The Theory/Observation connectionlecture 5

the theory behind (selected) observationsof structure formation

Will Percival

The University of Portsmouth

Lecture outline

Dark Energy and structure formation– peculiar velocities

– redshift space distortions

– cluster counts

– weak lensing

– ISW

Combined constraints– parameters

– the MCMC method

– results (brief)

Structure growth depends on dark energy

A faster expansion rate makes is harder for objects to collapse

– changes linear growth rate

– to get the same level of structure at present day, objects need to form earlier (on average)

– for the same amplitude of fluctuations in the past, there will be less structure today with dark energy

If perturbations can exist in the dark energy, then these can affect structure growth

– for quintessence, on large scales where sound speed unimportant

– scale dependent linear growth rate (Ma et al 1999) On small scales, dark energy can lead to changes in non-linear structure growth

– spherical collapse, turn-around does not necessarily mean collapse

Peculiar velocities

All of structure growth happens because of peculiar velocities

TimeTime

Initially distribution of matter is approximately homogeneous ( is small)

Final distribution is clustered

Linear peculiar velocities

Consider galaxy with true spatial position x(t)=a(t)r(t), then differentiating twice and splitting the acceleration d2x/dt2=g0+g into expansion (g0)and peculiar (g) components, gives that the peculiar velocity u(t) defined by a(t)u(t)=dx/dt satisfies

In conformal units, the continuity and Poisson equations are

Look for solutions of the continuity and Poisson equations of the form u=F(a)g

The peculiar gravitational acceleration is

So, for linearly evolving potential, u and g are in same direction

Linear peculiar velocities

Solution is given by

where

Zeld’ovich approximation: mass simply propagates along straight lines given by these vectors

The continuity equation can be rewritten

So the power spectrum of each component of u is given by

k-1 factor shows that velocities come from larger-scale perturbations than density field

Peculiar velocity observations

Obviously, can only hope to measure radial component of peculiar velocities

To do this, we need the redshift, and an independent measure of the distance (e.g. if galaxy lies on fundamental plane). Can then attempt to reconstruct the matter power spectrum

The 1/k term means that the velocity field probes large scales, but does directly test the matter field. However, current constraints are poor in comparison with those provided by other cosmological observations

So peculiar velocities constrain f.can we measure these directly?

Redshift-space distortions

We measure galaxy redshifts, and infer the distances from these. There are systematic distortions in the distances obtained because of the peculiar velocities of galaxies.

Large-scale redshift-space distortions

In linear theory, the peculiar velocity of a galaxy lies in the same direction as its motion. For a linear displacement field x, the velocity field is

Displacement along wavevector k is

The displacement is directly proportional to the overdensity observed (on large scales)

Kaiser 1987, MNRAS 227, 1

Line-of-sight

Redshift space distortions

At large distances (distant observer approximation), redshift-space distortions affect the power spectrum through:

Large-scale Kaiser distortion. Can measure this to constrain

On small scales, galaxies lose all knowledge of initial position. If pairwise velocity dispersion has an exponential distribution (superposition of Gaussians), then we get this damping term for the power spectrum.

Redshift space distortion observations

Therefore we usually quote (s) as the “redshift-space” correlation function, and (r) as the “real-space” correlation function.

We can compute the correlation function rp, ), including galaxy pair directions

“Fingers of God”

Infall around clusters

Expected

Cluster cosmology

Largest objects in Universe– 1014…1015Msun

– Discovery of dark matter

(Zwicky 1933)

– Can be used to measure

halo profiles

Cosmological test based on hypothesis that clusters form a fair sample of the Universe (White & Frenk 1991)

Cluster cosmology

Cluster X-ray temperature and profile give

• total mass of system

• X-ray gas mass

Can therefore calculate

If we know s and b, where

We can measure

Allen et al., 2007, MNRAS, astro-ph/0706.0033

Cluster cosmology

Saw in lecture 3 that the Press-Schechter mass function has an exponential tail to high mass

Number of high mass objects at high redshift is therefore extremely sensitive to cosmology

Borgani, 2006, astro-ph/0605575

Problem is defining and measuring mass. Determining whether halos are relaxed or not

Cluster observations

Short-term: Weak-lensing mass estimates used to constrain mass-luminosity relations Need to link N-bosy simulation theory to observations - will we ever be able to solve this?

Longer term:

Large ground based surveys will find large numbers of clusters in optical

– PanSTARRS, DES

SZ cluster searches

Weak-lensing

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Deflection of light, magnification, image multiplication, distortion of objects: directly depend on the amount of matter. Gravitational lensing effect is achromatic (photons follow geodesics regardless their energy)

General relativity: Curvature of spacetime locally modified by mass condensation

Weak-lensing

Assumptions– weak field limit v2/c2<<1

– stationary field tdyn/tcross<<1

– thin lens approximation Llens/Lbench<<1

– transparent lens

– small deflection angle

Weak-lensing

The bend angle depends on the gravitational potential through

So the lens equation can be written in terms of a lensing potential

The lensing will produce a first order mapping distortion (Jacobian of the lens mapping)

Weak-lensing

We can write the Jacobian of the lens mapping as

In terms of the convergence

And shear

represents an isotropic magnification. It transforms a circle into a larger / smaller circle

Represents an anisotropic magnification. It transforms a circle into an ellipse with axes

Weak-lensing

Galaxy ellipticities provide a direct measurement of the shear field (in the weak lensing limit)

Need an expression relating the lensing field to the matter field, which will be an integral over galaxy distances

The weight function, which depends on the galaxy distribution is

The shear power spectra are related to the convergence power spectrum by

As expected, from a measurement of the convergence power spectrum we can constrain the matter power spectrum (mainly amplitude) and geometry

Weak-lensing observations

Short-term: CFHT-LS finished

– 5% constraints on 8 from quasi-linear power spectrum amplitude. Split into large-scale and small-scale modes.

