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)aiknecnreaosveserwiathllwsapveancuem. 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. The window function is then
3 3 (x RW) 4πRW W (x; RW) = ≤ . (8) 0 (x > RW) Gravitational Instability
δ = (ρ-ρ0)/ρ0 • δ << 1: linear theory • δ >> 1 non-linear regime. • make assumptions (spherical sym.) • higher order perturbation theory • solve numerically • in general, Universe is lumpy on small scales and smoother on large scales -- can consider inhomogenieties as a perturbation to the homogeneous solution
competing gravitational & pressure terms the scale they are equal is called the Jeans’ length • in the radiation dominated case, modes inside the horizon are fixed but modes outside the horizon grow rapidly • the matter dominated case, all modes grow linearly 2 3 −1 δ δ x a δ = A(x)t / + B(x)t = 0( ) • in the lambda dominated case growth stops −2Ht δ = A(x) + B(x)e
• general case δ = δ0(x)ag(a, Ωm0)
g is constant at early times and scales as 1/a at late times for our cosmology, the action ended around z=0.5 The Power Spectrum • in CDM, in the matter dominated regime all scales grow equally P ~ k on large scales • scales smaller than the horizon: growth is stalled by the presence P(k) = |δ |2 of radiation pressure. growth k slows as a2, power spectrum P(k) = Ak nT(k)2 suppressed by k4. • power below the Jeans length is € suppressed. the transition scale P ~ k-3 on marks the scale of the horizon € small scales at matter-radiation equality fluctuation damping
• scales that have entered the horizon while dark matter particles are relativistic get erased by “free streaming” (fast random particle velocities disperse the fluctuations out of dense regions) • fluctuations smaller than the free-streeming scale can be completely erased • for CDM, this is way before zeq (this is why it’s “cold”) • for HDM, this happens at zeq, so only large scale perturbations survive. fluctuations in CDM
• δ ~ M-(n+3)/6 • for n=1 spectrum: δ ~ M-2/3 t2/3 • smaller fluctuations at bigger mass scales. Hierarchical Structure Formation • small structures form first and merge to form larger structures
what can we learn from measurements of the power spectrum?
• initial primordial power spectrum (tilt and running) • the nature of dark matter -- affects the transfer function (ratio of large to small scale power) • dark energy and curvature -- affects the growth function (normalization of CMB power spectrum compared to normalization today) z=10 z=7.2 z=4 a = 1/1+z
z=1 z=0.5 z=0 • structure formation in the non-linear regime • roughly, non linear is when the density fluctuations are ~ 1 ρ(x) − ρ¯ δ(x) = ρ¯ • equivalently, when the dimensionless power ~1 k3P (k) ∆2(k) = 2π2 • or , when the correlation function ~ 1 V 2 ξ(r) = P (k)sinkr/krk dk 2π2 ! • matter accumulates in dense regions; random motions halt the growth • most of the field of structure formation and galaxy formation concerns understanding the non-linear regime, and collapsed objects in the density field • some approximations possible, but accurate results must be obtained from simulations how to characterize this? very clumpy; structure on a wide range of scales • look at the evolution of P(k) into the non-linear regime • look at the distribution of density peaks • how many collapsed things of various masses? • how are they distributed in space? • what internal properties do they have? (internal density distribution? internal angular distribution? shapes? internal substructure?) evolution of the matter power spectrum
104 3 2 k P (k) non-linear power spectrum ∆ (k) = 103 2π2 z = 0.00 z = 0.98 102 largest scales are still in z = 3.05 the linear regime 101
) k 0 z = 7.02 ( 10 2 ! linear power spectrum finite volume box; -1 10 large modes have noise z = 14.87
10-2
10-3
10-4 0.01 0.10 1.00 10.00 100.00 k [ h / Mpc ] Springel et al 2005 Spherical Collapse the ‘tophat model’ to get a basic idea of what happens in the non-linear regime, use the approximation of spherical symmetry • consider a uniform, spherically-symmetric perturbation in an expanding background δ = ρ(t )/ρ (t )-1 i i b i δ M = (4 r 3/3)(1+ ) ρb π i δi • Can treat this as an independent scaled version of the Universe: non-linear collapse r x on small scales doesn’t change the linear evolution of the large-scale perturbations • investigate expansion, turnaround, collapse, and virialization (with a equation of motion for the perturbation cosmological (same as for a closed universe) GM constant) r¨ = − GM Λ r2 r¨ = − + r integrating r2 1 GM 3 r˙2 − = E = constant 2 r for EdS, solving this equation at turnaround, GM gives the density at E = − r turnaround as max ρmax ≈ 5.5 the perturbation will be in ρ¯ virial equilibrium when
2Kvir + Wvir = 0 GM GM E = Kvir + Wvir = −Wvir/2 ≈ = − 2rvir rmax which implies rvir = rmax/2 ie, 8 times denser than at turnaround turn- non-linear collapse the overdensity for around linear theory is δc ∼ 1.69
perturbation
ρvir ≈ 178 8 ¯ dlin=1.69 ρ
density 2 virial equilibrium universe 5.5 5.5* 8 * 4 = 178 ρmax ≈ 5.5 -3 ρ¯ a
scale factor universe expands by 22/3 less dense by factor of 4 now we can study the properties of these collapsed objects.
