Structure Formation Risa Wechsler SLAC Summer Institute 2007 August 2, 2007 • So far: (Dodelson) Background cosmological model; how fluctuations are generated; and how the smooth Universe grows • These lectures: how fluctuations grow into the non-linear regime • Next week: (Allen, Nichol) How to test the picture The Plan • Today: Quick review of the Linear Power Spectrum and Growth of Fluctuations in the Linear Regime Basics of Non-Linear Structure Formation; Spherical Collapse Abundance of Dark Matter Halos (The “Mass Function”) • Tomorrow: • Clustering, Growth, and Structure of Dark Matter Halos • Dark Matter Substructure • Galaxy Clustering The initial conditions for structure formation The final conditions for structure formation on large scales SDSS, 2dF, 2MASS, DEEP2, etc The final conditions for structure formation on smaller scales rotation curves/density profiles ; substructures go. The Goal Some understanding of what happens in this movie, how we might test this picture with observations, and where it depends on the amount and nature of dark matter Structure formation depends on: • The initial fluctuation spectrum (type; tilt: n, amplitude: σ8) • The amount of matter (Ωm) • The nature of matter (baryonic, dark, cold, warm, hot) • The expansion rate (ΩΛ, h) The Primordial Power Spectrum inflation predicts that the fluctuations are: • Gaussian • close to scale-invariant • “adiabatic” • pressure fluctuations ~ density fluctuations • an overdense region contains overdensities of all particle species 5 5 II. NOTATION AND CONVENTIONS II. NOTATION AND CONVENTIONS In the following, I consider fluctuations in the density field ρ("x) described by the density In the following, I consider fluctuations in the density field ρ("x) described by the density contrast δ("x) [ρ("x) ρM]/ρM, where ρM is the mean mass density in the universe and "x contrast δ("x)≡ [ρ("x)− ρ ]/ρ , where ρ is the mean mass density in the universe and "x is a comoving ≡spatial c−oorMdinaMte. In the Mstandard paradigm, the universe is endowed with 5 is a comoving spatial coordinate. In the standard paradigm, the universe is endowed with primordial density fluctuations during an epoch of cosmological inflation and the primordial primordial density fluctuations during an epoch of cosmological inflation and the primordial density contrast iIsIa. staNtisOtiTcaAllyThIoOmNogenAeoNusDanCd OisoNtroVpiEc NGaTusIsOianNrSandom field. This density contrast is a statistically homogeneous and isotropic Gaussian random field. This means that the joint probability distribution of the density contrast at a set of points in means that the joint probability distribution of the density contrast at a set of points in space is given by a multivariate Gaussian distribution. Homogeneity requires that the mean In the fospllaocwe isnggiv,eIn cboynasmidueltrivflaruiactteuGaatuisosniasn idnisttrhibeutdioenn. sHitoymofigeelndeitρy(r"xeq)udiresscthraibt ethde bmyeatnhe density δ("x) , of the distribution and the two-point function δ("x )δ("x ) ξ("x , "x ) be invariant # $ # 1 2 $ ≡ 1 2 contrast δ("xδ)("x) , [oρf (t"xhe) distρribu]t/ioρn a,nwd htheeretwoρ-poiinst tfuhnectmioneaδn("xm1)δa(s"xs2)denξs(i"xt1y, "xi2n) btehienvuarniainvterse and "x u#nde≡r$translatio−ns. MThe twMo-point functMion is then a fu#nction only$ o≡f the separation vector under translations. The two-point function is then a function only of the separation vector is a comovbientgwesenpatwtioapl ocinotos,rdξ(i"xn1a, "xte2). =Inξ("xth1 e s"xt2a).nIdsoatrrdopyparerqaudiriegsmth,athξe("x)unisivinevrasrieanitsuenndderowed with between two points, ξ("x1, "x2) = ξ("x1 − "x2). Isotropy requires that ξ("x) is invariant under primordialrodteatniosnitsyasfluwceltl,usaotitohne stwdou-proiningt caon−rreelpatoiocnhfounfcctioosnmisoolonglyicaafluinncfltioantiofnthaenddisttahneceprimordial rotations as well, so the two-point correlation function is only a function of the distance between two points, ξ("x1, "x2) = ξ( "x1 "x2 ). density conbtertawsetenitswao psotianttsi,sξt(i"xc1a,l"xly2) =hoξm(| "xo1g−en"x2e| )o.us and isotropic Gaussian random field. This The Fourier transform of the de|nsit−y co|ntrast is given by the convention means that tThhee jFooiunrtierptrroanbsafobrmilitoyf thdeisdternisbituytcioonntraostf itshgeivednebnystihtye cconovnetnrtiaosnt at a set of points in 3 i!k !x δ("k) = d x δ("x)e · (1) " 3 i!k !x space is given by a multivariate Gaussδi(akn) =d!istdrixbuδ(t"xi)oen.