The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for nonlinear structure formation, the spherical collapse moel. But we cannot observe irectly the process of structure formation. What we can see are the results, for example clusters of galaxies. We can count these objects an fin their ensity, an also further etails about their statistical istribution. One example we have encountere alreay is the galaxy luminosity function, which gives the number ensity of galaxies of a given luminosity. The quantity we will consier in the following is the mass function nm) of cosmic structures, efine by N = nm)m, 1) where N is the number of structures per unit volume with mass between M an M + M. It shoul be fairly obvious how the mass function is measure in principle: Just select a volume in space an count the number of structures of a given mass containe within it. In practice there will, of course, be complications an I will mention a couple of them when we look at an application of the theory that now follows. Having an analytical expression for the mass function woul be goo. In a classic paper publishe in 1974 Press an Schechter set out to provie us with that. We will now look at what they i. Consier the ensity fluctuation fiel δx) I have omitte to make the time epenence explicit, but it is of course there.) We can filter the fiel on a length scale R, which means throwing away information about the etaile behaviour of δx) on scales smaller than R. The filtering can be accomplishe by convolving δ with a winow function W R x x ): δx; R) = 3 x δx )W R x x ). ) A popular choice of W R, an the one we will use, is the top hat function Wx x ) = 1, x x < R, = 0, otherwise. 3) 1
The region of raius R contains a mass M = 4π 3 ρ 0R 3, 4) where ρ 0 is the uniform backgroun ensity. We introuce the notation δx; R) δ M. Press an Schechter assume that the ensity fiel has a Gaussian probability istribution: Pδ M )δ M = ) 1 exp δ M δ πσm σm M, 5) where σ M is the variance of the ensity fiel filtere on a scale R enclosing a mass M. The probability that at some point δ M excees some critical value is now given by P >δc M) = Pδ M )δ M. 6) This probability epens on the filter mass M, an also on the reshift z through the variance σm = σr = 1 kk P π m k, z) W Rk), 7) 0 where P m k, z) is the matter power spectrum an W R k) is the Fourier transform of the top hat filter function. It is also the case that P >δc is proportional to the number of cosmic structures characterize by a ensity perturbation >, regarless of whether they are isolate or containe within enser structures which collapse with them. A value of of great interest to us is 1.686, since we have seen that this is the linear-theory ensity contrast which correspons to virialize structures. To fin the number of regions with mass M which are isolate, in other wors surroune by unerense regions, we must subtract P >δc M + M). This ignores to so-calle clou-in-clou problem: At a given instant some object, which is nonlinear on a scale M, can later be containe within another object on a larger mass scale. In essence we assume that the only objects which exist on a given mass scale are those which have just collapse. Another problem is that we cannot treat unerense regions properly those with δ < 0), which means that half the mass is unaccounte for.
Presumably these unerensities will accrete onto overensities. Press an Schechter fixe this problem by brute force: They multiplie their mass function by a factor of. We can now set up an expression for the mass function nm). It is proportional to the ifference of the probabilities alreay mentione, but to convert probabilites to a quantity with units of per volume, we nee to multiply by the backgroun ensity ivie by the mass scale M. In aition we have the artificial factor of to account for unerense regions: nm)m = ρ 0 M [P > M) P >δc M + M)] = ρ 0 M = ρ 0 M P >δc We can use the funamental theorem of calculus x x a M M P >δc M. 8) M ft)t = fx) = x to evaluate the erivative, along with the substitution x = P >δc = = 1 π 1 a πσm exp / σ M e x = 1 π e /σ M = Finally we have nm)m = π x x σm ft)t, 9) δ M σ M ) δc σm : δ M 1 δ c e δ π σm c /σ M. 10) M ρ 0 Mσ M exp σ M ) M. 11) This mass function, with = 1.686, gives us the number ensity of collapse objects per unit mass. The normal proceure is to evaluate nm) by calculating σ M an its erivative from the linear theory matter power spectrum. 3