• collapsed things: “dark matter halos” • schematically: halos are self-gravitating systems in the universe • or: halos are non-linear peaks in the dark matter density field whose self-gravity has overcome the Hubble expansion • operationally: a halo is a non-linear peak in the dark matter density field with its boundary defined by a given density contrast. • let’s use the model just described to define it -- halos defined by a spherical overdensity, motivated by the spherical collapse model • lots of different definitions in the literature. sadly, halos don’t look like this: halos look like this: 9
variance σ2(R) as the smoothing scale R 0. Press & Schechter essentially assumed → ∞ → that objects will collapse on some small scale once the smoothed density contrast on this scale exceeds some threshold value, but that the nonlinearities introduced by these virialized objects do not affect the collapse of overdense regions on much larger scales. The collapsed objects act only as resolution elements that trace the larger-scale fluctuations which may collapse at some later time. Strictly speaking, this is not correct; however, it is approximately true when primordial power spectra are sufficiently shallow that the additional large-scale power generated by nonlinearities is small compared to the primordial fluctuations on these scales (a study in one-dimension is Ref. [27]). Moreover, this assumption leads to a simple parsing of the ingredients in the formation of nonlinear structure. The first ingredient is a characterization of the statistical properties of the primordial density fluctuations. In the standard picture this is set during the inflationary epoch. The second ingredient is the evolution of overdensities according to linear perturbation theory. This is encapsulated in the growth function D(a) = δ(k, a)/δ(k, a = 1) which is specified by the evolution of the background cosmology (e.g., Refs. [19, 28, 29]). The last ingredient is the threshold for collapse into a virialized object and is determined by examining the nonlinear collapse of spherical overdensities. Press & Schechter stated that the likelihood for collapse of objects of a specific size or mass (R M 1/3) could be computed by examining the density fluctuations on the desired ∝ scale. They continued by using a model for the collapse of a spherical tophat overdensity to argue that collapse on scale R should occur roughly when the smoothed density on that
scale exceeds a critical value δc, of order unity, independent of R. but we usuallThe implementyatio nassumeof the Press-Schechte r ptherescriptioyn is simple. The mass within a region in which the smoothed density fluctuation (dictated by the linear theory) is the
look clikritical evalu e thisδc, at some reandshift ywaz, corresponys:ds to an object that has just virialized with mass M(R). The relationship between mass and smoothing scale is set by the volume of the 3 window function. As an example, the relationship is M = 4πρMR /3 for a tophat window 3/2 3 and M = (2π) ρMR for a Gaussian window. Further, any region that exceeds the critical density fluctuation threshold, will meet that threshold when smoothed on some larger scale
R! > R. Consequently, the cumulative probability for a region to have a smoothed density above threshold gives the fractional volume occupied by virialized objects larger than the
then for a given overdensity, one 4π 2 M = ∆virρ¯ can define a mass 3 7 9
