Spherical Collapse and the Press-Schechter Formalism

Linear perturbation theory is exact only while the density contrast is small; the instant a fluctuation reaches $\delta\sim1$ its self-gravity takes over and the equations that carried structure from last scattering break down. Yet the objects we observe — galaxies, groups, clusters — have $\delta$ run to thousands. Spherical collapse is the bridge. It follows one idealized overdense sphere, a uniform 'top-hat', through its whole nonlinear history: because a homogeneous ball feels only its enclosed mass, each shell obeys the same Friedmann equation as a closed $k>0$ universe, and its radius traces a cycloid — expanding with the Hubble flow, decoupling, reaching a maximum (turnaround), then collapsing. The model is fully analytic, and its payoff is two pure numbers. The first is the linear collapse threshold $\delta_c=\tfrac{3}{20}(12\pi)^{2/3}\approx1.686$ : the value that linear theory, naively extrapolated, assigns to a perturbation at the moment the real object collapses. The second is the virial overdensity $\Delta_{\mathrm{vir}}=18\pi^2\approx178$ , the settled halo's density over the background. Press and Schechter (1974) saw that these constants turn a single-object model into a statistical theory: identify collapsed mass with the fraction of a Gaussian field exceeding $\delta_c$ when smoothed on the mass scale $M$ , and halo abundance falls out as an error function of the peak height $\nu=\delta_c/\sigma(M)$ . Differentiating gives the halo mass function — the comoving number of halos per unit mass — turning a primordial power spectrum into a prediction for cluster counts.

Key takeaways

A uniform overdense sphere collapses like a closed sub-universe; its radius follows a cycloid with turnaround at $\eta=\pi$ and collapse at $\eta=2\pi$ , so $t_{\mathrm{col}}=2\,t_{\mathrm{ta}}$ .
The model delivers two universal constants in an Einstein–de Sitter background: the collapse threshold $\delta_c\approx1.686$ and the virial overdensity $\Delta_{\mathrm{vir}}=18\pi^2\approx178$ .
Press–Schechter equates the mass fraction in halos above $M$ with the Gaussian probability that the linearly-extrapolated smoothed density exceeds $\delta_c$ , so abundances follow from $\nu=\delta_c/\sigma$ .
The mass function has a universal shape: a near power-law at low mass and an exponential cutoff $e^{-\nu^2/2}$ above the characteristic mass $M_\ast$ where $\sigma=\delta_c$ , making cluster counts a sharp cosmological probe.
The original argument counts only half the collapsed mass; the ad-hoc factor of 2 is derived by excursion-set first-crossing statistics and refined by Sheth–Tormen and N-body fits.
Key idea: reduce nonlinear collapse to one number $\delta_c\approx1.686$ , then read halo abundances off the Gaussian statistics of the smoothed linear density field through $\nu=\delta_c/\sigma(M)$ .

The core insight

Reduce nonlinear collapse to a single number $\delta_c\approx1.686$ , the linear density contrast at collapse, then read halo abundances off the Gaussian statistics of the smoothed linear field via the peak height $\nu=\delta_c/\sigma(M)$ .

Where this lives today

The mass function is now a precision cosmology tool. Cluster surveys count the most massive halos versus redshift and compare to $\mathrm{d}n/\mathrm{d}M(z)$ using Tinker or Sheth–Tormen calibrations built on the spherical-collapse skeleton: Sunyaev–Zel'dovich catalogues from ACT, SPT and Planck; X-ray samples from eROSITA; optical-richness and weak-lensing samples from the Dark Energy Survey, Euclid and Rubin's LSST. Because the high-mass tail depends exponentially on $\sigma_8$ and the growth history, cluster counts constrain the amplitude of structure and the dark-energy equation of state — and the mild tension with Planck's primary-CMB $\sigma_8$ is a live puzzle. On the theory side, extended Press–Schechter supplies the merger trees behind semi-analytic galaxy formation, while the collapsed fraction above the atomic-cooling mass sets the timing of reionization now mapped by 21-cm experiments and JWST's earliest galaxies. A half-page 1974 ansatz remains the connective tissue between the primordial power spectrum and nearly every count of structure.

