (k), (1.17)
where δ(3)(~k) is the 3D Dirac delta function and is not to be confused with δ(~k). P (k) is
the power spectrum is and related to the correlation function by P (~k) = F [ξ(~r)] (where
the vector signs over k and r remind us that, while P and ξ only depend on the magnitude
of their arguments, the Fourier transform must be performed in 3D).
The initial power spectrum of perturbations is set by physical processes in the first
instants of the Universe’s evolution, the dominant theory for which is cosmological in-
flation. Inflation predicts a nearly scale-invariant primordial power spectrum, where
Pp(k) ∼ k−3. This is in good agreement with observations.
Chapter 1. Introduction 11
The initial power spectrum can be related to that observed in the large-scale structure
through linear perturbation theory. Perturbation theory describes the evolution of the
perturbations over cosmic time. As mentioned above, when perturbation amplitude reach
order unity, perturbation theory ceases to be valid, and simulations must be employed.
Evolution equations
The governing equations for linear perturbation theory come from two sources. The first
is the conservation conditions, which derive from the fact that the stress-energy tensor
has zero divergence. The second is Einstein’s equations. Here we quote these equations
for an arbitrary perfect fluid, which includes most of the important cases in cosmology.
We use η as our primary time coordinate.
The fluid is described by three variables: the density perturbations, δ; the pressure
perturbations, π; and the velocity perturbations, θ. These are defined by
δ(~x, η) =ρ(~x, η)− ρ(η)
ρ(η)(1.18)
π(~x, η) =p(~x, η)− p(η)
p(η)(1.19)
−i~kkθ(~k, η) = ~v(k, η), (1.20)
where ~v is the 3D velocity field, and all quantities (e.g. ρ, p) refer to the fluid. We note
that the curling part of the velocity field does not couple to the density perturbations at
linear order and can thus be ignored.
In addition, metric perturbations are represented by the two fields Φ and Ψ, which
contribute to the metric in the following manner:
g00(~x, η) = −a(η)2[1 + 2Ψ(~x, η)] (1.21)
3∑i=1
gii(~x, η) = 3a(η)2[1 + 2Φ(~x, η)]. (1.22)
Ψ is recognizable as the Newtonian potential, and Φ is the spatial curvature perturbation.
With all the ingredients in place, we can now write down the equations of motion
governing the perturbations. The fact that the stress energy tensor has zero divergence
Chapter 1. Introduction 12
yields two conservation equations:
(a3ρδ) + a3(ρ+ p)(kθ + 3Φ) + 3pa2aπ = 0 (1.23)
θ + aH(1− 3 ˙p/ ˙ρ)θ − kΨ− k p
ρ+ Pπ = 0. (1.24)
The first of these equations is the mass continuity equation and the second is the Euler
fluid equation. Einstein’s equations yield an additional two equations:
k2Φ = 4πa2ρ
[δ + 3
aH
k
(ρ+ p
ρ
)θ
](1.25)
k2(Φ + Ψ) = 0. (1.26)
The first of these is recognizable as Poisson’s equation, while the second convenient
equation allows for the elimination of Φ.
When studying large-scale structure after recombination, when the baryonic matter
has decoupled from the radiation, the matter fluid can be treated as being pressureless.
The evolution is described by the three reduced equations:
δ + 3Φ + kθ = 0 (1.27)
θ + aHθ + kΦ = 0 (1.28)
k2Φ = 4πa2ρ
[δ + 3
aH
kθ
]. (1.29)
On scales much smaller than the Hubble scale, where k aH, the system can be easily
reduced to a single equation for the evolution of the density perturbations:
δ + aHδ − 4πa2ρδ = 0. (1.30)
The fact that k does not appear in this equation means that in this limit, structure
undergoes scale-independent growth. The linearity of this equation means that the frac-
tional growth of perturbations depends only on the background expansion, a(η). This is
an important result when studying structure at late times, say z . 50, when all modes
of interest are well within the horizon. This is also the regime in which all observations
of large-scale structure are made.
Chapter 1. Introduction 13
Summary
The evolution of an individual mode at late times is normally split into two components
such that
δ(~k, η) =
(a
aig(η)
)(9
10Ti(k)
)δp(~k) (η > ηi). (1.31)
Here, the subscript i refers to an intermediate time, when all modes of interest are
well within the horizon, but before we intend to make observations of the large-scale
structure. A reasonable choice would be zi = 20. g(η) is the growth function, which
describes the scale-independent growth of perturbations at late times (η > ηi), as given
by Equation 1.30. The factor of (a/ai) is removed such that g(η) is unity for a matter
dominated Universe (Ωm = 1). Ti(k) is the transfer function, which describes the scale-
dependent growth of the perturbations from the primordial value (δp(~k)) to the value at
the intermediate time. The factor is 9/10 is removed such that Ti(k) asymptotes to unity
at large scales, (k aiHi).
The transfer function is calculated using the full scale-dependent evolution equations.
The dark matter component is well described by the pressureless versions, but prior to
recombination at z ∼ 1000 the baryonic component of the matter is tightly coupled to the
photons. The perturbations of multiple coupled fluids must then be considered, greatly
complicating the calculation. This is generally done numerically.
With these definitions the matter power spectrum, which we hope to observe in our
redshift surveys, is then
P (k, η) =
(a
aig(η)
)2(9
10Ti(k)
)2
Pp(k) (η > ηi). (1.32)
1.3 Overview
Here each chapter is summarized in the broader context of this thesis. In addition I state
my contributions to each chapter within my collaborations.
1.3.1 Outline
This thesis is divided into two parts. Part I contains entirely theoretical work concern-
ing measurements that could in principle be performed using 21 cm intensity mapping.
Part II contains entirely observational work, where the Green Bank Telescope was used
to perform one of the first large-scale structure surveys using 21 cm intensity mapping.
This work represents a pioneering effort to establish intensity mapping as an efficient
Chapter 1. Introduction 14
technique for learning about the Universe.
Part I
In Chapter 2, originally published in Masui et al. [2010b], we considered the ability of
21 cm intensity mapping experiments to constrain modified gravity models. Modifications
to Einstein’s theory of gravity, General Relativity, are sometimes invoked as an alternative
to dark energy to explain the observed accelerating expansion of the Universe. We show
that experiments designed to measure the properties of dark energy are also able to
tightly constrain modified gravity models, through the observational probes of baryon
acoustic oscillations and weak lensing. This chapter involved a relatively straight forward
calculation, using well established techniques. The project represents my introduction to
statistical analysis in modern cosmology.
In Chapter 3, originally published in Masui and Pen [2010], we discovered a new effect
by which gravity waves created in cosmological inflation leave a distinct signature in the
large-scale structure of the Universe. The effect could be used to gain rare insight into
the inflationary era if observed. We considered the feasibility of making a detection of the
effect using 21 cm observations of the early Universe at redshift z ∼ 12. We concluded
that while such a detection would be very difficult, the reward would be sufficient that
searching for the effect using a futuristic experiment would still be very compelling.
Chapter 4, originally published in Masui et al. [2010a], considers what measurements
could in principle be made using instruments that either currently exist or will be con-
structed in the near future. In this way, it differs significantly from the previous two
chapters which each consider measurements that would be performed using ‘the ulti-
mate’ intensity mapping survey. We showed that even without building a dedicated
experiment, intensity mapping could be used to make interesting measurements. In par-
ticular, the Green Bank Telescope would be capable of performing a large-scale structure
survey that would have the sensitivity to detect the Kaiser red-shift space distortions,
settling a long standing controversy about the abundance of neutral hydrogen in the
Universe.
Part II
Chapter 5, which forms the basis of an intended future publication, gives an overview
of the analysis pipeline used to analyze survey data from the Green Bank Telescope.
It describes the formalism for the various parts of the data analysis including radio
frequency interference mitigation, calibration, noise estimation and map-making. It also
Chapter 1. Introduction 15
describes some details of the software modules that implement the analysis.
Chapter 6, originally published in [Masui et al., 2013], presents the cross-correlation
power spectrum of the intensity mapping survey at the Green Bank Telescope with a tra-
ditional galaxy survey. The cross correlation was detected with a statistical significance
of 7.4σ, far exceeding the significance of previous measurements and putting a lower limit
on the amplitude of the 21 cm brightness fluctuations.
Chapter 7, submitted to Monthly Notices of the Royal Astronomical Society: Letters
and available in pre-print as Switzer et al. [2013], represents the first use of the auto-
correlation power spectrum from the GBT survey to make an astrophysical measurement.
A Bayesian analysis is used to combine the lower limit from the cross-correlation and the
upper limit from the auto-correlation into a determination of the 21 cm signal amplitude.
We discuss future directions and conclude in Chapter 8.
1.3.2 Summary of contributions
In Chapter 2, Patrick McDonald calculated the BAO error bars for the 21 cm experiments
as well as the constraints from the Planck mission. I calculated the predicted BAO signal
for the modified gravity models and combined these two ingredients into the projected
constraints on the modified gravity models. Likewise, Fabian Schmidt calculated the weak
lensing spectrum for the modified gravity models, and the error bars for the intensity
mapping experiments were taken from a calculation performed by Ting Ting Lu [Lu
et al., 2010]. Again I combined these into the constraints on the models. I also lead the
project, produced the figures, and did the majority of the writing for its publication.
All calculations, figures and writing for Chapter 3 was prepared by myself, under the
guidance of Ue-Li Pen.
In Chapter 4, Patrick McDonald wrote the software that calculates the sensitivity for
a general 21 cm survey. I used this software to perform forecasts for the surveys under
consideration and to optimize the surveys. I also did the majority of the writing and
created all the plots.
For the observations and data analysis that form the basis of Chapters 5, 6 and 7, I
took the lead on the survey planning and data analysis. I made significant contributions
to all proposals to the GBT telescope allocation committees, lead the planning of all
observations, wrote the vast majority of the telescope control scripts, and performed
roughly one quarter of the observations. I designed the software framework for the
data analysis pipeline and made contributions to all the data analysis software up to
the map making. This includes preprocessing the data, radio frequency interference
Chapter 1. Introduction 16
mitigation (written principally by Liviu-Mihai Calin), and calibration (written principally
by Tabitha Voytek). I was the sole author of all noise estimation and map making
software. While I lead the development of the pipeline, the data was actually run through
the software by collaborators, mostly Tabitha Voytek.
In early versions of the data analysis, I wrote the software that performed the fore-
ground subtraction and power spectrum (then correlation function) estimation. Re-
sponsibility for this part of the pipeline has since been transfered to my collaborators,
principally Eric Switzer with contributions from Yi-Chao Li. While I have remained a
consulting party throughout, I have made no subsequent contributions to writing the
software for the parts of the pipeline subsequent to map-making. Eric Switzer and
Yi-Chao Li wrote all the software that dealt with the WiggleZ galaxy catalogues and
cross-correlating them with the intensity mapping survey.
In Chapters 6 and 7, most of the writing was roughly evenly split between myself
and Eric Switzer. In Chapter 7, I performed the Bayesian analysis to arrive at the final
conclusions of the paper and produced all plots.
Part I
The potential of 21 cm cosmology
17
Chapter 2
Projected Constraints on Modified
Gravity Cosmologies from 21 cm
Intensity Mapping
A version of this chapter was published in Physical Review D as “Projected constraints
on modified gravity cosmologies from 21 cm intensity mapping”, Masui, K. W., Schmidt,
F., Pen, U.-L. and McDonald, P., Vol. 81, Issue 6, 2010. Reproduced here with the
permission of the APS.
2.1 Summary
We present projected constraints on modified gravity models from the observational tech-
nique known as 21 cm intensity mapping, where cosmic structure is detected without re-
solving individual galaxies. The resulting map is sensitive to both BAO and weak lensing,
two of the most powerful cosmological probes. It is found that a 200 m×200 m cylindrical
telescope, sensitive out to z = 2.5, would be able to distinguish Dvali, Gabadadze and
Porrati (DGP) model from most dark energy models, and constrain the Hu & Sawicki
f(R) model to |fR0| < 9 × 10−6 at 95% confidence. The latter constraint makes exten-
sive use of the lensing spectrum in the nonlinear regime. These results show that 21 cm
intensity mapping is not only sensitive to modifications of the standard model’s expan-
sion history, but also to structure growth. This makes intensity mapping a powerful and
economical technique, achievable on much shorter time scales than optical experiments
that would probe the same era.
18
Chapter 2. Constraining modified gravity 19
2.2 Introduction
One of the greatest open questions in cosmology is the cause of the observed late time
acceleration of the universe. Within the context of normal gravity described by Einstein’s
General Relativity, this phenomena can only be explained by an exotic form of matter
with negative pressure. Another possible explanation is that on cosmological scales,
General Relativity fails and must be replaced by some theory of modified gravity.
Several approaches have been proposed to modify gravity at late times to explain the
apparent acceleration of the universe. The challenge in these modifications is to preserve
successful predictions of the CMB at z ≈ 1000, and also the precision tests at the present
epoch in the solar system.
A generic class of theories operates with the Chameleon effect, where at sufficiently
high densities General Relativity (GR) is restored, thus applying both in the solar system
and the early universe. To further understand the nature of gravity would require probing
gravity on cosmological scales. Large scales means large volume, requiring large fractions
of the sky. Gravity can be probed by gravitational lensing, which measures geodesics
and thus the gravitational curvature of space, and is a sensitive probe of the growth
of structure in the Universe [Knox et al., 2006, Jain and Zhang, 2008, Tsujikawa and
Tatekawa, 2008, Schmidt, 2008].
In working out predictions for cosmology, the theoretical challenge posed by these the-
ories are the nonlinear mechanisms in each model, necessary in order to restore Einstein
Gravity locally to satisfy Solar System constraints. We present quantitative results from
nonlinear calculations for a specific f(R) model, and forecasted constraints for future
21 cm experiments.
An upcoming class of experiments propose the observation of the 21 cm spectral line
at low resolution over a large fraction of the sky and large range of redshifts [Peterson
et al., 2009]. Large scale structure is detected in three dimensions without the detection
of individual galaxies. This process is referred to as 21 cm intensity mapping. These
experiments are sensitive to structures at a redshift range that is observationally difficult
to observe for ground-based optical experiments due to a lack of spectral lines. Yet these
experiments are extremely economical since they only require limited resolution and no
moving parts [Seo et al., 2010].
Intensity mapping is sensitive to both the Baryon Acoustic Oscillations (BAO) and to
weak lensing, two of the most powerful observational methods to determine cosmological
parameters. It has been shown that BAO detections from 21 cm intensity mapping are
powerful probes of dark energy, comparing favourably with Dark Energy Task Force
Chapter 2. Constraining modified gravity 20
Stage IV projects within the figure of merit framework [Chang et al., 2008, Albrecht
et al., 2006].
In this paper we present projected constraints on modified gravity models from 21 cm
intensity mapping. In Section 2.3 we describe the modified gravity models considered. In
Section 2.4 we discuss the observational signatures accessible to 21 cm intensity mapping,
and calculate the effects of modified gravity on these signatures. In Section 2.5 we present
statistical analysis and results and we conclude in Section 2.6.
We assume a fiducial ΛCDM cosmology with WMAP5 cosmological parameters:
Ωm = 0.258, Ωb = 0.0441, ΩΛ = 0.742, h = 0.719, ns = 0.963 and log10As = −8.65
[Komatsu et al., 2009]. We will follow the convention that ωx ≡ h2Ωx.
2.3 Modified Gravity Models
Here we describe some popular modified gravity models for which projected constraints
will later be derived. Throughout we will use units in which G = c = ~ = 1 and will be
using a metric with mostly negative signature: (+,−,−,−).
2.3.1 f(R) Models
In the f(R) paradigm, modifications to gravity are introduced by changing the standard
Einstein-Hilbert action, which is linear in R, the Ricci scalar. The modifications are
made by adding an additional non linear function of R [Starobinsky, 1980, Capozziello,
2002, Carroll et al., 2004]
S =
∫d4x√−g[R + f(R)
16π+ Lm
], (2.1)
where Lm is the matter Lagrangian. See Sotiriou and Faraoni [2010] for a comprehensive
review of f(R) theories of gravity.
The choice of the function f(R) is arbitrary, but in practice it is highly constrained by
precise solar system and cosmological constraints, as well as stability criteria [Nojiri and
Odintsov, 2003, Sawicki and Hu, 2007] (see below). In this paper, we choose parameter-
izations of f(R) such that it asymptotes to a constant for a certain choice of parameters
and thus approaches the fiducial ΛCDM.
In general, f(R) models have enough freedom to mimic exactly the ΛCDM expansion
history and yet still impose a significant modification to gravity [Nojiri and Odintsov,
2006, Song et al., 2007]. As such probes of the expansion history are less constraining
Chapter 2. Constraining modified gravity 21
than probes of structure growth, which will be evident in the constraints presented in
later sections.
Variation of the above action yields the modified Einstein Equations
Gµν + fRRµν −(f
2−fR
)gµν −∇µ∇νfR = 8π Tµν , (2.2)
where fR ≡ df(R)/dR, a convention that will be used throughout. f(R) gravity is
equivalent to a scalar-tensor theory [Nojiri and Odintsov, 2003, Chiba, 2003] with the
scalar field fR having a mass and potential determined by the functional form of f(R).
The field has a Compton wavelength given by its inverse mass
λC =1
mfR
=√
3fRR. (2.3)
The main criterion for stability of the f(R) model is that the mass squared of the fR
field is positive, i.e. fRR > 0. In most cases, this simply corresponds to a sign choice for
the field fR (specifically for the model we consider below, fR0 is constrained to be less
than 0).
On scales smaller than λC , gravitational forces are enhanced by 4/3, while they reduce
to unmodified gravity on larger scales. The reach of the modified forces λC generically
leads to a scale-dependent growth in f(R) models.
While the dynamics are significantly changed in f(R), the relation between matter
and the lensing potential is unchanged up to a rescaling of the gravitational constant by
the linear contribution in f . The fractional change is of order the background field value
fR ≡ fR(R) 1 where R is the background curvature scalar.
Proceeding further requires a choice of the functional form for f . A functional form
is considered which is representative of many other cases.
Hu and Sawicki [2007] (HS) proposed a simple functional form for f(R), which can
be written as
f(R) = −R0c1(R/R0)n
c2(R/R0)n + 1, (2.4)
where we have used the value of the scalar curvature in the background today, R0 ≡ R|z=0
for convenience. This three parameter model passes all stability criteria for positive n,
c1 and c2. One parameter can be fixed by demanding the expansion history to be close
(within observational limits) to ΛCDM. In this case, Equation 2.4 can be conveniently
Chapter 2. Constraining modified gravity 22
reparametrized and approximated by
f(R) ≈ −2Λ− fR0R0
n
(R0
R
)n. (2.5)
Here Λ and fR0—the value of the fR field in the background today—have been used
to parameterize the function in lieu of c1 and c2. This approximation is valid as long
as |fR0| 1, which is necessary to satisfy current observational constraints [Hu and
Sawicki, 2007, Schmidt et al., 2009b]. While Λ is conceptually different than vacuum
energy, it is mathematically identical and will thus be absorbed into the right hand side
of the Friedmann equation and parameterized by ΩΛ. In quoting constraints, we will
marginalize over this parameter as it is of no use in identifying signatures of modified
gravity. The parameter fR0 can be though of as controlling the strength of modifications
to gravity today, while higher n pushes these modifications to later times. The effects of
changing these parameters are discussed in greater detail in Hu and Sawicki [2007].
Allowed f(R) models exhibit the so-called chameleon mechanism: the fR field be-
comes very massive in dense environments and effectively decouples from matter. This ef-
fect is active whenever the Newtonian potential is of order the background fR field. Since
cosmological potential wells are typically of order 10−5 for massive halos, the chameleon
effect becomes important if |fR| . 10−5. If the background field is ∼ 10−7 or smaller, a
large fraction of the collapsed structures in the universe are chameleon-screened, so that
the model becomes observationally indistinguishable from ΛCDM.
Since the chameleon effect will affect the formation of structure, standard fitting
formulas based on ordinary GR simulations, such as those mapping the linear to the
nonlinear power spectrum, cannot be used for these models. Recently, however, self-
consistent N-body simulations of f(R) gravity have been performed which include the
chameleon mechanism [Oyaizu, 2008, Oyaizu et al., 2008, Schmidt et al., 2009a]. We will
use the simulation results for forecasts of weak lensing in the nonlinear regime below.
It should be noted that f(R) models are not without difficulties. In particular, an
open issue is the problem of potential unprotected singularities [Abdalla et al., 2005,
Frolov, 2008, Nojiri and Odintsov, 2008].
2.3.2 DGP Braneworld
A theory of gravity proposed by Dvali, Gabadadze and Porrati (DGP) assumes that our
four dimensional universe sits on a brane in five dimensional Minkowski space [Dvali et al.,
2000]. On small scales gravity is four dimensional but, on larger scales it becomes fully
Chapter 2. Constraining modified gravity 23
five dimensional. Here we parameterize DGP by rc, the scale at which gravity crosses
over in dimensionality. The DGP model has two branches depending on the embedding
of the brane in 5D space. In the self-accelerating branch, the universe accelerates without
need for a cosmological constant if rc ∼ 1/H0 [Deffayet, 2001, Deffayet et al., 2002]. In
this branch, assuming a spatially flat Universe for now, the modified Friedmann equation
is given by
H2 − H
rc=
8π
3ρ, (2.6)
which clearly differs from ΛCDM. Thus, in contrast to the other models considered here,
DGP without a cosmological constant does not reduce to ΛCDM and it is possible to
completely rule out this scenario (where the others can only be constrained). In fact
DGP (without a cosmological constant) has been shown to be in conflict with current
data [Fang et al., 2008]. It is presented here largely for illustrative purposes.
On scales much smaller than rc, gravity is four-dimensional but not GR. On these
scales, DGP can be described as an effective scalar-tensor theory [Koyama and Maartens,
2006, Koyama and Silva, 2007, Scoccimarro, 2009]. The massless scalar field, the brane-
bending mode, is repulsive in the self-accelerating branch of DGP. Hence, structure for-
mation is slowed in DGP when compared to an effective smooth Dark Energy model
with the same expansion history. While the growth of structure is thus modified in DGP
even on scales much smaller than rc, gravitational lensing is unchanged. In other words,
the relation between matter overdensities and the lensing potential is the same as in GR
[Lue et al., 2004].
As in f(R), the DGP model contains a nonlinear mechanism to restore GR locally.
This Vainshtein mechanism is due to self-interactions of the scalar brane-bending mode
which generally become important as soon as the density field becomes of order unity. In
the Vainshtein regime, second derivatives of the field saturate, and thus modified gravity
effects are highly suppressed in high-density regions [Lue et al., 2004, Koyama and Silva,
2007, Schmidt, 2009]. We will only consider linear predictions for the DGP model here.
2.4 Observational Signatures
In this Section we describe the observational signatures available to 21 cm intensity map-
ping. We also give details on calculating the observables within modified gravity models.
We consider two types of measurements: the Baryon Acoustic Oscillations and weak
gravitational lensing.
For the fiducial survey, we assume a 200 m× 200 m cylindrical telescope, as in Chang
Chapter 2. Constraining modified gravity 24
et al. [2008]. We will also present limited results for a 100 m×100 m cylindrical telescope
to illustrate effects of reduced resolution and collecting area on the results. This latter
case is representative of first generation projects [Seo et al., 2010]. In the 200 m case we
assume 4000 receivers, and in the 100 m case 1000 receivers. We assume either telescope
covers 15000 sq. deg. over 4 years. We assume neutral hydrogen fraction and the bias
remain constant with ΩHI = 0.0005 today and b = 1. The object number density is
assumed to be n = 0.03 per cubic h−1Mpc (effectively no shot-noise, as should be the
case in practice [Chang et al., 2008]).
2.4.1 Baryonic acoustic oscillation expansion history test
Acoustic oscillations in the primordial photon-baryon plasma have ubiquitously left a
distinctive imprint in the distribution of matter in the universe today. This process is
understood from first principles and gives a clean length scale in the universe’s large
scale structure, largely free of systematic uncertainties and calibrations. This can be
used to measure the global cosmological expansion history through the angular diameter
distance, dA, and Hubble parameter, H, vs redshift relation. The detailed expansion and
acceleration will differ between pure cosmological constant and modified gravity models.
