Large Scale Structure

Matter Perturbations Beyond Cold Matter Probes

Scott DodelsonPASI 2006

Comoving Distance

Coordinate difference (comoving distance) is not the same as physical distance

Can rewrite conformal horizon as integral over

Hubble radius (aH)-1

Perturbations outside the

horizon

During inflation, fluctuates quantum mechanically around a smooth background

The mean value of is zero, but its variance is

Get contributions from all scales equally if

with n=1 (scale-invariant spectrum)

Relation of potential to overdensity

32

2 ~~

aa

k

mG42 m

mx

)(

Then Poisson’s equation

in Fourier space becomes

Two results:

)()()( 24 kTkkPkkP n

~~a

Define

How do perturbations evolve?

Matter only:

042 mGH

where δ=(ρ- ρm)/ ρm

Non-Expanding Universe

04 mGH=0, so

If the universe is not expanding, the matter density is constant, so

mGte 4exponential growth, with time scale of order H0

Expanding, Matter Dominated Universe

042 mGH

)3/(22/34 22 tHG m

In this universe, H=(2/3t) and

So,

03

2

3

42

tt

Can solve this analytically

03

2

3

42

tt

03

2

3

4)1( ppp

Assume a solution of the form: δ=tp ; get an algebraic equation for p

with solutions:

3

8

9

1

2

1

6

1p

Two modes: growing and decaying

Growing mode scales as a~t2/3

Gravitational accretion fights against dilution due to the expansion of the universe: Exponential growth changed to power law growth

Recall: ~~a

Gravitational wells remain constant in a matter dominated universe

Beyond Cold Dark Matter

Dark Energy (important at late times z~1)

Radiation (important at early times z>1000)

Baryon Acoustic Oscillations (Remnant of Pre-recombination era)

Neutrino Mass (operates at all times)

Dark EnergySolve the same differential equation

042 mGH

accounting for the new H(t) relation

Suppression in growth due to smooth component of the universe

Radiation Dominated Era

03

4 2

k

Newton’s equations - with radiation as the source - reduce to

with analytic solution

33/

)3/cos()3/()3/sin()0(3)(

k

kkk

Expect less power on small scales

For scales that enter the horizon well before equality,

So, we expect the transfer function to fall off as

23/

3/cos)0()(

EQ

EQEQ

k

k

ConsequencesFor a scale invariant spectrum

On large scales,On small scales,

Log since structure grows slightly during radiation era when potential decays

Power Spectrum sensitive to matter density

The turnover scale is the one that enters the horizon atthe epoch of matter-radiation equality:

Therefore, measuring the shape of the power spectrumwill give a precise estimate of mh

Baryon Acoustic Oscillations

Eisenstein et al. 2006Baryons

Dark Matter

Apparent position of bump related to actual size (which is known!) and distance to galaxies at intermediate redshifts

Neutrinos affect large scale structure

Since we know the neutrino abundance, we cancompute the energy density of a massive neutrino

This fraction of the total density does not participatein collapse on scales smaller than the freestreaming scale

At the relevant time, this scale is 0.02 Mpc-1 for a 1eV

Qualitatively …

Colombi, Dodelson, & Widrow 1995

Structure is smoothed out in model with light neutrinos

CDM WarmDM C+HDM

Even for a small neutrino mass, get large impact on structure: power spectrum is excellent probe of neutrino mass!

Quantitatively …

Gravitational Lensing

Galaxy distribution

Lyman alpha forest

Probes of The Power Spectrum

Deflection of Light first proposed by Einstein in

1912!

Einstein writes to George Hale (Director of Mount Wilson Observatory) in 1913. He mentions the 0.84’’ (2GM/Rc2) deflection expected from the Sun.

Wambsganss 1998

The next total solar eclipse was August 21, 1914. An expedition was sent to observe in the region of greatest eclipse …

Russian Crimean Peninsula

1914 was not a good time to start a scientific expedition

in Europe

The astronomers were captured by Russian soldiers and released a month

later … with no data

… which in retrospect is a good thing. Einstein improved his theory over the next several years. He eventually concluded that the deflection should be twice as large as the Newtonian result … And this was confirmed by the famous expeditions in 1919.

Geodesic Equation

Affine parameter can be replaced by comoving distance

Since the transverse components are i , the geodesic equation becomes

Evaluate the Christoffel Symbol

Derivative wrt a on the left cancels the second term on the right

Consider the geometry

With this boundary condition

with kernel

Define the distortion tensor

Distortion Tensor

is the projected density, a measure of the convergence of light rays. I are the two components of shear.