The physical picture

Take a slightly overdense ball in an expanding universe. It carries extra mass, so its self-gravity is a touch stronger than the background's: it expands, but always slightly slower than the Hubble flow around it. The lag accumulates until the ball halts at a maximum radius — turnaround — and then, gravity unopposed, falls back and collapses. Birkhoff's theorem guarantees the outside universe exerts no net force on a shell, so the sphere is dynamically a closed Friedmann universe and its radius versus time is a cycloid. The first figure traces that trajectory, whose clock is fixed: collapse takes exactly twice as long as turnaround.

Figure

Radius of a top-hat sphere versus time as a cycloid: it expands, turns around at r = 2A when η = π, and collapses at t = 2πB = twice the turnaround time, while the background scale factor a ∝ t^(2/3) keeps growing. Virialization halts the fall at r = r_ta divided by 2, and the true overdensity at turnaround is 1 + δ = 9π²/16 ≈ 5.55

The trajectory is only half the story: what matters is density. Instead of chasing the true nonlinear contrast — which runs to infinity as the sphere shrinks to a point — we ask a linear-theory question: what would a naive linear extrapolation of $\delta$ read at the moment the real sphere collapses? The answer is a fixed universal number, $\delta_c\approx1.686$ , independent of the sphere's mass or amplitude. That is the pivot: a region has collapsed by today if its linear density, smoothed on its mass scale, has reached this line — a criterion we evaluate with nothing but Gaussian statistics. The second figure contrasts the two densities and their hero numbers.

Figure

Linear versus true nonlinear overdensity of the same sphere through collapse. The linearly-extrapolated contrast (blue) grows smoothly as t^(2/3) and reaches exactly δc = 1.686 at collapse, whereas the true nonlinear contrast (red) passes 1 + δ = 5.55 at turnaround and formally diverges — real halos are caught by virialization at Δvir ≈ 178 times the background

The Press–Schechter step is then a level-crossing count on the linear field. Smooth it on the scale enclosing mass $M$ ; the rms is $\sigma(M)$ , and the fraction above $\delta_c$ is collapsed. Shrink $M$ and $\sigma$ rises, so more of the field clears the threshold — low-mass halos common, high-mass halos (where $\sigma<\delta_c$ ) exponentially rare. That rarity, the Gaussian tail $e^{-\nu^2/2}$ , is why counting the most massive clusters is one of cosmology's sharpest instruments.

The core equations

Because a uniform sphere feels only its enclosed mass $M$ , its outer shell obeys $\ddot r=-GM/r^2$ with $M=\tfrac{4\pi}{3}\bar\rho_i(1+\delta_i)r_i^3$ conserved. Integrating gives $\tfrac12\dot r^2-GM/r=E$ , and for a bound sphere ( $E<0$ ) the solution is a cycloid in a development angle $\eta$ .

r=A(1-\cos\eta),\qquad t=B(\eta-\sin\eta),\qquad A^3=GMB^2 \qquad (1)

where: $r$ — physical radius; $\eta$ — development angle running $0\to2\pi$ ; $A,B$ — constants set by the sphere's energy and mass; $M$ — enclosed (conserved) mass. Reading it: $\eta$ is the natural clock, $A,B$ set the scale. The special epochs sit at fixed $\eta$ : maximum radius at $\eta=\pi$ , formal collapse at $\eta=2\pi$ , so collapse takes exactly twice as long as turnaround.

r_{\mathrm{ta}}=2A,\quad t_{\mathrm{ta}}=\pi B,\quad t_{\mathrm{col}}=2\pi B=2\,t_{\mathrm{ta}} \qquad (2)

where: $r_{\mathrm{ta}}$ — turnaround radius, at $\eta=\pi$ ; $t_{\mathrm{ta}}$ — turnaround time; $t_{\mathrm{col}}$ — formal collapse time, at $\eta=2\pi$ . Reading it: the factor of two is exact and cosmology-independent. Comparing $\rho_{\mathrm{sph}}\propto r^{-3}$ to the EdS background $\bar\rho\propto t^{-2}$ gives the true nonlinear overdensity at any $\eta$ ; at turnaround it is a pure number.

1+\delta_{\mathrm{ta}}=\frac{9}{2}\frac{(\eta-\sin\eta)^2}{(1-\cos\eta)^3}\bigg|_{\eta=\pi}=\frac{9\pi^2}{16}\approx5.55 \qquad (3)

where: $\delta_{\mathrm{ta}}$ — true nonlinear density contrast at turnaround. Reading it: at maximum expansion the sphere is already $5.55$ times the background — well beyond linear theory, whose $\delta$ there reads only $\approx1.06$ . Extrapolating instead the linear growing mode to the collapse time yields the central constant.