We use essentially the method of Seo and Eisenstein [2007] for estimating distance
errors obtainable from a BAO measurement, including 50% reconstruction of nonlinear
degradation of the BAO feature. We assume the frequency range corresponding to z < 2.5
is covered (the lower z end should be covered by equivalent galaxy redshift surveys if not
a 21cm survey). For the sky area and redshift range surveyed, the 200 m telescope is
nearly equivalent to a perfect BAO measurement. The limited resolution and collecting
area of the 100 m telescope substantially degrades the measurement at the high-z end.
The expansion history for modified gravity models can be calculated in an analogous
way to that in General Relativity. The Friedmann Equation in DGP, Equation 2.6 can
be written as
H2 = − k
a2+
(1
2rc+
√1
4rc2+
8πρ
3
)2
, (2.7)
where k is the curvature, and rc is the crossover scale. It is convenient to introduce the
parameter ωrc ≡ 1/4r2c which stands in for rc. This equation can be solved numerically
to calculate the observable quantities.
We now calculate the expansion history in the HS f(R) model using a perturbative
framework which is well suited for calculating constraints on fR0. Working in the confor-
mal gauge and mostly negative signature, we start with the modified Einstein’s Equation
(2.2). At zeroth order the left hand side of the 00 component contains the modified
Chapter 2. Constraining modified gravity 25
Friedmann equation
H2 =8πρ
3+ fR0gn(a, a, a,
...a ), (2.8)
where ρ is the average density (including contributions from Λ), the over-dot represents
a conformal derivative and
gn ≡−1
fR0a2
[(f + 2Λ)a2
6+ fR
(a2
a2− a
a
)+6fRR
( ...a a
a4− 3aa2
a5
)]. (2.9)
For verifiability we quote
g1 =a2R2
0(2aa2 − 7aa2 + 2...a aa)
36a3. (2.10)
Evaluating Equation 2.8 at the present epoch yields the modified version of the standard
constraint
h2 = ωm + ωr + ωk + ωΛ + fR0gn0. (2.11)
Note that the modified version of the Friedmann Equation is third order instead of
first order, however, it has been shown that the expansion history stably approaches
that of ΛCDM for vanishing fR0 [Hu and Sawicki, 2007]. For observationally allowed
cosmologies fR0 1 we expand
fR0gn = fR0gn(a, ˙a, ¨a,...a ) +O(f 2
R0), (2.12)
where a is the solution to the standard GR Friedmann equation.
By using Equation 2.12 in Equation 2.8 and keeping only terms linear in fR0, the ex-
pansion history can be calculated in the regular way, along with the observable quantities
dA(z) and H(z). For small fR0 this agrees well with the calculation in Hu and Sawicki
[2007] where the full third order differential equation was integrated
In calculating the Fisher Matrix, this treatment is exact because the Fisher Matrix
depends only on the first derivative of the observables with respect to the model param-
eters, evaluated at the fiducial model.
2.4.2 Weak Lensing
A second class of observables measures the spatial perturbations in the gravitational met-
ric. Modified gravity will change the strength of gravity on large scales and thus modify
Chapter 2. Constraining modified gravity 26
the growth of cosmological structure. Weak gravitational lensing, the gravitational bend-
ing of source light by intervening matter, is a probe of this effect.
Weak lensing measures the distortion of background structures as their light propa-
gates to us. Here, the background structure is the 21 cm emission from unresolved sources.
While light rays are deflected by gravitational forces, this deflection is not directly ob-
servable, since we don’t know the primary unlensed 21 cm sky. However, weak lensing
will induce correlations in the measured 21 cm background, since neighbouring rays pass
through common lens planes. While the deflection angles themselves are small (of order
arcseconds) the deflections are coherent over scales of arcminutes. In this way, the lensing
signal can be extracted statistically using quadratic estimators [Lu et al., 2010]. Given
the smallness of the lensing effect, a high resolution (high equivalent number density of
“sources”) is necessary to detect the effect.
The weak lensing observable that is predicted by theory is the power spectrum of the
convergence κ. It is given by
Cκκ(`) =
(3
2Ωm H
20
)2 ∫ χs
0
dχ
χ
WL(χ)2
χ a2(χ)ε2(χ)P (`/χ;χ), (2.13)
where χ denotes comoving distances, P (k, χ) is the (linear or nonlinear) matter power
spectrum at the given redshift, and we have assumed flat space. The lensing weight
function WL(χ) is given by:
WL(χ) =
∫ ∞z(χ)
dzsχ
χ(zs)(χ(zs)− χ)
dN
dz(zs). (2.14)
Here, dN/dz is the redshift distribution of source galaxies, normalized to unity. The factor
ε(χ) in Equation 2.13 encodes possible modifications to the Poisson equation relating the
lensing potential to matter (Section 2.3). In f(R), it is given by ε(χ) = (1 + fR(χ))−1,
while ε = 1 for GR as well as DGP. Note that for viable f(R) models, ε − 1 . 0.01, so
the effect of ε on the lensing power spectra is very small.
The CAMB Sources module [Lewis et al., 2000, Lewis and Challinor, 2007] was used
to calculate the lensing convergence power spectrum in flat ΛCDM models. The HALOFIT
[Smith et al., 2003] interface for CAMB was used for calculations that include lensing at
nonlinear scales.
For the modified gravity models in the linear regime, the convergence power spectra
were calculated using the Parametrized Post-Friedmann (PPF) approach [Hu and Saw-
icki, 2007] as in Schmidt [2008]. Briefly, the PPF approach uses an interpolation between
super-horizon scales and the quasi-static limit. On super-horizon scales (k aH), spec-
Chapter 2. Constraining modified gravity 27
ifying the background expansion history, together with a relation between the two metric
potentials, already determines the evolution of metric and density perturbations. On
small scales (k aH), time derivatives in the equations for the metric perturbations
can be neglected with respect to spatial derivatives, leading to a modified Poisson equa-
tion for the metric potentials. The PPF approach uses a simple interpolation scheme
between these limits, with a few fitting parameters adjusted to match the full calcu-
lations [Hu and Sawicki, 2007]. The full calculations are reproduced to within a few
percent accuracy. We use the transfer function of Eisenstein and Hu [1998] to calculate
the ΛCDM power spectrum at an initial redshift of zi = 40, were modified gravity effects
are negligible, and evolve forward using the PPF equations.
For the f(R) model, we also calculate predictions in the nonlinear regime. For these,
we use simulations of the HS model with n = 1 and fR0 values ranging from 10−6 to 10−4.
We use the deviation ∆P (k)/P (k) of the nonlinear matter power spectrum measured
in f(R) simulations from that of ΛCDM simulations with the same initial conditions
[Oyaizu et al., 2008]. This deviation is measured more precisely than P (k) itself. We
then spline-interpolate the measurements of ∆P (k)/P (k) for k = 0.04− 3.1 h/Mpc and
at scale factors a = 0.1, 0.2, ...1.0, and multiply the standard nonlinear ΛCDM prediction
(HALOFIT) with this value. For values of k > 3.1 h/Mpc, we simply set ∆P (k) = 0.
However, for the angular scales and redshifts considered here (` < 600, see below), such
high values of k do not contribute significantly.
One might be concerned that this mixing of methods, for calculating the lensing spec-
trum, might artificially exaggerate the effects of modified gravity if these methods do not
agree perfectly. While the spectra calculated for the fiducial ΛCDM model differed by up
to a percent between these methods, presumably due to slight differences in the transfer
function, this should have no effect on the results. Any direct comparison between spec-
tra (for example finite difference derivatives) are made between spectra calculated in the
same manner. Note that the Fisher Matrix depends only on the first derivative of the
observables with respect to the parameters and no cross derivatives are needed.
The lensing spectra were not calculated for non-flat models, but it is expected that
the CMB and BAO are much more sensitive to the curvature and as such the lensing
spectra are relatively unaffected. Formally we are assuming that
σωkσCκκ
∂Cκκ
∂ωk 1.
Reconstructing weak lensing from 21 cm intensity maps involves the use of quadratic
estimators to estimate the convergence and shear fields. The accuracy with which this
Chapter 2. Constraining modified gravity 28
can be done increases with information in the source maps, however, this information
saturates at small scales due to nonlinear evolution. As such, one cannot improve the
lensing measurement indefinitely by increasing resolution, and the experiments considered
here extract much of the available information within the redshift range considered.
The accuracy with which the convergence power spectrum can be reconstructed from
21 cm intensity maps was derived in Lu et al. [2010], where the effective lensing galaxy
density was calculated at redshifts 1.25, 3 and 5 (see Figure 7 and Table 2 therein). The
effective volume galaxy density was corrected for the finite resolution of the experiment
considered here. It was then interpolated, using a piecewise power law, and integrated
from redshift 1 to 2.5 to obtain an effective area galaxy density of ng/σ2e = 0.37arcmin−2.
The parameter σ2e is the variance in the intrinsic galaxy ellipticity, which is only used
here for comparison with optical weak lensing surveys. From the effective galaxy density
the error on the convergence power is given by
∆Cκκ(`) =
√2
(2`+ 1)fsky
(Cκκ(`) +
σ2e
ng
), (2.15)
where fsky is the fraction of the sky surveyed. The galaxy distribution function dN/dz
used to calculate the theoretical curves (from Equation 2.13) should follow the effective
galaxy density. Instead for simplicity, a flat step function was used, with this distribution
function equal from redshift 1 to 2.5 and zero elsewhere. While the difference between the
these distributions would have an effect on the lensing spectra, the effect on differences of
spectra when varying parameters is expected to be negligible. Our approximation is also
conservative, since the proper distribution function is more heavily weighted toward high
redshift. Rays travelling from high redshift will be affected by more intervening matter
and thus experience more lensing. This would increase the lensing signal, allowing a more
precise measurement.
Figure 2.1 shows the lensing spectra for the fiducial cosmology and a modified gravity
model, including both linear and nonlinear calculations. The linear regime is taken to
be up to ` = 140 for projected constraints. For calculations including weak lensing in
the nonlinear regime, Cκκ(`) up to ` = 600 is used for the larger telescope. Beyond this
scale the model used for lensing error-bars is not considered accurate at the shallowest
redshifts in the source window [Lu et al., 2010]. This cut off coincides with the scale
at which information in the source structures saturates due to non-linear evolution in
standard gravity (although it is also not far from the resolution limit of the experiment).
We speculate that a similar phenomena would occur in modified gravity and smaller
scales are not expected to carry significant additional information. Note that it is the
Chapter 2. Constraining modified gravity 29
101
102
103
10−6
10−5
10−4
l2C
(l)/
2π
l
Fiducial modelHS f(R) with f
R0 = 10
−4
Fiducial model, linear
f(R) linear
Figure 2.1: The Weak lensing convergence power spectra for ΛCDM and the HS f(R)model with n = 1 and fR0 = 10−4. Galaxy distribution function is flat between z = 1and z = 2.5.
source structures in which information saturates. At smaller scales the lensing spectrum
would continue to carry information [Dore et al., 2009] if it could be reconstructed. For
the smaller telescope the scale is limited to ` < 425 by the resolution at the high end
of the redshift window. If the redshift window were subdivided into narrower bins, it
would be possible to use information at scales down to ` ≈ 1000 in the centre bins as
at these redshifts the telescope resolutions are better and structures are less non-linear.
However, considering tomographic information is beyond the scope of this work. It is
noted that these scales are very large by weak lensing standards where optical surveys
typically make detections down to an ` of order 105.
2.4.3 External Priors from Planck
While the CMB is not sensitive to the late time effects of modified gravity (except by
the integrated Sachs-Wolfe effect), it is invaluable for constraining other parameters and
breaking degeneracies. As such, projected information from the Planck experiment is
included. The Planck covariance matrix used here is given in McDonald and Eisenstein
[2007, Table II]. All late time cosmological parameters (including the curvature) are
Chapter 2. Constraining modified gravity 30
marginalized over, removing information contained in the ISW effect, and ensuring that
sensitivity to f(R) is entirely from 21 cm tests below. The only remaining parameter
that is related to the late time expansion is θs, the angular size of the sound horizon,
which is then used as a constraint on the parameter sets of the modified gravity models.
2.5 Results
To quantify the projected constraints on f(R) models, the Fisher matrix formalism is
employed. The HS f(R) models reduces to the fiducial model for vanishing fR0 and any
value of n. Thus the Fisher Matrix formalism is used to project constraints on fR0 for
given values of n. In the case of DGP, which does not reduce to the fiducial model, it
is shown that a measurement consistent with the fiducial model can not be consistent
with DGP for any parameter set. Unless otherwise noted, we account for freedom in
the full cosmological parameter set: h, ωm, ωb, ωk, As and ns; representing the Hubble
parameter; physical matter, baryon and curvature densities; amplitude of primordial
scalar fluctuations and the spectral index; respectively.
Within the f(R) models, the fiducial model is a special point in the parameter space as
there are no modifications to gravity. As such, one cannot in general expect perturbations
to observables to be linear in the f(R) parameter fR0, an assumption implicit in the
Fisher Matrix formalism. This assumption does seem to hold for the expansion history,
where our first order perturbative calculation agrees with the full solution to the modified
Friedmann Equations calculated in Hu and Sawicki [2007]. However, this is not the case
for weak lensing. For each f(R) model, the lensing spectrum was calculated for several
values of fR0. It was observed that enhancements to the lensing power spectrum go as
Cκκ(`)− Cκκfiducial(`) ∼ (fR0)α(`),
with α(`) in the 0.5–0.7 range. This is because the reach of the enhanced forces in f(R) is
a power law in fR0 following Equation (2.3), and the enhancement of the power spectrum
for a given mode k roughly scales with the time that this mode has been within the reach
of the enhanced force. Because of this behaviour, the constraints derived within the
Fisher Matrix formalism depend on the step size in fR0 used for finite differences.
To correct for this, we use a step size that is dependent on the final constraint.
The weak lensing Fisher Matrices where calculated for fR0 step sizes of 10−3, 10−4 and
10−5. These were then interpolated—using a power law—such that the ultimate step
size used for finite differences is roughly the quoted constraint on the modified gravity
Chapter 2. Constraining modified gravity 31
95% confidence HS |fR0|upper limits n = 1 NL WL n = 2 n = 4BAO 1.5e-02 ∼ 1.8e-02 3.0e-02WL 2.3e-03 4.3e-05 4.0e-03 8.6e-03BAO+WL 5.0e-05 8.9e-06 9.7e-05 4.6e-04
Table 2.1: Projected constraints on f(R) models for various combinations of observa-tional techniques, for a 200 m telescope. Constraints are the 95% confidence level upperlimits and include forecasts for Planck. The non linear results (column marked NL WL)are for the HS model with n = 1. Results that make use of weak lensing with con-straints above 10−3 are only order of magnitude accurate. The linear regime is taken tobe ` < 140, with the nonlinear constraints extending up to ` = 600.
parameter. For instance when the 95% confidence constraint on fR0 is quoted, the step
size for finite differences is ∆fR0 ≈ 2σfR0, where σfR0
is calculated from the interpolated
Fisher matrix. This is expected to be valid down to step sizes at the 10−6 level where
the chameleon mechanism is important. As such, for constraints below 10−5 a step
size of 10−5 is always used. Note that this is conservative because an over sized finite
difference step always underestimates the derivative of a power law with an power less
than unity. For constraints above the 10−3 level a step size of 10−3 is used, which is the
largest modification to gravity simulated. These constraints are considered unreliable
due to these difficulties. We reiterate this this only affects results that include weak
lensing information. Likelihood contours remain perfect ellipses in this procedure (which
is clearly inaccurate), however the spacing between contours at different confidence levels
is altered.
Figure 2.2 shows the projected constraints on the HS f(R) model with n = 1 for
various combinations of observational techniques, and a (200m)2 telescope. The elements
in the lensing fisher matrix associated with the curvature are taken to be zero for the
reasons given in Section 2.4.2. While this assumption is not conservative, it is expected
to be valid, as the angular diameter distance as measured by the BAO is very sensitive
to the curvature. In total three f(R) models were considered: HS with n = 1, 2, 4. The
results are summarized in Table 2.1.
It was found that while weak lensing, in the linear regime, is very sensitive to the
modifications to gravity, it is only barely capable of constraining f(R) models without
separate information about the expansion history. Even with the inclusion of Planck
forecasts, degeneracies with h and ωk, the mean curvature, drastically increase the un-
certainties on the modified gravity parameters. Indeed these three parameters are more
than 95% correlated (depending on the exact model and confidence interval). This of
course brings into question the neglect of the ωk terms in the weak lensing Fisher Ma-
Chapter 2. Constraining modified gravity 32
0.65 0.7 0.7510
−6
10−5
10−4
10−3
10−2
h
|fR
0|
BAO
BAO+WL
NL WL
BAO+NL WL
Figure 2.2: Projected constraints on the HS f(R) model with n = 1 using several com-binations of observational techniques, for a 200 m telescope. All curves include forecastsfor Planck. Allowed parameter values are shown in the fR0 − h plane at the 68.3%, and95.4% confidence level. Results are not shown for “WL” which were calculated much lessaccurately (see text).
Chapter 2. Constraining modified gravity 33
0.65 0.7 0.7510
−6
10−5
10−4
10−3
10−2
h
|fR
0|
BAO
BAO+WL
NL WL
BAO+NL WL
Figure 2.3: Same as Figure 2.2 but for a 100 m cylindrical telescope.
trix. However it is noted that in these cases, the predicted limits on the curvature are
|ωk| < 0.025 at 95% confidence. The current, model independent, limits on the curvature
using WMAP, SDSS and HST data are approximately half this value [Komatsu et al.,
2011]. Our neglect of any direct probes of the expansion history for the Planck+WL con-
straints is clearly unrealistic; however, the constraints illustrate what is actually measured
by weak lensing. In any case these degeneracies are broken once BAO measurements are
included, and in this final case the modified gravity parameters are correlated with the
other parameters by at most 35%. Also, considering lensing in the nonlinear regime
breaks the degeneracy to a certain extent.
First generation cylindrical telescopes will likely be smaller than the one considered
above. To illustrate the differences in constraining ability, we now present a few results
for a cylindrical radio telescope that is 100m on the side. Reducing the resolution of the
experiment degrades measurements in a number of ways. BAO measurements become
less than ideal in the higher redshift bins. The smallest scale that can be considered for
weak lensing drops to about ` = 425. A more important effect is that the lensing spectra
can not be as accurately reconstructed, dropping the effective galaxy density down to
ng/σ2e = 0.22. Figure 2.3 shows analogous results to 2.2 but for a telescope with half the
resolution.
Chapter 2. Constraining modified gravity 34
0.95
1
1.05
dA
,DG
P/d
A,fid
ucia
l
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.95
1
1.05
Scale factor, a
HD
GP/H
fid
ucia
l
Figure 2.4: Ratio of the coordinate dA(z) (top) and the Hubble parameterH(z) (bottom)as predicted by the best fit DGP model to the fiducial model. Error bars are from 21 cmBAO predictions. Fit includes BAO data available from the 200 m telescope and CMBpriors on θs and ωm.
To show that a set of measurements consistent with the fiducial model would be
inconsistent with DGP we first fit DGP to the fiducial model’s CMB and BAO expansion
history by minimizing
χ2 = (rDGP − rfiducial)TC−1(rDGP − rfiducial), (2.16)
where rDGP and rfiducial are vectors of observable quantities as calculated in the DGP
and fiducial models, and C is the covariance matrix. r includes BAO dA(z) and H(z) as
well as Planck priors on ωm and θs. Note that χ2 is not truly chi-squared since rfiducial
contains fiducial model predictions and is not randomly distributed like a real data set.
Performing the fit yields DGP parameters: h = 0.677, ωm = 0.112, ωk = −0.0086 and
ωrc = 0.067. Figure 2.4 shows the deviation of H and dA respectively for the best fit DGP
model compared to the fiducial model. χ2 = 332.8 for the fit despite there only being 16
degrees of freedom, and as such a measurement consistent with the fiducial model would
thoroughly rule out DGP.
In the case that expansion history measurements are consistent with DGP, the ques-
Chapter 2. Constraining modified gravity 35
0 20 40 60 80 100 120 1400
0.5
1
1.5
2
l2C
(l)/
2π
l
Dark Energy
DGP
Figure 2.5: Weak lensing spectra in for DGP and a smooth dark energy model withthe same expansion history. DGP parameters are h = 0.665, ωm = 0.116, ωk = 0 andωrc = 0.06. Errorbars represent expected accuracy of the 200 m telescope.
tion arises as to whether DGP could be distinguished from a smooth dark energy model
that had the same expansion history. The additional information in linear perturbations
as measured by weak lensing allows DGP to be distinguished even from a dark energy
model with an identical expansion history. Figure 2.5 shows the lensing spectra for a
DGP cosmology similar to the best fit discussed above, as well as the dark energy model
with the same expansion history as in Fang et al. [2008].
In principle one should consider the small amount of freedom within the DGP pa-
rameter set that could be used to make the DGP spectrum better fit the dark energy
spectra. However this is unlikely to significantly change the spectrum as all relevant
parameters are tightly constrained by the CMB and BAO. For example it is clear from
Figure 2.5 that the lensing spectra of the two models would better agree if the amplitude
of primordial scaler perturbations was increased in the DGP model. However, Planck
measurements would only allow of order half a percent increase while the disagreement is
of order 10%. This is justified by the lack of correlations found in the f(R) fisher matrices
once all three observational techniques are included. In addition we have not considered
information from weak lensing in the nonlinear regime. Adding nonlinear scales would
Chapter 2. Constraining modified gravity 36
only make our conclusion that DGP and smooth dark energy are distinguishable with
these observations more robust.
2.6 Discussion
We have shown that the first generation of 21 cm intensity mapping instruments will
be capable of constraining the HS f(R) model (with n = 1) down to a field value of
|fR0| . 2×10−5 at 95% confidence (Figure 2.3). This is an order of magnitude tighter than
constraints currently available from galaxy cluster abundance [Schmidt et al., 2009b].
Furthermore, model parameters in this regime are not ruled out by Solar System tests.
In comparing Figures 2.2 and 2.3 it is clear that a more advanced experiment, with
resolution improved by a factor of two, would further half the allowed value of |fR0|. It
should be noted however, that halving of the allowed parameter space does not correspond
to a factor of four increase in information. Deviations in the lensing spectrum scale sub-
linearly in the f(R) parameters, enhancing the narrowing of constraints as information
is added (see Section 2.5).
While we have concentrated on a particular f(R) model, many viable functional forms
for f(R) have been proposed in the literature [Nojiri and Odintsov, 2007, Starobinsky,
2007, Appleby and Battye, 2007]. The predictions for the growth of structure in these
different models agree qualitatively: the gravitational force is enhanced by 4/3 within λC ,
enhancing the growth on small scales. However, there are quantitative differences in the
model predictions due to the different evolution of λC over cosmic time. Our results for
the HS model with different values of n should thus cover a range of different functional
forms for f(R). Table 2.1 shows that our constraints do not depend very sensitively on
the value of n. This is because the weak lensing measurements cover a wide range of
scales as well as redshifts. Furthermore, it is straightforward to map the enhancement
in the linear P (k, z) at given k and z from the HS model considered here to any other
given model, to obtain approximate constraints for that model.
Future cluster constraints will almost certainly improve on the current limits of
|fR0| . few10−4 [Schmidt et al., 2009b]. However, for smaller field values, the main effect
of f(R) gravity shifts to lower mass halos, since the highest mass halos are chameleon-
screened (see Fig. 2 in Schmidt et al. [2009a]). Hence, future cluster constraints will
depend on the ability to accurately measure the halo abundance at masses around few
1014M and less. Furthermore, the constraints from cluster abundances depend sensi-
tively on the knowledge of the cluster mass scale, and are already systematics-dominated
[Schmidt et al., 2009b]. Weak lensing constraints have a completely independent set of
Chapter 2. Constraining modified gravity 37
observational systematics, and are in principle less sensitive to baryonic or astrophysical
effects. Thus, the forecasted constraints on modified gravity presented here are quite
complementary to constraints from cluster abundances.
