Higher orders and ISW
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Transcript of Higher orders and ISW
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Cross-correlation of CMB & LSS :recentmeasurements, errors and prospects
astro-ph/0701393 WMAP vs SDSS
Enrique Gaztañaga Consejo Superior de Investigaciones Cientificas, CSIC
Instituto de Ciencias del Espacio (ICE), www.ice.csic.es (Institute for Space
Studies)Institut d'Estudis Espacials de Catalunya, (IEEC-CSIC)
Santiago, 21-23rd March , 2007
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Higher orders and ISW
I- Perturbation theory and Higher order correlations
II- CMB & LSS: ISW effect
III- Error analysis in CMB-LSS cross-correlation
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Tiempo Energia
atomsHOW DID WE GET HERE?
Two driving questions in Cosmology:
- Background: Evolution of scale factor a(t).
+ Friedman Eq. (Gravity?) + matter-energy content H2(z) = H2
0 [ M (1+z)3 + R (1+z)4 + K(1+z)2 + DE (1+z)3(1+w) ]
r(z) = dz/H(z) Dark Matter and Dark Energy!
- Structure Formation:
origin of structure (IC) + gravitational instability + matter-energy content
’’ + H ’ - 3/2 m H2 = 0 + galaxy formation (SFR)
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Where does Structure in the Universe come From?
How did galaxies/star/molecular clouds form? time
Small Initial overdensed seed
background
Overdensed region
Collapsed region
Perturbation theory:
= b( 1 + ) => = ( - b ) = b
b V /M =
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Jeans Instability (linear regime) LxD0x
Lxa0x
EdS
a = 0.1
EdS
Open
a = 1/(1+z) a = 1 (now) a = 0.01 a = 10
z = 0 (now)
z = 9
Another handle on Dark Energy (DE):-Friedman Eq. (Expansion history) can not separate gravity from DE
-Growth of structure could: models with equal expansion history yield difference D(z) (EG & Lobo 2001), astro-ph/0303526 & 0307034)
-how do you measure D(z) from observations?
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Problem IArgue that the linear growth equation:
Has the following solutions:
Show that:
(2)
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Non-linear evolution
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Spherical collapse model:In this case we can solvefully the non-linear evolution: resultsIn a strongly non-linear collapse
Critical densityc = 1.68
Another handle on DE:
-Models with equal expansion history yield
difference D(z) and difference c (EG. &
Lobo, astro-ph/0303526 & 0307034)
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Weakly non-linear Perturbations: Solved problem!? RPT (Crocce & Sccocimarro 2006)
vertices
LL
angular average
Leading order contribution in corresponds to the spherical collapse.
EdS
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Observations require an statistical approach:
Evolution of (rms) variance 2 = < 2> instead ofinstead of
Or power spectrum P(k)= < 2(k)> => 2 = ∫ dk P(k) k2 W(k) dk
Initial Gaussian distribution of density fluctuations:
p p (V) = < (V) = < PP>> == 0 0 for allfor all p ≠2p ≠2
Perturbations due to gravity generate non-Gaussian statistics pp
3
= S3
22 with S3(m)= 34/7 (time & Cosmo invariant)
IC problem: Linear Theory a
2 = < 2> = D2 < 2>
Normalization8 2 < 2(R=8)>
To find D(z) -> Compare rms at two times or find evolution invariants
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Predictions of Inflation
- Flat universe
- scale invariance IC: n~1
+ CDM transfer funcion: P(k) = kn T(k)
=> Gaussian IC
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Local spectral index P(k) ~ kn (initial spectrum + transfer function)
2[r]= ∫ dk P(k) k2 W(k) dk ~ r-(n+3)
n ~ -2 => 2 [r] ~ r -1 (1D fractal ) equal power on all scales (m~0.2)
n ~ -1 => 2 [r] ~ r -2 (2D fractal ) less power on large scales (m~1.0)
n ~ 1
n ~ -1
CMB Superclusters Clusters Galaxies
8
m
SCDM
CDM
n ~ -2
n ~ -1
n ~ -2
Horizon @ Equality
m~0.2 m~1.0
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Interest of Higher order PT or correlations:
- Gaussian IC?
- non-linearities: mode coupling
- non-linearities= non-gaussianities
- cosmic time invariants: do not depend much on cosmic history (cosmological parameters)
- bias: how light traces mass => measure mass
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Weakly non-linear Perturbation Theory: Solved problem!
vertices
LL
angular average
Leading order contribution in corresponds to the spherical collapse.