· Homogeneity requires th(a1t) the mean ! δ("x) , of twhiteh dthiestinrviberusetitornansafonrdm the two-point function δ("x )δ("x ) ξ("x , "x ) be invariant # $ with the inverse transform # 1 2 $ ≡ 1 2 1 3 i!k !x under translations. The two-point function is then "a f−un· ction only of the separation vector δ("x) = 1 3 d 3k δ(k)e i!k !x. (2) δ("x) = (2π) d k δ("k)e− · . (2) (2π)3 ! betweenThetwo po iPntos, wξ("xer1, "x2 )Spectrum= ξ("x1 "x2). !Isotropy requires that ξ("x) is invariant under δ "k δ "x Notice that the ( ") have dimensio−ns of volume and that for a real-valued field ( ), the rotations aNsowticeellt,hastothtehδe(kt)whoa-vpe odiinmtencsioornrseolaf tvioolunmefuanncdttihoant fiosr oanrelyal-vaalufuedncfiteildonδ("xo)f, tthhee distance Fourier coefficients obey the relation δ( "k) = δ∗("k). We have implicitly assumed that there Fourier coefficients obfluctuationsey the relation inδ (−the"k) density= δ∗("k). fieldWe have implicitly assumed that there 1/3− 3 between twisosopmoeinvetrsy, laξr(g"xe1c,u"xt-2o)ff =scaξle(L"x1 V ρ"x(2xth,)ta.)t−reρnd (et)rs the integral δ("x) d x finite and that is some very large cut-off scaδl(ex|L,t≡) −=V 1/3 |that renders the integral |δ("x)|d3x finite and that this scale is much larger than any≡other scaρle (to)f interest so that it"p|lays n| o meaningful role. The Foutrhiies rsctarleainssmfourchmlaorgferththeandaennysiottyhercoscnatleraosf tintiseregsitvseonthbayt itt"hpleaycsonnovmeneatniionngful role. Using these conventionsthe, on Feouriercan com transfpute tormhe tw iso- pgivoinent fubnyction ξ("r) δ("x)δ("x + "r) in Using these conventions, one can compute the two-point function ξ("r) ≡ #δ("x)δ("x + "r)$ in terms of the Fourier coefficients, where the a3verage is tia!kke!xn over all spa≡ce#. The two-po$int terms of the Fourier coefficientδs,("kw)he=re thedavxeraδg(e"xi)setak·en over all space. The two-point (1) function is a func€ti on only of the amplitu!de of "r due to isotropy, and the result is function is a function othenly twof toh epointampl icortuderelationof "r due functionto isotrop yis:, and the result is with the inverse transform 1 3 1 2 sin(kr) ξ(r) = 1 k 3V − 1 δ(k) 2 sin(kr) d ln k. (3) ξ(r) = 2π2 k V − | δ(k)| kr d ln k. (3) 2 ! 6 2π ! | | kr 1 3 i!k !x The correlation function is the Fourier transform of th"e po−wer· spectrum The correlation function iδs (t"xhe)theF=ou rpoierwtrer3an sspectrumfodrmkofδt(hk eis:)pe ower s.pectrum (2) where the average is over an ensembl(e2oπf )uni!verses with the same statistical properties. The 1 2 P (k) V − 1 δ(k) 2 , (4) power spectrum has dimensions of vPol(ukm)≡e aVnd− s#o| δa(kq)|ua$n, tity that lends itself more easily(4t)o Notice that the δ("k) have dimensions of v≡olum#|e an| d$ that for a real-valued field δ("x), the direct interpretation iswhichthe dim ise noftensionle sgivs cenom binin dimensionlessation units: Fourier coefficients obey the relation δ( "k) = δ∗("k). We have implicitly assumed that there ∆2−(k) k3P (k)/2π2. (5) is some very large cut-off scale L V 1/3 th≡at renders the integral δ("x) d3x finite and that ≡ | | The correlation function δ2(#x) is simply the mass variance. From Eq. (3), ∆2(k) is the this scale is much larger than "any o#ther scale of interest so that it"plays no meaningful role. contribution to the mass variance from modes in a logarithmic interval in wavenumber, so Using these conventions, one can compute the two-point function ξ("r) δ("x)δ("x + "r) in that ∆2(k) 1 indicates order unity fluctuations in density on scales of order ≡k#. $ ∼ ∼ terms of theInFtohue rsitearndcaorde,fficoclidendtasrk, mwahteterre(CthDeMa)vmeordaegl,e∆i2s(kt)aiknecnreaosevserwiathllwsapveancuem. bTerh(eattwo-point function ilseaastfunntciltsioomneoenxcleyedoinfgtlyhesmaamll psclaitleuddeeteormf i"rnedubey tthoe ipshoytsricospoyf,thaenpdrotdhuectrioensuofltthies CDM in the early universe), but we observe the density field smoothed with some resolution. 1 3 1 2 sin(kr) Therefore, a quantitξy(orf)p=hysical intekresVt i−s thδe(dke)nsity field smodotlhnedko.n a particular scale (3) 2π2 | | kr RW, ! 3 The correlation function is theδ(#xF;oRuWr)ier trdanxs! fWor(m#x! of#xt;hReWp)δo(w#x!e)r spectrum (6) ≡ ! | − | The function W (x; RW) is the window function that weights the density field in a manner 1 2 P (k) V − δ(k) , (4) that is relevant for the particular applic≡ation. A#c|cordin|g$to the convention used in Eq. (6), the window function (sometimes called filter function) has units of inverse volume by di- mensional arguments. It is also useful to think of a window as having a particular window volume VW. The window volume can be obtained operationally by normalizing W (x) such that it has a maximum value of unity and is dimensionless. Call this new dimensionless 3 window function W !(x). The volume is given by integrating to give VW = d xW !(x). In this way, one thinks of the window weighting points in the space by differe"nt amounts. It should be clear that W (x) = W !(x)/VW. Roughly speaking, the smoothed field is the av- erage of the density fluctuation in a region of volume V R3 . The Fourier transform of W ∼ W the smoothed field is δ(#k; R ) W (#k; R )δ(#k), (7) W ≡ W # where W (k; RW) is the Fourier transform of the window function. The most natural choice of window function is probably a simple sphere in real space.