The processes that produce the BAO feature in the matter power spectrum are un-
derstood from first principles. In addition the BAO length scale can be extracted even in
the presence of large uncertainties in biases and mass calibrations. Likewise, weak lensing
on large scales is well understood, with baryonic physics being much less important than
on smaller scales [Zhan and Knox, 2004]. In addition the dominant systematics present
in optical weak lensing surveys are instrumental in nature and not intrinsic to the quan-
tities being measured. While 21 cm intensity mapping is as yet untested, instrumental
systematics will be very different from those that affect the optical.
In the case of this study, and more generally for cosmological models which sub-
stantially modify structure formation, the motivation for higher resolution comes not
from improved BAO measurements but from better weak lensing reconstruction. Higher
resolution not only makes weak lensing information available at higher multi-poles, but
improves the accuracy at which lensing can be reconstructed on all scales.
The inclusion of lensing information in the nonlinear regime was crucial, and largely
responsible for the competitiveness of these forecasts. As seen in Figure 2.1, much of
the constraints come from multi-poles in the nonlinear regime. It should be noted that
for the higher resolution experiment considered, the minimum scale is limited not by
the resolution at high redshift, but by the saturation of information in nonlinear source
structures at low redshift [Lu et al., 2010].
Our constraints from lensing are conservative since only one wide source redshift bin
was considered, limited to ` < 600 as described above. To maximize information, the
source redshift range could be split into multiple bins, properly considering the correlation
in the lensing signal between them; a process known as lensing tomography. The low
redshift bin would be limited as above, and the high redshift bin would be limited by the
resolution to ` ≈ 850 at z = 2.5. However in intermediate bins, the lensing signal could
be reliably reconstructed above ` ≈ 1000.
Unlike most smooth dark energy models, such as quintessence, constraints on the mod-
els considered here are chiefly sensitive to structure formation, as is clear from Figure 2.2.
These forecasts show that 21 cm intensity mapping is not only sensitive to a cosmology’s
expansion history through the BAO, but also to structure growth through weak lens-
ing. The weak lensing measurements cannot compete with far off space based surveys
like Euclid or JDEM, which will have galaxy densities of order 100 arcmin−2[Albrecht
et al., 2006] and resolution to far greater `. However, cylindrical 21 cm experiments are
Chapter 2. Constraining modified gravity 38
realizable on a much shorter time scale and at a fraction of the cost. In addition, the
measurements considered here are approaching the limit at which f(R) models can be
tested. For |fR0| much less than 10−6 the chameleon mechanism becomes important be-
fore there are observable modifications to structure growth, reducing the motivation to
further study these models.
It has also been shown that, for these experiments, a BAO measurement consistent
with ΛCDM would definitively rule out DGP without a cosmological constant as a cosmo-
logical model. Even in the case that a BAO measurement consistent with DGP is made,
the model is still distinguishable from an exotic smooth dark energy model through struc-
ture growth. The former result is not surprising given that DGP is now in conflict with
current data [Fang et al., 2008]. However it is illustrative that a single experiment can
precisely probe both structure formation and expansion history. Even a dark energy
model that conspires to mimic DGP is, to a large extent, distinguishable.
We have studied the effects of modified gravity theories on observational quantities for
future 21 cm surveys. Because these surveys measure the distribution of galaxies on large
angular scales over large parts of the sky, they are well suited to measure the expected
deviations relative to standard general relativity. We have computed the predictions of
modified gravity in the linear and nonlinear regimes, and compared to the sensitivity of
future surveys. We find that a large part of parameter space can be tested.
Acknowledgements
We would like to thank Tingting Lu for helpful discussions. KM is supported by an
NSERC Canadian Graduate Scholars-M scholarship. FS is supported by the Gordon
and Betty Moore Foundation at Caltech. PM acknowledges support of the Beatrice D.
Tremaine Fellowship.
Chapter 3
Primordial gravity waves fossils and
their use in testing inflation
A version of this chapter was published in Physical Review Letters as “Primordial Gravity
Wave Fossils and Their Use in Testing Inflation”, Masui, K. W. and Pen, U.-L., Vol. 105,
Issue 16, 2010. Reproduced here with the permission of the APS.
3.1 Summary
A new effect is described by which primordial gravity waves leave a permanent signature
in the large scale structure of the Universe. The effect occurs at second order in per-
turbation theory and is sensitive to the order in which perturbations on different scales
are generated. We derive general forecasts for the detectability of the effect with future
experiments, and consider observations of the pre-reionization gas through the 21 cm line.
It is found that the Square Kilometre Array will not be competitive with current cosmic
microwave background constraints on primordial gravity waves from inflation. However,
a more futuristic experiment could, through this effect, provide the highest ultimate sen-
sitivity to tensor modes and possibly even measure the tensor spectral index. It is thus
a potentially quantitative probe of the inflationary paradigm.
3.2 Introduction
It has been proposed that redshifted 21 cm radiation, from the spin flip transition in
neutral hydrogen, might be a powerful probe of the early universe. The era before
the first luminous objects reionized the universe–around redshift 10–contains most of
39
Chapter 3. Detecting primordial gravity waves 40
the observable volume of the universe, and 21 cm radiation is the only known probe of
these so called dark ages (see Furlanetto et al. [2006] for a review). The density of the
hydrogen could be mapped in 3D analogous to how the cosmic microwave background
(CMB) is mapped in 2D. The wealth of obtainable statistical information may allow for
the detection of many subtle effects which could probe the early universe. In particular,
the primordial gravity wave background, also referred to as tensor perturbations, are of
considerable cosmological interest.
Inflation robustly predicts the production of tensor perturbations with a nearly scale-
free spectrum, however, their amplitude is essentially unconstrained theoretically. The
amplitude of the tensor power spectrum is quantified by r, the tensor to scalar ratio.
The current upper limit is r < 0.24 at 95% confidence [Komatsu et al., 2011], however
upcoming CMB measurements will be sensitive down to r of a few percent [Burigana
et al., 2010]. The current limits on r correspond to characteristic primordial shear on the
order of 10−5 per logarithmic interval of wavenumber.
Several probes of gravity waves using the pre-reionization 21 cm signal have been
proposed. These include polarization [Lewis and Challinor, 2007] and redshift space
distortions [Bharadwaj and Sarkar, 2009]. Dodelson et al. [2003] considered the weak
lensing signature of gravity waves and found that the signal is sensitive to the so called
metric shear. This is closely related to the present work.
Here we describe a mechanism by which primordial gravitational waves may leave an
imprint in the statistics of the large scale structure (LSS) of the universe. This signature
becomes observable when the gravity wave enters the horizon and begins to decay.
3.3 Mechanism
In the following, Greek indices run from 0 to 3 and lower case Latins from 1 to 3. Latin
indices are always raised and lowered with Kronecker deltas. Commas denote partial
derivatives, and an over-dot (#) represents a derivative with respect to the cosmological
conformal time. Finally, we adopt a mostly positive metric signature (−1, 1, 1, 1).
We start with an inflating universe with some distribution of previously generated
tensor modes that are now super horizon (have wave-length much longer than the horizon
scale). Scalar, vector and smaller scale tensor modes may exist but their contribution to
the metric is ignored. The line element is given by
ds2 = a(η)2[−dη2 + (δij + hij)dx
idxj]. (3.1)
Chapter 3. Detecting primordial gravity waves 41
where a is the scale factor, η the conformal time and a spatially flat background geometry
has been assumed. The metric perturbations hij are assumed to be transverse and
traceless and thus contain only tensor modes. The elements of hij are also assumed
to be small such that only leading order terms need be retained. The assumption that
all tensor modes under consideration are super horizon implies that kh a/a, where kh
denotes the wave numbers of tensor modes. The frame in which the line element takes
the form in Eq. 3.1 will hereafter be referred to as the cosmological frame (CF).
By the equivalence principle, it is possible to perform a coordinate transformation
such that the space-time appears locally Minkowski at a point. New coordinates are
defined in which the tensor modes are gauged away at the origin:
xα = (xα +1
2hαβx
β), (3.2)
where the elements h0α are taken to be zero. The metric now takes the form (up to first
order in hij)
ds2 = a2[−dη2 + δijdx
idxj − xc∂αhβcdxαdxβ]. (3.3)
This frame will be loosely referred to as the locally Friedmann frame (LFF), because in
these coordinates the metric is locally that of an unperturbed FLRW Universe. We will
give quantities in these coordinates a tilde (#) to distinguish them from their counterparts
in the CF. It is seen from Eq. 3.3 that the local effects of gravity waves are suppressed
not only by the smallness of hij but also by kh/k where k = L−1 and L is some length
scale of interest. This will be important in justifying some later assumptions. Note that
for super horizon gravity waves, temporal derivatives are much smaller than spacial ones.
On small scales, inflation generates scalar perturbations which are then carried to
larger scales by the expansion. By the equivalence principle, physical processes on small
scales can not know about the long wavelength tensor modes. As such these small scale
scalar modes must be uncorrelated with the long wavelength tensor modes. We assume
statistical homogeneity and isotropy in the LFF as would be expected from inflation.
The power spectrum of scalar perturbations can then be written as a function of only
the magnitude of the wave number, i.e., P (ka) = P (k). This applies only within the
local patch near the point where the tensor mode was gauged away. The average in the
definition of the scalar power spectrum is over realizations of the scalar map, but not the
tensor map.
In the CF, the isotropy is broken. Transforming back to cosmological coordinates
Chapter 3. Detecting primordial gravity waves 42
maps ki → ki − kjh ji /2. The power spectrum becomes sheared:
P (ka) = P (k)− kikjhij
2k
dP
dk+O(
khkhij) +O(hij
2). (3.4)
If the metric perturbations are not assumed to be traceless, the right hand side of this
equation gains an additional term proportional to this trace. This deviation from isotropy
is not observable since any possible observation would take place in the LFF.
It is noted that the leading order correction to CF power spectrum is not suppressed
by kh/k. It is therefore not expected that the residual terms in the LFF metric (Eq. 3.3)
can break isotropy to undo CF anisotropy. However it was the CF in which the power
spectrum should be isotropic, then there would be observable anisotropy in the LFF.
This would be a violation of the equivalence principle, since an experiment local in both
space and time would be able to detect the super horizon tensor modes by measuring the
power spectrum of the locally generated scalar perturbations.
We would now like to evolve the system to some later time when observations can
be made. Ignoring the internal dynamics of the scalar perturbations, we solve for their
evolution as if they were embedded in a sea of test particles. This is trivial since an
object at coordinate rest in the CF will remain at rest for any time dependence of hij
(this is true at all orders). At some point well after inflation, when the universe is
in its deceleration phase, the horizon will become larger than the length scale of the
tensor modes. The tensor modes will then decay by redshifting, and after some period
of time the metric perturbations hij become negligible. The CF and LFF then become
equivalent and both correspond to the frame in which observations can be made. The
distribution of test particles is the same as it initially was in the CF. As such, the initially
physically isotropic power spectrum now contains a measurable local anisotropy given by
Eq. 3.4. The values of the initial metric perturbations can be determined by measuring
this distortion at any time in the future, constituting a fossil of the initial tensor modes.
The scalar perturbations remain Gaussian but become non-stationary, and the trispec-
trum gains the corresponding terms. This is analogous to the apparent distortions ex-
pected in the CMB and 21 cm fields induced by gravitational lensing. Similarly the
bispectra of mixed scalars and tensors were calculated in Maldacena [2003], employing
similar methodology to that presented here.
The effect described here is a second order perturbation theory effect, in that it is a
small effect due to tensor modes on the already small scalar perturbations. This coupling
occurs in the initial conditions, not between the dynamics of the scalars and tensors. The
simple argument presented above avoided the complication of a full second order calcula-
Chapter 3. Detecting primordial gravity waves 43
tion, but it is expected that such calculations would yield the same results. Specifically,
an expression agreeing with Eq. 3.4, to relevant order, was derived in Giddings and Sloth
[2011a, Eq. 4.5] as part of a longer calculation.
3.4 Tests of inflation
The above arguments relied on perturbations on large scales being generated before per-
turbations on small scales. This is the case in any conceivable model of inflation, however
it is not be the case in all scenarios. As an illustrative example, in the cosmic defect
scenario perturbations are generated on small scales and then causally transported to
larger scales as the universe evolves. It is argued that in this scenario, tensor perturba-
tions leave no fossils, i.e. the described effect does not occur. A detection of primordial
tensors by another means (CMB B-modes for example) with an observed lack of the
corresponding fossils would provide a serious challenge to inflation.
The most specific prediction of single field inflation is the power spectrum of tensor
modes, defined by
(2π)3δ(ka − k′a)Ph(ka) ≡ 〈hij(ka)hij(k′a)〉. (3.5)
Given the amplitude of the scalar power spectrum As, the tensor power spectrum is fixed
by a single parameter, the tensor to scalar ratio r. The shape of the spectrum is then
nearly scale-free:
Ph =2π2rAsk3
(k
k0
)nt. (3.6)
We follow the WMAP conventions for defining Ph, As and r [Komatsu et al., 2009]. The
spectral index fixed by the consistency relation, nt = −r/8 [Liddle and Lyth, 2000]. The
pivot scale is taken to be k0 = 0.002 Mpc−1 and we assume the WMAP7 central value
for As of 2.46× 10−9.
Because r is likely small, any deviation from a scale-free spectrum will be difficult
to measure, making the verification of the consistency relation correspondingly difficult.
The CMB is sensitive primarily to large scale tensor modes, with smaller scale modes
having decayed by recombination. Cosmic variance and lensing contamination will likely
prevent a measurement of nt from the CMB, unless the lensing can be cleaned from the
signal [Zhao and Baskaran, 2009]. Conversely, the amplitude of the fossil signal does not
decay as the universe expands. It may thus be possible to make a measurement of the
spectral index, provided r is sufficiently large.
Chapter 3. Detecting primordial gravity waves 44
3.5 Statistical detection in LSS
In practice, the tensor gravity wave fossils could be reconstructed by applying quadratic
estimators to the density field. Aside from the increased dimensionality, this is identical
to the manner in which lensing shear is reconstructed [Zaldarriaga and Seljak, 1999, Lu
and Pen, 2007]. Rather than considering the statistics of such estimators, here we follow
a simpler line of reasoning to approximate the accuracy to which the tensor parameter
can be measured.
We begin by asking how well a long wavelength, tensor mode can be reconstructed
from its effects on the scalar power spectrum (Eq. 3.4). The metric perturbations are
assumed to be spatially constant and take the form
hij = h+e+ij(z) + h×e
×ij(z) (3.7)
where e+ij and e×ij are the polarization tensors and the z direction of propagation is chosen
for convenience. The uncertainty on the scalar power spectrum is
[∆P (ka)]2 = 2 [P (ka) +N ]2 , (3.8)
where N is the noise power. We use a Fisher Matrix analysis to sum this information over
all ka to determine the corresponding uncertainty on the shear h+ and h×. Assuming
an experiment whose noise is sub dominant to sample variance (N P ), the resulting
variance is inversely proportional to the number of modes surveyed:
(∆hC
)2 ∼[V (kmax/2π)3
]−1, (3.9)
where h stands for either h+ or h× (the superscript C indicates that the formula applies
for spatially constant h), V is the volume of the survey and kmax is set by the resolution
of the survey. The constant of proportionality depends on the shape of the unsheared
power spectrum P (k), but to within a few tens of percent it is unity. 21 cm emission will
be difficult to observe on large scales [Furlanetto et al., 2006], however it is small scales
that dominate the number of modes and thus the reconstruction. It is only the coherence
of small scale anisotropy that must be measured on large scales.
Given the reconstruction uncertainty on a spatially constant shear, and the fact that
reconstruction noise is scale independent (white) [Zaldarriaga and Seljak, 1999], the noise
Chapter 3. Detecting primordial gravity waves 45
power spectrum for spatially varying tensor modes is then
Nh = 4V (∆hC)2 = 4
(2π
kmax
)3
. (3.10)
The factor of four comes from the definition of the power spectrum in Eq. 3.5, noting
that 〈hijhij〉 = 4〈h2〉.We now sum over ka
1 to determine the signal to noise as a function of tensor power
spectrum amplitude r. The signal to noise ratio squared is then
SNR2 =∑
ka,+,×
P 2h
2(Nh + Ph)2(3.11)
≈ V
∫ kupper
klower
dk k2
2π2
P 2h (k)
(Nh + Ph)2. (3.12)
It is seen from the redness of the spectrum Ph (Eq. 3.6) that the result is completely
independent of the upper limit of integration. The same redness makes the final result
extremely sensitive to the lower limit. As described above, the fossil of a primordial
tensor mode can only be observed once the mode has decayed. This begins to happen
when the scale of the gravity wave becomes comparable to the horizon scale, and as such,
the largest scale observable mode has wavelength klower ≈ aH.
For an initial detection, we assume that noise dominates sample variance at each ka,
i.e., Nh Ph. Setting the signal to noise ratio to be 2, for a 95% confidence detection,
yields a minimum detectable amplitude of
rmin =32π2
Ask3max
(6
V VH(z)
)1/2
(3.13)
where VH ≡ (aH)−3.
While the observability of 21 cm radiation depends on the reionizaton model, one
regime in which a strong signal may exist is near redshift 15 [Furlanetto et al., 2006]. The
planned Square Kilometer Array (SKA) will aim to probe this era with 10 km baselines
[Dewdney et al., 2009]. Assuming a survey volume of 200 (Gpc/h)3 and a noiseless
measurement, the limit on r achievable with SKA will be
rmin ≈ 7.3
(1.2 Mpc/h
kmax
)3 [200 (Gpc/h)3
V
3.3 (Gpc/h)3
VH
]1/2
. (3.14)
1From this point forward, ka will refer to the wave number of a tensor mode, not a scalar mode. The
exception will be kmax which is the smallest scale at which a scalar can be resolved.
Chapter 3. Detecting primordial gravity waves 46
While this constraint is not competitive with current constraints from the CMB, it is a
strong function of the resolution of the experiment. The Low Frequency Array (LOFAR)
for instance, has baselines extending to 400 km. However LOFAR will not have the
sensitivity to probe the dark ages [Rottgering et al., 2003]. It is the physical shear due
to gravity waves at the source that is being measured, and all light propagation effects,
such as the lensing considered in Dodelson et al. [2003], have been ignored.
Similar arguments are used to find the achievable error on the spectral index nt.
Properly considering the degeneracy with r, the error on nt is:
∆nt = F
[(2π
kmax
)31
rAsV
]1/2
, (3.15)
where F is a function of the combination of parameters VH/(k3maxrAs). In the limit that
Ph(k = aH) Nh, which is the limit in which a measurement of nt is possible, F is
approximately 6. Assuming the same volume and redshift as above, and that r = 0.1,
the consistency relation is tested at the 2 sigma level for kmax = 168h/Mpc. The tensor
power spectrum and error bars for this scenario are shown in Fig. 3.1.
Such a measurement is very futuristic indeed, requiring a nearly filled array with
greater than thousand kilometre baselines. Note that such an experiment would be
sensitive to r down to the 10−6 level. Also, higher redshifts contain even more information,
though their observation is technically more challenging.
3.6 Discussion
Aside from the technical challenge of mapping the 21 cm signal over hundreds of cubic
gigaparsecs and down to scales smaller than a megaparsec, there may be other competing
effects that could hinder a detection. Of primary concern is weak lensing which also
shears observed structures, creating apparent local anisotropies. The weak lensing shear
is of order a few percent, and is thus many orders of magnitude greater than gravity
wave shear. However, the 3D map of gravity wave shear will be transverse, transforming
intrinsically as a tensor. To linear order, the lensing pattern is the gradient of a scalar.
Even at higher order, lensing always maps one point in space to another and is thus
at most vector like. This test does not exist for the CMB or lensing due to the lower
dimensionality of these probes.
Also of concern is the preservation of the anisotropy on small scales. The scale
corresponding to k = 168h/Mpc is still larger than the Jeans length at these redshifts,
Chapter 3. Detecting primordial gravity waves 47
10−2
10−1
2.2
2.25
2.3
2.35
2.4
2.45
2.5
k (h/Mpc)
10
10k
3P
h/(
2π
2)
Figure 3.1: Primordial tensor power spectrum obeying the consistency relation forr = 0.1. The solid line is the tensor power spectrum. Error bars represent the recon-struction uncertainty on the binned power spectrum for a noiseless experiment, surveying200 (Gpc/h)3 and resolving scalar modes down to kmax = 168h/Mpc. The dashed, nearlyvertical, line is the reconstruction noise power. The non-zero slope of the solid line is thedeviation from scale-free.
and as such hydrogen should trace the dark matter. However, the evolution of scalar
perturbations is mildly nonlinear, and it is possible that this evolution will erase the
anisotropy. Detailed analysis of the nonlinear erasure of the anisotropy is deferred to
future investigation.
There has been much recent interest in searching for anisotropy, and this has some
implications for the fossil signal. The constraints on quadrupolar isotropy in LSS by
Pullen and Hirata [2010] should already imply a weak constraint at the r . 106 level.
Constraints from the CMB are not relevant however, since modes spanning the surface
of last scatter remain super horizon today.
CMB B-modes will be the most sensitive probe of primordial gravity waves in the
next generation of experiments. However, fossils may eventually be sensitive well below
the limits of the CMB.
3.7 Addendum
At the time of writing of this thesis, the observability of the effect presented in this
chapter was disputed by Pajer et al. [2013]. Their central argument is that if there is
a separation of scales (in this chapter’s notation kh k) only tidal fields are locally
observable. In this case all observables must be proportional to (k/kh)2. This is in
Chapter 3. Detecting primordial gravity waves 48
contrast to the leading order fossil effect, given in Equation 3.4, where the anisotropy is
proportional to the amplitude of the tensor mode (hij) and with kh not appearing until
higher order.
The source of the discrepancy is the extra information gained by having a prolonged
observation rather than an instantaneous one. We will use a simplified version of the
Laser Interferometer Gravitational Wave Observatory (LIGO) [Abramovici et al., 1992]
to illustrate this argument. Our simplified LIGO has two mirrors with separation (d)
of roughly a kilometre, and who follow geodesics in one dimension by being suspended
vertically on wires. The distance between the mirrors is precisely measured by setting up
an interferometer between them. In the presence of a gravitational wave, with a wave-
length of several thousand kilometres, the proper distance between the mirrors oscillates,
allowing for the detection of the gravity wave. It is true that instantaneously only the
acceleration of the mirrors is observable, leading to a suppressed signal whose amplitude
isd2d
dt2∼ ω2
hhijd ∼ c2k2hhijd, (3.16)
which is indeed suppressed by k2h. However LIGO does not make an instantaneous
measurement of acceleration, but a measurement of distance extended in time. If the
measurement is performed over several periods of the gravitational wave, the signal is
∆d(t) ∼ hijd, no longer suppressed by the separation of scales.
The situation is analogous for the fossil signal in which changes in position are mea-
sured through the local power spectrum of the scalar perturbations. The integral over
time, which we just argued is necessary to avoid the kh/k suppression, is provided by the
geodesic motions of matter throughout the evolution and decay of the gravitational wave.
Our observation of the scalar modes is instantaneous, however we have an additional piece
of information: the initial isotropic shape of the power spectrum. This is because ini-
tially during inflation, before the scalar perturbations have evolved, all observables for
the super-horizon tensor modes should be suppressed by separation of scales.
Acknowledgements
We would like to thank Patrick McDonald, Latham Boyle, Adrian Erickcek, Neil Barnaby,
Neal Dalal, Chris Hirata and Eiichiro Komatsu for helpful discussions. KM is supported
by NSERC Canada.
Chapter 4
Near term measurements with 21 cm
intensity mapping: neutral hydrogen
fraction and BAO at z < 2
A version of this chapter was published in Physical Review D as “Near-term measure-
ments with 21 cm intensity mapping: Neutral hydrogen fraction and BAO at z < 2”,
Masui, K. W., McDonald, P. and Pen, U.-L., Vol. 81, Issue 10, 2010. Reproduced here
with the permission of the APS.