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Spherical collapse model:In this case we can solvefully the non-linear evolution: resultsIn a strongly non-linear collapse
Critical densityc = 1.68
Another handle on DE:
-Models with equal expansion history yield
difference D(z) and difference c (EG. &
Lobo, astro-ph/0303526 & 0307034)
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LL
Weakly non-linear Perturbation Theory (Spherical average)
LL
L
L
La
<0
LL
Gaussian Initial conditions
L
S
High order statistics -> vertices of non-linear growth!
gravity?
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Test in N-body simulations
3-pt funct N3 = (106)3 !!
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Weakly non-linear Perturbation Theory
Loops(higher order corrections): F2 F3
Tree level= dominant
Tree level: F2 F3
=
=
Gaussian Initial conditions: connected correlations are zero, except 2-pt=> All correlations are built from 2-pt!
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Weakly non-linear Perturbation Theory
Tree level
P(k) ~ kn
r12
r23
1
2
3
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n ~ -2
n ~ -1
n ~ -2
n ~ -1
Depends on local spectral index P(k) ~ kn (not on m)
2[r]= ∫ dk P(k) k2 W(k) dk ~ r-(n+3)
n ~ -2 => 2 [r] ~ r -1 (1D fractal ) equal power on all scales (m~0.2)
n ~ -1 => 2 [r] ~ r -2 (2D fractal ) less power on large scales (m~1.0)
n ~ -1
n ~ -2
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Where does Structure in the Universe come From?
How did galaxies/star/molecular clouds form?
IC + Gravity+ Chemistry = Star/Galaxy (tracer of mass?)
time
Initial overdensed seed
background
Overdensed region
Collapsed region
H2
dust
STARS
D.Hughes
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Hogg & Blanton
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QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
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Bias: lets take a very simple model.
rare peaks in a Gaussian field (Kaiser 1984, BBKS)
Linear bias “b”: (peak) = b (peak) = b (mass)(mass) with b= with b=
SC:c
2 2 (peak) = b(peak) = b22 2 2 (m) (m)
Threshold
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Biasing: does light trace mass?
On large scales 2-ptStatistics is linear
gb m
gbm
bD0
m L D0
Gravity vs Galaxy formation
Gravity
Bias
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Biasing: does light trace mass?
Local approximation gF[ m]
gb
mbm
gbm
bL
g
bmbb
2
m
gbbbg
mLL
Gravity vs Galaxy formation
c
2bb
c
3bb
Lis Gaussian
m is not
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Bias: rare peaks in a Gaussian field (Kaiser 1984. BBKS)
Linear bias “b”: (peak) = b (peak) = b (mass) with b= (mass) with b= for SCc
2 2 (peak) = b(peak) = b22 2 2 (m) (m)
Non-linear bias: b2= b2 ( bk= bk )
Bias S33 S416 (Skkk-2 ) -> Close to DM!!
Gravity S334/7-(n+3) ~ 3 S420Threshold
How to separate one from the other?
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How to separate Bias from Gravity? QG= (Qm+C)/B
Using scale or shape (configurational) dependence of 3-pt function:
Fry & EG 1993; EG & Frieman 1994; Frieman & EG 1994; Fry 1994; Scoccimarro 1998; Verde etal 2001
B>1
B<1
C
CGF model: Bower etal 1993
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- Gravity @ work (astro-ph/0501637 & astro-ph/0506249)
-3pt correlation can be used to understand biasing: this is independent of normalization or cosmological parameters
-1st mesurement of galaxy bias (c2 and b) with 3pt function (away from b=1 and c2=0, Verde etal 2001)
b1= 0.95 0.12 b2 = -0.3 0.1 ( -0.4 <c2< -0.2) -0.4 <c2< -0.2)
Work in progress (by galaxy type and color)
-measure of normalization: 0.8 < 0.8 < 88 < 1.0 < 1.0
=> Future applications?
Comparison with 2dfGRS
Gravity vs Galaxy formation
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Bias & Higher: conclusionLocal approximation works on larrge scales gF[ m]
For P(k) or 2-pt statistics:Linear theory works on scales > 10 MpcBut amplitude (b1) is unknown: degeneracy between D(z) or sigma8 and b1!
For 3-pt statistics:Need higher bias coeffcients (b1, b2, b3…)But can define invariables (S3, Q3) that do notDepend on D(z). Can separate b1 from b2!