Example: Magnification

But

So

Move beyond point images (QSOs) to extended objects

(galaxies)

HST CL0024

Lensing producing elliptical images

Move from Strong Lensing (multiple images) to Weak Lensing (small changes in shapes of extended

objects)

Jain, Seljak, & White (2000)

Cosmic Shear field depends on cosmology: one of these has more

matter than the other

Apply Limber formula for the Power Spectrum

One of these combinations - the B mode - vanishes. The other - the E mode

Has a power spectrum

We can compute this power spectrum with knowledge of the nonlinear 3D

power spectrum

Dodelson, Shapiro, & White 2005

Points from ray tracing through a numerical simulation

Curve from integrating nonlinear power spectrum

Need to measure Amplitude of Fluctuations

in Shear

Van Waerbeke & Mellier 2003

Constraints on parameters

Contaldi, Hoekstra, Lewis: astro-ph/0302435

Matter Density

Amplitude of Matter fluctuations

Several Upcoming Surveys

Panstarrs

LSSTSNAP

Dark Energy Survey

What can we expect?

Hu & Tegmark 1999

Tomography

Hu 2002

Interesting Degeneracies

Abazajian & Dodelson 2003

Sloan Digital Sky Survey

2.5 meter telescope in Apache Point, New Mexico

Collaboration of: Fermilab, Princeton, U. Chicago, U.Washington, Johns Hopkins, New Mexico State, Max Planck, Japan, Pittsburgh, …

Scheduled to end in 2005; has been extended until 2008; will cover ¼ of the sky

Two surveys in one

Photometric survey: hundreds of millions of objects in 5 bands

Spectroscopic survey: ~1 million objects with spectra

Spectroscopic survey targets objects found in photometric survey. Reduces systematic effects (typically objects targeted for redshifts are found in different survey, leads to complicated selection function).

5 Filters very efficient

Ultimately will get redshifts for ~750,000 galaxies; 100,000 QSOs

i’ and z’ bands especially important for high redshift QSOs. Lyman alpha line (1215Ang) redshifted to 1215*(1+z) Ang. Can get z>6 QSOs.

SDSS Galaxy Power Spectrum

Tegmark et al. 2004

Corrects for luminosity bias

In these probes [and all others], the observables arecomplicated functionals of the easy-to-predict lineardensity field, L.

N-Body interactions in Newtonian gravity

Galaxy formation including hydro, feedback from SN, star formation, …

Simple biasing scheme valid on large scales

Assumed to hold on scales k≤0.2 h Mpc-1

Bias unknown so must be fit for: give up hope of determining amplitude of the power spectrum Cosmological constraints come from power spectrum shape

Constraints on Neutrino Mass

Use as variables:

Cmbgg OmOlCMB

Tegmark et al. 2004

Cmbgg OmOlCMB

+

LSS

Lyman alpha forestPhotons with energy > (n=1 to n=2 transition energy) get absorbed along the line of sight as they lose energy due to cosmic redshift.

Every absorption line corresponds to cloud of neutral hydrogen.

Fluctuations in forest trace fluctuations in density

Gnedin & Hui, 1997

Flux

Baryon Density

Position along line of Sight

Lyman alpha observes universe at early times

Sloan Digital Sky Survey (SDSS)

At high redshift, even small scales were linear!

Redshifts of Absorbers

Num

ber

of S

pect

ra

SDSS Spectra of 3300 Quasars

McDonald et al. (2004)

11 redshift bins1D Power Spectrum of the Flux

This is only half the battle!

Want to test cosmology Need to run simulations

which generate 1D flux spectra for every parameter set

Do likelihood analysis to see which simulations are closest to observations

Constraints on running

WMAP+ACBAR+CBI+ SDSS Lyman alpha 7 cosmological parameters Consistent with no running

Abazajian et al (March 19, 11:10 CST)

Hoping to resolve the issue, researchers are once again turning to quasars. So far, the results have been inconclusive: Two groups analyzing the same quasar data

have come up with starkly different answers. One, represented by Fermilab's Hui, sees no deviation from scale invariance. The other team, which included

Princeton University's Uros Seljak, claims to have spotted not only a significant deviation from scale invariance but also a change in the spectral index over

different scales, a quantity known as the "running" of the spectral index, far larger than most inflation models predict. If Seljak's team is correct, almost all inflationary

theories can be ruled out right away. Most physicists, however, are skeptical. Hui suggests that the differences between the two groups' conclusions arise from

differing assumptions about the properties of the telescopes as well as assumptions that went into the computer models that contribute to the analysis. "We're trying to

get to the bottom of it," he says.

Science, Vol 300, Issue 5620, 730-731 , 2 May 2003

Conclusions

Coherent/Beautiful picture of formation and evolution of lumpy universe from smooth origins Requires Dark Matter Strong Constraints on Neutrino Mass Comparing observations with theory is very complex; Weak lensing is promising

Notice the difference between these 2 pictures

How can we extract information from the non-Gaussianity?

• Compute N-point functions: e.g. Bispectrum is

• Several groups have shown that there is much cosmological information stored in the bispectrum (e.g., Hui 1999; Takada & Jain 2004)

• Bispectrum vanishes at zeroth order• Need to be careful when computing

perturbatively (Dodelson, Kolb, Mataresse, Riotto, & Zhang 2005)

We theorists have work to do!

Dodelson, Huterer, & Zhang 2005

Super-horizon modes remain constant

Small decay through the transition era: radiation domination to matter domination

The time parameter y=a/aEQ