\delta_c=\frac{3}{20}(12\pi)^{2/3}\approx1.686 \qquad (4)

where: $\delta_c$ — linearly-extrapolated density contrast at collapse. Reading it: not a real density but a linear-theory bookmark, which is why it can be compared with the linear $\sigma(M)$ . Requiring the sphere to virialize at half its turnaround radius, then diluting the background, fixes the halo's overdensity.

\Delta_{\mathrm{vir}}\equiv\frac{\rho_{\mathrm{halo}}}{\bar\rho}=18\pi^2\approx178 \qquad (5)

where: $\rho_{\mathrm{halo}}$ — mean density of the virialized halo; $\bar\rho$ — background density at collapse. Reading it: the physical overdensity defining a halo boundary — origin of the ' $\sim200\times$ critical' radii used in simulations. To go statistical we need the linear field's amplitude on each mass scale; smoothing with a top-hat of radius $R$ (enclosing mass $M$ ) gives the variance.

\sigma^2(M)=\frac{1}{2\pi^2}\int_0^\infty P(k)\,\tilde W^2(kR)\,k^2\,\mathrm{d}k,\qquad M=\frac{4\pi}{3}\bar\rho R^3 \qquad (6)

where: $\sigma^2(M)$ — variance of the smoothed field; $P(k)$ — linear matter power spectrum; $\tilde W(kR)$ — Fourier transform of the top-hat; $\bar\rho$ — mean comoving matter density. Reading it: all cosmology enters the abundance through this one function $\sigma(M)$ . The ansatz identifies collapsed mass with the Gaussian tail beyond $\delta_c$ , giving the fraction in halos above $M$ .

F(>M)=\frac{2}{\sqrt{2\pi}\,\sigma}\int_{\delta_c}^{\infty}e^{-\delta^2/2\sigma^2}\,\mathrm{d}\delta=\mathrm{erfc}\!\left(\frac{\delta_c}{\sqrt2\,\sigma}\right) \qquad (7)

where: $F(>M)$ — mass fraction in halos of mass $\ge M$ ; $\mathrm{erfc}$ — complementary error function; the factor $2$ is the famous correction. Reading it: as $M\to0$ , $\sigma\to\infty$ and $F\to1$ — the factor $2$ makes all mass end in halos. Differentiating $F$ and multiplying by $\bar\rho/M$ gives the mass function.

\frac{\mathrm{d}n}{\mathrm{d}M}=\sqrt{\frac{2}{\pi}}\,\frac{\bar\rho}{M^2}\,\frac{\delta_c}{\sigma}\left|\frac{\mathrm{d}\ln\sigma}{\mathrm{d}\ln M}\right|e^{-\delta_c^2/2\sigma^2} \qquad (8)

where: $\mathrm{d}n/\mathrm{d}M$ — comoving number of halos per unit mass; $\bar\rho$ — mean comoving matter density. Reading it: a mild power-law prefactor times a decisive exponential. All redshift dependence collapses into one self-similar variable, the peak height.

\nu\equiv\frac{\delta_c(z)}{\sigma(M)}=\frac{\delta_c}{D(z)\,\sigma_0(M)} \qquad (9)

where: $\nu$ — dimensionless peak height; $D(z)$ — linear growth factor with $D(0)=1$ ; $\sigma_0(M)$ — present-day rms. Reading it: a fixed mass grows rarer at high $z$ as $D$ shrinks, driving $\nu$ up and the exponential down.

Derivation AThe linear collapse threshold from the cycloid solution

The goal is to extract the universal number $\delta_c\approx1.686$ without solving the full nonlinear problem: expand the parametric cycloid for small development angle to isolate the linearly growing mode, confirm it grows as $t^{2/3}$ as EdS linear theory demands, then extrapolate that mode to the collapse epoch $\eta=2\pi$ . Although the true density diverges at collapse, the linear extrapolation is finite and universal — and this bookkeeping value is what the statistical theory needs.

Step 1. Write the exact overdensity as the sphere-to-background ratio, $1+\delta=\tfrac{9}{2}(\eta-\sin\eta)^2/(1-\cos\eta)^3$ , using $\rho_{\mathrm{sph}}\propto r^{-3}\propto(1-\cos\eta)^{-3}$ and $\bar\rho\propto t^{-2}\propto(\eta-\sin\eta)^{-2}$ . Why: casting both in $\eta$ removes $A,B$ and leaves a pure function.

