The Theory/Observation connection lecture 5 the theory behind (selected) observations of 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

Initially distribution of matter is approximately homogeneous (  is small) Final distribution is clustered Time

Linear peculiar velocities

Consider galaxy with true spatial position

x(t)=a(t)r(t)

, then differentiating twice and splitting the acceleration

d 2 x/dt 2 =g 0 +g

into expansion (

g 0

)and peculiar (

g

) components, gives that the peculiar velocity

u(t)

defined by

a(t)u(t)=dx/dt

satisfies The peculiar gravitational acceleration is So, for linearly evolving potential,

u

and

g

are in same direction 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

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

So peculiar velocities constrain

f

.can we measure these directly? 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

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 Line-of-sight The displacement is directly proportional to the overdensity observed (on large scales) Kaiser 1987, MNRAS 227, 1

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

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

We can compute the correlation function  r

p,

 ), including galaxy pair directions Infall around clusters

Cluster cosmology

 Largest objects in Universe – 10

14

…10

15

Msun – 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

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

Cluster cosmology

Borgani, 2006, astro-ph/0605575

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

General relativity: Curvature of spacetime locally modified by mass condensation 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)

Weak-lensing

 Assumptions – weak field limit v

2

/c

2

<<1 – stationary field t

dyn

/t

cross

<<1 – thin lens approximation L

lens

/L

bench

<<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

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 W

tot (=1?)

matter density W W W

m

baryon density

b

neutrino density n

(=0?)

Neutrino species

f

n dark energy eq n of state or

w(a) (=-1?) w 0 ,w 1 perturbations after inflation

scalar spectral index

n s (=1?)

normalisation running a s

= dn

tensor spectral index

n 8 t s /dk (=0?) (=0?)

tensor/scalar ratio

r (=0?)

Assume Gaussian, adiabatic fluctuations

evolution to present day

Hubble parameter

h

Optical depth to CMB t

marginalised and ignored

galaxy bias model or

parameters usually b(k) (=cst?) b,Q

CMB beam error

B

CMB calibration error

C

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) given a chain at parameter x, and a candidate for the next step x’, then x’ is accepted with probability 1 p(x’)

>

p(x) p(x’)/p(x) otherwise -ln(p(x)) an example chain starting at x

1

A.) accept x

2

B.) reject x

3

C.) accept x

4

A B C x

1

x

2

x

4

x

3

for any symmetric proposal distribution q(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).

CHAIN: x

1

, x

2

, x

2

, x

4

, ...

x

MCMC problems: jump sizes

q(x) too broad chain lacks mobility as all candidates are unlikely -ln(p(x)) x

1

x q(x) too narrow chain only moves slowly to sample all of parameter space -ln(p(x)) x

1

x

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) ) Approx. end of burn-in Approx. end of burn-in 2 chains – jump size adjusted to be large initially, then reduce as chain grows 2 chains – jump size too large for too long, so chain takes time to find high likelihood region

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   1  1

N



B R



N N

 1

W W

Then R is the ratio of two estimates of the variance. 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

Resulting constraints

From Tegmark et al (2006)

Supernovae + BAO constraints

SNe WMAP-3 6-7% measure of BAO SNLS+BAO (No flatness) SNe BAO SNLS + BAO + simple WMAP + Flat (relaxing flatness: error in 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