Will Percival The University of Portsmouth
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Transcript of Will Percival The University of Portsmouth
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
QuickTime™ and aTIFF (Uncompressed) decompressor
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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
QuickTime™ and aTIFF (Uncompressed) decompressor
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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