4.1 Summary
It is shown that 21 cm intensity mapping could be used in the near term to make cos-
mologically useful measurements. Large scale structure could be detected using existing
radio telescopes, or using prototypes for dedicated redshift survey telescopes. This would
provide a measure of the mean neutral hydrogen density, using redshift space distortions
to break the degeneracy with the linear bias. We find that with only 200 hours of
observing time on the Green Bank Telescope, the neutral hydrogen density could be
measured to 25% precision at redshift 0.54 < z < 1.09. This compares favourably to
current measurements, uses independent techniques, and would settle the controversy
over an important parameter which impacts galaxy formation studies. In addition, a
4000 hour survey would allow for the detection of baryon acoustic oscillations, giving a
cosmological distance measure at 3.5% precision. These observation time requirements
could be greatly reduced with the construction of multiple pixel receivers. Similar results
are possible using prototypes for dedicated cylindrical telescopes on month time scales,
49
Chapter 4. Forecasts for near term experiments 50
or SKA pathfinder aperture arrays on day time scales. Such measurements promise to
improve our understanding of these quantities while beating a path for future generations
of hydrogen surveys.
4.2 Introduction
An upcoming class of experiments propose the observation of the 21 cm spectral line over
large volumes, detecting large scale structure in three dimensions [Peterson et al., 2009].
This method is sensitive to a redshift range of z & 1, which is observationally difficult
for optical experiments, due to a dearth of spectral lines in the atmospheric transparency
window. Such experiments require only limited resolution to resolve the structures of
primary cosmological interest, above the non-linear scale. At z ≈ 1 this corresponds to
tenths of degrees. There is no need to detect individual galaxies, and in general each
pixel will contain many. This process is referred to as 21 cm intensity mapping. A first
detection of large scale structure in the 21 cm intensity field was reported in Pen et al.
[2008].
Intensity mapping is sensitive to the large scale power spectra in both the transverse
and longitudinal directions. From this signal, signatures such as the baryon acoustic
oscillations (BAO), weak lensing and redshift space distortions (RSD) can be detected
and used to gain cosmological insight.
To perform such a survey dedicated cylindrical radio telescopes have been proposed
[Peterson et al., 2006, Seo et al., 2010], which could map a large fraction of the sky
over a wide redshift range and on timescales of several years. These experiments are
economical since their low resolution requirements imply limited size and they have no
moving parts. It has been shown that BAO detections from 21 cm intensity mapping
are powerful probes of dark energy, comparing favourably with Dark Energy Task Force
Stage IV projects within the figure of merit framework [Chang et al., 2008, Albrecht
et al., 2006]. Additionally, in Masui et al. [2010b] it was shown that such experiments
could tightly constrain theories of modified gravity, making extensive use of weak lensing
information. The f(R) theory, for instance, could be constrained nearly to the point
where the chameleon mechanism masks any deviations from standard gravity before
they first appear.
While such experiments will be very powerful probes of the Universe, it is useful to ex-
plore how 21 cm intensity mapping could be employed in the short term. Here we discuss
surveys that could be performed using existing radio telescopes, where the Green Bank
Telescope will be used as an example. Prototypes for the above mentioned cylindrical
Chapter 4. Forecasts for near term experiments 51
telescopes (for which some infrastructure already exists) are also considered. Finally, the
Square Kilometre Array Design Studies (SKADS) focused on developing aperture array
technology for the Square Kilometre Array [Faulkner et al., 2010]. Motivated by the pro-
posed Aperture Array Astronomical Imaging Verification (A3IV) project1, we consider
pathfinders for such arrays. These aperture arrays could share many characteristics in
common with cylindrical telescope prototypes, but would be capable of much greater
survey speed.
While these limited resources would not have nearly the statistical power required
to detect effects like weak lensing, a detection of the RSD would be possible. This
would give a measure of the mean density of neutral hydrogen in the Universe. This has
been an important and controversial parameter in galaxy formation studies and a precise
measurement would be invaluable in this field [Putman et al., 2009]. In addition BAO
are considered, a detection of which would yield cosmologically useful information about
the Universe’s expansion history.
In this paper we first describe the RSD and BAO and the information that can be
achieved with their detection. We then present forecasts for 21 cm redshift surveys as a
function of telescope time, followed by a brief discussion of these results.
We assume a fiducial ΛCDM cosmology with parameters: Ωm = 0.24, Ωb = 0.042,
ΩΛ = 0.76, h = 0.73, ns = 0.95 and log10As = −8.64; where these represent the
matter, baryon and vacuum energy densities (as fractions of the critical density), the
dimensionless Hubble constant, spectral index and logarithm of the amplitude of scalar
perturbations.
4.3 Redshift Space Distortions
In spectroscopic surveys, radial distances are given by redshifts. However redshift does
not map directly onto distance as matter also has peculiar velocities, which Doppler
shifts incoming photons. On large scales, these velocities are coherent and result in
additional apparent clustering of matter in redshift space. In linear theory, the net effect
is an enhancement of power for Fourier modes with wave vectors along the line of sight
[Kaiser, 1987],
P sX(~k, z) = b2
[1 + β(z)µ2
k
]2P (k, z). (4.1)
Here P sX is the power spectrum of tracer X (assumed to be linearly biased) as observed in
redshift space, P is the matter power spectrum and µk = k · z. The bias, b, quantifies the
1[van Ardenne, 2009], http://www.ska-aavp.eu/
Chapter 4. Forecasts for near term experiments 52
degree to which the density perturbation of the tracer follows the density perturbation of
the underlying dark matter. The redshift space distortion parameter β is equal to f/b in
linear theory, where f is the dimensionless linear growth rate. To a good approximation
f(z) = Ωm(z)0.55.
We define the signal neutral hydrogen power spectrum as
PHI(~k, z) ≡ x2HIP
sHI(~k, z), (4.2)
where xHI(z) is the mean fraction of hydrogen that is neutral. It is the parameter xHI that
we wish to measure. The above definition is useful because it defines the quantity that is
most directly measured in a 21 cm redshift survey. Note that in discussions of the more
standard galaxy redshift surveys one usually takes a measurement of the mean density
for granted; however, 21 cm intensity mapping will make differential measurements and
will not measure the mean signal directly. One cannot divide out the mean to define a
measurement of fluctuations around the mean. Using the fact that at late times, on large
scales, structure undergoes scale independent growth, the above quantity can be written
as
PHI(~k, z) = b2x2HI
(g(z)a
a0
)2 [1 + βµ2
k
]2P (k, z0), (4.3)
where subscript 0 refers to some early time well after recombination and g is the growth
factor (relative to an Einstein-de Sitter Universe).
Because the power spectrum at early times, P (k, z0), can be inferred from the Cosmic
Microwave Background (to good enough accuracy for our purposes in this paper), the
observed power spectrum can be parameterized by just two redshift dependent numbers,
β and the combination of scale independent prefactors in Equation 4.3,
AH ≡ b2x2HIg
2. (4.4)
It is these two parameters that can be determined from a 21 cm redshift survey using the
RSD. In general AH will be better measured than β and does not significantly contribute
to the uncertainty in xHI.
The factors f(z) and g(z) depend on the expansion history. If one allows for a
general expansion history (for instance, the WMAP allowed CDM model with arbitrary
curvature and equation of state, OWCDM) these factors are poorly determined since
there is currently little data that directly probes the expansion in this era. However, if
one is willing to assume a flat ΛCDM expansion history, then the current uncertainty in
the late time expansion is attenuated at the redshifts of interest (since dark energy is
Chapter 4. Forecasts for near term experiments 53
sub-dominant to matter at z = 1) and uncertainties in theses parameters can be ignored.
As such, a measurement of β gives a measurement of the bias b which in turn gives a
measurement of xHI . We have
∆xHI
xHI
≈ ∆β
β(ΛCDM assumed). (4.5)
If one is not willing to assume an expansion history, it is a simple matter to propagate
the corresponding uncertainties in the expansion parameters.
To estimate errors on β, we assume that the primordial power spectrum is essentially
known from the cosmic microwave background, and we fix all parameters except for the
amplitude As, spectral index ns, and running of the spectral index αs. The observable
power spectrum is then parameterized by β, As, ns and αs. We then use the Fisher
matrix formalism to determine how precisely β can be measured from the 21 cm survey,
marginalizing over the other three parameters and using no other information. The pa-
rameter As is used as a stand-in for the other parameters that affect the overall amplitude
of the power spectrum: the bias and xHI. Its marginalization is critical to account for
the fact that we have no a priori information about these parameters. The spectral index
marginalization is not strictly necessary but allows for some degradation due to concern
about scale dependence of bias. We have restricted the numerator of the Fisher matrix
to include only the linear theory power as suppressed by the non-linear BAO erasure
kernels of Seo and Eisenstein [2007] (all the linear power is included though, not BAO
only), so non-linearity should not be a significant issue at the level of precision discussed
here. This treatment of the non-linearity cutoff is also motivated by the propagator work
of [Crocce and Scoccimarro, 2006b,a, 2008].
4.4 Baryon Acoustic Oscillations
Acoustic oscillations in the primordial photon-baryon plasma have ubiquitously left a
distinctive imprint in the distribution of matter in the Universe today. This process is
understood from first principles and gives a clean length scale in the Universe’s large scale
structure, largely free of systematic uncertainties, and calibrations. This can be used to
measure the global cosmological expansion history through the angular diameter distance
vs redshift relation. The detailed expansion will differ between a pure cosmological
constant and the various other cosmological models.
We use essentially the method of Seo and Eisenstein [2007] for estimating distance
errors obtainable from a BAO measurement, including 50% reconstruction of nonlinear
Chapter 4. Forecasts for near term experiments 54
0.05 0.1 0.15 0.2
−0.1
−0.05
0
0.05
0.1
k, (h/MPc)
rela
tive
wig
gle
ampl
itude
Figure 4.1: Baryon acoustic oscillations averaged over all directions. To show the BAOwe plot the ratio of the full matter power spectrum to the wiggle-free power spectrum ofEisenstein and Hu [1998]. The error bars represent projections of the sensitivity possiblewith 4000 hours observing time on GBT at 0.54 < z < 1.09.
degradation of the BAO feature (although this is unimportant since experiments consid-
ered here have low resolution). The BAO feature is isolated by dividing the total power
spectrum [Eisenstein and Hu, 1998] by the wiggle-free power spectrum and subtracting
unity, as illustrated in Figure 4.1. The wiggles are then parameterized by an overall
amplitude, and a length scale dilation (here Aw and D respectively), which control the
vertical and horizontal stretch of the theoretical curve shown in Figure 4.1. Our errors
on Aw come from a straightforward extension of the Seo and Eisenstein [2007] method
for estimating BAO errors. In addition to the BAO distance scale as a free parameter
in our Fisher matrix, we include Aw as a free parameter. This is similar to what one
sometimes tries to do by including the baryon/dark matter density ratio as a parameter,
but more straightforward to interpret.
The ability to measure Aw (which is zero in the absence of the BAO) represents
the ability to detect the presence these wiggles. A measurement of D allows one to
associate a comoving distance to length scales on the sky. This gives a measurement of
the angular diameter distance (dA) for detections in the transverse direction, and the
Hubble parameter (H) if the wiggles are detected in the longitudinal direction.
Chapter 4. Forecasts for near term experiments 55
4.5 Forecasts
We present forecasts for the Green Bank Telescope and prototypes for two classes of
telescope: cylindrical telescopes and SKA aperture arrays. The signal available for 21 cm
experiments is proportional to the neutral hydrogen fraction and bias. For estimating
telescope sensitivity we assume that the product of the bias and the neutral hydrogen
density ΩHIb = 0.0004 today [Chang et al., 2008], and that the neutral hydrogen fraction
and bias do not evolve. These assumptions only affect the sensitivity of the telescopes
and not the translation from uncertainty on P sHI to the uncertainty on xHI. Also, as
in galaxy surveys, there is expected to stochastic shot noise component and we assume
Poisson noise with an effective object number density n = 0.03 per cubic h−1Mpc. Note
that stochastic noise at this level is negligible, as should be the case in practice.
The 21 cm intensity mapping technique is expected to be complicated by a variety
of contaminating effects. These include diffuse foregrounds (predominantly galactic syn-
chrotron), radio frequency interference and bright point sources. The degree to which
these contaminants will limit future surveys has yet to be quantified, and here we simply
ignore them. As such these forecasts are theoretical idealizations. Methods for dealing
with these contaminants are discussed in Chang et al. [2008].
The Green Bank Telescope is a 100 m diameter circular telescope with a system
temperature of 25 K. It has interchangeable single pixel receivers at the frequencies of
interest with bandwidths of approximately 200 MHz. For extended surveys, multiple
pixel receivers could be implemented. The construction of a four pixel receiver is within
reason and would reduce the required telescope time by a factor of four. In planning a
survey on GBT, it is important to choose an appropriate survey area. As illustrated in
Figure 4.2, at fixed observing time, there is a survey area that best measures the desired
parameters. For all results the survey area has been roughly optimized for the quantity
being measured. The optimized areas are shown in Figure 4.3. Results are essentially
insensitive to this area within a factor of 2 of the optimum.
Prototyping for dedicated cylindrical telescopes is in its early stages. We present
forecasts for a hypothetical not-too-far-off telescope, composed of two cylinders. The
total array measures 40 m×40 m with 300 dipole receivers with 200 MHz bandwidth, and
a system temperature of 100 K. Such telescopes have no moving parts and point solely
by the rotation of the earth. As such, the area of the survey is set by latitude, receiver
spacing, and obstruction by foregrounds; we assume 15 000 square degrees. The survey
area scales with the number of receivers leaving the noise per pixel unchanged. Thus the
squared errors on measured quantities scale inversely with the number of receivers. Note
Chapter 4. Forecasts for near term experiments 56
100
101
102
103
0
0.1
0.2
0.3
0.4
survey area (sq. deg.)
frac
tiona
l err
or
∆β/β, ∆x
HI/x
HI
∆Aw/10A
w
∆D/D
Figure 4.2: Ability of GBT to measure the BAO and redshift space distortions as afunction of survey area at fixed observing time. Presented survey is between z = 0.54and z = 1.09 and observing time is 1440 hours. A factor of 10 has been removed fromthe Aw curve.
102
103
101
102
observing time (hours)
optim
al s
urve
y ar
ea (
sq. d
eg.)
Figure 4.3: Roughly optimized survey area as a function of telescope time on GBT.Redshift range is between z = 0.54 and z = 1.09.
Chapter 4. Forecasts for near term experiments 57
that this is a slightly different scaling than the area optimized case of GBT, where it is
the time axis that is scaled by the number of receivers. However, in the area optimized
case variances also scale inversely with time as area optimization effectively fixes the
noise per pixel. As such our results for GBT follow both scalings.
The forecasts for cylindrical telescopes are also applicable to pathfinder aperture
arrays [Faulkner et al., 2010]. Where-as cylinders use optics to form a beam in one
dimension and interferometry in the other, aperture arrays directly sample the incoming
radio waves without optics. Interferometry is used in both dimensions, forming a two
dimensional array of beams, instead of a two pixel wide strip for a two cylinder telescope.
In principle it would be possible for an aperture array to monitor essentially the whole sky
simultaneously, and thus form thousands of beams. In practise digitizing every antenna
is costly and many antennas must be added in analogue in such a way that some beams
are preserved but many are cancelled. We consider a compact aperture array that has the
same area and resolution as the cylinder considered here. This is almost identical in scale
as the proposed A3IV [van Ardenne, 2009]. We assume preliminary A3IV specifications,
with 700 effective receivers, 300 MHz bandwidth, and 50 K system temperature [de Bruyn,
2010]. The aperture array could thus perform the same survey as the cylinder but a
factor of (300 MHz/200 MHz)(700/300)(100 K/50 K)2 = 14 faster. Note however, that
for aperture arrays there is added freedom in which beams are preserved when antennas
are added. It would be thus possible to optimize the area of the survey as in the GBT
case.
Figures 4.4 and 4.5 show the obtainable fractional errors on RSD parameter β and
BAO parameters Aw, the amplitude of wiggles, and D the overall dilation factor. D is
defined as a simultaneous dilation in both the longitudinal and transverse directions, i.e.
both H and dA are proportional to D. There is another “skew” parameter which trades
one for the other, such that the Hubble parameter and the angular diameter distance
can be independently determined, however, this parameter is generally not as precisely
measured [Padmanabhan and White, 2008]. D is the mode that contains most of the
information available in the BAO and we marginalize over the skew parameter. The
marginalized error on D is independent of the exact definition of the skew. Fractional
errors on H and dA are of order twice the fractional error on D.
Referring to Figure 4.1 above, it can be seen intuitively that the parameters D and Aw
are weakly correlated. Furthermore, since the signal is linear in the parameter Aw, our
linear Fisher analysis applies even for large errors. This however cannot be said about
D. Indeed any uncertainty on D, that brings the smallest scale peak we resolve more
than ∼ π/2 out of phase, cannot be trusted. It can be seen from Figure 4.1 that this
Chapter 4. Forecasts for near term experiments 58
102
103
10−1
100
frac
tiona
l err
or
observing time (hours)
∆β/β, ∆xHI
/xHI
∆Aw/A
w
∆D/D
0.18 < z < 0.480.54 < z < 1.091.06 < z < 1.79
Figure 4.4: Forecasts for fractional error on redshift space distortion and baryon acous-tic oscillation parameters for intensity mapping surveys on the Green Bank Telescope(GBT). Frequency bins are approximately 200 MHz wide and correspond to availableGBT receivers. Uncertainties on D should not be trusted unless the uncertainty on Awis less than 50% (see text).
Chapter 4. Forecasts for near term experiments 59
101
102
103
104
10−3
10−2
10−1
100
frac
tiona
l err
or
observing time (days)
∆β/β, ∆xHI
/xHI
∆Aw/A
w
∆D/D
0.14 < z < 0.360.53 < z < 0.971.06 < z < 1.94
Figure 4.5: Forecasts for fractional error on redshift space distortion and baryon acousticoscillation parameters for intensity mapping surveys on a prototype cylindrical telescope.Frequency bins are 200 MHz wide corresponding to the capacity of the correlators whichwill likely be available. These result also apply to the aperture telescope but with theobserving time reduced by a factor of 14. Uncertainties on D should not be trusted unlessthe uncertainty on Aw is less than 50% (see text). Observing time does not account forlost time due to foreground obstruction.
Chapter 4. Forecasts for near term experiments 60
corresponds to a fractional error of order 10%, which of course depends on resolution.
For this reason, we require that the uncertainty on Aw must be at most 50% (a 95%
confidence detection of the BAO) before we have any faith in the uncertainty in D. We
note that Fisher analysis is not the optimal tool for determining when an effect can first
be measured; however, it applies for any cosmologically useful measurements of the BAO.
4.6 Discussion
Typically measurements of the neutral hydrogen density at high redshift are made using
damped Lyman-α (DLA) absorption lines; current measurement uncertainties being at
the 25% statistical precision level in the 0.5 < z < 2 redshift range [Prochaska et al.,
2005, Rao et al., 2006]. However, it has been argued that these measurements are biased
high [Prochaska and Wolfe, 2009] rendering the quantity effectively uncertain by a factor
of 3. Hydrogen is the main baryonic component in the universe and it becomes neutral
after falling into galaxies and becoming self shielded from ionizing radiation. As such, the
abundance of neutral hydrogen is linked to the availability of fuel for star formation [Wolfe
et al., 1986, Pei and Fall, 1995]. Understanding how the neutral hydrogen evolves over
cosmic time is key to understanding the star formation history and feedback processes in
galaxy formation studies [Wolfe et al., 2005, Shen et al., 2009]. Additionally, the linear
bias, and any scale dependence it might have, gives valuable information about how the
gas is distributed. Finally, these quantities are crucial for estimating the sensitivity of
future 21 cm redshift surveys since the signal is proportional to the product of the bias
and the mean neutral density [Chang et al., 2008].
We have shown that even with existing telescopes, it is possible to use 21 cm intensity
mapping to make useful measurements of large scale structure at high redshift. As seen
in Figure 4.4, a 4σ detection of redshift space distortions could be made at z ≈ 0.8
with only 200 hours of telescope time at GBT. This would provide a 25% measurement
of the neutral hydrogen fraction in the Universe using methods independent of DLA
absorption lines. A longer survey using 1000 hours of telescope time could make ∼12%
measurements.
Surveys extending to this level of precision become cosmologically useful. With 4000
observing hours on GBT (1000 hours with a four pixel receiver), the BAO overall distance
scale could be measured to 3.5% precision at the same redshift. This is approaching the
precision of the WiggleZ survey, which will make a ∼2% measurement of this scale over
a similar redshift range [Blake et al., 2009, Drinkwater et al., 2010]2. Because WiggleZ
2The projected uncertainties that include reconstruction in these references are on H and dA. The
Chapter 4. Forecasts for near term experiments 61
will make a similar measurement, such a survey would not have a dramatic effect on
cosmological parameter estimations. However, it would provide an excellent verification
of these measurements, using completely different methods, in different regions of the
sky, and at low cost.
Prototypes for cylindrical telescopes could perform similar science to existing tele-
scopes except—with dedicated resources—longer integration times would be feasible.
This in part makes up for the limited resolution as for 40 m telescopes there is a sub-
stantial loss of information, and only the first wiggles are resolved. The measurements
described here would constitute a proving ground for this technique. The success of
these prototypes would be a clear indicator of the power and future success of full scale
cylindrical telescopes.
The most powerful telescope considered is the aperture array, which would be capable
of making sub present level BAO measurements with only a few weeks of dedicated
observing. The design for demonstrator telescope A3IV has yet to be finalized but the
proposed telescope is of nearly the same scale as the aperture array considered here.
Depending on its eventual configuration, the A3IV could be a powerful probe of the
Universe.
We have shown that 21 cm intensity mapping surveys could be employed in the short
term to make useful measurements with large scale structure. With relatively small initial
resource allocations, requisite techniques such as foreground subtraction can be tried and
tested while performing valuable science. Such short term applications of this promising
method will lay the trail for future dark energy surveys.
Acknowledgements
We thank Ger de Bruyn for preliminary specifications of the A3IV. KM is supported
by NSERC Canada. PM acknowledges support of the Beatrice D. Tremaine Fellowship.
uncertainty on D is inferred from these.
Part II
Pioneering 21 cm cosmology
62
Chapter 5
A data analysis pipeline for 21 cm
intensity mapping with single dish
telescopes: data to maps
This chapter forms the basis of an intended future publication with authorship lead by
myself and Tabitha Voytek. It is intended to be submitted simultaneously with a second
article describing the latter parts of the analysis—maps to power spectra—which will be
lead by Eric Switzer and Yi-Chao Li.
5.1 Introduction
Single dish telescopes are not be the most powerful instruments for performing 21 cm
large-scale structure surveys. Generally they have a small number of simultaneous pixels
on the sky, which limits their survey speed. In addition, in order to obtain sufficient
angular resolution, a single monolithic structure must be constructed, which tends to be
expensive. However, they do have some advantages. Firstly, several appropriately spec-
ified single dishes are already in existence, with time allocated through public telescope
allocation competitions. This means that pilot 21 cm surveys may commence immedi-
ately upon being granted telescope time. In addition, single dishes have far simpler
beams than interferometers, greatly simplifying the data analysis. So while single dishes
will not be used to perform the ultimate large-scale structure survey, they are ideal for
pilot surveys and early science.
Here we describe part of a software data analysis pipeline designed for performing
21 cm surveys with single dish telescopes. The pipeline was built specifically for the
63
Chapter 5. Data analysis pipeline 64
21 cm intensity mapping effort at the Green Bank Telescope in West Virginia, however,
it could be—and is—being adapted for other instruments such as the L-Band Multibeam
Receiver on the Parkes Telescope in Australia.
We describe the data analysis up to and including converting the time ordered data
into maps on the sky. The latter half of the pipeline; which includes foreground subtrac-
tion, power spectrum estimation, and compensating the power spectrum for signal lost
in foreground subtraction; will be described in detail in a separate publication.
The first detection of large-scale structure using 21 cm intensity mapping was achieved
in Chang et al. [2010]. While many of the methods and algorithms used in that analysis
were carried over to the pipeline described here, the current analysis represents a complete
overhaul of the earlier software.
Our analysis software is publicly available at https://github.com/kiyo-masui/
analysis_IM.