Step 2. Taylor-expand for small $\eta$ : $\eta-\sin\eta\approx\tfrac{\eta^3}{6}(1-\tfrac{\eta^2}{20})$ and $1-\cos\eta\approx\tfrac{\eta^2}{2}(1-\tfrac{\eta^2}{12})$ , which combine to $1+\delta\approx1+\tfrac{3}{20}\eta^2$ . Why: the $\tfrac{3}{20}\eta^2$ term is the linearly growing mode; higher terms are the nonlinear ones we discard.

Step 3. Check the growth law: since $t=B(\eta-\sin\eta)\approx B\eta^3/6$ , $\eta^2\propto t^{2/3}$ , so $\delta_{\mathrm{lin}}=\tfrac{3}{20}\eta^2\propto t^{2/3}$ . Why: $t^{2/3}$ is exactly the EdS linear growth $D\propto a\propto t^{2/3}$ , so this is the genuine linear mode, extrapolable by linear rules.

Step 4. Extrapolate to collapse at $\eta=2\pi$ , where $t_{\mathrm{col}}=2\pi B$ , so $6t_{\mathrm{col}}/B=12\pi$ and $\delta_c=\tfrac{3}{20}(6t_{\mathrm{col}}/B)^{2/3}=\tfrac{3}{20}(12\pi)^{2/3}$ . Why: we feed the collapse time into the linear growing-mode expression, not the divergent exact one — the entire content of "linearly extrapolated to collapse". *Numerically, $\tfrac{3}{20}(12\pi)^{2/3}=1.6865$ ; the same small- $\eta$ expansion evaluated at turnaround ( $\eta=\pi$ ) gives $\delta_{\mathrm{lin}}(\eta=\pi)=\tfrac{3}{20}(6\pi)^{2/3}=1.06$ , confirming that linear theory already underestimates the true contrast of $4.55$ at maximum expansion.*

Go deeper · Derivation B · 1 minThe virial overdensity of a collapsed halo

Derivation BThe virial overdensity of a collapsed halo

The goal is the second universal constant, $\Delta_{\mathrm{vir}}=18\pi^2\approx178$ , the density of a settled halo relative to the background at the moment its parent sphere would formally have collapsed to a point. The real sphere never reaches a singularity: violent relaxation converts infall energy into random motion, and the virial theorem pins the equilibrium radius at half the turnaround radius — a factor-of-eight density boost we combine with the fourfold dilution of the background.

Step 1. Apply the virial theorem $2K+U=0$ with energy conservation $E=U(r_{\mathrm{ta}})$ (all energy is potential at turnaround, where $\dot r=0$ ). With $E=U/2$ and $U\propto-1/r$ , one finds $r_{\mathrm{vir}}=\tfrac12 r_{\mathrm{ta}}$ . Why: halving the radius at fixed mass is the one input that makes the sphere a finite-density object.

Step 2. Halving the radius raises density by $2^3=8$ , so $\rho_{\mathrm{vir}}=8\,\rho_{\mathrm{ta}}=8\cdot\tfrac{9\pi^2}{16}\,\bar\rho_{\mathrm{ta}}$ using equation (3). Why: density scales as radius $^{-3}$ at fixed mass, an exact factor of eight.

Step 3. Dilute the background: in EdS $\bar\rho\propto t^{-2}$ and $t_{\mathrm{col}}=2\,t_{\mathrm{ta}}$ , so $\bar\rho_{\mathrm{col}}=\bar\rho_{\mathrm{ta}}(t_{\mathrm{ta}}/t_{\mathrm{col}})^2=\bar\rho_{\mathrm{ta}}/4$ . Why: the comparison must be at collapse, when the background has thinned by four.

