Large Scale Structure of the Universe

Evolution of the LSS – a brief history 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, d>>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. Picture credit: A. Kravtsov, http://cosmicweb.uchicago.edu/filaments.html

Matter Density Fluctuation Power Spectrum P(k)~kn Harrison-Zel’dovich n=1 A different convention: plot P(k)k3

Evolution of density fluctuations: the set-up z=1200 z=4 x 103 z=1 z>>1010 log(t) log(rcomov) lambda-matter equality recombination; production of CMB matter-radiation equality end of inflation Planck time domination matter radiation lambda infla- tion sub-horizon super-horizon P(k) k P(k) k 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.

Linear growth of density perturbations: Super-horizon, w comp 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(rcomov) MD CMB MRE inflation log(t) log(rcomov) RD

Linear growth of density perturbations: Sub-horizon, radiation dominated, pre recombination 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: log(t) zero CMB radiation dominates, and because radiation does not cluster  all dk=0… MRE inflation log(rcomov) …but the rate of change of dk’s can be non-zero growing “decaying” mode mode

Linear growth of density perturbations: Sub-horizon, matter dominated, pre & post recomb. 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: log(t) zero also, can assume that total density is the critical density at that epoch: CMB MRE inflation log(rcomov) Two linearly indep. solutions: growing mode always comes to dominate; ignore decaying mode soln. growing decaying mode mode

Linear growth of density perturbations: Sub-horizon, lambda dominated, pre & post recomb. 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: log(t) can assume the amplitude of perturbations is zero, because lambda, which dominates, does not cluster: zero CMB MRE inflation log(rcomov) Two linearly indep. solutions: growing mode always comes to dominate; ignore decaying mode soln. “growing” decaying mode mode

Linear growth of density perturbations: Sub-horizon, curvature dominated, pre & post recomb. 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: log(t) can assume the amplitude of perturbations is zero, because curvature, which dominates, does not cluster: zero CMB MRE inflation log(rcomov) Two linearly indep. solutions: growing mode always comes to dominate; ignore decaying mode soln. “growing” decaying mode mode

Linear growth of density perturbations: dark matter, baryons, and photons log(t) CMB MRE inflation log(rcomov)

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

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

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

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

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

Late Integrated Sachs-Wolfe (ISW) Effect If a potential well evolves as a photon transverses it, the photon’s energy will change Sachs & Wolfe (1967) ApJ 147, 73 Crittenden & Turok (1996) PRL 76, 575 Energy Energy Energy Energy photon gains energy after crossing a potential well potential well 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

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

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

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

Recombination affects the matter power spectrum too sound horizon size at recombination Luminous SDSS red galaxies, z ~ 0.35 Wmh2=0.12, 0.13, 0.14 galaxy correlation function Wmh2=0.130+/-0.011 Eisenstein et al. astro-ph/0501171

Quantifying LSS on linear and non-linear scales The power spectrum quantifies clustering on spatial scales larger than the sizes of individual collapsed halos 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 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) 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. Picture credit: A. Kravtsov, http://cosmicweb.uchicago.edu/filaments.html

Correlation functions 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? 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. best fit line 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. r dV r dV1 dV2

Estimating 2pt correlation function How does one calculate the 2pt correlation function given a distribution 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 # pairs separation r clustered random random 1 -1 Correlation functions measure the fractional excess of pairs compared to a random distrib. separation r

Correlation fcn and correlation length linear vertical scale 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.

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

Mass function of collapsed halos: Press-Schechter Smoothly fluctuating density field; randomly scattered equal volume spheres, each has some overdensity d. Some of these volumes will have a large enough overdensity (dc>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? rms dispersion in mass, or, equivalently, overdensity d, in spheres of radius R large R medium R small R 0 1.69 d fraction of volumes Press-Schechter (1974) main assumption: the fraction of spheres with volume V having overdensity d is Gaussian distributed these spheres collapse

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

Press-Schechter halo mass function large R medium R small R 0 1.69 d fraction of volumes power law exponential cut-off 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.) small R medium R large R

Collapse of individual DM halos Hubble expansion 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. local overdensity time rm constant time rm Halos collapse from inside out.

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

Collapse of individual halos: the algebra leading to d=4 Collapse of individual halos: the algebra leading to d=4.5 at turn-around

Why are there no galaxies with M>1013Msun ? So far we have been mostly concerned with dark matter halos. The distribution DM halos in mass is continuous from ~109 to ~1015 Msun. But, DM halos with M>few x 1012Msun are not observed to host galaxies, only clusters of galaxies. Why? about 1/10 of virial radius, r200 for both Cooling curve diagram galaxies tcool < tdyn gas has cooled gas has not cooled galaxy clusters tcool > tdyn Whether a galaxy forms in a given halo is determined by the rate of gas cooling.

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