Large Scale Structure   Matter Perturbations Beyond Cold Matter  Probes

Scott Dodelson PASI 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 Define    ( 

x

 ) 

m



m

Then Poisson’s equation in Fourier space becomes  2  

k

2

a

2 4 

G



m

   ~

a

3 Two results:  ~ 

a



P

 (

k

) 

k

4

P

 (

k

) 

k n T

2 (

k

)

How do perturbations evolve?

Matter only:     2

H

   4 

G



m

  0 where δ=(ρ- ρ m )/ ρ m

Non-Expanding Universe

H=0, so     4 

G



m

  0 If the universe is not expanding, the matter density is constant, so  

e



t

4 

G



m

exponential growth, with time scale of order H

0

Expanding, Matter Dominated Universe     2

H

   4 

G



m

  0 In this universe, H=(2/3t) and 4 

G



m

 3

H

2 / 2  2 /( 3

t

2 ) So,     4 3

t

   2 3

t

2   0

Can solve this analytically

    4 3

t

   2 3

t

2   0 Assume a solution of the form: δ=t

p

; get an algebraic equation for p

p

(

p

 1 )  4 3

p

 2 3  0 with solutions:

p

  1 6  1 2 1 9  8 3

Two modes: growing and decaying Growing mode scales as a~t

2/3

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 Energy

Solve the same differential equation     2

H

   4 

G



m

  0 accounting for the new H(t) relation Suppression in growth due to smooth component of the universe

Radiation Dominated Era Newton’s equations - with radiation as the source reduce to  4  

k

2 3   0 with analytic solution  (  )  3  ( 0 ) sin(

k

 / 3 )   (

k



k

 / / 3 3 3  ) cos(

k

 / 3 )

Expect less power on small scales For scales that enter the horizon well before equality,  ( 

EQ

)   ( 0 ) cos 

k

 

k



EQ EQ

/ / 3 2  3  So, we expect the transfer function to fall off as

Consequences

For 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 at the epoch of matter-radiation equality: Therefore, measuring the shape of the power spectrum will give a precise estimate of 

m h

Baryon Acoustic Oscillations Dark Matter Baryons

Eisenstein et al. 2006

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 can compute the energy density of a massive neutrino This fraction of the total density does not participate in collapse on scales smaller than the freestreaming scale At the relevant time, this scale is 0.02 Mpc -1 for a 1eV 

Qualitatively …

CDM WarmDM C+HDM

Colombi, Dodelson, & Widrow 1995

Structure is smoothed out in model with light neutrinos

Quantitatively …

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

Probes of The Power Spectrum  Gravitational Lensing  Galaxy distribution  Lyman alpha forest

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  /R  c 2 ) 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 images) to Weak Lensing (multiple (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 Points from ray tracing through a numerical simulation Curve from integrating nonlinear power spectrum

Dodelson, Shapiro, & White 2005

Need to measure Amplitude of Fluctuations in Shear

Van Waerbeke & Mellier 2003

Constraints on parameters Amplitude of Matter fluctuations Matter Density Contaldi, Hoekstra, Lewis: astro-ph/0302435

Several Upcoming Surveys

Dark Energy Survey Panstarrs SNAP LSST

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

Corrects for luminosity bias

Tegmark et al. 2004

In these probes [and all others], the observables are complicated functionals of the easy-to-predict linear density 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:

CMB

Cmbgg OmOl

Tegmark et al. 2004

CMB + LSS

Cmbgg OmOl

Photons 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.

Lyman alpha forest

Fluctuations in forest trace fluctuations in density Flux Baryon Density Position along line of Sight

Gnedin & Hui, 1997

Lyman alpha observes universe at early times Sloan Digital Sky Survey (SDSS) At high redshift, even small scales were linear!

Redshifts of Absorbers

SDSS Spectra of 3300 Quasars

11 redshift bins 1D Power Spectrum of the Flux

McDonald et al. (2004)

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

 + SDSS Lyman alpha   WMAP+ACBAR+CBI 7 cosmological parameters Consistent with no running Abazajian et al (March 19, 11:10 CST)

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

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/a EQ