The Theory/Observation connectionlecture 5
the theory behind (selected) observationsof structure formationWill PercivalThe 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 velocitiesAll of structure growth happens because of peculiar velocitiesTimeInitially distribution of matter is approximately homogeneous ( is small)Final distribution is clustered
Linear peculiar velocitiesConsider 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 satisfiesIn conformal units, the continuity and Poisson equations are Look for solutions of the continuity and Poisson equations of the form u=F(a)gThe peculiar gravitational acceleration isSo, for linearly evolving potential, u and g are in same direction
Linear peculiar velocitiesSolution is given bywhereZeldovich approximation: mass simply propagates along straight lines given by these vectors The continuity equation can be rewrittenSo the power spectrum of each component of u is given byk-1 factor shows that velocities come from larger-scale perturbations than density field
Peculiar velocity observationsObviously, can only hope to measure radial component of peculiar velocitiesTo 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 spectrumThe 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 observationsSo peculiar velocities constrain f.can we measure these directly?
Redshift-space distortionsWe 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 distortionsIn 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 isDisplacement along wavevector k isThe displacement is directly proportional to the overdensity observed (on large scales)Kaiser 1987, MNRAS 227, 1Line-of-sight
Redshift space distortionsAt 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 observationsTherefore 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 directionsFingers of GodInfall around clustersExpected
Cluster cosmology Largest objects in Universe 10141015Msun Discovery of dark matter (Zwicky 1933) Can be used to measurehalo profiles
Cosmological test based on hypothesis that clusters form a fair sample of the Universe (White & Frenk 1991)
Cluster cosmologyCluster X-ray temperature and profile give total mass of system X-ray gas massCan therefore calculateIf we know s and b, whereWe can measureAllen et al., 2007, MNRAS, astro-ph/0706.0033
Cluster cosmologySaw in lecture 3 that the Press-Schechter mass function has an exponential tail to high massNumber of high mass objects at high redshift is therefore extremely sensitive to cosmologyBorgani, 2006, astro-ph/0605575Problem is defining and measuring mass. Determining whether halos are relaxed or not
Cluster observationsShort-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-lensingDeflection 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-lensingThe bend angle depends on the gravitational potential throughSo the lens equation can be written in terms of a lensing potentialThe lensing will produce a first order mapping distortion (Jacobian of the lens mapping)
Weak-lensingWe can write the Jacobian of the lens mapping asIn terms of the convergenceAnd 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-lensingGalaxy 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 isThe shear power spectra are related to the convergence power spectrum byAs expected, from a measurement of the convergence power spectrum we can constrain the matter power spectrum (mainly amplitude) and geometry
Weak-lensing observationsShort-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 modelsLonger 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 Wtot (=1?)matter densityWmbaryon densityWb neutrino densityWn (=0?)Neutrino speciesfndark energy eqn of statew(a) (=-1?) orw0,w1perturbations after inflation
scalar spectral indexns (=1?)normalisations8running a = dns/dk (=0?)tensor spectral indexnt (=0?)tensor/scalar ratior (=0?)evolution to present dayHubble parameterhOptical depth to CMBtparameters usually marginalised and ignoredgalaxy bias modelb(k) (=cst?)orb,QCMB beam errorBCMB calibration errorC
Assume 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 methodMCMC 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 x1A.) accept x2B.) reject x3C.) accept x4CHAIN: x1, x2, x2, x4, ...x1 x2 x4 x3ABCgiven a chain at parameter x, and acandidate for the next step x, thenx is accepted with probability p(x) > p(x)
p(x)/p(x) otherwisefor 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 sizesq(x) too broad
chain lacks mobility as all candidates are unlikelyq(x) too narrow
chain only moves slowly to sample all of parameter space
MCMC problems: burn inChain 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 grows2 chains jump sizetoo large for too long, so chain takes time to find high likelihood regionApprox. end of burn-inApprox. end of burn-in
MCMC problems: convergenceHow 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
Resulting constraintsFrom Tegmark et al (2006)
Supernovae + BAO constraintsSNLS+BAO (No flatness)SNLS + BAO + simple WMAP + FlatBAOBAOSNeSNeWMAP-36-7% measure of (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 athttp://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