In this case, the smoothed field is the average density in spheres of radius R about point variance σ2(R) as the smoothing scale R W 0. Press & Schechter essentially assumed 3 → ∞ → !x. The window volume is simplythVaWt o=bj4ecπtRs Wwi/ll3.coHllaopwsevoenr,stohmise csmhoailclescoaflewoincdeowthehasms othotehed density contrast on this undesirable property that the shasrcpaletreaxncseietdios nsoimnectohnrfiesghuorladtvioanlues,pbauctetlheaatdtshteonopnolwineeraroitnieaslilntroduced by these virialized scales in Fourier space. Thereforoeb,jeitctissdooftneont acffoencvtenthienctoltlaopsemoofootvhertdheensbeoruengidoanrsyoninmruecahl larger scales. The collapsed space. As I discuss below, it is ooftbejnectcsonacvtenoinelnytatsoreinsotrluotdiounceelpemaretnictsultahratwtinradcoewthfeunlacrtgioern-scale fluctuations which may to ensure that the smoothed fieldchoallsappsaeratticsuolmare plartoepr etirmtiee.s.StTrihcetlythsrpeeeakminogs,tthcoismismnoont lcyorurescetd; however, it is approximately window functions are the real-spatcreuetowphheantpwriimndoordwiaol fpEowqe.r(8sp),ecwtriatharFeousurffiiecrietnratlnysfsohramllow that the additional large-scale power generated by nonlinearities is small compared to the primordial fluctuations on these 3[sin(kRRW) kRW cos(kRW)] W (k; RW) = − 3 , (9) scales (a study i(nkRonWe-)dimension is Ref. [27]). Moreover, this assumption leads to a simple the Fourier-space tophat windowp, aarnsidngthoef tGheauinsgsiraendiewntinsdinowth. eTfohremFatoiuonrieorf-snpoanclienetaorpshtarutcitsure. The first ingredient is a defined in Fourier space as characterization of the statistical properties of the primordial density fluctuations. In the standard picture this is 1set during the inflationary epoch. The second ingredient is the 1 (k RW− ) W (k; RW) = ≤ , (10) evolution of overdensities1 according to linear perturbation theory. This is encapsulated in 0 (k > RW− ) the growth function D(a) = δ(k, a)/δ(k, a = 1) which is specified by the evolution of the and is 1 1 1 bac1kgrou(nsidn(cxoRsm−ol)ogyx(Re.−g.,cRose(fsx.R[1−9,))28, 29]). The last ingredient is the threshold for W (x; R ) = W W W (11) W 2 3 − 1 3 c2oπllaRpWse into a virial(ixzeRdW−ob)ject and is determined by examining the nonlinear collapse of in real space. A disadvantage of tshpihsewriicnadl oowveridsetnhsaittieist. does not have a well-defined volume. This concern creeps up repeatedly inPrweshsa&t fSocllhoewchs.teTr hsteatGedautshsaitanthwe ilnikdeoliwhoiosd for collapse of objects of a specific size or mass (R Mex1p/3() cxo2u/ld2Rbe2 c)omputed by examining the density fluctuations on the desired ∝ W W (x; RW) = − 3/2 3 (12) scale. They con(2tiπn)uedRbWy using a model for the collapse of a spherical tophat overdensity with a Fourier transform that alsotohaarsgutehethfaotrmcololafpaseGoanusscsaialenR should occur roughly when the smoothed density on that scale exceeds a critical value δ , of order unity, independent of R. Press-Schechter2 2 cTheory W (k; RW) = exp( k /2RW− ), (13) The implemen−tation of the Press-Schechter prescription is simple. The mass within a with a width that is the reciprocarelgoiofnitisnwwidhtichh inthereasml sopoathceed. Tdehnesivtyoluflumcetuoaftitohne(Gdiactuastseidanby the linear theory) is the The variance3/2 3of the mass fluctuations3 is given by: window•is V = (2π) R . criticdakl vkalPue kδc, at some redshift z, corresponds to an object that has just virialized with W σW2 R ( )W 2 kR ( ) = 2 ( ) The density fluctuation field is ams!assusmkMed(Rt2o)π.bTehaeGrealautsisoinasnhirpanbdetowmeevnamriasbsleansod tshmeosomthoinogthsecdale is set by the volume of the 3 density flucassumetuation fiherelde