=> Need to find b1, b2, b3….
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Higher orders and ISW
I- Perturbation theory and Higher order correlations
II- CMB & LSS: ISW effect
III- Error analysis in CMB-LSS cross-correlation
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Observations require an statistical approach:
Evolution of (rms) variance 2 = < 2> instead ofinstead of
IC problem:
Linear Theory a
2 = < 2> = D2 < 2>
Normalization8 2 < 2(R=8)>
To find D(z) -> Compare rms at two times or find evolution invariants
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Where does Structure in the Universe come From?
Perturbation theory:
= b( 1 + ) => = ( - b ) = b b V /M =
With’’ + H ’ - 3/2 m H2 = 0in EdS linear theory:
a
Gravitation potential:
= - G M /R => = G M / R = GM/R
in EdS linear theory: a => = GM (RGM (R
is constant even when fluctuations grow linearly!
We can mesure today an at CMB: should be the same!
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T/T=(SW)=/c2
PRIMARY CMB ANISOTROPIES
Sachs-Wolfe (ApJ, 1967)
T/T(n) = [(n) ]if
Temp. F. = diff in N.Potential (SW)
i
f
= GM (R/c2
CMB & LSS
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Problem II
Calculate the rms temperature fluctuation in the CMB due to the Sachs-Wolfe effect as a function sigma_8 (the linear rms density fluctuations on a sphere of radius 8 Mpc/h) and the value of Omega_m (fraction of matter over the critical density). Does the result depend on the cosmological constant (ie Omega_Lambda)?
i
f
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PRIMARY & SECONDARY CMB ANISOTROPIES
Sachs-Wolfe (ApJ, 1967)
T/T(n) = [ 1/4 (n) + v.n + (n) ]if
Temp. F. = Photon-baryon fluid AP + Doppler + N.Potential (SW)
i
f
In EdS (linear regime) D(z) = a , and therfore dd
Not in dominated universe !
SZ- Inverse Compton Scattering -> Polarization
+ Integrated Sachs-Wolfe (ISW) + lensing + Rees-Sciama + SZ
2 ∫if d dd(n)
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CMB Noise
ISW map, z< 4 Early map, z~1000
Primary CMB signal becomes a contaminant when looking for secondary (ISW, SZ, lensing) signal.
The solution is to go for bigger area. But we are limited by having a single sky.
Noise!
Signal
Crittenden
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Cross-correlation ideaCrittenden & Turok
(PRL, 1995)
Both Both T T and and (g) (g) are proportional to local mass fluctuations are proportional to local mass fluctuations
(m) (m)
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Problem III
(1) Assuming that galaxies trace the mass, demostrate that in the linear regime and for small angles (~<10 deg), the angular galaxy-galaxy correlation and the galaxy-temperature correlation (induced by ISW effect) are:
sight
(2) How does the above expressions change with linear bias?
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ISW in equations...
Limber approximation
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APM
WMAPAPM
APM
WMAP
WMAPAPM
WMAP
0.7 deg FWHM
0.7 deg FWHM
5.0 deg FWHM
5.0 deg FWHM
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Possible ISW contaminants:
-Primary CMB (noise)-Extincion/Absorption (of dust) in our galaxy(CMB and LSS contaminants)-Dust emission in galaxies/clusters-SZ effect-RS effect-CMB lensing by LSS structures-Magnification bias- … ?
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APMSignificance:
P= 1.2% null detection
-> wTG = 0.35 ± 0.13 K (68% CL) @ 4-10 deg
-> = 0.53-0.86 ( 2-sigma)
Pablo Fosalba & EG, (astro-ph/0305468)
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Significance (null detection):
SDSS high-z:
P= 0.3% for < 10 deg.
