Will Percival The University of Portsmouth

1. The Theory/Observation connectionlecture 5the theory behind (selected) observationsof structure formation Will Percival The University of Portsmouth

2. 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)

3. 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

4. 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

5. 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 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

6. 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

7. 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

8. 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.

9. 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

10. 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.

11. 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” Expected Infall around clusters

12. 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)

13. 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

14. 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 Problem is defining and measuring mass. Determining whether halos are relaxed or not Borgani, 2006, astro-ph/0605575

15. 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

16. 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)

17. 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

18. 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)

19. 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

20. 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

21. 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

22. Integrated Sachs-Wolfe effect

23. 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

24. Now quickly look at combining observations …

25. Model parameters (describing LSS & CMB) content of the Universe total energy density Wtot (=1?) matter density Wm baryon density Wb neutrino density Wn(=0?) Neutrino species fn dark energy eqn of state w(a) (=-1?) or w0,w1 perturbations after inflation scalar spectral index ns (=1?) normalisation s8 running a=dns/dk (=0?) tensor spectral index nt (=0?) tensor/scalar ratio r (=0?) evolution to present day Hubble parameter h Optical depth to CMB t parameters usually marginalised and ignored galaxy bias model b(k) (=cst?) or b,Q CMB beam error B CMB calibration error C Assume Gaussian, adiabatic fluctuations

26. WMAP3 parameters used

27. 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

28. 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) an example chain starting at x1 A.) accept x2 B.) reject x3 C.) accept x4 given a chain at parameter x, and a candidate for the next step x’, then x’ is accepted with probability • p(x’) > p(x) • p(x’)/p(x) otherwise -ln(p(x)) A B C x x1 x2 x4 x3 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: x1, x2, x2, x4, ...

29. -ln(p(x)) -ln(p(x)) x x x1 x1 MCMC problems: jump sizes q(x) too broad chain lacks mobility as all candidates are unlikely q(x) too narrow chain only moves slowly to sample all of parameter space

30. 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

31. 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 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

32. Resulting constraints From Tegmark et al (2006)

33. Supernovae + BAO constraints SNe WMAP-3 6-7% measure of <w> SNe BAO BAO SNLS+BAO (No flatness) SNLS + BAO + simple WMAP + Flat (relaxing flatness: error in <w> goes from ~0.065 to ~0.115)

34. 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