5.2 Time ordered data
Here, we describe the raw data from the Green Bank Telescope for which our pipeline was
designed to process. When making observations using GBT, one gets to choose both a
front-end and back-end instrument. The front-end instrument is called the receiver which
sits at one of the telescope’s focal points and collects light reflected off of the telescope
mirror. The choice of receiver is generally set by the desired observing frequency. We
have chosen to use the 800 MHz receiver which is sensitive to a band between 700 MHz
and 900 MHz, corresponding to a redshift between z = 1.0 and z = 0.58. This band is
scientifically interesting because it is at a higher redshift than the largest galaxy redshift
surveys, and because the band has been relatively free of radio frequency interference
(RFI) since the switch to digital television in the United States. The 800 MHz receiver’s
beam full width half max (angular resolution) at 700 MHz is 0.314, and at 900 MHz
it is 0.250. The system temperature, which is a measure of the noise power in the
receiver that is unrelated to the radiative power from the sky, is roughly 25 K. It has
contributions from the receiver’s thermal noise as well as radiation from the ground
entering the receiver.
The back-end instrument is responsible for sampling the voltage stream from the
receiver, converting this to a measurement of power (by squaring and integrating over
some period of time) and then writing the data to disk. The traditional back-end that
would be used for our survey is the spectrometer, which gives a measure of power as a
function of both time and frequency. However, the spectrometer at GBT is quite old and
Chapter 5. Data analysis pipeline 65
nearly obsolete. Its analogue to digital converter (ADC) resolves only three levels of the
input voltage (i.e. it is only a∼ 1.5 bit ADC) and thus frequently saturates in the presence
of excessive power, which can be caused by strong RFI. Also, it is limited to relatively long
minimum integration times of one second, which precludes the possibility of scanning the
telescope across the sky at a high rate. The GBT Ultimate Pulsar Processing Instrument
or GUPPI is a new back-end, launched in 2008, designed for observations of pulsars
[DuPlain et al., 2008]. It has an 8-bit analogue to digital converter, and has a minimum
integration time of 1µs, far shorter than what is required by our survey. The disadvantage
to using GUPPI is, having been designed for pulsar observations, the output data is in a
very inconvenient format for standard spectroscopy observations. Nevertheless, GUPPIs
advantages far outweigh this one disadvantage, which was overcome by writing a front-
end to our analysis pipeline that automatically converts the GUPPI data to a more
tractable format.
Our data, once reformatted, are a function of frequency and time. Natively the
data has 4096 frequency bins across the 200 MHz bandwidth. This is far finer frequency
resolution than is required for our large-scale structure science, with each ≈ 50 kHz
wide bin spanning a line-of-sight distance of about 0.2 Mpc/h, however the resolution is
helpful for identifying RFI as described below. The time bins initially correspond to 1 ms
integrations, although these are rebinned to ∼ 0.1 s in the early stages of the pipeline.
The data also has four polarization channels. The 800 MHz receiver has two linearly
polarized antenna, one oriented in the horizontal, x, direction and the other in the ver-
tical, y, direction. GUPPI calculates all four correlation products of the voltages from
each antenna:
P XX ≡ 〈V XV X〉 (5.1)
P XY ≡ < [〈V XV Y〉] (5.2)
P YX ≡ = [〈V XV Y〉] (5.3)
P YY ≡ 〈V YV Y〉. (5.4)
Here, these formula a understood to apply on a spectral channel-by-channel basis, and
the angled brackets, 〈〉, represent the average over an integration time bin. Modulo some
calibration factors, these products are trivially related to the familiar Stokes parameters
Chapter 5. Data analysis pipeline 66
for polarized flux:
P I = (P XX + P YY) /2 (5.5)
PQ = (P XX − P YY) /2 (5.6)
P U = P XY (5.7)
P V = P YX. (5.8)
Finally, the data has two noise-cal state channels. The 800 MHz receiver has a noise
diode capable of injecting a small amount of noise directly into the signal path. Some
care has been taken to insure that the power injected by the noise-cal is stable over time
scales spanning several days. During observations the noise-cal switches on and off with a
period of 64 ms. Our data is separated into noise-cal on and noise-cal off channels before
the time axis is rebinned. The switching of the noise cal allows for the characterization
of the system gain on short time scales, greatly improving the accuracy of subsequent
calibrations.
The data is split into individual scans of the telescope, generally one to four minutes
in length. The scans are subsequently grouped into sessions of a few hundred scans,
corresponding to a single night of observing.
5.2.1 Pipeline design
The data analysis pipeline was designed to be, above all things, modular. As long as the
data is time ordered, i.e. until map-making, it never changes format. Individual pipeline
modules generally read the data in, do a minimal set of operations on the data, and then
immediately write the data back out to a new file, but in the same format.
This design, while I/O intensive, allows individual pipeline elements to be dropped in
and out of the pipeline, or reordered within the pipeline, trivially. The pipeline also allows
for nonlinear flow, where data paths can diverge along different branches re-converging
later, or looping back on itself in an iterative process.
This versatility has been invaluable in the development of the analysis, where different
procedures for analyzing the data can be tested with minimal effort. It has also proved
to be useful in ancillary science projects, where a different procedure can be developed
without diverging code bases.
Chapter 5. Data analysis pipeline 67
5.2.2 Radio frequency interference
RFI is a major concern for all radio astronomy observations. While the 800 MHz band is
cleaner than the adjacent bands, and while Green Bank is a protected radio quite zone,
RFI still dominates thermal noise in our data in a subset of our spectral channels. This
is especially true between 850 MHz and 900 MHz, which contains some of the cell phone
bands. Complicating the flagging of RFI is the presence of bright foregrounds in our
data, which dominate the over the thermal noise and create a floor at which RFI can be
identified unless some measure is taken to remove them.
Early versions of the RFI flagging module were based on the notion of flagging anoma-
lous data points which stand out some number of standard deviations, say 4, above the
sky signal and thermal noise in the time ordered data. Having fine spectral resolution is
helpful since RFI tends to be localized in frequency. In addition, the polarized channels
of the data P XY and P YX are sensitive probes, since the RFI tends to be polarized and the
sky signal much less so [Chang et al., 2010]. The fundamental issue with these early flag-
gers is that they had the tendency to bias the data. Data that contained a bright point
source on the sky was preferentially flagged, creating a catastrophic non-linearity in the
telescope’s response. Also, this flagging algorithm responded differently to the noise-cal
on and noise-cal off channels (see Section 5.2)—the noise-cal being highly polarized—
biasing our determination of the noise-cal power required for calibration. The bias can
be reduced by raising the threshold for flagging, but this resulted in large amounts of
RFI being missed.
The solution is to flag entire frequency channels based on their variance over the
time axis (for a single scan), instead of flagging individual outlying data points. More
data ends up flagged, but the algorithm is still relatively efficient since narrow band RFI
tends to contaminate all the data in an individual spectral channel. The bias is eliminated
because the flagger has far less freedom. For data with 4096 spectral channels, the flagger
makes 4096 decisions whether or not to flag the channel. In contrast, the initial algorithm
needed to make almost a million decisions for a 60 second scan.
This initial round of flagging eliminates the worst of the RFI and allows for the
construction of an initial map. The map can be used to get a rough, unbiased, estimate
of the sky signal (dominated by foregrounds) in the time ordered data, which can be
subtracted, yielding a version of the data that is dominated by noise and RFI. The
process of using a preliminary map to estimate the signal in the time ordered data is
described in more detail in Section 5.3.2. With the bulk of the sky signal removed from
the data, a less conservative RFI flagging can be applied—based on flagging ∼ 4 sigma
excursions—without concern for biasing the data, and escaping the foreground induced
Chapter 5. Data analysis pipeline 68
750 800 850spectral channel frequency (MHz)
0
50
100
150
200
250
300
350
400
time (s)
750 800 850spectral channel frequency (MHz)
0
50
100
150
200
250
300
350
400ti
me (
s)
−0.010
−0.008
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006
0.008
0.010
Figure 5.1: Data before (left) and after (right) RFI flagging. Colour scale representsperturbations in the power, P/〈P 〉t−1. Frequency axis has been rebinned from 4096 binsto 256 bins after flagging, which fills in many of the gaps in the data left by the flagging.Any remaining gaps are assigned a value of 0 for plotting.
floor for RFI identification. The effect of flagging the data for RFI is shown in Figure 5.1.
It should be noted that our RFI flagging algorithm is highly non-linear and could
in principal bias our determination of the 21 cm signal if the flagging were somehow
correlated with the signal. However, within an individual scan, the RFI and thermal
noise dominate the signal by over an order of magnitude and as such the RFI flagging is
unlikely to correlate with the signal. For this reason we have not attempted to simulate
the signal loss induced by flagging.
5.2.3 Calibration
The power recorded by GUPPI and stored in its output files has completely arbitrary
units and must be calibrated. Not only is an absolute calibration necessary to interpret
our data in terms of cosmic hydrogen, but relative calibrations are necessary, since the
telescope gain can, and will, have some time dependence. This would affect the internal
consistency of the data and prevent the extraction of the signal. Naively, since the
foregrounds are of order 104 times as bright as the signal, one might expect that relative
Chapter 5. Data analysis pipeline 69
calibration errors better than 10−4 are required. In practise this is not the case since
gain drifts tend to be highly correlated across frequency channels, and our foreground
subtraction algorithms have some robustness against this type of gain drift. The precise
requirement on calibration precision is not known.
Preliminaries
The basic concepts used in this section are described in detail in Rybicki and Lightman
[1979] and Wilson et al. [2009]. Here we give a brief introduction to these concepts,
before detailing our calibration procedure.
There are two relevant measures of radiative power. The first is flux, S with units [Jy],
a measure of the power emitted by a source on the sky. Flux is the most relevant measure
of radiative power for point sources, i.e. sources that are much smaller than the beam
of the telescope. This is because for extended sources, the measured flux will depend
on the detailed structure of the telescope’s beam. For intensity mapping, brightness
temperature, T with units [K], is a more appropriate measure of radiative power. It
gives a measure of the flux per unit angular area on the sky. In intensity mapping we do
not identify individual sources and instead make maps of emitted radiative intensity. As
such, it is clear that we wish to obtain a calibration in terms of brightness temperature.
It is not possible to completely ignore flux however, since our calibration sources are
all point sources and small compared to the GBT beam. Thus, it is necessary to relate
these two measures. This is done through the notion of the forward gain Gf . The forward
gain, with units [K/Jy], compares the telescope’s response to a point source to that of a
spatially constant brightness. In other words, the gain is the spatially constant brightness
temperature that causes the same increase in measured power as a 1 Jy point source. It
is therefore a measure of the peakedness of the beam. In our definition, the forward gain
only depends on the beam shape.
For a beam with shape b(~θ) the forward gain is
Gf =
(c2
2ν2kB
)b(0)∫b(~θ)dΩ
, (5.9)
where ν is the observing frequency and b(0) is the centre/peak of the beam where a point
source would be measured. For a perfectly Gaussian beam, the gain is given by
Gf =
(c2
2ν2kB
)1
2πσ2, (5.10)
where σ is the Gaussian width parameter. It is seen that the forward gain of an instrument
Chapter 5. Data analysis pipeline 70
depends only on the beam shape and as such is measured by mapping the beam using a
point source. In practise the primary gain is calculated for GBT using a Gaussian fit to
the measured beam function.
Flux calibration
Our model for the raw data is
P = gT (5.11)
Pcal-on = g(Tsys + Tsky + Tcal) (5.12)
Pcal-off = g(Tsys + Tsky), (5.13)
for some system gain g. Here we have distinguished between the channel where the
noise-cal (described above) is on, and off. The system temperature, Tsys, parameterizes
the base-line amount of noise power that unavoidably exists in the receiver. It is assumed
that g and Tsys are essentially stable over times scales of at least a few minutes, but that
Tcal is stable on time scales of at least several hours. The first step of the calibration is
to eliminate the system gain from the data,
T/Tcal = P/Pcal (5.14)
= P/〈Pcal-on − Pcal-off〉t. (5.15)
Here, P and T without a subscript stand for either the cal-on or cal-off state. The time
average is over a period of time in which the gain is assumed to be relatively stable,
typically a scan of length up to a few minutes. The average is necessary since the noise-
cal power is weak compared to the system temperature, and as such its determination is
noisy. We refer to data with this calibration as being in Tcal units.
To fully calibrate the data, we now need to convert from T cal units to physical temper-
ature in Kelvin. This is done by observing a calibration point source of known flux. We
observe point sources which have flux data from a wide range of telescopes and frequen-
cies. Using the NASA/IPAC Extragalactic Database (NED) catalogue1, we consider all
data points for a given source within or near our frequency range and calculate a power-
law fit. This fit provides data for Ssrc at all frequencies. Because we are interested in
the full Stokes parameters for polarized flux (T I,TQ,T U and T V), we have to know the
polarization angle of the point source as well. We have used both unpolarized sources,
1http://ned.ipac.caltech.edu
Chapter 5. Data analysis pipeline 71
and polarized sources, which we take to have a frequency independent polarization angle
within our band.
During each data collection session, we collect a set of on-off scans of the known point
source. Each scan is a series of four tracking scans with a 56.5 second duration. We track
on the point source, then move slightly off the source and track. The second two scans
are a repeat of the first, but with the off position on the opposite side of the source. We
collect these on-off sets at least twice during each data collection session, or roughly four
hours apart for the longer sessions.
Using each on-off set we can then calculate:
Tsrc/Tcal = 〈Ton-src/Tcal〉 − 〈Toff-src/Tcal〉. (5.16)
Again, here we have used a temporal average to maximize the signal to noise on the source
and thus minimize the error on the calibration. Even if the gain g changes between the
on-source and off-source observations due to non-linearity, our determination of Tsrc/Tcal
is not affected since the gain is already cancelled in each observation.
We can compare Tsrc/Tcal that we measure to the expected value of SsrcGf to get a
conversion factor:
Cflux = (Tcal/Tsrc)GfSsrc. (5.17)
Note that since Cflux is the conversion factor from Tcal units to Kelvin, Cflux is really just
a measurement of Tcal.
To get the most accurate value for Cflux, we calculate it for each linear polarization
independently (CXflux, and CY
flux). This automatically provides a first order calibration for
the polarized brightness T I and TQ. We also investigated the stability of Cflux on longer
time scales. We found that it is stable over individual sessions, and is usually stable
between sessions. However, there is occasionally a dramatic change in Cflux. This change
is associated with the physical rotation of the low frequency receivers at GBT, and only
occurs a few times in our overall dataset. The times of these changes correspond with
the transition between observation of the different observational fields, so we are able to
temporally average over the full set of sessions for a given field to minimize the error in
Cflux.
Differential polarization calibration
If we are interested in the full Stokes data as opposed to just the absolute intensity,
an additional calibration component is necessary. Here we follow the formalism and
notation in Heiles et al. [2001], but only perform the first order polarization calibration.
Chapter 5. Data analysis pipeline 72
This operation resolves a known phase ambiguity in the GUPPI data, which mixes Stokes
U and V. It also rotates the measured polarizations from telescope coordinates to sky
coordinates based on the telescope’s orientation. For an ideal instrument, this calibration
is sufficient for performing full polarimetry, however, GBT has additional instrumental
leakages between the polarized channels at the 10% level. Implementing a full polarization
calibration to deleak the Stokes parameters is an area of active research, however it is
complicated by the fact that the polarization leakage is a function of location within the
GBT primary beam.
The correction factors are most easily calculated and applied in X-Y space, so the
data is calibrated while we are still working with T XX,T XY, T YX, and T YY, later converting
to the Stokes parameters (T I,TQ,T U,T V).
Above, we calculated the magnitudes of the calibration factors CXflux and CY
flux, however,
these factors have phases as well. These phases will affect the other Stokes components
(T U and T V). We can write this influence as a series of matrices:
Tsky = GfSsky = Gf
SXX
sky
SXYsky
SYXsky
SYYsky
= M−1PA ∗Mflux ∗
T XX
meas
T XYmeas
T YXmeas
T YYmeas
, (5.18)
where MPA is the parallactic angle (θ) rotation matrix used to go from the telescope
frame to the sky frame of reference:
MPA =
0.5(1 + cos 2θ) sin 2θ 0 0.5(1− cos 2θ)
−0.5 sin 2θ cos 2θ 0 0.5 sin 2θ
0 0 1 0
0.5(1− cos 2θ) − sin 2θ 0 0.5(1 + cos 2θ)
. (5.19)
The flux matrix encodes the complex correction factors for flux, with the complex
phase φ(ν):
Mflux =
CX
flux 0 0 0
0√CX
fluxCYflux cosφ −
√CX
fluxCYflux sinφ 0
0√CX
fluxCYflux sinφ
√CX
fluxCYflux cosφ 0
0 0 0 CYflux
. (5.20)
The ambiguity in the value of φ(ν) results from a known software issue in GUPPI where
the analogue to digital sampling of the X and Y voltages can be offset in time by up to
Chapter 5. Data analysis pipeline 73
two samples. This issue causes a non zero phase that takes the form φ(ν) = p0ν + p1, for
unknown parameters p0 and p1.
To correct for this issue, the complex phase φ parameters p0 and p1 must be deter-
mined using a fit, after which the data can be corrected by applying the Mflux matrix.
The parameters p0 and p1 can be measured from the noise-cal. The noise-cal should have
a Stokes PV equal to zero, but a non-zero phase φ causes some of the signal from PU to
be seen in PV . We calculate the phase parameters by fitting to
Rcal = P U
cal/√
(P Ucal)
2 + (P Vcal)
2 = cos (p0ν + p1). (5.21)
Here P Ucal and P V
cal are 〈P Ucal-on − P U
cal-off〉t and 〈P Vcal-on − P V
cal-off〉t respectively. Unlike the flux
correction Cflux, the phase correction φ changes on a session time scale, or rather it resets
every time the software system resets (which is usually once a session but can be more
frequently). However, p0 and p1 only assume four sets values other than zero.
5.3 Map-making
Formally, map-making is the linear process of solving for the temperature on the sky,
given time ordered data. It is a well established subject in the cosmic microwave back-
ground (CMB) studies [Smoot et al., 1992, Bennett et al., 1996, Tegmark, 1997], with
the important caveat that CMB maps are two dimensional while our maps have the third
redshift (spectral frequency ν) dimension. Here we lay out the map-making formalism
before going into the details of the map making module used in the data analysis pipeline.
5.3.1 Formalism
Here we review the standard CMB map-making formalism as laid out in Dodelson [2003],
with some straight forward adaptations for the extra spectral frequency dimension. This
section also serves to introduce our notation.
The time ordered data is modelled by the equation
dνt = Atimνi + nνt, (5.22)
where the index t runs over the time bins, ν over the spectral bins, and i over the
angular pixels in the map. t and ν are understood to have units of time and frequency
respectively and as such are more labels than indexes. Repeated indexes are summed
over unless otherwise stated. In the above equation, dνt is the time ordered data; mνi is
Chapter 5. Data analysis pipeline 74
the map of the sky, including both 21 cm signal and foregrounds; nνt is the noise; and
Ati is the pointing operator, indicating how the sky signal enters the telescope at time t.
There are two possible treatments of the telescope’s beam. If the beam is stationary,
i.e. it alway couples adjacent pixels in an identical manner independent of time, then the
beam can be interpreted as being part of the map. In this case, mνi is a beam convolved
map relating to the true sky by
mνi = Bνii′mνi (no sum on ν), (5.23)
where mνi is the true sky map and Bii′ is the beam operator. Alternately, the beam can
be included in the pointing operator:
Ati = Ati′Bνti′i (no sum on ν, t), (5.24)
where Ati′ encodes where the telescope is pointing and Bti′i applies the beam. This later
treatment is necessary if one needs to treat time variability in the beam, or for treating
beam anisotropies (since the telescope’s angle relative to the sky is a function of sidereal
time). While the second treatment is more general, it has at least one major algorithmic
difficulty. Since the goal of map-making is to solve for mνi, the map-maker will need to
deconvolve the beam from data, which is numerically difficult. As such we treat the beam
as being part of the map, noting that the map-making module is in principal extensible
to include the beam in the pointing operator.
The noise, nνt, are assumed to be correlated Gaussian random numbers with a co-
variance 〈nνtnν′t′〉 = Nνt,ν′t′ . Given an estimate for the map, mνi, we can construct
chi-squared:
χ2 = (dνt − Atimνi)N−1νt,ν′t′(dν′t′ − At′i′mν′i′), (5.25)
where ()−1 is the matrix inverse. The optimal estimator for the map minimizes χ2,
∂χ2
∂mνi
= 0 (5.26)
= −2AtiN−1νt,ν′t′(dν′t′ − At′i′mν′i′) (5.27)
= −2AtiN−1νt,ν′t′dν′t′ + 2AtiN
−1νt,ν′t′At′i′mν′i′ . (5.28)
Solving for mνi yields the map-making equation:
(AtiN−1νt,ν′t′At′i′)mν′i′ = AtiN
−1νt,ν′t′dν′t′ , (5.29)
Chapter 5. Data analysis pipeline 75
which in principal has a solution
mνi = (AtiN−1νt,ν′t′At′i′)
−1At′′i′N−1ν′t′′,ν′′t′′′dν′′t′′′ . (5.30)
The quantity AtiN−1νt,ν′t′dν′t′ in Equation 5.29 is known as the dirty map, and can be
thought of as a noise weighted map. The matrix in the brackets turns out to be the
inverse noise covariance matrix in map space:
(C−1N )νi,ν′i′ = AtiN
−1νt,ν′t′At′i′ , (5.31)
(CN)νi,ν′i′ ≡ 〈(mνi −mνi)(mν′i′ −mν′i′)〉. (5.32)
As such, map-making can be thought of as a two step process. The first is constructing
the dirty map and the inverse noise covariance. The second step is applying the inverse
of this matrix to obtain the estimate for the map, also known as the clean map.
The map-making equation is an unbiased estimate for the map given the data as long
as 〈Atimνi nν′t′〉 = 0, i.e. as long as the noise is uncorrelated with the sky map. This
assumption is thought to be true of our data but could be broken by strong receiver
non-linearity. In particular, the map will be highly biased if
d2P
dT 2
∣∣∣∣Tsys
RMS(Tnoise)RMS(Tsky) ∼ dP
dT
∣∣∣∣Tsys
RMS(Tsky), (5.33)
where RMS() is the root mean square operation, P is the power recorded by the telescope,
and Tsys, Tnoise, and Tsky are the antenna temperatures from the system temperature, noise
and sky respectively. For our data, taking RMS(Tnoise) ≈ Tsys, we have estimated the
RHS of the above to be ∼ 500 times larger than the LHS, meaning the noise correlates
with the foregrounds at one part in 500. Assuming the foregrounds and the 21 cm signal
are uncorrelated, the noise also correlates with the 21 cm signal at same one part in 500.
All map-making codes, with the exception of some of the maps produced by the
Planck Collaboration [Planck Collaboration et al., 2013c], are based on Equation 5.29
or some approximation there of [Tegmark, 1997]. It should be noted that Equation 5.29
remains an unbiased estimator for the map even if Nνt,ν′t′ is not perfectly representative
of the noise covariance. In this case it is the optimality of the map estimator that suffers,
not its validity as an unbiased estimator [Dunner et al., 2013]. This is comparable to
using a non-optimal set of noise weights when taking a weighted average.
Even if Nνt,ν′t′ is known perfectly, taking the required inverse, N−1νt,ν′t′ , by brute force
is computationally intractable. Data sets from the GBT approach 109 individual time-
Chapter 5. Data analysis pipeline 76
frequency data points even after coarse binning the data, and as such the inverse would
require many millennia of computing time on a large cluster. Map-making codes must
therefore find a shortcut for performing this inverse by either exploiting some symmetry
of the noise, approximating the noise, or both.
5.3.2 Noise model and estimation
The first way in which the noise is approximated is by dividing the data into discreet
chunks in time, each assumed to have independent noise. That is, the noise is assumed to
be block diagonal. The use of this simplification greatly reduces the computational task of
inverting the noise matrix and is well established in CMB map-making (see e.g. [Jarosik
et al., 2007, Dunner et al., 2013, Planck Collaboration et al., 2013b]). This approximation
is born out of the fact that the noise tends to decorrelate between far separated times,
and as such very little information is lost when the correlations between these times are
ignored. The block length is chosen to correspond to a single scan of the telescope (of
order a few minutes) out of convenience, although no part of our map-maker precludes
using longer blocks, thus preserving more information.