δthat(!x; R W) i swisti nhade sphericalonwafuGnacutisos ntophati.anArsaann dwindooexmamvpawlrei,a function.bthlee areslawtieolnl sbheipcaiussMe it=r4eπpρ- MR /3 for a tophat window • 3/2 3 resents a sum of Gaussian randomandvaMria=bl(e2sπ. )ThρeMRvarfioarncaeGoafuδss(i!xa;nRw)incdaonw.beFucrotmhepr,uatnedy riengion that exceeds the critical Press-Schechter theory: The fraction of mass in collapsed objects • density fluctuation threshold, will meet that threshold when smoothed on some larger scale the same wmoray ase bmassivefore [ese ethanEq. M(3 )is] arnelatedd is to the fraction of the volume in which the smoothed Rinitial! > R .densityConseq ufieldently ,ist haboe cuvme usomelative pdensityrobabili ty for a region to have a smoothed density 2 2 2 2 threshold.σ (R) = δ a(b!xo;vRe )thr=eshodldlngikve∆s th(ke)frWac(tkio;nRal) vo.lume occupied by v(i1ri4a)lized objects larger t1h0an the # $ ! | | Thus, t•he pTheroba prbilobabilityity of att aofin iobtainingng a value ao valuef δ(!x; Rbetw) beeentwee nd δandan dd+dδ +d dis:δ is smoothing scale F (M). Integrating Eq. (15), this probability is 1 2 2 P (δ; R)dδ = exp[ δ /2σ (R)]dδ. 10 (15) 2πσ2(R) − 1 ν F (M) = ∞ P (δ; R)dδ = erfc , (16) So the fraction of mass in" collapsed objects is: smoothing scale F•(M). Integrating Eq. (15), this probabi!litδyc is 2 "√2# ∞ 1 ν where erfc(x) Fi(sMt)h=e coPm(δp; Rle)dmδ e=ntearfrcy err,or fuwhernctione, and(16ν) δc/σ(M) is the height of the !δc 2 "√2# ≡ where erfc(x) is the complementary error function, and ν δc/σ(M) heightis the hofe itheght throf tesholdhe in units of the standard threshold in units of the standard d≡eviation of the smoothed density distribution. In this threshold in units of the standard deviation of the smoothed density distribdeutiviationon. In ofth theis density distribution model, collapse of mass M is defined so that it occurs when the smoothed density fluctuation model, collapse of mass M is defined so that it occurs when the smoothed density fluctuation is δc oinsthδecaoppnroptrhiaeteaspcapler. oTphursiathteeresics aa ltey.picTalhscuasle thhaet risecoilslapasintgypaticthaelpsrecsaenlte that is collapsing at the present epoch, M", when the variance is σ(M") = δc. epoch, M", when the variance is σ(M") = δc. In the hierarchical power spectra that we consider, σ(R) becomes arbitrarily large as R becomes aIrbnitrtahriely hsmiealrl.arTchhusi,cFal(0)pionwEeq.r(1s6p)eschoturlad gtihveatthewfreacctionnosfidalel rm,asσs(inR) becomes arbitrarily large as R virializbedecoobmjectess; haowrbevietrr, aerrfcil(y0) =sm1 asollt.haTt Ehqu. s(1,6)Fst(a0te)s tihnatEonqly. h(a1lf6o)f tshhe omuaslsd give the fraction of all mass in density of the universe is contained in virialized objects. Press & Schechter noted this as a probvleimriaslsiozceiadtedowbitjhecntots;couhnotiwngeuvnedre,rdenrsfecr(e0gi)on=s in1thesointetghraltEqE. q(1.6)(. 1T6h)esestates that only half of the mass authordseanrgsuietdythoatf utnhdeerduennseivreegriosnes wisill ccoollnaptsaeionnetodovienrdevnisreiraelgiioznesdanodbmjueltciptlsie.d Press & Schechter noted this as F (M) in Eq. (16) by a factor of two in order to account for all mass. Though the sense of thisaeffpecrtoibs clermtainalyssuocchitahtaet dmowreimthassnwoiltl bceocuonntatiinnedginubnodunedrdobejnecstse, trheagt itohins s in the integral Eq. (16). These shouldalueatdhtoorpsrecaisreglyuaedfactohraofttwuonidncerreadse ninsFe(Mre)gisiofanrsfrowmilcloncvoinlcliangp.se onto overdense regions and multiplied Proceeding with this extra factor of two, the number of virialized objects with