Step 4. Assemble: $\Delta_{\mathrm{vir}}=\rho_{\mathrm{vir}}/\bar\rho_{\mathrm{col}}=8\cdot\tfrac{9\pi^2}{16}\cdot4=18\pi^2\approx178$ . Why: turnaround overdensity $\tfrac{9\pi^2}{16}$ , virial boost $8$ , and dilution $4$ multiply to the clean $18\pi^2$ . *Numerically, $18\pi^2=177.65$ ; the ΛCDM Bryan–Norman fit $\Delta_{\mathrm{vir}}\approx18\pi^2+82x-39x^2$ ( $x=\Omega_m(z)-1$ ) gives $\approx101$ times critical, i.e. $\approx337$ times mean matter for $\Omega_m=0.3$ today.*

Go deeper · Derivation C · 1 minThe Press–Schechter mass function and the factor of two

Derivation CThe Press–Schechter mass function and the factor of two

The goal is to turn the single-sphere threshold into a number density of halos, $\mathrm{d}n/\mathrm{d}M$ , and confront the notorious factor of two. The ansatz is a leap: identify the mass fraction bound in halos above $M$ with the probability that the linear field, smoothed on scale $M$ , exceeds $\delta_c$ . For a Gaussian field that is a tail; differentiating gives the abundance, but the naive normalization holds only half the mass — so we see where the factor of two enters and why excursion sets make it honest.

Step 1. Write the collapsed fraction as a Gaussian exceedance, $F(>M)=\tfrac{2}{\sqrt{2\pi}\sigma}\int_{\delta_c}^{\infty}e^{-\delta^2/2\sigma^2}\mathrm{d}\delta=\mathrm{erfc}(\delta_c/\sqrt2\sigma)$ , inserting the factor $2$ by hand. Why: without the $2$ , the integral gives $\tfrac12$ as $\sigma\to\infty$ — absurdly, only half of all matter ever collapses; the factor forces $F\to1$ on small scales.

Step 2. Convert mass fraction to number: each halo of mass $M$ holds mass $M$ , so $\tfrac{\mathrm{d}n}{\mathrm{d}M}M\,\mathrm{d}M=\bar\rho\,|\mathrm{d}F/\mathrm{d}M|\,\mathrm{d}M$ . Why: $\bar\rho\,\mathrm{d}F$ is the mass per volume crossing into collapse in the bin; dividing by $M$ counts halos.

Step 3. Differentiate the erfc through $\sigma(M)$ with $\tfrac{\mathrm{d}}{\mathrm{d}x}\mathrm{erfc}(x)=-\tfrac{2}{\sqrt\pi}e^{-x^2}$ (and $\sigma$ decreases with $M$ ): $\tfrac{\mathrm{d}F}{\mathrm{d}M}=\sqrt{\tfrac{2}{\pi}}\tfrac{\delta_c}{\sigma^2}\tfrac{\mathrm{d}\sigma}{\mathrm{d}M}e^{-\delta_c^2/2\sigma^2}$ . Why: all mass dependence lives in $\sigma(M)$ , routed by the chain rule through $\mathrm{d}\ln\sigma/\mathrm{d}\ln M$ .

Step 4. Repackage with $\nu=\delta_c/\sigma$ into $\tfrac{\mathrm{d}n}{\mathrm{d}M}=\sqrt{\tfrac{2}{\pi}}\tfrac{\bar\rho}{M^2}\nu\,|\mathrm{d}\ln\sigma/\mathrm{d}\ln M|\,e^{-\nu^2/2}$ , i.e. $\tfrac{M^2}{\bar\rho}\tfrac{\mathrm{d}n}{\mathrm{d}M}=\nu f(\nu)|\mathrm{d}\ln\sigma/\mathrm{d}\ln M|$ with universal multiplicity $f(\nu)=\sqrt{\tfrac{2}{\pi}}e^{-\nu^2/2}$ . Why: in $\nu$ the cosmology-independence is manifest — cosmology enters only through $\sigma(M)$ and $\delta_c(z)$ .

Step 5. Justify the factor of two: treat $\delta$ as a random walk in variance $S=\sigma^2$ (sharp- $k$ filtering makes increments independent) with an absorbing barrier at $\delta_c$ . The first-crossing distribution $f(S)=\tfrac{\delta_c}{\sqrt{2\pi}S^{3/2}}e^{-\delta_c^2/2S}$ integrates to $F=\mathrm{erfc}(\delta_c/\sqrt{2S})$ — exactly twice the naive tail. Why: first crossings on all larger scales capture the cloud-in-cloud mass the tail misses, delivering the factor of two. *As a check, $\int_0^\infty f(\nu)\,\mathrm{d}\nu=\sqrt{\tfrac{2}{\pi}}\int_0^\infty e^{-\nu^2/2}\mathrm{d}\nu=1$ , confirming that the multiplicity function integrates to unity and all mass is accounted for in halos.*