From this point t will index the time bins within a block of data as opposed to all
available time bins. The index s will be used to index the block (e.g. dsνt), however
much of the subsequent analysis refers to a single block and as such the index will be
suppressed unless explicitly needed.
The noise is most effectively measured in the frequency domain, obtained by Fourier
transforming the time axis of the data. In this chapter, we always use the discrete Fourier
transform operator, denoted by the symbol Fωt using the conventions
xω = Fωtxt =∑t
e−iωtxt (5.34)
xt = F−1tω xω =
∑ω
eiωtxω/ct, (5.35)
where ct = δtt is the number of time samples, or the cardinality of the set t2.
To clarify some terminology, there are two separate frequencies that will be used in the
following sections. The first is the spectral radiation frequency, ν, which is obtained by
Fourier transforming the voltage stream as sampled from the receiver. The transform is
performed by the back-end, GUPPI, and the time sampling rate is the analogue to digital
converter rate, on order GHz, resulting in a spectrum with frequencies corresponding to
2This notation is slightly awkward, but is necessary to not conflict with our use of n to denote the
noise and to ensure that t is not mistaken for an index
Chapter 5. Data analysis pipeline 77
the frequency of the radiation being measured by the receiver, in our case 700 MHz to
900 MHz. The second frequency is the frequency at which the power in the telescope
changes, ω (actually an angular frequency). The time sampling rate corresponds to
the length of the power integration bins, ∼ 10 Hz, resulting in a spectrum covering the
∼ 0.01 Hz to ∼ 5 Hz range.
The key property of the noise, that makes the frequency domain so convenient, is
that noise is assumed to be stationary. That is
Nνt,ν′t′ = ξνν′(t−t′), (5.36)
where ξνν′(t−t′) is the noise spectral covariance correlation function. This symmetry results
in a noise matrix that is diagonal in the frequency domain,
Nνω,ν′ω′ = ctδωω′Pνν′ω (no sum on ω), (5.37)
where δωω′ is the Kronecker delta and Pνν′ω is the noise spectral covariance power spec-
trum. Pνν′ω is related to ξνν′(t−t′) by Pνν′ω = Fω(t−t′)ξνν′(t−t′). The frequency domain
noise matrix is defined by
Nνω,ν′ω′ = 〈nνωn∗ν′ω′〉, (5.38)
with nνω ≡ Fωtnνt. Combining Equations 5.37 and 5.38 gives
Pνν′ω = 〈nνωn∗ν′ω〉/ct (no sum on ω). (5.39)
An estimate of the noise matrix could be obtained by fitting a model for Pνν′ω to
Equation 5.39 if we had an estimate of the noise, nνt. Clearly we can never know the
noise exactly, since if we did, we could simply eliminate all noise by subtracting it from
the data. The standard assumption that is used for many CMB experiments [Jarosik
et al., 2007, Dunner et al., 2013] is that in a single scan, the signal is far sub-dominant to
the noise, that is Aωimνi nνω, and thus dνω can be taken as an estimate of the noise.
This is not the case for 21 cm mapping, where the foregrounds (which we have included
in the signal in this chapter), dominate the thermal noise even for short integrations.
However, since the foregrounds are so bright, it is easy to get a rough estimate of the
map, even with noise weights which are far from optimal, e.g. uniform diagonal noise,
setting Nνt,ν′t′ ∼ δtt′δνν′ . This yields a rough map, mνi, which is far from optimal but
is good enough that Aωi(mνi − mνi) nνω and the quantity dνt − Atimνi is dominated
by noise. This procedure can be performed iteratively with map estimation such that
successively better noise estimates are obtained by subtracting successfully better maps.
Chapter 5. Data analysis pipeline 78
This is similar to the procedure used in in CMB map-making to eliminate the signal bias
in noise estimation [Dunner et al., 2013].
Pνν′ω′ can thus be measured by fitting to Equation 5.39, replacing the expectation
value with a either a single realization (generally a scan’s worth of data) or several
realizations assumed to have similar noise properties. A minor issue is that the discrete
Fourier transform assumes that data is periodic, which is not the case. The data must
thus be windowed by an appropriate function, and the effects of the windowing must
also be applied to the power spectrum model prior to fitting. Likewise, there is data
missing due to being flagged for RFI. This also contributes to the window function.
Before windowing it is helpful to remove the time-mean as well as a linear function of
time, since these are the modes that are most contaminated by noise fluctuations on time
scales greater than those spanned by the data, which are more difficult to estimate.
The only remaining question is how to model Pνν′ω. This is dependant on the in-
strument. The noise will certainly have the thermodynamically guaranteed thermal part
with
Nνt,ν′t′ = δνν′δtt′T 2
sys(ν)
∆t∆ν, (5.40)
where ∆t and ∆ν are the bin sizes in integration time and spectral channel respectively.
Investigations of the noise for GBT’s 800 MHz receiver led to the conclusion that
non-thermal parts of the noise occupy small number of modes in the spectral channel
covariance. That is, the noise is well modelled by
Pνν′ω = VνqVν′q′Pqq′ω, (5.41)
where the total number of modes required to describe the noise, cq, is small—of order
5. The columns of Vνq are mutually orthogonal and are extracted from the data by
performing an eigenvalue decomposition on nνωn∗ν′ω. This leaves Pqq′ω diagonal in (q, q′),
with the ω dependence well fit by a power-law. This noise term is similar to a term found
in the Atacama Cosmology Telescope noise discussed in Dunner et al. [2013] where noisy
eigenmodes are found in correlations between individual detectors instead of the spectral
channels discussed here. The success of this model is demonstrated in Figure 5.2, with
the modes Vνq shown in Figure 5.3. Several of the observed noise modes have a straight-
forward physical interpretation, such as achromatic gain variations (first and second
modes) and individual noisy channels due to RFI contamination (fifth mode). The RFI
in our band worsens above 850 MHz, which is visible in the modes.
Our noise model should include one more ingredient. Recall that before taking the
power spectrum of the noise, we had to subtract off the time mean and time slope in each
Chapter 5. Data analysis pipeline 79
10-2 10-1 100
frequency (Hz)
100
101
102
noise power
Figure 5.2: Noise power spectrum, averaged over all spectral channels (δνν′Pνν′ω/cν),as measured in the GBT 800 MHz receiver. Units of the vertical axis are normalized suchthat pure thermal noise would be a horizontal line at unity. In individual time samplesare 0.131 s long and spectral bins are 3.12 MHz wide. The telescope is pointing at thenorth celestial pole to minimize changes in the sky temperature. Descending the variouscoloured lines corresponds to removing additional noise eigenmodes, Vνq, from the noisepower spectrum. It is seen that after removing 7 of the 64 possible modes the noise issignificantly reduced and is approaching the thermal value on all time scales. The modesremoved from each subsequent line are shown in Figure 5.3.
Chapter 5. Data analysis pipeline 80
700 750 800 850 900spectral channel frequency (MHz)
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
mode a
mplit
ude (
norm
aliz
ed)
Figure 5.3: The modes Vνq removed from the noise power spectra to produce the curvesin Figure 5.2. Each mode is offset vertically for clarity, with mode number increasingfrom bottom to top. The nth mode in this figure is the dominant remaining mode in thenth curve in Figure 5.2.
Chapter 5. Data analysis pipeline 81
channel due to long time-scale noise. The fact that these modes are noisy and contain
little information should be included in the noise model such that they are deweighted
by the map-maker. This last part of the noise matrix takes the form
Nνt,ν′t′ = δνν′δpp′UtpUt′p′T2large, (5.42)
with cp = 2 and two columns of Utp being orthonormal constant and linear functions of
time respectively. Tlarge is a number in units of K, large enough to deweight the mode, but
not so large that it causes numerical instability. Discarding a mode in this manner does
not lead to biases in the map and discards only a small amount of the total information.
In principle Utp could be expanded to deweight additional noisy modes over the time axis,
for instance to discard a particular noisy ω or to deweight the longest time-scale modes
which tend to have persistent noise (this can be seen in the lower curves in Figure 5.2).
Putting these all together yields the full noise model:
Nνt,ν′t′ = δνν′δtt′T 2
sys(ν)
∆t∆ν+ VνqVν′q′ξqq′(t−t′) + δνν′δpp′UtpUt′p′T
2large, (5.43)
where ξqq′(t−t′) = F−1(t−t′)ωPqq′ω. We see that the noise model takes the form of a diagonal
matrix (the first term) plus two relatively low rank updates. The first is of rank cqct
and the second of rank cpcν, both of which are orders of magnitude smaller than the
full rank of ctcν. This structure greatly reduces the computational expense of taking
the noise inverse, as will be shown in the next section.
5.3.3 An efficient time domain map-maker
Traditionally, in CMB map-making a significant amount of the computation is performed
in the frequency domain [Jarosik et al., 2007, Dunner et al., 2013]. We have already seen
that going to the frequency domain is useful for estimating the noise. Unfortunately the
structure of our data is not conducive to actually performing the map-making operation
in the frequency domain. As previously noted, the advantages of working in the fre-
quency domain rely on the noise being stationary. This may be intrinsically true of our
noise, however our data is recorded only in small scans of a minute or so in length, and
with interruptions in between. To realize the advantages of the frequency domain it is
necessary for the total length of a single uninterrupted block of data to be much longer
than the longest time over which we wish to consider correlations in the noise. This is
not true of the data in our survey and as such we work in the time domain.
Chapter 5. Data analysis pipeline 82
Inverting the noise matrix
In the last section, we developed our model for the noise of an individual block of data
(typically a scan of length a few minutes), with each block of data assumed to be inde-
pendent. The form of our final noise model is diagonal, uncorrelated noise plus a highly
correlated component of dramatically lower rank. This form greatly facilitates the cal-
culation of the inverse noise matrix through the use of the well known Binomial Inverse
Theorem (also known as the matrix inversion lemma, Woodbury matrix identity, and the
Sherman-Morrison-Woodbury formula). In the form we will be using, the theorem states
that for matrices D, G, and W, with dimensions ch × ch, ch × cg and cg × cg,
(D + WGWT )−1 = D−1 −D−1W(G−1 + WTD−1W)−1WTD−1, (5.44)
where we have switched to index free matrix notation for convenience. This formula is
useful when cg ch and matrix D has a simple form, such as being block or fully
diagonal, allowing it to be inverted and multiplied by other matrices quickly. In the case
where D is diagonal and WGWT is dense, the left-hand-side requires c3h floating point
operations, where the right-hand-side requires O(c2hcg) floating point operations.
Applying this to our noise model in Equation 5.43, we take the first term to be D,
with
Dhh′ = δνν′δtt′T 2
sys(ν)
∆t∆ν, with (5.45)
h = (t, ν), (5.46)
which is indeed diagonal with ch = ctcν. The second two terms constitute WGWT
and are written
Ggg′ = ξqq′(t−t′) + δνν′δpp′T2large, (5.47)
Whg′ = δνν′Utp′ + δtt′Vνq′ , with (5.48)
g = (t, q)+ (ν, p). (5.49)
Here it is understood that if g = (t, q) then all elements with subscript (ν, p), such as
δνν′δpp′T2large, are zero and vice-versa. We note that W is very sparse providing another
opportunity to shorten the calculation.
In practise it is never necessary to calculate the full inverse noise matrix, since what
we really want is the inverse noise matrix times another quantity (such as dνt or Ati).
In general, it is much more efficient (in both operation count and memory usage) to
Chapter 5. Data analysis pipeline 83
calculate the constituent parts of the right-hand-side of Equation 5.44 (D, W and the
inverse of the term in brackets, Q ≡ (G−1 + WTD−1W)−1), and apply this directly to
another quantity. The computational complexity of the preparation phase is dominated
by calculating Qs for each block of data s, which requires O[cs(ctcq + cνcp)3]
operations for the full data set.
A word of caution about the use of the Binomial Inverse Theorem, in that it can
be numerically much less stable than standard algorithms for calculating the inverse
directly. This is caused in part by the differencing of the two terms on the right-hand-
side of Equation 5.44, which for certain modes in the matrix can be extremely close. This
occurs when an eigenvalue of WGWT is much larger than the corresponding value in D.
Pointing operator
For each block of data, the pointing operator is in principle a ct×ci matrix. However,
it is very sparse. In its simplest form, at each time t, the telescope is pointing at a single
pixel on the sky, and so the matrix representation of Ati has exactly one non-zero entry in
each row with value unity. Ati can thus be represented by a list of angular pixel indexes
of length ct. If we want to be a little bit more sophisticated, we can use linear or cubic
interpolation between pixels, since the telescope will never be pointed exactly at pixel
centre. In this case four and sixteen times more numbers are required to represent the
operator, but in all cases the operator can be applied in O(ct) operations.
Dirty map
The dirty map is defined by the quantity∑s
AstiN−1sνt,sνt′dsνt′ , (5.50)
where we have been explicit about the sum over the blocks of data indexed by s. With the
methods described above, this operation is not particularly challenging computationally.
It can be shown that forming this quantity can be done in O[cs(ctcq+ cνcp)2] op-
erations. This is much less work than the preparation phase of calculating the constituent
parts of Equation 5.44.
Inverse noise covariance
Calculating the inverse noise covariance matrix, (C−1N )νi,ν′i′ =
∑sAstiN
−1sνt,sνt′Ast′i′ , is
among the most challenging aspects of map-making. Most CMB map-making codes do
Chapter 5. Data analysis pipeline 84
not explicitly calculate this matrix. Instead, they calculate the action of the matrix on
a map estimate and use iterative methods, such as the conjugate gradient method, to
solve for the clean map [Wright et al., 1996, Hinshaw et al., 2003, Jarosik et al., 2007,
Dunner et al., 2013]. Our choice to explicitly calculate the matrix is born out of a
desire for simplicity, and is feasible at the current size of our maps. In addition, there
is in principle a significant amount of information that can be obtained about the noise
properties of the map from this matrix. We note that as our survey size increases, it
may become necessary to switch to iterative methods. This would require rewriting the
core engine of our map-maker. However, the infrastructure modules, which constitute
the majority of the code base, would be reusable.
When constructing this matrix, efficient memory management is essential. The matrix
generally does not fit into memory of an individual machine and as such some portion
of the matrix must be considered at a given time, making sure that input and output
is minimized. The most convenient and efficient way to do this is to hold the all the
constituent parts of the time domain noise inverse matrix, Ds, Ws and Qs for all s,
in memory and calculate a single row of the large matrix at a time. That is, calculate∑s(C
−1N )sνi,ν′i′ for a single (ν, i) at a time, and write it to disk before starting on the next
pair. This structure also lends itself well to being parallelized over a computer cluster
should this ever need to be implemented.
It can be shown that with the methods described in the previous sections, constructing
this matrix requires O[csctcν(cq+cp)(ctcν+ctcq+cνcp)] operations. The
first term in the second bracket dominates over the others.
Clean map
Solving for the clean map, mνi, is straight forward as Equation 5.29 is a linear equation.
In practise, more efficient and numerically stable algorithms exist than calculating the
required matrix inverse directly. In particular, since the noise inverse covariance ma-
trix is symmetric and positive definite by definition, a Cholesky decomposition can be
used. Solving equations in this form is well established, with libraries such as SCALAPACK
available for porting the process to a cluster for large maps. Solving for the clean map
requires O[(cicν)3] operations.
Computational problem size
To give a rough idea of the number of operations required in each step of map making,
we take the our Green Bank Telescope observations of the 15 hr field as an example. This
Chapter 5. Data analysis pipeline 85
field has roughly 100 hours of data spread over 10 sq. deg. This yields rough set sizes of
cν ∼ 100, ci ∼ 103, cs ∼ 103, ct ∼ 103, cq ∼ 5, and cp ∼ 2.
Under these assumptions, preparing to invert the noise matrices which is dominated
by calculating the matrices Qs, requires of order 1013 operations. Calculating the dirty
map requires a pithy 109 operations. Calculating the inverse noise covariance matrix
requires roughly 1013 operations. Calculating the clean map then takes of order 1015
operations.
These figures should not be mistaken for the amount of time required for each oper-
ation, since other factors can be very important. Calculating the matrices Qs is trivially
parallelized with each individual calculation dominated by taking a matrix inverse which
is a standard, and thus highly optimized, operation. The whopping 1015 operations re-
quired for the clean map is a matrix linear solve. This is similar to the operation that
computer clusters are benchmarked against, meaning that highly optimized parallel soft-
ware packages exist for performing the operation. Forming the inverse covariance matrix
is on the other hand highly specialized and not yet implemented on a cluster. As such
this takes a comparable amount of time as the solving for the clean map, despite the
difference in the operation count.
Simplification
Earlier versions of the map-maker, notably the versions used to produce the maps used in
the analyses in the subsequent chapters, used a simplified version of the map-maker. Here
the second term in the noise model, the one defined in Equation 5.41, was neglected. This
undoubtedly had a negative impact on the optimality of the maps, but greatly simplified
the map making problem. This is because the neglected term was the only one that
coupled the various frequency channels together, and without that term, each spectral
frequency slice could be treated as an independent map. As such, the operational count
of most steps in the map-maker was reduced by a factor of roughly cν, and solving for
the clean map became a factor of c2ν faster.
5.4 Conclusions
Arguably, the most critical part of the data analysis for a 21 cm large-scale structure
survey comes after map-making, in the foreground subtraction. This is because fore-
grounds are the greatest challenge for 21 cm surveys. However, having high quality maps
is essential for being able to subtract foregrounds. Foreground subtraction works because
Chapter 5. Data analysis pipeline 86
the foregrounds are confined to a limited number of modes within the map. If the map-
making is not done carefully, this will not be true and foreground subtraction will be
impossible.
The infrastructure described here will enable efficient and relatively optimal analyses
of 21 cm intensity mapping data from single dish telescopes. As will be shown in the
following chapters, this software has been highly successful in the analysis of the Green
Bank Telescope intensity mapping survey data. In addition, at the time of writing there
is an ongoing effort to adapt these methods to the data taken at the Parkes Telescope.
Chapter 6
Measurement of 21 cm brightness
fluctuations at z ∼ 0.8 in
cross-correlation
A version of this chapter was published in The Astrophysical Journal Letters as “Mea-
surement of 21 cm Brightness Fluctuations at z ∼ 0.8 in Cross-correlation”, Masui, K.
W., Switzer, E. R., Banavar, N., Bandura, K., Blake, C., Calin, L.-M., Chang, T.-C.,
Chen, X., Li, Y.-C., Liao, Y.-W., Natarajan, A., Pen, U.-L., Peterson, J. B., Shaw, J.
R., Voytek, T. C., Vol. 763, Issue 1, 2013. Reproduced here with the permission of the
AAS.
6.1 Summary
21 cm intensity maps acquired at the Green Bank Telescope are cross-correlated with
large-scale structure traced by galaxies in the WiggleZ Dark Energy Survey. The data
span the redshift range 0.6 < z < 1 over two fields totaling ∼ 41 deg. sq. and 190 hr of
radio integration time. The cross-correlation constrains ΩHIbHIr = [0.43 ± 0.07(stat.) ±0.04(sys.)] × 10−3, where ΩHI is the neutral hydrogen (H i) fraction, r is the galaxy–
hydrogen correlation coefficient, and bHI is the H i bias parameter. This is the most
precise constraint on neutral hydrogen density fluctuations in a challenging redshift range.
Our measurement improves the previous 21 cm cross-correlation at z ∼ 0.8 both in its
precision and in the range of scales probed.
87
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 88
6.2 Introduction
Measurements of neutral hydrogen are essential to our understanding of the universe.
Following cosmological reionization at z ∼ 6, the majority of hydrogen outside of galaxies
is ionized. Within galaxies, it must pass through its neutral phase (H i) as it cools and
collapses to form stars. The quantity and distribution of neutral hydrogen is therefore
intimately connected with the evolution of stars and galaxies, and observations of neutral
hydrogen can give insight into these processes.
Above redshift z = 2.2, the Ly-α line redshifts into optical wavelengths and H i can
be observed, typically in absorption against distant quasars [Prochaska and Wolfe, 2009].
Below redshift z = 0.1, H i has been studied using 21 cm emission from its hyperfine
splitting [Zwaan et al., 2005, Martin et al., 2010]. There, the abundance and large-
scale distribution of neutral hydrogen are inferred from large catalogs of discrete galactic
emitters. Between z = 0.1 and z = 2.2 there are fewer constraints on neutral hydrogen,
and those that do exist [Meiring et al., 2011, Lah et al., 2007, Rao et al., 2006] have large
uncertainties.
While the 21 cm line is too faint to observe individual galaxies in this redshift range,
one can nonetheless pursue three-dimensional (3D) intensity mapping [Chang et al., 2008,
Loeb and Wyithe, 2008, Ansari et al., 2012a, Mao et al., 2008, Seo et al., 2010, Mao, 2012].
Instead of cataloging many individual galaxies, one can study the large-scale structure
(LSS) directly by detecting the aggregate emission from many galaxies that occupy large
∼ 1000 Mpc3 voxels. The use of such large voxels allows telescopes such as the Green
Bank Telescope (GBT) to reach z ∼ 1, conducting a rapid survey of a large volume.
Aside from being used to measure the hydrogen content of galaxies, intensity mapping
promises to be an efficient way to study the large-scale structure of the Universe. In
particular, the method could be used to measure the baryon acoustic oscillations to high
accuracy and constrain dark energy [Chang et al., 2008]. However, intensity mapping
is a new technique which is still being pioneered. Ongoing observational efforts such as
the one presented here are essential for developing this technique as a powerful probe of
cosmology.
Synchrotron foregrounds are the primary challenge to this method, because they are
three orders of magnitude brighter than the 21 cm signal. However, the physical process
of synchrotron emission is known to produce spectrally smooth radiation [Oh and Mack,
2003, Seo et al., 2010]. If the calibration, spectral response and beam width of the
instrument are well-controlled and characterized, the subtraction of foregrounds should
be possible because the foregrounds have fewer degrees of freedom than the cosmological
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 89
signal. We find that this allows the foregrounds to be cleaned to the level of the expected
signal. The auto-correlation of intensity maps is biased by residual foregrounds, and
minimizing and constraining these residuals is an active area of work. However, because
residual foregrounds should be uncorrelated with the cosmological signal, they only boost
the noise in a cross-correlation with existing surveys. This makes the cross-correlation
a robust indication of neutral hydrogen density fluctuations in the 21 cm intensity maps
[Chang et al., 2010, Vujanovic et al., 2012].
The first detection of the cross-correlation between LSS and 21 cm intensity maps at
z ∼ 1 was reported in Chang et al. [2010], based on data from GBT and the DEEP2
galaxy survey. Here we improve on these measurements by cross correlating new intensity
mapping data with the WiggleZ Dark Energy Survey [Drinkwater et al., 2010]. Our
measurement improves on the statistical precision and range of scales of the previous
result, which was based on 15 hr of GBT integration time over 2 deg. sq.
Throughout, we use cosmological parameters from Komatsu et al. [2009], in accord
with Blake et al. [2011].
6.3 Observations
The observations presented here were conducted with the 680–920 MHz prime-focus re-
ceiver at the GBT. The unblocked aperture of GBT’s 100 m offset paraboloid design
results in well-controlled sidelobes and ground spill, advantageous to minimizing radio-
frequency contamination and overall system temperature (∼ 25 K). The receiver is sam-
pled from 700 MHz (z = 1) to 900 MHz (z = 0.58) by the Green Bank Ultimate Pulsar
Processing Instrument (GUPPI) pulsar back-end systems [DuPlain et al., 2008].
The data used in this analysis were collected between 2011 February and November as
part of a 400 hr allocation over four fields. This allocation was specifically to corroborate
previous cross-correlation measurements [Chang et al., 2010] over a larger survey area,
and to search for auto-power of diffuse 21 cm emission. The analysis here is based on
a 105 hr integration of a 4.5 × 2.4 “15 hr deep field” centered at 14h31m28.5s right
ascension, 20′ declination and an 84 hr integration on a 7.0 × 4.3 “1 hr shallow” field
centered at 0h52m0s right ascension, 29′ declination. The beam FWHM at 700 MHz is
0.314 and at 900 MHz it is 0.25. At band-center, the beam width corresponds to a
comoving length of 9.6h−1Mpc. Both fields have nearly complete angular overlap and
good redshift coverage with WiggleZ.
