1. Large Scale Structure of the Universe

2. Evolution of the LSS – a brief history Picture credit: A. Kravtsov, http://cosmicweb.uchicago.edu/filaments.html Somewhat after recombination -- density perturbations are small on nearly all spatial scales. Dark Ages, prior to z=10 -- density perturbations in dark matter and baryons grow; on smaller scales perturbations have gone non-linear,  >>1; small (low mass) dark matter halos form; massive stars form in their potential wells and reionize the Universe. z=2 -- Most galaxies have formed; they are bright with stars; this is also the epoch of highest quasar activity; galaxy clusters are forming. In LCDM growth of structure on large (linear) scales has nearly stopped, but smaller non-linear scales continue to evolve. z=0 -- Small galaxies continue to get merged to form larger ones; in an open and lambda universes large scale (>10-100Mpc) potential wells/hill are decaying, giving rise to late ISW.

3. Matter Density Fluctuation Power Spectrum P(k)~k n Harrison-Zel’dovich n=1 A different convention: plot P(k)k 3

4. P(k) gives rms density fluctuations on various spatial scales 2dF

5. Peacock; astro-ph/0309240 P(k) gives rms density fluctuations on various spatial scales

6. Evolution of density fluctuations: the set-up z=1200 z=4 x 10 3 z=1 z>>10 10 log(t) log(r comov ) lambda-matter equality recombination; production of CMB matter-radiation equality end of inflation Planck time matter domination radiation domination lambda domination infla- tion Growth rate of a density perturbation depends on epoch (i.e. what component dominates global expansion dynamics at that time), and whether a perturbation k-mode is super- or sub-horizon. P(k) k k

7. Linear growth of density perturbations: Super-horizon, w comp. dominated, pre & post recomb. fluid pressure is not important on super-horizon scales, so it makes no difference whether recombination has taken place or not. Friedmann eq: different patches of the Universe will have slightly different average densities and curvatures – at a fixed H: CMB MRE inflation log(t) log(r comov ) MD CMB MRE inflation log(t) log(r comov ) RD

8. Linear growth of density perturbations: Sub-horizon, radiation dominated, pre recombination growing “decaying” mode mode zero dark matter has no pressure of its own; it is not coupled to photons, so there is no restoring pressure force. Jeans linear perturbation analysis applies: inflation CMB MRE log(t) log(r comov ) radiation dominates, and because radiation does not cluster  all  k =0… …but the rate of change of  k ’s can be non-zero

9. Linear growth of density perturbations: Sub-horizon, matter dominated, pre & post recomb. growing decaying mode mode zero also, can assume that total density is the critical density at that epoch: dark matter has no pressure of its own; it is not coupled to photons, so there is no restoring pressure force. Jeans linear perturbation analysis applies: Two linearly indep. solutions: growing mode always comes to dominate; ignore decaying mode soln. CMB MRE inflation log(t) log(r comov )

10. Linear growth of density perturbations: Sub-horizon, lambda dominated, pre & post recomb. “growing” decaying mode mode zero dark matter has no pressure of its own; it is not coupled to photons, so there is no restoring pressure force. Jeans linear perturbation analysis applies: Two linearly indep. solutions: growing mode always comes to dominate; ignore decaying mode soln. CMB MRE inflation log(t) log(r comov ) can assume the amplitude of perturbations is zero, because lambda, which dominates, does not cluster:

11. Linear growth of density perturbations: Sub-horizon, curvature dominated, pre & post recomb. “growing” decaying mode mode zero can assume the amplitude of perturbations is zero, because curvature, which dominates, does not cluster: dark matter has no pressure of its own; it is not coupled to photons, so there no restoring pressure force. Jeans linear perturbation analysis applies: Two linearly indep. solutions: growing mode always comes to dominate; ignore decaying mode soln. CMB MRE inflation log(t) log(r comov )

12. Linear growth of density perturbations: dark matter, baryons, and photons CMB MRE log(t) log(r comov ) inflation

13. CMB MRE z=1 log(t) log(r comov ) Evolution of matter power spectrum log(k) EoIn Now On sub-horizon scales growth of structure begins and ends with matter domination