Limiting cases

The two constants are exact only in Einstein–de Sitter, and the abundance splits into a common power-law regime and a rare exponential tail:

Case	Result	Meaning
Einstein–de Sitter	$\delta_c=\tfrac{3}{20}(12\pi)^{2/3}\approx1.686,\ \Delta_{\mathrm{vir}}=18\pi^2\approx178$	Exact analytic constants; only weakly cosmology-dependent.
ΛCDM correction	$\Delta_{\mathrm{vir}}(z)\approx18\pi^2+82x-39x^2,\ x\equiv\Omega_m(z)-1$	Bryan–Norman fit; $\approx340$ times mean matter today for $\Omega_m\approx0.3$ ; $\delta_c$ drifts only to $\approx1.675$ .
High-mass tail $\nu\gg1$	$\dfrac{\mathrm{d}n}{\mathrm{d}\ln M}\propto\nu\,e^{-\nu^2/2}$	Exponentially suppressed; sensitivity to $\sigma_8$ and growth makes clusters a probe.
Low-mass tail $\nu\ll1$	$\dfrac{\mathrm{d}n}{\mathrm{d}M}\propto M^{-2}\left\vert\dfrac{\mathrm{d}\ln\sigma}{\mathrm{d}\ln M}\right\vert$	Nearly scale-free; total mass dominated by small halos in hierarchical CDM.

Common trapReading δc ≈ 1.686 as a real overdensity

The threshold $\delta_c\approx1.686$ is not the density of anything. A real collapsing sphere passes $\delta\approx4.55$ at turnaround and its true contrast formally diverges at collapse, settling near $\Delta_{\mathrm{vir}}\approx178$ after virialization. The number $1.686$ is what a purely linear extrapolation — the growing mode continued past its validity — would read at collapse. Its purpose is to live in the same linear-theory world as $\sigma(M)$ , so $\nu=\delta_c/\sigma$ consistently measures how far a fluctuation lies above threshold. Comparing $\delta_c$ with the $\approx178$ of a virialized halo, or plugging a nonlinear $\delta$ into the Gaussian tail, is the classic error.

Worked examples

1 · The two universal constants, from scratch (confidence builder)

Find the linear collapse threshold, the turnaround overdensity, and the virial overdensity for an EdS sphere, then convert the last to a ΛCDM value for $\Omega_m=0.3$ today.

Step 1. Linear threshold. With $12\pi=37.699$ and $37.699^{2/3}=11.243$ , $\delta_c=0.15\times11.243=\mathbf{1.686}$ — the line every fluctuation must reach to count as collapsed.

Step 2. Turnaround overdensity. From equation (3) at $\eta=\pi$ , $1+\delta_{\mathrm{ta}}=9\pi^2/16=\mathbf{5.55}$ , so $\delta_{\mathrm{ta}}=4.55$ — already far beyond linear theory (which reads only $1.06$ here).

Step 3. Virial overdensity. Multiply turnaround overdensity by the virial boost and background dilution: $\Delta_{\mathrm{vir}}=\tfrac{9\pi^2}{16}\times8\times4=18\pi^2=\mathbf{177.7}$ — a halo $\approx178$ times denser than the mean background at formation.

Step 4. ΛCDM correction. With $\Omega_m=0.3$ , $x=-0.7$ , the Bryan–Norman fit gives $\Delta_{\mathrm{vir}}=177.7-57.4-19.1=\mathbf{101}$ relative to critical density; dividing by $\Omega_m$ gives the mean-matter reference $101/0.3\approx\mathbf{337}$ — the familiar " $\sim340\times$ mean, $\sim100\times$ critical" of today's universe.

2 · Why massive clusters vanish at high redshift

A rich cluster corresponds to a mass scale with present-day rms $\sigma_0=0.5$ . Using $\delta_c=1.686$ and growth factor $D(z{=}1)\approx0.61$ , find the peak height $\nu$ and the abundance factor $e^{-\nu^2/2}$ at $z=0$ and $z=1$ , and interpret the ratio.

Step 1. Present-day peak height. At $z=0$ , $D=1$ , so $\nu(0)=\delta_c/[D\,\sigma_0]=1.686/0.5=\mathbf{3.37}$ — a rare $3.4\sigma$ peak even today.

Step 2. Present-day suppression. The exponential factor is $e^{-\nu^2/2}=e^{-5.68}=\mathbf{3.4\times10^{-3}}$ : rare, but rich clusters exist in appreciable numbers today.