Our observing strategy consists of sets of azimuthal scans at constant elevation to
control ground spill. We start the set at the low right ascension (right hand) side of the
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 90
field and allow the region to drift through. We then re-point the telescope to the right
side of the field and repeat the process. For the 15 hr field, this set of scans consists of
8 one-minute scans each with a stroke of 4. For the 1 hr field, a set of scans consists
of 10 two-minute scans, each 8 in length. Note that since we observe over a range of
local sidereal times, our scan directions cover a range of angles with respect to the sky.
This range of crossing angles makes the noise more isotropic, and allows us to ignore the
directional dependence of the noise in the 3D power spectrum. The survey regions have
most coverage in the middle due to the largest number of intersecting scans. Observations
were conducted at night to minimize radio-frequency interference (RFI).
The optical data are part of the WiggleZ Dark Energy Survey [Drinkwater et al., 2010],
a large-scale spectroscopic survey of emission-line galaxies selected from UV and optical
imaging. It spans redshifts 0.2 < z < 1.0 across 1000 sq. deg. The selection function
[Blake et al., 2010] has angular dependence determined primarily by the UV selection, and
redshift coverage which favors the z = 0.6 end of the radio band. The galaxies are binned
into volumes with the same pixelization as the radio maps and divided by the selection
function, so that we consider the cross-power with respect to optical over-density.
6.4 Analysis
Here we describe our analysis pipeline, which converts the raw data into 3D intensity
maps, then correlates these maps with the WiggleZ galaxies.1
6.4.1 From data to maps
The first stage of our data analysis is a rough cut to mitigate contamination by terrestrial
sources of RFI. Our data natively have fine spectral resolution with 4096 channels across
200 MHz of bandwidth. This facilitates the identification and flagging of RFI. In each
scan, individual channels are flagged based on their variance. Any RFI not sufficiently
prominent to be flagged in this stage is detected as increased noise later in the pipeline
and subsequently down-weighted during map-making. Additional RFI is detected as
frequency-frequency covariance in the foreground cleaning and subtracted in the map
domain. While RFI is prominent in the raw data, after these steps, it was not found to
be the primary limitation of our analysis.
In addition to RFI, we also eliminate channels within 6 MHz of the band edges (where
aliasing is a concern) and channels in the 800 MHz receiver’s two resonances at roughly
1Our analysis software is publicly available at https://github.com/kiyo-masui/analysis IM
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 91
798 MHz and 817 MHz. Before mapping, the data are re-binned to 0.78 MHz-wide bands
(corresponding to roughly 3.8h−1Mpc at band-center).
For a time-transfer calibration standard, we inject power from a noise diode into the
antenna. The noise diode raises the system temperature by roughly 2 K and we switch
it at 16 Hz so that the noise power can be cleanly isolated. Calibration is performed by
first dividing by the noise diode power (averaged over a scan) in each channel, and then
converting to flux using dedicated observations of 3C286 and 3C48. The gain for X and
Y polarizations may differentially drift and so these are calibrated independently. Our
absolute calibration uncertainty is dominated by the calibration of the reference flux scale
(5%, Kellermann et al. [1969]), measurements of the calibration sources with respect to
this reference (5%, see also Scaife and Heald [2012]), and uncertainty of our measurement
of these fluxes (5%). Receiver nonlinearity, uncertainty in the beam shape and variations
in the diffuse galactic emission in the on- and off-source measurements are estimated to
contribute of order 1% each. These are all assumed to be uncorrelated errors and give
9% total calibration systematic error.
Gridding the data from the time ordered data to a map is done in two stages. We
follow cosmic microwave background (CMB) map-making conventions as described in
Tegmark [1997]. The map maker treats the noise to be uncorrelated except for deweight-
ing the mean and slope along the time axis for each scan. Each frequency channel is
treated independently. In the first round of map-making, the noise is estimated from
the variance of the scan. This is inaccurate because the foregrounds dominate the noise.
This yields a sub-optimal map which nonetheless has high a signal-to-noise ratio on the
foregrounds. This map is used to estimate the expected foreground signal in the time
ordered data and to subtract this expected signal, leaving time ordered data which are
dominated by noise. After flagging anomalous data points at the 4σ level, we re-estimate
the noise and use this estimate for a second round of map-making, yielding a map which
is much closer to optimal. In reality, it is a bad assumption that the noise is uncorrelated.
We have observed correlations at finite time lag and between separate frequency channels
in our data. Exploiting these correlations to improve the optimality of our maps is an
area of active research. For all map-making, we use square pixels with widths of 0.0627,
which corresponds to a quarter of the beam’s FWHM at the high frequency edge of our
band. Fig. 6.1 shows the 15 hr field map.
In addition to the observed maps, we develop signal-only simulations based on Gaus-
sian realizations of the non-linear, redshift-space power spectrum using the empirical-NL
model described by Blake et al. [2011].
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 92
6.4.2 From maps to power spectra
The approach to 21 cm foreground subtraction in literature has been dominated by the
notion of fitting and subtracting smooth, orthogonal polynomials along each line of
sight. This is motivated by the eigenvectors of smooth synchrotron foregrounds [Liu
and Tegmark, 2011, 2012]. In practice, instrumental factors such as the spectral cali-
bration (and its stability) and polarization response translate into foregrounds that have
more complex structure. One way to quantify this structure is to use the map itself
to build the foreground model. To do this, we find the frequency-frequency covariance
across the sample of angular pixels in the map, using a noise inverse weight. We then
find the principal components along the frequency direction, order these by their singular
value, and subtract a fixed number of modes of the largest covariance from each line of
sight. Because the foregrounds dominate the real map, they also dominate the largest
modes of the covariance.
There is an optimum in the number of foreground modes to remove. For too few
modes, the errors are large due to residual foreground variance. For too many modes,
21 cm signal is lost, and so after compensating based on simulated signal loss (see below),
the errors increase modestly. We find that removing 20 modes in both the 15 hr and 1 hr
field maximizes the signal. Fig. 6.1 shows the foreground-cleaned 15 hr field map.
We estimate the cross-power spectrum using the inverse noise variance of the maps
and the WiggleZ selection function as the weight for the radio and optical survey data,
respectively. The variance is estimated in the mapping step and represents noise and
survey coverage. The foreground cleaning process also removes some 21 cm signal. We
compensate for signal loss using a transfer function based on 300 simulations where
we add signal simulations to the observed maps (which are dominated by foregrounds),
clean the combination, and find the cross-power with the input simulation. Because the
foreground subtraction is anisotropic in k⊥ and k‖, we estimate and apply this transfer
function in 2D. The GBT beam acts strictly in k⊥, and again we develop a 2D beam
transfer function using signal simulations with the beam.
The foreground filter is built from the real map which has a limited number of inde-
pendent angular elements. This causes the transfer function to have components in both
the angular and frequency direction [Nityananda, 2010], with the angular part dominat-
ing. This is accounted for in our transfer function. Subtleties of the cleaning method will
be described in a future methods paper.
We estimate the errors and their covariance in our cross-power spectrum by calcu-
lating the cross-power of the cleaned GBT maps with 100 random catalogs drawn from
the WiggleZ selection function [Blake et al., 2010]. The mean of these cross powers is
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 93
consistent with zero, as expected. The variance accounts for shot noise in the galaxy
catalog and variance in the radio map either from real signal (sample variance), resid-
ual foregrounds or noise. Estimating the errors in this way requires many independent
modes to enter each spectral cross-power bin. This fails at the lowest k values and so
these scales are discarded. In going from the two-dimensional power to the 1D powers
presented here, we weight each 2D k-cell by the inverse variance of the 2D cross-power
across the set of mock galaxy catalogs. The 2D to 1D binning weight is multiplied by
the square of the beam and foreground cleaning transfer functions. Fig. 6.2 shows the
resulting galaxy-H i cross-power spectra.
6.5 Results and discussion
To relate the measured spectra with theory, we start with the mean 21 cm emission
brightness temperature [Chang et al., 2010],
Tb = 0.29ΩHI
10−3
(Ωm + (1 + z)−3ΩΛ
0.37
)− 12(
1 + z
1.8
) 12
mK. (6.1)
Here ΩHI is the comoving H i density (in units of today’s critical density), and Ωm and
ΩΛ are evaluated at the present epoch. We observe the brightness contrast, δT = TbδHI,
from fluctuations in the local H i over-density δHI. On large scales, it is assumed that
neutral hydrogen and optically-selected galaxies are biased tracers of the dark matter,
so that δHI = bHIδ, and δopt = boptδ. In practice, both tracers may contain a stochas-
tic component, so we include a galaxy-H i correlation coefficient r. This quantity is
scale-dependent because of the k-dependent ratio of shot noise to large-scale structure,
but should approach unity on large scales. The cross-power spectrum is then given by
PHI,opt(k) = TbbHIboptrPδδ(k) where Pδδ(k) is the matter power spectrum.
The large-scale matter power spectrum is well-known from CMB measurements [Ko-
matsu et al., 2011] and the bias of the optical galaxy population is measured to be
b2opt = 1.48 ± 0.08 at the central redshift of our survey [Blake et al., 2011]. Simula-
tions including nonlinear scales (as in Sec. 6.4.1) are run through the same pipeline as
the data. We fit the unknown prefactor ΩHIbHIr of the theory to the measured cross-
powers shown in Fig. 6.2, and determine ΩHIbHIr = [0.44±0.10(stat.)±0.04(sys.)]×10−3
for the 15 hr field data, and ΩHIbHIr = [0.41 ± 0.11(stat.) ± 0.04(sys.)] × 10−3 for the
1 hr field data. The systematic term represents the 9% absolute calibration uncer-
tainty from Sec. 6.4.1. It does not include current uncertainties in the cosmological
parameters or in the WiggleZ bias, but these are sub-dominant. Combining the two
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 94
fields yields ΩHIbHIr = [0.43 ± 0.07(stat.) ± 0.04(sys.)] × 10−3. These fits are based
on the range 0.075hMpc−1 < k < 0.3hMpc−1 over which we believe that errors are
well-estimated (failing toward larger scales where there are too few k modes in the vol-
ume) and under the assumption that nonlinearities and the beam/pixelization (failing
toward smaller scales) are well-understood. A less conservative approach is to fit for
0.05hMpc−1 < k < 0.8hMpc−1 where the beam, model of nonlinearity and error esti-
mates are less robust, but which shows the full statistical power of the measurement, at
7.4σ combined. Here, ΩHIbHIr = [0.40± 0.05(stat.)± 0.04(sys.)]× 10−3 for the combined,
ΩHIbHIr = [0.46± 0.08]× 10−3 for the 15 hr field and ΩHIbHIr = [0.34± 0.07]× 10−3 for
the 1 hr field.
To compare to the result in Chang et al. [2010], ΩHIbrelr = [0.55± 0.15(stat.)]× 10−3,
we must multiply their relative bias (between the GBT intensity map and DEEP2) by
the DEEP2 bias b = 1.2 [Coil et al., 2004] to obtain an expression with respect to bHI.
This becomes ΩHIbHIr = [0.66± 0.18(stat.)]× 10−3, and is consistent with our result.
The absolute abundance and clustering of H i are of great interest in studies of galaxy
and star formation. Our measurement is an integral constraint on the H i luminosity
function, which can be directly compared to simulations. The quantity ΩHIbHI also de-
termines the amplitude of 21 cm temperature fluctuations. This is required for forecasts
of the sensitivity of future 21 cm intensity mapping experiments. Since r < 1 we have
put a lower limit on ΩHIbHI.
To determine ΩHI alone from our cross-correlation requires external estimates of the
H i bias and stochasticity. The linear bias of H i is expected to be ∼ 0.65 to ∼ 1 at
these redshifts [Marın et al., 2010, Khandai et al., 2011]. Simulations to interpret Chang
et al. [2010] find values for r between 0.9 and 0.95 [Khandai et al., 2011], albeit for a
different optical galaxy population. Measurements of the correlation coefficient between
WiggleZ galaxies and the total matter field are consistent with unity in this k-range
(with rm,opt & 0.8) [Blake et al., 2011]. These suggest that our cross-correlation can be
interpreted as ΩHI between 0.45× 10−3 and 0.75× 10−3.
Measurements with Sloan Digital Sky Survey [Prochaska and Wolfe, 2009] suggest
that before z = 2, ΩHI may have already reached ∼ 0.4 × 10−3. At low redshift, 21 cm
measurements give ΩHI(z ∼ 0) = (0.43± 0.03)× 10−3 [Martin et al., 2010]. Intermediate
redshifts are more difficult to measure, and estimates based on Mg-II lines in DLA systems
observed with Hubble Space Telescope find ΩHI(z ∼ 1) ≈ (0.97±0.36)×10−3 [Rao et al.,
2006], in rough agreement with z ≈ 0.2 DLA measurements [Meiring et al., 2011] and
21 cm stacking [Lah et al., 2007]. This is in some tension with a model where ΩHI falls
monotonically from the era of maximum star formation rate [Duffy et al., 2012]. Under
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 95
the assumption that bHI = 0.8, r = 1, the cross-correlation measurement here suggests
ΩHI ∼ 0.5× 10−3, in better agreement, but clearly better measurements of bHI and r are
needed. Redshift space distortions can be exploited to break the degeneracy between ΩHI
and bias to measure these quantities independently of simulations [Wyithe, 2008, Masui
et al., 2010a]. This will be the subject of future work.
Our measurement is limited by both the number of galaxies in the WiggleZ fields and
by the noise in our radio observations. Simulations indicate that the variance observed in
our radio maps after foreground subtraction is roughly consistent with the expected levels
from thermal noise. This is perhaps not surprising, our survey being relatively wide and
shallow compared to an optimal LSS survey, however, this is nonetheless encouraging.
Acknowledgements
We thank John Ford, Anish Roshi and the rest of the GBT staff for their support;
Paul Demorest and Willem van-Straten for help with pulsar instruments and calibration;
and K. Vanderlinde for helpful conversations.
K.W.M. is supported by NSERC Canada. E.R.S. acknowledges support by NSF
Physics Frontier Center grant PHY-0114422 to the Kavli Institute of Cosmological Physics.
J.B.P. and T.C.V. acknowledge support under NSF grant AST-1009615. X.C. acknowl-
edges the Ministry of Science and Technology Project 863 (under grant 2012AA121701);
the John Templeton Foundation and NAOC Beyond the Horizon program; the NSFC
grant 11073024. A.N. acknowledges financial support from the Bruce and Astrid McWilliams
Center for Cosmology.
Computations were performed on the GPC supercomputer at the SciNet HPC Con-
sortium.
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 96
1
1.5
2
2.5
3
215.5 216 216.5 217 217.5 218 218.5 219 219.5 220
De
c
RA
GBT 15hr field (800.4 MHz, z = 0.775)
-1000
-500
0
500
1000
Te
mp
era
ture
(m
K)
1
1.5
2
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215.5 216 216.5 217 217.5 218 218.5 219 219.5 220
De
c
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GBT 15hr field, cleaned, beam convolved (800.4 MHz, z = 0.775)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Te
mp
era
ture
(m
K)
Figure 6.1: Maps of the GBT 15 hr field at approximately the band-center. The purplecircle is the FWHM of the GBT beam, and the color range saturates in some placesin each map. Top: The raw map as produced by the map-maker. It is dominated bysynchrotron emission from both extragalactic point sources and smoother emission fromthe galaxy. Bottom: The raw map with 20 foreground modes removed per line of sightrelative to 256 spectral bins, as described in Sec. 6.4.2. The map edges have visiblyhigher noise or missing data due to the sparsity of scanning coverage. The cleaned mapis dominated by thermal noise, and we have convolved by GBT’s beam shape to bringout the noise on relevant scales.
Chapter 6. 21 cm cross-correlation with an optical galaxy survey 97
0.001
0.01
0.1
1
0.1
∆(k
)2 (
mK
)
k (h Mpc-1
)
15 hr
1 hrΩHI bHI r = 0.43 10
-3
Figure 6.2: Cross-power between the 15 hr and 1 hr GBT fields and WiggleZ. Negativepoints are shown with reversed sign and a thin line. The solid line is the mean ofsimulations based on the empirical-NL model of Blake et al. [2011] processed by thesame pipeline.
Chapter 7
Determination of z ∼ 0.8 neutral
hydrogen fluctuations using the
21 cm intensity mapping
auto-correlation
A version of this chapter was submitted to Monthly Notices of the Royal Astronomical
Society: Letters as “Determination of z ∼ 0.8 neutral hydrogen fluctuations using the
21 cm intensity mapping auto-correlation”, Switzer, E. R., Masui, K. W., Bandura, K.,
Calin, L.-M., Chang, T.-C., Chen, X., Li, Y.-C., Liao, Y.-W., Natarajan, A., Pen, U.-L.,
Peterson, J. B., Shaw, J. R., Voytek, T. C. It is available as a preprint as Switzer et al.
[2013].
7.1 Summary
The large-scale distribution of neutral hydrogen in the universe will be luminous through
its 21 cm emission. Here, for the first time, we use the auto-power spectrum of 21 cm
intensity fluctuations to constrain neutral hydrogen fluctuations at z ∼ 0.8. Our data
were acquired with the Green Bank Telescope and span the redshift range 0.6 < z <
1 over two fields totaling ≈ 41 deg. sq. and 190 hr of radio integration time. The
dominant synchrotron foregrounds exceed the signal by ∼ 103, but have fewer degrees
of freedom and can be removed efficiently. Even in the presence of residual foregrounds,
the auto-power can still be interpreted as an upper bound on the 21 cm signal. Our
previous measurements of the cross-correlation of 21 cm intensity and the WiggleZ galaxy
98
Chapter 7. 21 cm auto-correlation 99
survey provide a lower bound. Through a Bayesian treatment of signal and foregrounds,
we can combine both fields in auto- and cross-power into a measurement of ΩHIbHI =
[0.62+0.23−0.15] × 10−3 at 68% confidence with 9% systematic calibration uncertainty, where
ΩHI is the neutral hydrogen (H i) fraction and bHI is the H i bias parameter. We describe
observational challenges with the present dataset and plans to overcome them.
7.2 Introduction
There is substantial interest in the viability of cosmological structure surveys that map
the intensity of 21 cm emission from neutral hydrogen. Such surveys could be used to
study large-scale structure (LSS) at intermediate redshifts, or to study the epoch of
reionization at high redshift. Surveys of 21 cm intensity have the potential to be very
efficient since the resolution of the instrument can be matched to the large scales of
cosmological interest [Chang et al., 2008, Loeb and Wyithe, 2008, Seo et al., 2010, Ansari
et al., 2012a]. Several experiments, including BAOBAB [Pober et al., 2013b], BAORadio
[Ansari et al., 2012b], BINGO [Battye et al., 2012], CHIME1, and TianLai [Chen, 2012]
propose to conduct redshift surveys from z ∼ 0.5 to z ∼ 2.5 using this method.
The principal challenges for 21 cm experiments are astronomical foregrounds and ter-
restrial radio frequency interference (RFI). Extragalactic sources and the Milky Way
produce synchrotron emission that is three orders of magnitude brighter than the 21 cm
signal. However, the physical process of synchrotron emission is known to produce
spectrally-smooth radiation, occupying few degrees of freedom along each line of sight. In
the absence of instrumental effects, these degrees of freedom are thought to be separable
from the signal [Liu and Tegmark, 2011, 2012, Shaw et al., 2013]. RFI can be minimized
through site location, sidelobe control, and band selection. In the Green Bank Telescope
data analyzed here, RFI is not found to be a significant challenge or limiting factor.
Subtraction of synchrotron emission has proven to be challenging in practice. In-
strumental effects such as passband calibration and polarization leakage couple bright
foregrounds into new degrees of freedom that need to be removed from each line of sight
to reach the level of the 21 cm signal. The spectral functions describing these systemat-
ics can not all be modeled in advance, so we take an empirical approach to foreground
removal by estimating dominant modes from the covariance of the map itself. This
method requires more caution because it also removes cosmological signal, which must
be accounted for.
1http://chime.phas.ubc.ca/
Chapter 7. 21 cm auto-correlation 100
Large-scale neutral hydrogen fluctuations above redshift z = 0.1 have been unam-
biguously detected only in cross-correlation with existing surveys of optically-selected
galaxies [Lah et al., 2009, Chang et al., 2010, Masui et al., 2013]. Here, residual 21 cm
foregrounds boost the errors but do not correlate with the optical galaxies. The density
fluctuations traced by survey galaxies may not correlate perfectly with the emission of
neutral hydrogen, so their cross-correlation can be interpreted as a lower limit on the
fluctuation power of 21 cm emission.
Several efforts have used the 21 cm line to place upper bounds on the reionization
era [Bebbington, 1986, Bowman and Rogers, 2010, Paciga et al., 2013, Pober et al.,
2013a] and z ∼ 3 (see e.g., Wieringa et al. [1992], Subrahmanyan and Anantharamaiah
[1990]) without the need to cross-correlate with an external data set. This is the first
work to describe similar bounds for z ∼ 0.8, using two fields totaling ≈ 41 deg. sq. and
190 hr of radio integration time with the Green Bank Telescope. Unlike the bounds from
reionization, for which there is currently no cross-correlation, we are able to combine the
auto- and cross-powers in a novel way, making a Bayesian inference of the amplitude of
neutral hydrogen fluctuations, parameterized by ΩHIbHI.
Throughout, we use cosmological parameters from Komatsu et al. [2009].
7.3 Observations and Analysis
The analysis here is based on the same observations used for the cross-correlation mea-
surement in Masui et al. [2013]. We flag RFI in the data, calculate 3D intensity map
volumes, clean foreground contamination, and estimate the power spectrum. Here we
will summarize essential aspects of the observations and analysis in Masui et al. [2013],
and describe the auto-power analysis in more detail.
Observations were conducted with the 680 − 920 MHz prime-focus receiver at the
Green Bank Telescope (GBT), sampled from 700 MHz (z = 1) to 900 MHz (z = 0.58) in
256 uniform spectral bins. The analysis here uses a 105 hr integration of a 4.5 × 2.4
15 hr “deep” field centered on 14h31m28.5s right ascension, 20′ declination and an 84 hr
integration on a 7.0 × 4.3 1 hr “wide” field centered on 0h52m0s right ascension, 29′
declination.
The beam FWHM at 700 MHz is 0.314, and at 900 MHz it is 0.250. At band-
center, the beam width corresponds to a comoving length of 9.6h−1Mpc. Both fields
have nearly complete angular overlap and good redshift coverage with the WiggleZ Dark
Energy Survey [Drinkwater et al., 2010]. Our absolute calibration is determined from
radio point sources and is accurate to 9% [Masui et al., 2013]. For clarity, this remains
Chapter 7. 21 cm auto-correlation 101
as a separately quoted systematic error throughout, and plotted posterior distributions
are based on statistical errors only.
7.3.1 Foreground Cleaning
In this section, we develop the map cleaning formalism and discuss its connection to
survey strategy. Begin by packing the three-dimensional map into an Nν ×Nθ matrix M
by unwrapping the Nθ RA, Dec pointings. For the moment, ignore thermal noise in the
map. The empirical ν− ν ′ covariance of the map is C = MMT/Nθ, and it contains both
foregrounds and 21 cm signal. This can be factored as C = UΛUT , where Λ is diagonal
and sorted in descending value. From each line of sight, we can then subtract a subset of
the modes U that describe the largest components of the variance through the operation
(1−USUT )M, where S is a selection matrix with 1 along the diagonal for modes to be
removed and 0 elsewhere.
In reality, M also contains thermal noise. To minimize its influence on our foreground
mode determination, we find the noise-inverse weighted cross-variance of two submaps
from the full season of observing. Here, CAB = (WA MA)(WB MB)T/Nθ, where
A and B denote sub-season maps, WA is the noise inverse-variance weight per pixel of
map A (neglecting correlations), and is the element-wise matrix product. CAB is no
longer symmetric, and we take its singular value decomposition (SVD) instead, using
the left and right singular vectors to clean maps A and B respectively. The weights
are calculated in the noise model developed in the map-maker, but roughly track the
map’s integration depth and weigh against RFI. The weight is nearly separable into
angle (through integration time) and frequency (through Tsys(ν)), but we average to
make it formally separable and so rank-1, so that it does not increase the map rank. The
weighted removal for map A becomes (1/WA) (1−UASUTA)WA MA, where 1/WA is
the element-wise reciprocal.