masses betweeFn (MMan)d iMn +EdqM. i(s16) by a factor of two in order to account for all mass. Though the sense dn ρ dF (M) of this effect is certaidnMly= suMch thatdMm. ore mass will be co(1n7t)ained in bound objects, that this dM M $ dM $ $ $ $ $ In terms of the mass variance, this is $ $ should lead to precisely a fa$ ctor o$ f two increase in F (M) is far from convincing. dn 2 ρ δ d ln σ δ2 dM = M c exp c dM ProceeddiMng with%tπhMis2 σe$xdtlnraM $fact"o−r 2oσf2 #two, the number of virialized objects with masses $ $ $ $ 2 ρ $d ln ν $ ν2 between M and M= + dMMν $ is $exp dM. (18) %π M 2 d ln M " − 2 # Without regard to the details of the shape of the powedr nspectrum, σ(MρM) ordνF(M()M, th)e mass 2 dM = dM. (17) function is close to a power law dn/dM M − for M M and is exponentially cut-off ∝ dM% " M $ dM $ > $ $ for M M". $ $ I∼n terms of the mass variance, this is $ $ $ $ dn 2 ρ δ d ln σ δ2 IV. EXCURSION SET THEOdRMY O=F THE MAMSS FcUNCTION exp c dM dM %π M 2 σ $d ln M $ " − 2σ2 # A weakness of the Press-Schechter approach is that it does not acco$unt for the$ fact that $ $ 2 2 ρM $d ln ν $ ν at a particular smoothing scale δ(&x; R) may be les=s than δc, yet itνm$ay be larg$eerxtphan δc dM. (18) %π M 2 d ln M " − 2 # Without regard to the details of the shape of the power spectrum, σ(M) or ν(M), the mass 2 function is close to a power law dn/dM M − for M M and is exponentially cut-off ∝ % " > for M M". ∼
IV. EXCURSION SET THEORY OF THE MASS FUNCTION
A weakness of the Press-Schechter approach is that it does not account for the fact that at a particular smoothing scale δ(&x; R) may be less than δc, yet it may be larger than δc • several ways to charachterize the clumpiness of the fluctuations. power spectrum, cf, • rms density flucuations dk k3P (k) σ2(R) = W 2(kR) ! k 2π2
• these fluctuations are roughly 1 on a scale of 8 Mpc/h is roughly 1 today -- this is roughly the non-linear mass • people often talk about the number -1 σ8 = σ(R=8 h Mpc): the mass variance of the primoridial density field w/in a sphere with R=8 h-1 Mpc projected forwards to z=0 using linear theory • this probes the PS on scales comparable to rich clusters mass variance
Pk here is the linear power spectrum, and W(kR) is the Fourier transform of the same real-space ‘tophat’ window function 10
smoothing scale F (M). Integrating Eq. (15), this probability is 10 1 ν F (M) = ∞ P (δ; R)dδ = erfc , (16) !δc 2 "√2# where erfc(x) is the complementary error function, and ν δ /σ(M) is the height of the smoothing scale F (M). Integrating Eq. (15), this probability is ≡ c threshold in units of the standard deviation of the smoothed density distribution. In this 1 ν F (M) = ∞ P (δ; R)dδ = erfcmodel, coll,apse of mass M is defined so tha(t1it6o)ccurs when the smoothed density fluctuation δc 2 "√ # ! is δc on2the appropriate scale. Thus there is a typical scale that is collapsing at the present where erfc(x) is the complementary error function, and eνpoch, Mδ "/, σwh(eMn t)heisvartihanecehiseσig(Mht") o=fδct.he ≡ c threshold in units of the standard deviation of the smoothInedthedheinersairtcyhicdalisptowriebr uspteicotnra. thIant wtehciosnsider, σ(R) becomes arbitrarily large as R collapsing becomes arbitrarily small. Thtodayus, F (0) in Eq. (16) should give the fraction of all mass in model, collapse of mass M is defined so that it occurs whevnirrmsitah lmassieze dsmobojeocttsh; ehdowdeveenr,sietrfyc(0fl)u=ct1usaottihoant Eq. (16) states that only half of the mass fluctuations, σ δ∼1 for collapse density of the uncollapsediverse is contained