How many halos? — the Press–Schechter mass function

A halo of mass $M$ formed from matter that was originally spread out at the mean density. The radius of the sphere that held it is

where $\rho_c\Omega_m = \bar\rho$ is the mean matter density. To ask whether that mass has collapsed, we smooth the density field over regions of size $R$, averaging $\delta$ with a window function $W_R$ (for example a sphere of radius $R$):

[The notes write $\delta(\vec x)$ inside the integral; $\delta(\vec y)$ is meant.] Halos of mass $M$ form where $\bar\delta$ crosses $\delta_c = 1.69$. The smoothed field has variance

— the variance of the power spectrum page, with each mode weighted by the Fourier transform $\hat W_R$ of the window, which removes waves much shorter than $R$. [The notes print the prefactor as $1/2\pi$ and the window as $\hat W_R$; with eq. (405) the prefactor is $1/2\pi^2$, and the window enters squared because $\bar\delta$ is a convolution in real space, which becomes the product $\hat\delta\,\hat W_R$ in Fourier space.] Larger $M$ means larger $R$, more smoothing, and a smaller $\sigma_R$.

Inflation produces a Gaussian random field, so the smoothed contrast at any point follows a normal distribution with zero mean:

where $\sigma_R(z)$ is the linear $\sigma_R$ evolved with the growth factor $D$ ($D = 1$ today).

The central assumption: the probability that the smoothed contrast exceeds $\delta_c$ equals the fraction of volume (equivalently, of mass) in halos of mass $M$ or more. Integrating the Gaussian tail,

The fraction in halos with masses between $M$ and $M + dM$ is the change of $F$ with $M$. Since $M$ enters only through $\sigma_R$, use the chain rule, $\partial/\partial M = (d\sigma_R/dM)\,\partial/\partial\sigma_R$ (eq. 453; the notes print a "−" where "=" is meant), together with $\frac{d}{dx}\text{erfc}(x) = -\frac{2}{\sqrt\pi}e^{-x^2}$:

so that $(\partial F/\partial M)\,dM$ is the fraction of volume in halos of mass $[M, M + dM]$ (eq. 455). [In the third line of eq. (454) the notes write $\sqrt{2\pi\sigma_R^2}$, dropping a factor $1/\sigma_R$; the last line, used from then on, is right. Also, $\sigma_R$ falls as $M$ grows, so $d\ln\sigma_R/dM < 0$ and $F$ decreases with $M$; the fraction in the interval is $|\partial F/\partial M|\,dM$. Here and below the magnitude is meant.]

A volume fraction is not yet a number of halos. Each halo of mass $M$ came from a volume $M/\bar\rho = M/\rho_c\Omega_m$, so dividing by that volume gives the comoving number density:

(the notes keep the label $\partial F/\partial M$ on the left of eq. 456; it is now a number density per unit mass). Now a check: summed over all masses, the volume fractions should add up to 1 — all matter is in some halo, however small:

[The notes print the upper limit as 1.] But the calculation gives only ½: as $M \to 0$, $\sigma_R \to \infty$ and $F \to \frac12\text{erfc}(0) = \frac12$. Half the mass is missing. The notes simply multiply by 2 — the famous Press–Schechter "fudge factor" — to obtain the mass function:

The missing half is the cloud-in-cloud problem. A point can have $\bar\delta < \delta_c$ when smoothed on scale $R$, yet $\bar\delta > \delta_c$ on some larger scale — it sits inside a bigger collapsed halo, but the simple argument doesn't count it. The notes refer to Problem Set 6 for the extended Press–Schechter (excursion-set) formalism, which follows $\bar\delta$ as the smoothing scale shrinks, like a random walk, and counts the first crossing of $\delta_c$; for a sharp-$k$ window it gives exactly the factor 2. (This approach is due to Bond, Cole, Efstathiou and Kaiser, ApJ 379, 440, 1991; the original argument is Press and Schechter, ApJ 187, 425, 1974. Both references are added here; the notes don't give them.)

Everything depends on the single combination $\nu = \delta_c/(\sigma_R D)$, the height of the collapse threshold in units of the typical fluctuation. Halos with $\nu \lesssim 1$ are common: their scale has typically already collapsed. Halos with $\nu \gg 1$ need rare, high peaks of the Gaussian, and the factor $e^{-\nu^2/2}$ makes them exponentially rare. Since $\sigma_R$ falls with mass, the mass function is roughly a power law at low masses and drops off exponentially above a characteristic mass $M_*$, where $\sigma(M_*)\,D = \delta_c$. As the universe evolves, $D$ grows, $M_*$ moves to larger masses, and ever bigger halos form — hierarchical growth.

• A halo of mass $M$ comes from a region of radius $R(M) = (3M/4\pi\bar\rho)^{1/3}$; smoothing the linear field on that scale gives a Gaussian $\bar\delta$ with variance $\sigma_R^2 = \frac{1}{2\pi^2}\int k^2 P(k)|\hat W_R|^2 dk$, growing as $D(z)$.
• Press–Schechter: the fraction of space with $\bar\delta > \delta_c \approx 1.69$, $F = \frac12\text{erfc}(\delta_c/\sqrt2\sigma_R D)$, is the fraction of mass in halos heavier than $M$.
• Differentiating and dividing by $M/\bar\rho$ gives the number density; a factor 2 is needed to put all the mass in halos (the cloud-in-cloud problem, explained by the excursion-set approach).
• The result depends on $\nu = \delta_c/\sigma_R D$: a power law at low masses, an exponential cutoff above $M_*$ that moves to larger masses with time. Massive clusters are exponentially rare, especially at high redshift.