14. CMB MRE z=1 log(t) log(r comov ) P(k) k k k k k k Evolution of matter power spectrum log(k) EoIn Now sub-horizon perturb. do not grow during radiation dominated epoch baryonic oscillations appear – the P(k) equivalent of CMB T power spectrum Harrison-Zeldovich spectrum P(k)~k from inflation high-k small scale perturbations grow fast, non-linearly

15. Transfer Functions Transfer function is defined by this relation: Peacock; astro-ph/0309240

16. Growth of large scale structure the Virgo Collaboration (1996) 350 Mpc Standard spatially flat  matter =1.0 fractional overdensity ~1/(1+z) Dark Matter density maps from N-body simulations Recent epoch: dark matter or dark energy dominated? dominated? During matter dominated epoch, fractional overdensity grows as the scale factor. The corresponding potential fluctuations stay constant, because decrease in average density and increase in linear size combined compensate for  ~ a

17. Growth of large scale structure 350 Mpc Lambda (DE) spatially flat  matter =0.3 Standard spatially flat  matter =1.0 fractional overdensity ~const fractional overdensity ~1/(1+z) Dark Matter density maps from N-body simulations the Virgo Collaboration (1996)

18. Growth of large scale structure 350 Mpc Lambda (DE) spatially flat  matter =0.3 Standard spatially flat  matter =1.0 In linear theory gravitational potential decays if DE or negative curvature dominate late time expansion gravitational potential ~(1+z) gravitational potential ~const the Virgo Collaboration (1996)

19. Late Integrated Sachs-Wolfe (ISW) Effect Energy Sachs & Wolfe (1967) ApJ 147, 73 Crittenden & Turok (1996) PRL 76, 575 Look for correlation between CMB temperature fluctuations and nearby structure. Detection of late ISW effect in a flat universe is direct evidence of Dark Energy If a potential well evolves as a photon transverses it, the photon’s energy will change potential well photon gains energy after crossing a potential well

20. Detecting late ISW Boughn & Crittenden (2005) NewAR 49, 75, astro-ph/0404470 HEAO1 hard X-rays full sky median z~0.9 NVSS 1.4 GHz nearly full sky radio galaxies; median z~0.8 Late ISW is detected as a cross-correlation, CCF on the sky between nearby large scale structure and temperature fluctuations in the CMB. Lines are LCDM predictions, not fits to data Note: points are highly correlated

21. lambda dom. log(t) log(r comov ) lambda-matter equality CMB MRE end of inflation Planck time matter domination radiation domination inflation Coles & Lucchin Growth of baryonic perturbations without DM: given that the observed fluctuations in the potential at the CMB (z=1000) on horizon scales are ~10 -5, and assuming linear growth of perturbations gives  now =  CMB (1+z) = 0.01. However, today on these scales we see rms overdensities ~10-100 times larger. with dark matter Growth of perturbations with and without DM without dark matter

22. Baryonic Acoustic Oscillations One wave around one center: Wave starts propagating at Big Bang; end at recombination. The final length is the sound crossing horizon at recomb. (Change of color means recombination.) Many waves superimposed

23. Matter power spectrum - observations Baryonic Acoustic Oscillations (BAO) SDSS and 2dF galaxy surveys from k-space to real space gal. corr. fcn. comoving r (Mpc/h) BAO bump Percival et al. astro-ph/0705.3323 Eisenstein et al. astro-ph/0501171 Narrow feature: standard ruler standard ruler (sound crossing horizon at recombination)

24. Recombination affects the matter power spectrum too Eisenstein et al. astro-ph/0501171 sound horizon size at recombination Luminous SDSS red galaxies, z ~ 0.35 galaxy correlation function  m h 2 =0.12, 0.13, 0.14  m h 2 =0.130+/-0.011

25. Observed fluctuations in temperature and matter density Tegmark et al. Angular size of the sound crossing horizon at recombination Comoving size of the horizon at matter radiation equality These two power spectra are the main statistical descriptors of the large scale structure of the Universe. Both are fully consistent with what is usually known as the concordance model:     DM =   bar =  flat, n i = 

26. Quantifying LSS on linear and non-linear scales Picture credit: A. Kravtsov, http://cosmicweb.uchicago.edu/filaments.html The mass function of discrete objects is the number density of collapsed dark matter halos as a function of mass - n(M)dM. This was evaluated analytically by Press & Schechter (1974) The power spectrum quantifies clustering on spatial scales larger than the sizes of individual collapsed halos Internal structure of individual collapsed halos: one can use an analytical description for mildly non- linear regimes, but numerical N-body simulations are needed to deal with fully non-linear regimes. The 2pt correlation fcn is another way to quantify clustering of a continuous fluctuating density field, or a distribution of discrete objects, like collapsed DM halos. these are Fourier transforms of each other