An application of Press-Schechter theory to galaxy clusters Determining nm) from galaxy clusters has been a popular job for cosmologists. One thing that allows one to o is to fin the normalization of the matter power spectrum through the parameter sigma 8, that is, σ R evaluate for R = 8 h 1 Mpc at z = 0. But in orer to o this, one must be able to etermine the mass of galaxy clusters. Since the ominant contribution to the mass of a cluster is in the form of ark matter, this is a non-trivial task. At least three ifferent approaches may be taken: 1. Determine the line-of-sight velocity ispersion, assume isotropy, an use the virial theorem.. Determine the temperature of the X-ray emitting hot gas in the cluster. The temperature is relate to the epth of the graviational potential well in the cluster, so there shoul be a corresponence between temperature an mass. The problem here is to moel the gas accurately. 3. Gravitational lensing. This is in principle the least ambigious an most assumption free metho, but it is time-consuming. Increasingly, though, this is the metho of choice. We will consier a simplifie version of how the cluster abunance can be use to fin σ 8, assuming for simplicity that the backgroun universe is escribe by the Einstein-e Sitter moel. The number ensity of clusters at temperature corresponing to k B T = 7 kev at the present epoch has been measure to be n7 kev, z = 0) =.0 +.0 1.0 10 7 h 3 Mpc 3 kev 1. 1) In simulations of structure formation in an ES backgroun one fins that a cluster with X-ray temperature T = 7.5 kev has a mass within a raius 1.5 h 1 Mpc this is often calle the Abell raius) of M Abell = 1.1 ± 0.) 10 15 h 1 M. 13) So the simulation gives us useful, but not immeiately applicable information. To compare with the theoretical mass function, we nee to know what virial 4
mass a temperature of 7 kev correspons to. Fortunately the simulation can also give us the ensity profile of the cluster, allowing us to convert the Abell mass to the mass within the virial raius. An a scaling relation like ) /3 k B T 0.07 kev = M 1 + z vir) 14) 10 1 h 1 M can be use to scale the mass from 7.5 to 7 kev. The en result turns out to be M vir = 1. ± 0.3) 10 15 h 1 M. 15) With the top-hat filter this correspons to a sphere of raius R = 3Mvir 4πρ c0 ) 1/3 10 h 1 Mpc, 16) where the critical ensity at the present epoch is ρ c0 =.78 10 11 h M Mpc 3. We now nee to relate M to k B T) using equation 14) with z vir = 0: k B T) 0.07 kev = 1 3 M ) /3 M M 17) 10 1 h 1 M which can be rewritten as k B T) = k B T M. 18) 3 M The next thing we nee is the erivative of σ M. In accurate work we woul have to evaluate the erivative numerically from the efinition of σ M in terms of the matter power spectrum. Here we will settle for something simpler. A ecent fit to σ M = σ R for the ES moel with the scalar spectral inex n s = 1 is ) 0.8 R σ R = σ 8. 19) 8 h 1 Mpc This an the relation M = 4πρ c0 R 3 /3 between mass an raius allows us to calculate the erivative we nee: M = σ R R 5 R M
= 0.8 σ ) 1 R M R R = 0.8 σ ) R 3M 1 R R = 0.8 σ R 3 M. 0) Inserting this in expression 11) for the mass function gives ) ρ c0 0.8 nm)m = exp M. 1) π M 3 σ R σr To get something we can compare with the observe cluster abunance we use equation 18) an nm) = N ) M to go from nm) to nt): so that N M = N k B T) k B T) M = nt)k BT) M, 3) nt) = 3M nm). 4) k B T Inserting equation 0) an substituting M = M vir = 1. 10 15 h 1 M, k B T = 7 kev finally leaves us with ) nk B T = 7 kev) = 1.06 10 5 h 3 Mpc 3 kev 1 exp. 5) σ R σr Equating this to the observe value.0 10 7 h 3 Mpc 3 kev 1 leaves us with the equation ) x exp x = 0.019, 6) with x = /σ R. This equation must be solve numerically. A simple way of oing this is to rewrite it as ) x x = ln 7) 0.019 6
an iterate on your calculator. Make a first guess for x an plug it in on the right han sie. Use the output as the input for the next iteration. When the result changes little from one iteration to the next, a solution has been foun. Starting with the guess x =, I foun the solution x = 3.0 after five iterations. So Since = 1.686 we get x = σ R=10 h 1 Mpc = 3.. 8) σ R=10 h 1 Mpc = 1.686 3. 0.53. 9) Finally, we can now use equation 19) to fin σ 8 : ) 10 0.8 σ R=10 h 1 Mpc = 0.53 = σ 8, 30) 8 an we receive the fruit of all our labour in the form of the value σ 8 = 0.63. 31) There are several things we have neglecte to o here. We have not taken the uncertainties in the values of n an M into account an seen how they propagate to an uncertainty in σ 8. An we shoul not, unless we have strong reason to o so, assume an Einstein-e Sitter universe, but rather allow Ω m0 to vary. But still this example illustrates an important application of the mass function an some of the steps that must be taken to relate observations to theory. 7