Step 3. Peak height at $z=1$ . Growth has weakened to $D(1)=0.61$ , so $\nu(1)=1.686/(0.61\times0.5)=\mathbf{5.53}$ — the same mass is now a $5.5\sigma$ fluctuation, having had less time to grow.

Step 4. High- $z$ suppression and ratio. Now $e^{-\nu^2/2}=e^{-15.3}=\mathbf{2.3\times10^{-7}}$ , so the abundance ratio is $3.4\times10^{-3}/2.3\times10^{-7}\approx\mathbf{1.5\times10^{4}}$ : rich clusters were roughly fifteen thousand times rarer at $z=1$ than today. A factor $\sim2$ change in $D$ moving the count by four orders of magnitude is exactly why cluster abundance versus redshift probes $\sigma_8$ , $\Omega_m$ , and dark energy.

Physics of the portrait

A one-page 1974 paper, built on an idealization everyone knew was too simple, predicted cosmic structure well enough to still underlie cluster cosmology fifty years on.

Origins · 1 minpress–schechter formalism and how this was discovered

press–schechter formalism

halo abundances from spherical collapse and Gaussian statistics, 1974 · nonlinear cosmological structure formation

Counting galaxies and clusters from first principles began with a short, audacious 1974 paper by William H. Press and Paul Schechter at Caltech. They combined two ingredients everyone already had — the spherical-collapse threshold $\delta_c$ from the closed-universe cycloid (dynamics traceable to Gunn and Gott's 1972 infall study) and the Gaussian statistics of the density field — and made the daring identification of collapsed mass with the high-density tail. The derivation carried the now-infamous factor of two: their reasoning held only half the mass, so they doubled it, intuition ahead of justification. For fifteen years the formula was used and distrusted in equal measure, until Bond, Cole, Efstathiou and Kaiser (1991) recast it as an excursion-set problem — the smoothed density as a random walk in resolution, halos as first crossings of the barrier $\delta_c$ — producing the factor of two honestly. Sheth and Tormen (1999) then replaced the spherical barrier with an ellipsoidal one, and N-body-calibrated fits (Tinker and others) followed — but each corrects the skeleton rather than replacing it. The 1974 ansatz remains one of the most quoted approximations in all of cosmology.

History

1972

— Gunn and Gott model spherical infall onto an overdensity, formalizing turnaround and collapse

1974

— Press and Schechter derive the halo mass function and insert the factor of two by hand

1985

— Davis, Efstathiou, Frenk and White cold dark matter N-body simulations test the mass function

1991

— Bond, Cole, Efstathiou and Kaiser derive the excursion-set justification of the factor of two

1999

— Sheth and Tormen fit an ellipsoidal-collapse mass function to simulations

2008

— Tinker and collaborators publish a high-precision N-body-calibrated mass function

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 7 · Binney and Tremaine, Galactic Dynamics, 2nd ed., §9.3–9.4 · Ryden, Introduction to Cosmology, 2nd ed., Ch. 12 · Dodelson, Modern Cosmology, 2nd ed., Ch. 12 · Press and Schechter, ApJ 187, 425 (1974) · Bond, Cole, Efstathiou and Kaiser, ApJ 379, 440 (1991) · Sheth and Tormen, MNRAS 308, 119 (1999). Standard values used throughout: $\delta_c=\tfrac{3}{20}(12\pi)^{2/3}=1.6865$ , $\Delta_{\mathrm{vir}}=18\pi^2=177.65$ (EdS), $1+\delta_{\mathrm{ta}}=9\pi^2/16=5.55$ , $\delta_{\mathrm{lin}}(\text{turnaround})=1.06$ . Key intermediate results: the virial chain uses $r_{\mathrm{vir}}=r_{\mathrm{ta}}/2$ , density boost $8$ , and background dilution $4$ ; the multiplicity normalization is $\int_0^\infty f(\nu)\,\mathrm{d}\nu=1$ . Worked cluster numbers: $\nu=3.37\to e^{-\nu^2/2}=3.4\times10^{-3}$ at $z=0$ ; $\nu=5.53\to2.3\times10^{-7}$ at $z=1$ ; abundance ratio $\approx1.5\times10^4$ . Bryan–Norman ΛCDM correction gives $\Delta_{\mathrm{vir}}\approx340$ times mean matter for $\Omega_m=0.3$ .