Theory develops– improvements in systematics - intrinsic alignments, power spectrum models

Longer term:

Large ground based surveys

– PanSTARRS, DES

Large space based surveys

– DUNE, JDEM

Will measure 8 at a series of redshifts, constraining linear growth rate Will push to larger scales, where we have to make smaller non-linear corrections

Integrated Sachs-Wolfe effect

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Integrated Sachs-Wolfe effect

line-of-sight effect due to evolution of the potential in the intervening structure between the CMB and us affects the CMB power spectrum (different lecture) can also be measured by cross-correlation between large-scale structure and the CMB detection shows that the potential evolves and we do not have this balance between linear structure growth and expansion

– need either curvature or dark energy

Now quickly look at combining observations …

Model parameters (describing LSS & CMB)

content of the Universe

total energy density tot (=1?)

matter densitym

baryon densityb

neutrino density (=0?)

Neutrino speciesf

dark energy eqn of statew(a) (=-1?)

or w0,w1

perturbations after inflation

scalar spectral indexns (=1?)

normalisation8

running = dns/dk (=0?)tensor spectral index

nt (=0?)tensor/scalar ratio

r (=0?)

evolution to present day

Hubble parameterh

Optical depth to CMB

parameters usually marginalised and

ignoredgalaxy bias model

b(k) (=cst?)or b,QCMB beam error

BCMB calibration error

CAssume Gaussian, adiabatic fluctuations

WMAP3 parameters used

Multi-parameter fits to multiple data sets

Given WMAP3 data, other data are used to break CMB degeneracies and understand dark energy Main problem is keeping a handle on what is being constrained and why

– difficult to allow for systematics

– you have to believe all of the data! Have two sets of parameters

– those you fix (part of the prior)

– those you vary Need to define a prior

– what set of models

– what prior assumptions to make on them (usual to use uniform priors on physically motivated variables)

Most analyses use the Monte-Carlo Markov-Chain technique

Markov-Chain Monte-Carlo method

MCMC method maps the likelihood surface by building a chain of parameter values whose density at any location is proportional to the likelihood at that location p(x)

x

-ln(p(x))

an example chainstarting at x1

A.) accept x2

B.) reject x3

C.) accept x4

CHAIN: x1, x2, x2, x4, ...

x1 x2 x4 x3

A B

C

given a chain at parameter x, and acandidate for the next step x’, thenx’ is accepted with probability

1 p(x’) > p(x)

p(x’)/p(x) otherwise

for any symmetric proposal distributionq(x|x’) = q(x’|x), then an infinite number of steps leads to a chain in which the density of samples is proportional to p(x).

MCMC problems: jump sizes

q(x) too broad

chain lacks mobility as all candidates are unlikely

x

-ln(p(x))

x1

x

-ln(p(x))

x1

q(x) too narrow

chain only moves slowly to sample all of parameter space

MCMC problems: burn in

Chain takes some time to reach a point where the initial position chosen has no influence on the statistics of the chain (very dependent on the proposal distribution q(x) )

2 chains – jump sizeadjusted to be large initially, then reduceas chain grows

2 chains – jump sizetoo large for too long, so chain takes time to find high likelihood region

Approx. end of burn-in

Approx. end of burn-in

MCMC problems: convergence

How do we know when the chain has sampled the likelihood surface sufficiently well, that the mean & std deviation for each parameter are well constrained?

Gelman & Rubin (1992) convergence test:

Given M chains (or sections of chain) of length N, Let W be the average variance calculated from individual chains, and B be the variance in the mean recovered from the M chains. Define

Then R is the ratio of two estimates of thevariance. The numerator is unbiased if the chains fully sample the target, otherwise it is an overestimate. The denominator is an underestimate if the chains have not converged. Test: set a limit R<1.1

( )W

BNWN

N

R

111

++−

=

Resulting constraints

From Tegmark et al (2006)

Supernovae + BAO constraints

SNLS+BAO (No flatness) SNLS + BAO + simple WMAP + Flat

BAO BAO

SNe

SNe

WMAP-3

6-7% measure of <w>

(relaxing flatness: error in <w> goes from ~0.065 to ~0.115)

Further reading

Redshift-space distortions

– Kaiser (1987), MNRAS, 227, 1 Cluster Cosmology

– review by Borgani (2006), astro-ph/0605575

– talk by Allen, SLAC lecture notes, available online at

http://www-conf.slac.stanford.edu/ssi/2007/lateReg/program.htm Weak lensing

– chapter 10 of Dodelson “modern cosmology”, Academic Press

Combined constraints (for example)

– Sanchez et al. (2005), astro-ph/0507538

– Tegmark et al. (2006), astro-ph/0608632

– Spergel et al. (2007), ApJSS, 170, 3777