in virialized objects. Press & Schechter noted this as is δc on the appropriate scale. Thus there is a typical scale that is collearlyapsing at the present a problem associated with not counting underdense regions in the integral Eq. (16). These epoch, M", when the variance is σ(M") = δc. mass scale authors argued that underMdense regions will collapse onto overdense regions and multiplied In the hierarchical power spectra that we consider, σ(FR(M) )biencEoqm. (e1s6)abrybaitfraactroirlyof ltawrogien aorsdeRr to account for all mass. Though the sense •low mass halos form early, when the universe was denser of this effect is certainly such that more mass will be contained in bound objects, that this becomes arbitrarily small. Thus, F (0) in Eq. (16) should M*gi vis edefinedthe asfr theac ttypicalion collaof psingall mmassass in should lead to precisely a factor of two increase in F (M) is far from convincing. virialized objects; however, erfc(0) = 1 so that Eq. (16) stParotceeseditnhgawtitohnthlyis ehxatrlaf foacftotrhoef tmwoa, stshe number of virialized objects with masses density of the universe is contained in virialized objectsb. etPwereensMs &andSMch+ecdhMteisr noted this as dn ρ dF (M) dM = M dM. (17) a problem associated with not counting underdense regions in the integral Eq. (d1M6). ThMes$e dM $ $ $ $ $ In terms of the mass variance, this is $ $ authors argued that underdense regions will collapse onto overdense regions and multiplied$ $ dn 2 ρ δ d ln σ δ2 dM = M c exp c dM F (M) in Eq. (16) by a factor of two in order to account for all mass.dMThough%tπhMe 2sσen$dslen M $ " − 2σ2 # $ $ $ $ 2 ρ $d ln ν $ ν2 of this effect is certainly such that more mass will be contained in bound objec=ts, thaMt νt$his $exp dM. (18) %π M 2 d ln M " − 2 # should lead to precisely a factor of two increase in F (M)Wiisthfoaurt rfergoarmd tcoothnevdientcaiilns gof.the shape of the power spectrum, σ(M) or ν(M), the mass 2 function is close to a power law dn/dM M − for M M and is exponentially cut-off ∝ % " Proceeding with this extra factor of two, the number of >virialized objects with masses for M M". between M and M + dM is ∼ dn ρ dF (M) IV. EXCURSION SET THEORY OF THE MASS FUNCTION dM = M dM. (17) dM M $ dM $ $ $ $ $ A weakness of the Press-Schechter approach is that it does not account for the fact that In terms of the mass variance, this is $ $ $ $ at a particular smoothing scale δ(&x; R) may be less than δc, yet it may be larger than δc dn 2 ρ δ d ln σ δ2 dM = M c exp c dM dM %π M 2 σ $d ln M $ " − 2σ2 # $ $ $ $ 2 ρ $d ln ν $ ν2 = M ν $ $exp dM. (18) %π M 2 d ln M " − 2 # Without regard to the details of the shape of the power spectrum, σ(M) or ν(M), the mass 2 function is close to a power law dn/dM M − for M M and is exponentially cut-off ∝ % " > for M M". ∼
IV. EXCURSION SET THEORY OF THE MASS FUNCTION
A weakness of the Press-Schechter approach is that it does not account for the fact that at a particular smoothing scale δ(&x; R) may be less than δc, yet it may be larger than δc lin σ 10
smoothing scale F (M). Integrating Eq. (15), this probability is 1 ν F (M) = ∞ P (δ; R)dδ = erfc , (16) !δc 2 "√2# where erfc(x) is the complementary error function, and ν δ /σ(M) is the height of the ≡ c threshold in units of the standard deviation of the smoothed density distribution. In this model, collapse of mass M is defined so that it occurs when the smoothed density fluctuation
is δc on the appropriate scale. Thus there is a typical scale that is collapsing at the present
epoch, M", when the variance is σ(M") = δc.