Prerequisites. Density Perturbations and the Density Contrast · Linear Growth of Structure · The Primordial Power Spectrum · Transfer Function and the Matter Power Spectrum · Jeans Instability and Cosmological Perturbation Regimes. Leads to. The ΛCDM Concordance Model · Cold Dark Matter · Baryon Acoustic Oscillations.

Ask the tutor: Ask me to walk through the small- $\eta$ cycloid expansion line by line, or to show how changing $\sigma_8$ moves the cluster count at a mass and redshift you choose.

Why does a homogeneous sphere collapse exactly like a closed universe?

Birkhoff's theorem — the relativistic shell theorem — says a shell's dynamics inside a spherically symmetric distribution depend only on the enclosed mass; the exterior exerts no net force. A uniform overdense sphere therefore evolves as an isolated ball of matter, and such a ball with more than escape energy is precisely a closed ( $k>0$ ) Friedmann universe, whose bound scale factor is the cycloid $r=A(1-\cos\eta)$ — so the model is exactly solvable by reusing the closed-FRW solution. The one idealization is that shells at different radii turn around at different times, which the single-density top-hat sidesteps.

What is the difference between δc and Δvir, since both describe collapse?

They live in different worlds. $\Delta_{\mathrm{vir}}\approx178$ is a real, physical overdensity — a virialized halo's density over the background, the number a simulation uses to draw a halo boundary. $\delta_c\approx1.686$ is a linear-theory value — what the extrapolated linear density would reach at collapse, which never physically occurs because the true density has already diverged. Use $\delta_c$ to decide whether a region collapsed, and $\Delta_{\mathrm{vir}}$ for what it looks like afterward. Comparing $1.686$ with $178$ as if they measured the same thing is the most common error in this subject.

Where does the factor of two really come from?

The tail integral counts only mass directly above $\delta_c$ on scale $M$ . But a point can sit below threshold on scale $M$ while lying inside a larger region that is above threshold — already swept into a bigger collapsed structure. The naive count misses exactly this cloud-in-cloud mass, and misses precisely half, so the collapsed fraction is $\tfrac12$ instead of $1$ . Excursion sets fix it: follow the smoothed density versus resolution and declare a region collapsed if it ever crossed $\delta_c$ on any larger scale. Counting first crossings against an absorbing barrier recovers exactly twice the naive tail — turning the hand-inserted $2$ into a theorem.

What sets the characteristic mass M⋆ in the mass function?

$M_\ast$ is defined by $\sigma(M_\ast,z)=\delta_c$ , i.e. peak height $\nu=1$ . Below $M_\ast$ the field routinely clears the threshold, so halos are common with a mild power-law abundance; above it crossing needs a rare peak and the count plunges as $e^{-\nu^2/2}$ . So $M_\ast$ is the knee separating abundant small halos from suppressed giants, and it grows with time: as $\sigma$ rises through $D(z)$ , ever larger masses reach $\nu=1$ — hierarchical, bottom-up assembly.

How does the primordial power spectrum enter the abundance of halos?

Entirely through $\sigma(M)$ . Equation (6) filters $P(k)$ with a top-hat enclosing mass $M$ , so $\sigma(M)$ measures how much fluctuation power sits on each mass scale. A steeper or higher-amplitude $P(k)$ raises $\sigma$ , lowers $\nu=\delta_c/\sigma$ , and multiplies massive-halo abundance exponentially. Everything upstream — the spectral index $n_s$ , the transfer function, the growth factor $D(z)$ — reaches the mass function only by shaping $\sigma(M,z)$ , compressing all initial conditions into one function that hands back a countable prediction.

Why are cluster counts such a sensitive probe of cosmology?

Because clusters are the rare exponential tail. A modest change in $\sigma_8$ or the growth history shifts $\nu=\delta_c/\sigma$ only a little, but the abundance responds as $e^{-\nu^2/2}$ , so a small shift in the exponent moves the count by orders of magnitude — worked example 2 shows a factor $\sim2$ change in growth altering the abundance by $10^4$ . The leverage cuts both ways: it makes cluster counts a razor-sharp measure of structure growth and hence dark energy, but systematic errors in halo mass (lensing calibration, the mass–observable relation) propagate strongly — so much of modern cluster cosmology is calibrating mass well enough to exploit that sensitivity rather than be defeated by it.