27. Alternative definition: take two small volumes distance r apart; the joint probability that you will find a galaxy in either one of the two dV volumes a distance r apart is given by. If you are sitting on a galaxy, the probability dP that you will find another galaxy in a volume dV a distance r away from you is given by where n = average number density of galaxies. dP is the number of galaxies you expect to find in a volume dV. Correlation functions r dV r dV1 dV2 Two-point correlation function is a measure of the degree of clustering. It is a function of distance r only,. Suppose we are told that. What does that mean? best fit line

28. Estimating 2pt correlation function How does one calculate the 2pt correlation function given a distribution of galaxies is space? of galaxies is space? – Count the number of pairs of galaxies for every value of separation r. Then divide this histogram by the number of pairs expected if the spatial distribution of galaxies were random, and subtract 1. clustered random # pairs separation r 1 0 Correlation functions measure the fractional excess of pairs compared to a random distrib. clustered random

29. Correlation fcn and correlation length linear vertical scale Correlation length Correlation length is defined as the scale where so expect twice the number of galaxies compared to random. For galaxies, correlation length is ~5 Mpc, for rich galaxy clusters it is ~25 Mpc.

30. 2pt correlation function and power spectrum linear vertical scale Power spectrum is a Fourier transform of the correlation function:

31. Mass function of collapsed halos: Press-Schechter rms dispersion in mass, or, equivalently, overdensity , in spheres of radius R Press-Schechter (1974) main assumption: the fraction of spheres with volume V having overdensity  is Gaussian distributed Smoothly fluctuating density field; randomly scattered equal volume spheres, each has some overdensity  Some of these volumes will have a large enough overdensity (  c >1.69) that they will eventually collapse and form gravitationally bound objects. What is the mass function of these objects at any given cosmic epoch? large R medium R small R 0 1.69  fraction of volumes these spheres collapse

32. The fraction of spheres that will eventually collapse is The fraction of spheres that have just collapsed ( of all possible M, but same  c ) Mass function of collapsed halos: Press-Schechter How much mass in every unit of volume is contained in these objects? How many of the collapsed objects are there? power exponential law

33. Press-Schechter halo mass function power law exponential cut-off large R medium R small R large R medium R small R 0 1.69  fraction of volumes Press-Schechter vs. numerical simulations: solid red lines: simulations blue dotted: Press-Schechter green dashed: extended Press-Schechter (takes into account non-sphericity of proto halos.)

34. Collapse of individual DM halos rmrm time In comoving coordinates a sphere, centered on a local overdensity shrinks in time; Hubble expansion is getting retarded by the overdensity. At some point, the sphere’s expansion stops (turn-around), and the sphere starts to collapse. Hubble expansion local overdensity rmrm Halos collapse from inside out. constant time

35. Collapse of individual DM halos time radius turn-around; overdensity decouples from the Hubble flow reaches asymptotic radius parametric equations apply non-linear evolution, shell-crossing, relaxation at smaller radii (larger overdensities) halo is virialized shell-crossing turn-around radius moves out with time; halos collapse and virialize from inside out. at turn- around internal density increase external density decrease KE+0.5PE=0

36. Collapse of individual halos: the algebra leading to  =4.5 at turn-around

37. Why are there no galaxies with M>10 13 M sun ? Whether a galaxy forms in a given halo is determined by the rate of gas cooling. galaxies t cool < t dyn So far we have been mostly concerned with dark matter halos. The distribution DM halos in mass is continuous from ~10 9 to ~10 15 M sun. But, DM halos with M>few x 10 12 M sun are not observed to host galaxies, only clusters of galaxies. Why? galaxy clusters t cool > t dyn gas has cooled gas has not cooled Cooling curve diagram about 1/10 of virial radius, r 200 for both

38. Cosmological Parameters From the number density of galaxy clusters can obtain: Measurements of global geometry: std candles – Supernova Type Ia std rulers – Baryonic Acoustic Oscillations: CMB – a test of flatness a test for Lambda – late ISW effect