In the hierarchical power spectra that we consider, σ(R) becomes arbitr1a0 rily large as R
becomes arbitrarismlyootshmingaslcla.le TF (hMu).s,IntFegr(a0ti)ngiEnq. E(1q5)., t(h1is6p)robsahbiolituylids give the fraction of all mass in 1 ν virialized objects; however, erfc(0)F (M=) =1 s∞oPt(δh;aRt)dδE=q.er(fc16) s,tates that only(1h6)alf of the mass !δc 2 "√2# where erfc(x) is the complementary error function, and ν δ /σ(M) is the height of the density of the universe is contained in virialized object≡s. c Press & Schechter noted this as threshold in units of the standard deviation of the smoothed density distribution. In this a problem associamtoeddel, cwolilatphse nofomtascsoMuins dteifinngedusonthdaetritdoeccnusrsewhrengtihoensmsoointhetdhdensiitnytfleugctruaatlioEn q. (16). These authors argued thisaδtc oun nthde earppdreopnrisaete rsceaglei.oTnhsus wthielrle ics oaltlyappicaslescoalne thoatoisvceorlldapesinngseat rtheegpiroensesnt and multiplied epoch, M", when the variance is σ(M") = δc. F (M) in Eq. (16) Ibnytheahifearacrcthoicral opofwetrwspoectirna tohartdweercotnosidearc, cσ(oRu)nbtecofmoers aarbliltramrilayslasr.ge Tas hRough the sense of this effect is cebretcoamineslyarbsituracrhily tsmhallt. mThousr, eF (m0)ainssEqw. (i1l6l) bsheoucldognivteatihneefrdactiinonbofoaullnmdassoibn jects, that this virialized objects; however, erfc(0) = 1 so that Eq. (16) states that only half of the mass should lead to prdeecnissiteylyof tahefuancivteorsre ios fcotnwtaionedinincrveiraiasliezedinobFject(sM. P)resiss&faScrhefcrhotemr nocteodntvhisnacsing. a problem associated with not counting underdense regions in the integral Eq. (16). These thePr ochaloeeding wit hmassthis extra fa cfunctiontor of two, the number of virialized objects with masses authors argued that underdense regions will collapse onto overdense regions and multiplied between M and MF (M•+) idn MEtheq. (i1s6 )nbumbery a factor o foftwo virializedin order to acco uobjectsnt for all mass. Though the sense of this effect is certainly such that more mass will be contained in bound objects, that this dn ρM dF (M) should lead to precisely a factdorMof t=wo increase in F (M) isdfMar f.rom convincing. (17) dM M $ dM $ Proceeding with this extra factor of two,$the number $of virialized objects with masses PS overpredicts number of $ $ In terms of the mbaetswseevn aMriaannd cMe,+tdhMisis is $ $ M
(ie, this formula worksat atop ~15%artic uforlar a swidemoo trangehing ofsc acosmologiesle δ(&x; R) m anday rbedshifts)e less than δc, yet it may be larger than δc − . f(M) = 0.315exp(−|lnσ 1 + 0.61|0 38) Jenkins et al. 2001 the halo mass function Press & Schechter 1974 Sheth & Tormen 1999, 2001 Jenkins et al 2001 Reed et al 2005 Warren at al 2005 ...
note: this is for friends of for a specific cosmology, friends halo finder, but similar results for other choices. Warren et al 2004 calibrated to ~5% the halo mass function
the halo mass function evolves strongly with time. Text mass function is a strong function of cosmological parameters −1 3.8 dn ρ¯ dlnσ f(σ) = −0.315 exp[−|0.61 − ln(DzσM)| ] = f(σ) dlnM M CldulnstMer Redshift Distributiongrowth isfunction Sensitive to the overall Darkpow Energyer Equation of State Parameter normalization spectrum w constraints:
Raising w at fixed !E: ! decreases volume surveyed σ =0.77,! decreases 0.9, 1.05 growth rate of 8 density perturbations
dN(z) dV = n(z) dzd" dz d"
dV 2 c 2 z = H z dA (1+ z) dz' dzd" ( ) d " A # 0 E (z') d 1+ z is proper distance ! A ( ) H(z) = HoE(z) is the Hubble parameter
volume growth effect Volumeeffect effect Growthdepends effect on H(z) abundance of massive clustersAugust sensitiv '04 - Joee toMohr the normalizationBlanco and Instrument Review shape of power spectrum ( at ~10 Mpc): constrains mass evolution of the cluster abundance sensitive to density, inflation (tilt), neutrino mass dark energy
!
! on small scales, this mass variance can depend on the nature of dark matter
Warm Dark Matter
Zentner & Bullock can also depend on details of inflation... mass variance
Zentner & Bullock Next Time:
• Halo Growth & Internal Structure • Halo Clustering & Galaxy Clustering