(P=1.4% for 4-10 deg)
SDSS all: P= 4.8%
Combined: P=0.1 - 0.03%
(3.3 - 3.6 sigma)
P. Fosalba, EG, F.Castander
(astro-ph/0307249, ApJ 2003)
= 0.69-0.87 ( 2-sigma)
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Data CompilationEG, Manera, Multamaki (astro-ph/ 0407022, MNRAS 2006)
RADIO (NVSS) &X-ray (HEAO)
Boughm & Crittenden (astro-ph/0305001). WMAP team Nolta et al., astro-ph/0305097
z =0.8-1.1 (tentative < 2.5 )
APM Fosalba & EG astro-ph/0305468
z=0.15-0.3 (tentative < 2.5 )
SDSS Fosalba, EG, Castander, astro-ph/0307249
SDSS team Scranton et al 0307335
Pamanabhan (2005)
Cabre etal 2006 z=0.3-0.5 (detection > 4 )
2Mass Afshordi et al 0308260
Rassat etal 06
z=0.1 (tentative < 2. )
QSO Giannantonio etal 06 (tentative < 2.5)
Coverage: z= 0.1 - 1.0
Area 4000 sqrdeg to All sky
Bands: X-ray,Optical, IR, Radio
Sytematics: Extinction
& dust in galaxies.
m= 0.20 8=0.9
High!?
LSS!?
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b=1
S/N^2 = fsky*(2l+1) /[1+ Cl(TT)*Cl(GG)/Cl(TG)^2] ~ 8=0.9
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S/N^2 = fsky*(2l+1) /[1+ Cl(TT)*Cl(GG)/Cl(TG)^2]
8=0.9
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CompilationEG, Manera, Multamaki (MNRAS 2006)
Marginalized over:
-h=0.6-0.8
-relative normalization of P(k)
Normalize to sigma8=1 for CM
Bias from Gal-Gal correlation
With SNIa:
= 0.71 +/- 0.13
m= 0.29 +/- 0.04
Prob of NO detection: 3/100,000
With SNIa+ flat prior:
= 0.70 +/- 0.05
w= 1.02 +/- 0.17
= 0.4-1.2
m= 0.18- 0.34
Corasantini, Giannantonio, Melchiorri 05
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• has info about structure growth at redshift of sample • galaxy bias
• tells about growth rates at lens redshifts• (2.5s-1) s = d log(N(m))/dm
Relative magnitude of the two terms is redshift, scale and galaxy population dependent
Cosmic Magnification and the ISW effect
EG
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More Information
The total signal to noise remains large at high redshifts
but
The high redshiftsignal is strongly correlated with the low redshift signal
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Higher orders and ISW
I- Perturbation theory and Higher order correlations
II- CMB & LSS: ISW effect
III- Error analysis in CMB-LSS cross-correlation
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Error Analysis
Consider 4 methods:
1. Gaussian errors in Harmonic space (TH) + transform into configurational space
2. New errors in Configurational space (TC)
3. Jack-Knife errors (JK)
4. Simulations (MC1 and MC2)
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Error AnalysisConsider 4 methods:1. Gaussian errors in Harmonic space (TH)
transform into configurational space
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Problem IV
(1) Assuming that both the galaxy (G) and temperature (T) CMB fluctuations in the sky are Gaussian random fields show that for an all sky survey (f_sky=1) the expected variance in the galaxy-temperature angular cross-correlation spectrum (C^TG) at multipole “l” is:
Where C^TT and C^GG are the corresponding temperature-temperature and galaxy-galaxy angular spectrum.
(2) Argue under what approximations the above expression is valid when we only have measurements over a fraction f_sky of the whole sky.
(3) Argue why the above expression is dominated by the second term.How does the S/N change with bias in this case? And with sigma_8?
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Error AnalysisConsider 4 methods:
2. New errors in Configurational space (TC)
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Poors-man Boostrap?
EACH SIMULACIONPRODUCES AJK ERROR ANDJK Cij
3.
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4. All sky Montecarlo simulations
Simulate both CMB and LSS as gaussian fields with the corresponding c_l spectrum for TT, GG and also TG:
Boughn, Crittenden & Turok 1998
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Input vs1000 sim
10% sky z=0.33
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All sky z=0.33
Input vssim
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10% sky z=0.33
Input vssim
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JK= 0.207 ± 0.041 (true=0.224)
JK= 0.193 ± 0.045 (true=0.202)
JK= 0.170 ± 0.049 (true=0.167)
JK= 0.113 ± 0.039 (true=0.107)
Comparison of JK errors with MC errors
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Error in the error
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ERROS in C_L
This wildly used Eq. only works forBinned data!
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ERROS in C_L-Can propagate diagonal errors in C_l to w()
-Thid is surprising for f<1: transfer to off-diagonal elements-Bin C_l data to get diagonal errors.