Our empirical approach to foreground removal is limited by the amount of information
in the maps. The fundamental limitation here surprisingly is not from the number of
degrees of freedom along the line of sight, but is instead the number of independent
angular resolution elements in the map [Nityananda, 2010]. To see why this is the case,
notice that in the absence of noise, our cleaning algorithm is equivalent to taking the
SVD of the map directly: M = UΣVT and thus C ∝MMT = UΣ2UT , with the same
set of frequency modes U appearing in both decompositions. The rank of C coincides
with the rank of M and is limited by the number of either angular or frequency degrees
of freedom.
Chapter 7. 21 cm auto-correlation 102
Taking the foreground modes to all have comparable spurious overlap with the signal,
one arrives at a transfer function rule of thumb T = Psig. out/Psig. in ∼ (1 − Nm/Nν)(1 −Nm/Nres), where Nm is the number of modes removed, Nν = 256 is the number of
frequency channels and Nres is the number of resolution elements (roughly the survey
area divided by the beam solid angle). A limited number of resolution elements can
greatly reduce the efficacy of the foreground cleaning at the expense of signal.
The wide and deep fields respectively have central low-noise regions of ∼ 10 deg. sq.
and ∼ 3 deg. sq., giving roughly 90 and 30 independent resolution elements at the largest
beam size. The rank of C is then less than the number of available spectral bins in
both cases. To increase the effective number of angular degrees of freedom that enter the
weighted ν−ν ′ covariance, we saturate the angular weight maps at their 50’th percentile.
The choice of the number of modes to remove is a trade-off between bias from residual
foregrounds (for too few modes removed), and increasing errors (from signal lost as too
many modes are removed). We find that 30 (10) modes for the 1 hr (15 hr) field shows a
good balance of minimal foregrounds and errors.
7.3.2 Instrumental Systematics
The physical mechanism of synchrotron radiation suggests that it is described by a hand-
ful of smooth modes along each line of sight [Liu and Tegmark, 2012]. Instrumental
response to bright foregrounds, however, can convert these into new degrees of freedom.
An imperfect and time-dependent passband calibration will cause intrinsically spectrally
smooth foregrounds to occupy multiple modes in our maps with non-trivial spectral
structure. We control this using a pulsed electronic calibrator, averaged for each scan.
We believe that the most pernicious spectral structure is caused by leakage of po-
larization into intensity. Our observed polarization mixing is ∼ 10% between Stokes
parameters and has both angular and spectral structure. The spectral structure converts
spectrally smooth polarization into new degrees of freedom. Faraday rotation of the
polarization introduces further spectral degrees of freedom. The majority of the polar-
ization leakage is observed to be an odd function on the sky, slightly broader than the
primary intensity response beam. To mitigate this leakage, we convolve the maps to a
common resolution corresponding to 1.4 times the beam size at 700 MHz (the largest
beam). This also treats the leakage of spatial structure into spectral structure from fre-
quency dependence of the beam. Such a convolution is viable because GBT has roughly
twice the resolution needed to map large-scale structure in the linear regime. However,
this convolution reduces the number of independent resolution elements in the map by a
Chapter 7. 21 cm auto-correlation 103
factor of two, increasing the challenges discussed in Sec. 7.3.1.
The present results are limited largely by the area of the regions and our understand-
ing of the instrument. With a factor of roughly ten more area, the resolution could be
degraded at less expense to the signal. This requires significant telescope time because
the area must also be covered to roughly the same depth as our present fields. It would
however provide a significant boost in overall sensitivity for scientific goals such as mea-
surement of the redshift-space distortions. In addition, we are investigating map-making
that would unmix polarization using the Mueller matrix as a function of offset from the
boresight, as determined from source scans.
7.3.3 Power Spectrum Estimation
Our starting point for power spectral estimation is the optimal quadratic estimator de-
scribed in [Liu and Tegmark, 2011]. To avoid the thermal noise bias, we only con-
sider cross-powers between four sub-season maps [Tristram et al., 2005], labeled here
as A,B,C,D. Thermal noise is uncorrelated between these sections, which we have
chosen to have similar integration depth and coverage. The foreground modes are de-
termined separately for each side of the pair using the SVD of Sec. 7.3.1. Up to a
normalization, the resulting estimator for the pair of submaps A and B is P (ki)A×B ∝(wAΠAmA)TQiwBΠBmB. Here, we have unwrapped the map matrix MA into a one-
dimensional map vector mA and written the foreground cleaning projection (1/WA) (1 − UASUT
A)WA MA as ΠAmA. The weighted mean of each frequency slice of the
map is also subtracted. The map weight wA is the matrix WA used in the SVD, but
unwrapped, and along the diagonal. Procedurally, the estimator amounts to weight-
ing both foreground-cleaned maps, taking the Fourier transform, and then summing
the three-dimensional cross-pairs to find power in annuli in two-dimensional k space,
ki = k⊥,i, k‖,i. The Fourier transform and binning are performed by Qi here. We
calculate six such crossed pairs from the four-way subseason split of the data, and let the
average over these be the estimated power P (ki).
We calculate transfer functions to describe signal lost in the foreground cleaning
and through the finite instrumental resolution. These are functions of k⊥ and k‖. The
beam transfer function is estimated using Gaussian 21 cm signal simulations that have
been convolved by the beam model. The foreground cleaning transfer function can be
efficiently estimated through Monte Carlo simulations as
T (ki) =
⟨[wAΠA+s(mA + ms)−wAΠAmA]TQims
(wAms)TQims
⟩2
, (7.1)
Chapter 7. 21 cm auto-correlation 104
where the A+s subscript denotes the fact that the foreground cleaning modes have been
estimated from a ν− ν ′ covariance that has added 21 cm simulation signal, ms. The lim-
ited number of angular resolution elements (Sec. 7.3.1) results in an anticorrelation of the
cleaned foregrounds with the signal itself, represented by the term (wAΠA+smA)TQims.
To reduce the noise of the simulation cross-power, note that we subtract wAΠAmA in
the numerator. Finally, we find the weighted average of these across the four-way split of
maps. We find that 300 signal simulations are sufficient to estimate the transfer function.
After compensating for lost signal using transfer functions for the beam and fore-
ground cleaning, we bin the two-dimensional powers onto one-dimensional band-powers.
We weight bins by their two-dimensional Gaussian inverse noise variance∝ N(ki)T (ki)2/Pauto(ki)
2,
where Pauto(ki) is the average of PA×A, PB×B, PC×C , PD×D (pairs which contain the
thermal noise bias), and N(ki) is the number of three-dimensional k cells that enter a
two-dimensional bin ki. In addition to the Gaussian noise weights, we impose two addi-
tional cuts in the two-dimensional k power. For k‖ < 0.035h/Mpc, k⊥ < 0.08h/Mpc for
the 15 hr field, and k⊥ < 0.04h/Mpc for the 1 hr field, there are few harmonics in the
volume, resulting in “straggler” strips in the two-dimensional power spectrum where the
errors are poorly estimated. For k⊥ > 0.3h/Mpc, the instrumental resolution produces
significant signal loss, so this is truncated also.
Foregrounds in the input maps and the 21 cm signal itself are non-Gaussian, but
after cleaning, the thermal noise dominates both contributions in an individual map, and
Gaussian errors (see, e.g. Das et al. [2011]) provide a reasonable approximation. These
take as input the auto-power measurement itself (for sample variance) and PA×A terms
that represent the thermal noise. Sample variance is significant only in the 15 hr field
in the lower 1/3 of the reported wavenumbers. Gaussian errors agree with the standard
deviation of the six crossed-pairs that enter the spectral estimation in the regime where
sample variance is negligible.
The finite survey size and weights result in correlations between adjacent k bins. We
apodize in the frequency direction using a Blackman window, and in the angular direction
using the map weight itself (which falls off at the edges due to scan coverage). The bin-
bin correlations are estimated using 3000 signal plus thermal noise simulations assuming
Tsys = 25 K. To construct a full covariance model, these are then re-calibrated by the
outer product of the Gaussian error amplitudes for the data relative to the thermal noise
simulation errors.
The Bayesian method developed in the next section assumes that adjacent bins are
uncorrelated. To achieve this, we take the matrix square-root of the inverse of our
covariance model matrix and normalize its rows to sum to one. This provides a set
Chapter 7. 21 cm auto-correlation 105
of functions which decorrelates [Hamilton and Tegmark, 2000] the pre-whitened power
spectrum and boosts the errors. At large scales (k = 0.1h/Mpc) where these effects are
relevant, decorrelation and sample variance increase the errors by a factor of 1.5 in the
1 hr field and 4 in the 15 hr field.
The methods that we have developed for calculating maps from data and power
spectra from maps will be described and more fully motivated in a future paper.
7.4 Results
The auto-power spectra presented in Figure 7.1 will be biased by an unknown positive
amplitude from residual foreground contamination. These data can then be interpreted
as an upper bound on the neutral hydrogen fluctuation amplitude, ΩHIbHI. In addition,
we have also measured the cross-correlation with the WiggleZ Galaxy Survey [Masui
et al., 2013]. This finds ΩHIbHIr = [0.43 ± 0.07(stat.) ± 0.04(sys.)] × 10−3, where r
is the WiggleZ galaxy-neutral hydrogen cross-correlation coefficient (taken here to be
independent of scale). Since |r| < 1 by definition and is measured to be positive, the
cross-correlation can be interpreted as a lower bound on ΩHIbHI. In this section, we
will develop a posterior distribution for the 21 cm signal auto-power between these two
bounds, as a function of k. We will then combine these into a posterior distribution on
ΩHIbHI.
The probability of our measurements given the 21 cm signal auto-power and fore-
ground model parameters is
p(dk|θk) = p(dc|sk, r)p(d15 hrk |sk, f 15 hr
k )p(d1 hrk |sk, f 1 hr
k ). (7.2)
Here, dk = dc, d15 hrk , d1 hr
k contains our cross-power and 15 hr and 1 hr field auto-power
measurements, while θk = sk, r, f 15 hrk , f 1 hr
k contains the 21 cm signal auto-power, cross-
correlation coefficient, and 15 hr and 1 hr field foreground contamination powers, respec-
tively. The cross-power variable dc represents the constraint on ΩHIbHIr from both fields
and the range of wavenumbers used in Masui et al. [2013]. The band-powers d15 hrk and
d1 hrk are independently distributed following decorrelation of finite-survey effects. We
assume that the foregrounds are uncorrelated between k bins and fields, also. This is
conservative because knowledge of foreground correlations would yield a tighter con-
straint. We take p(dc|sk, r) to be normally distributed with mean proportional to r√sk,
and p(d15 hrk |sk, f 15 hr
k ) to be normally distributed with mean sk + f 15 hrk and errors deter-
mined in Sec 7.3.3 (and analogously for the 1 hr field). Only the statistical uncertainty
Chapter 7. 21 cm auto-correlation 106
0.1 0.2 0.3 0.4 0.5 0.6k(h/Mpc)
10-3
10-2
10-1
100
101
102
103
∆2(m
K2)
deep auto-power
deep noise
deep foreground
wide auto-power
wide noise
ΩHIbHI=0.43×10−3
Figure 7.1: Temperature scales in our 21 cm intensity mapping survey. The top curveis the power spectrum of the input 15 hr field with no cleaning applied (the 1 hr field issimilar). Throughout, the 15 hr field results are green and the 1 hr field results are blue.The dotted and dash-dotted lines show thermal noise in the maps. The power spectraavoid noise bias by crossing two maps made with separate datasets. Nevertheless, thermalnoise limits the fidelity with which the foreground modes can be estimated and removed.The points below show the power spectrum of the 15 hr and 1 hr fields after the foregroundcleaning described in Sec. 7.3.1. Negative values are shown with thin lines and hollowmarkers. Any residual foregrounds will additively bias the auto-power. The red dashedline shows the 21 cm signal expected from the amplitude of the cross-power with theWiggleZ survey (for r = 1) and based on simulations processed by the same pipeline.
Chapter 7. 21 cm auto-correlation 107
is included in the width of the distributions, as the systematic calibration uncertainty is
perfectly correlated between cross- and auto-power measurements and can be applied at
the end of the analysis.
We apply Bayes’ Theorem to obtain the posterior distribution for the parameters,
p(θk|dk) ∝ p(dk|θk)p(sk)p(r)p(f 15 hrk )p(f 1 hr
k ). For the nuisance parameters, we adopt con-
servative priors. p(f 15 hrk ) and p(f 1 hr
k ) are taken to be flat over the range 0 < fk < ∞.
Likewise, we take p(r) to be constant over the range 0 < r < 1, which is conservative
given the theoretical bias toward r ≈ 1. Our goal is to marginalize over these nuisance
parameters to determine sk. We choose the prior on sk, p(sk), to be flat, which translates
into a prior p(ΩHIbHI) ∝ ΩHIbHI. The data likelihood adds significant information, so the
outcome is robust to choices for the signal prior. The signal posterior is
p(sk|dk) =
∫p(sk, r, f
15 hrk , f 1 hr
k |dk) dr df 15 hrk df 1 hr
k . (7.3)
This involves integrals of the form∫ 1
0p(dc|s, r)p(r) dr which, given the flat priors that we
have adopted, can generally be written in terms of the cumulative distribution function
of p(dc|s, r). Figure 7.2 shows the allowed signal in each spectral k-bin.
Taking the analysis further, we combine band-powers into a single constraint on
ΩHIbHI. Following Masui et al. [2013], we consider a conservative k range where er-
rors are better estimated (k > 0.12 h/Mpc, to avoid edge effects in the decorrelation
operation) and before uncertainties in nonlinear structure formation become significant
(k < 0.3 h/Mpc). Figure 7.3 shows the resulting posterior distribution.
Our analysis yields ΩHIbHI = [0.62+0.23−0.15]× 10−3 at 68% confidence with 9% systematic
calibration uncertainty. Note that we are unable to calculate a goodness-of-fit to our
model because each measurement is associated with a free foreground parameter which
can absorb any anomalies.
7.5 Discussion and Conclusions
Through the measurement of the auto-power, we extend our previous cross-power mea-
surement of ΩHIbHIr [Masui et al., 2013] to a determination of ΩHIbHI. This is the first
constraint on the amplitude of 21 cm fluctuations at z ∼ 0.8, and circumvents the de-
generacy with the cross-correlation r. The 21 cm auto-power yields a true upper bound
because it derives from the integral of the mass function. In the future, redshift distor-
tions [Wyithe, 2008, Masui et al., 2010a] can be used to further break the degeneracy
between bHI and ΩHI, and complement challenging HST measurements of ΩHI [Rao et al.,
Chapter 7. 21 cm auto-correlation 108
0.1 0.2 0.3 0.4 0.5 0.6k(h/Mpc)
10-2
10-1
∆2(m
K2)
statistical limit
ΩHIbHI=0.43×10−3
Figure 7.2: Comparison with the thermal noise limit. The dark and light shadedregions are the 68% and 95% confidence intervals of the measured 21 cm fluctuationpower. The dashed line shows the expected 21 cm signal implied by the WiggleZ cross-correlation if r = 1. The solid line represents the best upper 95% confidence level we couldachieve given our error bars, in the absence of foreground contamination. Note that theauto-correlation measurements, which constrain the signal from above, are uncorrelatedbetween k bins, while a single global fit to the cross-power (in Masui et al. [2013]) is usedto constrain the signal from below. Confidence intervals do not include the systematiccalibration uncertainty, which is 18% in this space.
Chapter 7. 21 cm auto-correlation 109
10-4 10-3
ΩHIbHI
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
p[ ln
(ΩHIb
HI)]
cross-power
deep auto-power
wide auto-power
combined
Figure 7.3: The posterior distribution for the parameter ΩHIbHI coming from the WiggleZcross-power spectrum, 15 hr field and 1 hr field auto-powers, as well as the joint likelihoodfrom all three datasets. The individual distributions from the cross-power and auto-powers are dependent on the prior on ΩHIbHI while the combined distribution is essentiallyinsensitive. The distributions do not include the systematic calibration uncertainty of9%.
Chapter 7. 21 cm auto-correlation 110
2006]. Our present survey is limited by area and sensitivity, but we have shown that
foregrounds can be suppressed sufficiently, to nearly the level of the 21 cm signal, us-
ing an empirical mode subtraction method. Future surveys exploiting the auto-power of
21 cm fluctuations must develop statistics that are robust to the additive bias of residual
foregrounds, and control instrumental systematics such as polarized beam response and
passband stability.
Acknowledgements
We would like to thank John Ford, Anish Roshi and the rest of the GBT staff for their
support; and Paul Demorest and Willem van-Straten for help with pulsar instruments
and calibration.
The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities, Inc. This
research is supported by NSERC Canada and CIFAR. JBP and TCV acknowledge NSF
grant AST-1009615. XLC acknowledges the Ministry of Science and Technology Project
863 (2012AA121701); The John Templeton Foundation and NAOC Beyond the Hori-
zon program; The NSFC grant 11073024. AN acknowledges financial support from the
McWilliams Center for Cosmology. Computations were performed on the GPC super-
computer at the SciNet HPC Consortium. SciNet is funded by the Canada Foundation
for Innovation.
Chapter 8
Conclusions and outlook
In this thesis we have made significant progress toward establishing the method of 21 cm
intensity mapping as a powerful probe of the Universe. In Chapters 2 and 3 we gave two
quantitative examples of how observations of large-scale structure using the 21 cm line
could be used to study the late-time accelerated expansion and initial conditions of the
Universe, respectively. The central result of Chapter 4 is that cosmologically interesting
measurements are possible using 21 cm intensity mapping with existing instruments. As
such, surveys using, for example, the Green Bank Telescope would not only act as pi-
lots for dedicated experiments but could also perform measurements interesting in their
own right. The calculations in that chapter were invaluable in the survey planning and
sensitivity forecasts for the observations performed in Part II.
Part II of this thesis presents the analysis and results from the ongoing 21 cm intensity
mapping survey at the Green Bank Telescope. After describing some details of the
analysis in Chapter 5, we presented the results from the survey in Chapters 6 and 7. In
Chapter 6 we put a lower limit on the amplitude of the 21 cm signal power by correlating
our noisy survey maps with a traditional galaxy survey. In Chapter 7 we considered the
auto-correlation of our survey maps. This auto-correlation contains both 21 cm signal
and residual foregrounds, however, since the residual foreground must contribute positive
power to the maps, the auto-correlation allows the signal power to be constrained from
above. Combining the lower limit from the cross-correlation and upper limit from the
auto-correlation, we were able to constrain the amplitude of the 21 cm signal to within a
factor of two.
111
Chapter 8. Conclusions and outlook 112
8.1 Conclusions
In Part I we showed that due to the high efficiency with which intensity mapping experi-
ments can perform large surveys of large-scale structure (thus enabling many interesting
measurements), these experiments are perhaps the most promising future probes of the
Universe. In particular, in Chapter 3 we discovered a new effect, through which 21 cm
intensity mapping may eventually make the most precise measurements of primordial
gravity waves and as such provide firm evidence that inflation is the correct theory for
the early Universe.
Our observational program at the Green Bank Telescope (Part II) is currently the
leading experiment in the use of the 21 cm line to map large-scale structure at inter-
mediate redshift (0.5 . z . 3). While we have yet to make a detection of the signal,
in Chapter 7 we were able to constrain the amplitude of the signal to within a factor
of two. Our measurement directly constrains the product of two parameters: the total
neutral hydrogen abundance and the clustering of neutral hydrogen. Individually these
parameters are important for galaxy formation studies, however more importantly for
intensity mapping, this amplitude gives the total amount of 21 cm signal available for
future measurements. In particular several studies have simulated the H i signal for the
planned Square Kilometre Array (SKA)1 using a signal amplitude that is inconsistently
high with our measurement [Abdalla and Rawlings, 2005, Abdalla et al., 2010]. As such
their forecasts are overly optimistic, affecting either the sensitivity or cost of the exper-
iment. SKA is a large scale radio telescope whose projected cost is well over a billion
US dollars. Performing a large-scale structure survey using the 21 cm line is one if its
primary science goals.
8.2 Future work
The potential of idealized intensity mapping experiments at intermediate redshift is now
well established, decreasing the interest in studies such as those presented in Chapters 2
and 4. Instead the focus is now shifting from generic idealized experiments to studying
specific experiments and including effects such as foregrounds, coupled with systematic
beam inideality and uncertainty. An example of such a study is Shaw et al. [2013],
which will be invaluable to the planning, design, and analysis of the Canadian Hydrogen
Intensity Mapping Experiment2 (CHIME). In the future, such studies must be extended
1http://www.skatelescope.org2http://chime.phas.ubc.ca/
Chapter 8. Conclusions and outlook 113
by adding additional effects and uncertainties.
On the other hand, the high redshift effects that could be studied using 21 cm intensity
mapping, such as the effect presented in Chapter 3, is now an area of active research.
There are now several studies that address similar effects [Giddings and Sloth, 2011b,
Lewis, 2011, Book et al., 2012, Pen et al., 2012, Jeong and Kamionkowski, 2012, Schmidt
and Jeong, 2012b, Jeong and Schmidt, 2012, Schmidt and Jeong, 2012a, Pajer et al.,
2013]. Indeed the literature is not yet resolved as to whether ‘fossil’ effects are even
observable (see Pajer et al. [2013] and Section 3.7). It is now a priority to resolve this
controversy, as well as consider additional secondary effects that could be used to study
the early Universe.
We intend to extend the observational work, presented in Part II, on many fronts.
Our primary goal moving forward will be to make a convincing detection of the auto-
correlation in our survey maps in which the cosmological signal dominates over the resid-
ual foregrounds. There are several potential improvements to the data analysis pipeline,
described in Chapter 5, that may enable this. In particular, improving our time depen-
dant spectral calibration, as well as treating the polarization leakage within the GBT
primary beam (discussed in Chapter 7), is expected to yield is significant improvement
in foreground subtraction. In addition, we hope to extend our survey to a factor of
∼ 10 more area than our 15 hr deep field but at the same integration depth. This will
greatly improve the sensitivity of our survey and may additionally improve foreground
subtraction, as discussed in Chapter 7.
Beyond the simple detection of the auto-correlation signal, we also intend to break the
degeneracy between the total neutral hydrogen abundance parameter and the clustering
parameter through a detection of the Kaiser redshift space distortions, as described in
Chapter 4. This will require additional telescope time to achieve the desired sensitivity.
If we obtain a sufficient improvement in foreground subtraction, this could be performed
with the 21 cm auto-correlation, however even if this is not achieved, the measurement
could be performed by cross-correlating with an existing galaxy survey. The VIMOS
Public Extragalactic Redshift Survey (VIPERS), whose results were recently released in
de la Torre et al. [2013], Guzzo et al. [2013], has a much higher galaxy density than
WiggleZ and a redshift range well matched to our survey. As such, observations of the
VIPERS fields would accumulate sensitivity to the cross-correlation at a much faster
rate than the WiggleZ fields, greatly reducing the observing time required to detect the
redshift space distortions.
In addition to the extension of our survey there are several upcoming surveys using
new instruments with which I plan to be involved. The first is the GBT multi-beam
Chapter 8. Conclusions and outlook 114
project which intends to improve the mapping speed of the GBT by building a new
800 MHz receiver with multiple antennas. It is expected that the improved sensitivity of
this receiver will allow for the detection of the BAO at the GBT. At the time of writing, a
single pixel prototype for this receiver has been completed and testing for this prototype
on the telescope has been scheduled for summer 2013.
Finally, CHIME is an under-development, dedicated, intensity mapping experiment
whose primary science goal will be a precise measurement of the BAO. Currently a
CHIME pathfinder is under construction at the Dominion Radio Astronomical Observa-
tory in British Columbia. This pathfinder will have many times the sensitivity of even
the GBT multi-beam (detailed forecasts in Chapter 4). The full sized CHIME will have a
sensitivity to the BAO comparable to the next generation of LSS experiments, but should
be completed on a shorter time scale and at a fraction of the cost of its competitors.
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