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CMB data LSS data
WMAP 3rd year SDSS DR4
-5200 sq deg (13% sky)-Selection of subsamples with different redshift distribution-3 magnitude subsamples with r=18-19, r=19-20 and r=20-21 with 106 – 107 galaxies-high redshift Luminous Red Galaxy (Eisentein et al. 2001)-Mask avoids holes, trails, bleeding, bright stars and seeing>1.8
V-band (61 Hz) HEALPix tessellationKp0 mask
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Jack-knife errors
Covariance matrix
distribution
Singular Value Decomposition (SVD)
Redshift selection function
r=20-21 zc=0 z
0=0.2 z
m=0.3
LRG zc=0.37 z
0=0.45 z
m=0.5
20-21 LRG
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r=20-21 S/N=3.6 LRG S/N=3.
S/N total=4.7
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For a flat universe, with bias, sigma8 and w=-1 fix....
dark energy must be...
68% 0.80-0.85
95% 0.77-0.86
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Can we obtain information about w?
Contour: 1, 2 sigma1 dof
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The Science Case for the Dark Energy Survey
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The Dark Energy Survey
• We propose to make precision measurements of Dark Energy
– Cluster counting, weak lensing, galaxy clustering and supernovae
– Independent measurements
• by mapping the cosmological density field to z=1
– Measuring 300 million galaxies– Spread over 5000 sq-degrees
• using new instrumentation of our own design.
– 500 Megapixel camera– 2.1 degree field of view corrector– Install on the existing CTIO 4m
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DARK ENERGY SURVEY (DES)DARK ENERGY SURVEY (DES)
Science Goal: measure w=p/, the dark energy equation of state, to a precision of w ≤ 5%, with
• Cluster Survey
• Weak Lensing
• Galaxy Angular Power Spectrum
• Supernovae
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Science Goals to Science Objective
• To achieve our science goals:– Cluster counting to z > 1– Spatial angular power spectra of galaxies to z = 1– Weak lensing, shear-galaxy and shear-shear – 2000 z<0.8 supernova light curves
• We have chosen our science objective:– 5000 sq-degree imaging survey
• Complete cluster catalog to z = 1, photometric redshifts to z=1.3• Overlapping the South Pole Telescope SZ survey• 30% telescope time over 5 years
– 40 sq-degree time domain survey• 5 year, 6 months/year, 1 hour/night, 3 day cadence
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DES Dark Energy Constraints
Method/Prior Uniform WMAP Planck
Galaxy Clusters: abundance w/ WL mass calibration
0.130.09
0.100.08
0.040.02
Weak Lensing: shear-shear (SS) galaxy-shear (GS) +
galaxy-galaxy (GG) SS+GS+GG SS+bispectrum
0.150.080.030.07
0.050.050.030.03
0.040.030.020.03
Galaxy angular clustering 0.36 0.20 0.11
Supernovae Ia 0.34 0.15 0.04
Forecast statistical constraints on constant equation of state parameter w models(DES DETF white paper, astro-ph/0510346)● 4 Dark Energy Techniques
– Galaxy clusters– Weak lensing– Angular power spectrum– Type Ia supernovae
● Statistical errors on constant w models typically σ(w) = 0.05-0.1
● Complementary methods– Constrain different
combinations of cosmological parameters
– Subject to different systematic errors
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DES Instrument Project
OUTLINE• Science and Technical
Requirements• Instrument Description• Cost and Schedule
Prime Focus Cage of the Blanco Telescope
We plan to replace this and everything inside it
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Zmax=2Dz=0.08
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8=1.0
8=0.9ISW predictions
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Detailed CONCLUSIONS- #>800 simulations for 5% error accuracy- Diagonal errors in w() are accurate to <20 deg -Survey geometry important for deg (f<0.1): useTC method!-MC1 is 10% low-JK is OK within 10%-Uncertainty in error is about 20% because of sampling-S/N and fit in harmonic space equivalent to configuration space. -Can propagate diagonal errors in C_l to w()
-The above is surprising for f<1: transfer to off-diagonal elements-Bin C_l data to get diagonal errors.-Bias to large Omega_DE for large errors-S/N is quite model depended.
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GENERIC CONCLUSION
-Cross-correlation povides a new observational tool to challenge understanding of DE
-4-5 sigma detection of the effect (prospers are not so much better than this: up to 11 sigma). This is higher than previously forcasted (JK errors).
-need to improve on current analysis tools and simlations to get more realistic.
-Signal is very hard to explain with EDS.
- LCDM is OK: on low side even with large 8 or large .