Press-Schechter Formalism

The idea: count noise, predict galaxies

Imagine the universe a few hundred thousand years after the Big Bang as a nearly featureless sea of matter, ruffled only by faint density ripples of about one part in a hundred thousand. Those ripples are random — to extraordinary precision, they are drawn from a Gaussian distribution, a fact confirmed by Planck's measurements of the cosmic microwave background. Gravity then amplifies them: slightly dense regions pull in more matter, get denser still, and eventually decouple from the cosmic expansion and collapse into gravitationally bound clumps. Those clumps are dark matter halos, and galaxies, groups, and clusters live inside them.

The Press-Schechter formalism asks a deceptively simple question. If the early density field is just Gaussian noise, can we predict — without simulating anything — how many halos of each mass will form? The answer is yes, and the trick is almost embarrassingly direct: smooth the field on a chosen mass scale, then count the fraction of it that sits above a fixed collapse threshold. Bigger smoothing scales correspond to bigger halos. Tracking how the above-threshold fraction changes as you vary the smoothing scale gives you the abundance of halos at every mass. That single idea, published by William Press and Paul Schechter in 1974, is the founding analytic theory of cosmic structure formation.

The two ingredients

The formalism rests on two physical inputs and nothing else.

1. The variance of the smoothed field, σ²(M). Take the linear density contrast δ(x) = (ρ − ρ̄)/ρ̄, convolve it with a filter (usually a real-space top-hat sphere) enclosing mass M, and compute the variance of the result. This is an integral over the matter power spectrum P(k):

σ²(M) = (1 / 2π²) ∫ P(k) W²(kR) k² dk

where R is the radius enclosing mass M = (4/3)πρ̄R³ and W(kR) is the Fourier transform of the top-hat filter. Because P(k) is larger on large scales for the standard cold-dark-matter spectrum, σ(M) is a decreasing function of mass: small halos form out of high-variance, easily-exceeded fluctuations; cluster-scale halos require rare, large-scale fluctuations. At the present epoch the field is normalised by σ₈ ≡ σ(R = 8 h⁻¹ Mpc) ≈ 0.81 (Planck 2018).

2. The collapse threshold, δ_c. Spherical-collapse theory says a uniform overdense sphere turns around and collapses when its linearly extrapolated overdensity reaches a critical value. In an Einstein-de Sitter universe this is exactly

δ_c = (3/5)(3π/2)^(2/3) ≈ 1.68647

This is a yes/no barrier. If the smoothed linear overdensity at a point exceeds δ_c, that point is declared to live in a collapsed halo of mass at least M.

The mass function

For a Gaussian field with variance σ²(M), the probability that the smoothed overdensity exceeds δ_c is the tail of a Gaussian:

F(>M) = (1/2) erfc( δ_c / (√2 · σ(M)) )

This is the fraction of mass in collapsed objects more massive than M. Differentiate with respect to M, multiply by the cosmic mean matter density ρ̄ to convert mass fraction into number density, and — after multiplying by the notorious factor of two — you get the differential halo mass function:

dn/dM = √(2/π) · (ρ̄ / M²) · (δ_c / σ) · |d ln σ / d ln M| · exp( −δ_c² / 2σ² )

The exponential is the headline. For small halos σ is large, the exponent is near zero, and the abundance is set by the power-law prefactor. For massive halos σ is small, the exponent δ_c²/2σ² is large, and the abundance plummets — this is why galaxy clusters are exponentially rare and exponentially sensitive to cosmology. The whole thing is often written in the self-similar form dn/dM = (ρ̄/M²) f(ν) |d ln σ/d ln M| ν, where ν ≡ δ_c/σ is the peak height and f(ν) ∝ exp(−ν²/2) is a "universal" multiplicity function.

The numbers behind 1.686 and 178

It is worth seeing where the two famous constants come from, because they are easy to confuse. Spherical collapse tracks a top-hat sphere through three phases:

Stage	What happens	Linear δ (extrapolated)	True overdensity
Turnaround	Sphere stops expanding, begins to fall back	1.062	(3π/4)² ≈ 5.55× background
Collapse (formal)	Sphere collapses to a point in the idealised model	1.686	→ ∞ (unphysical)
Virialisation	Halo settles into equilibrium at half the turnaround radius	1.686	Δ_c ≈ 178× background

So 1.686 is a fictitious linear-theory bookkeeping value — the real perturbation never reaches infinite density; it virialises. And 178 (often quoted as Δ_vir ≈ 18π² ≈ 177.7 for Einstein-de Sitter) is the actual physical overdensity of a virialised halo relative to the background. Press-Schechter applies the linear value 1.686 to the linearly evolved field precisely so it can stay in linear theory and never has to follow the messy nonlinear collapse. In a ΛCDM universe δ_c drifts only weakly with redshift — from 1.686 at high z to about 1.675 today — and Δ_vir grows to roughly 350 times the background (or ~100 times critical) at z = 0, which is why halo catalogues often quote masses as M₂₀₀ or M₅₀₀ (mass within 200 or 500 times the critical density).

The factor of two and the cloud-in-cloud problem

Here is the conceptual scandal at the heart of the original paper. For a Gaussian field, the fraction of mass with δ > δ_c approaches one-half as σ → ∞ (the Gaussian is symmetric about zero, so half of it is positive). Yet physically all matter must eventually end up inside some halo. Press and Schechter simply multiplied their answer by two to make the collapsed fraction integrate to unity. For 17 years this factor sat there as an unjustified fudge.

The resolution is the cloud-in-cloud problem. A patch of matter can look underdense when smoothed on a small scale, while the larger region it belongs to is above threshold on a big scale. That patch is genuinely part of a collapsed halo — just a more massive one than its local small-scale value would suggest. The naive count never assigns it to anything, so it loses half the mass. In 1991 Jehoshua Bond, Shaun Cole, George Efstathiou and Nick Kaiser reframed the whole problem as a random walk: as you shrink the smoothing scale, the smoothed overdensity at a point traces a trajectory, and you ask for the first up-crossing of the barrier δ_c. Assigning each point to the largest scale on which it first crossed the barrier correctly resolves the cloud-in-cloud ambiguity and recovers the factor of two from first principles. This excursion-set theory, plus the related extended Press-Schechter (EPS) formalism, also gives halo merger rates and progenitor mass functions — the analytic backbone of semi-analytic galaxy formation models.

Worked example: how rare is a Coma-mass cluster?

Let us estimate the comoving number density of clusters as massive as Coma, M ≈ 10¹⁵ M☉, at z = 0. Take ρ̄ = Ω_m ρ_crit ≈ 0.31 × 2.775 × 10¹¹ h² M☉ Mpc⁻³ ≈ 4.0 × 10¹⁰ M☉ Mpc⁻³ (with h = 0.67). For a Planck-normalised power spectrum, σ(M = 10¹⁵ M☉) ≈ 0.59 at z = 0.

The peak height is

ν = δ_c / σ = 1.686 / 0.59 ≈ 2.86

and the exponential suppression is

exp(−ν²/2) = exp(−4.09) ≈ 0.0167

So a cluster this massive sits nearly 3σ out in the tail of the fluctuation field. Folding in the prefactors of the mass function gives a comoving number density of order 10⁻⁵ to 10⁻⁶ Mpc⁻³ for halos above 10¹⁵ M☉ — meaning roughly one such cluster per (50–100 Mpc)³ of volume, consistent with the observed Coma–Virgo spacing of tens of megaparsecs. Now push to z = 1: the growth factor shrinks σ by roughly 40 percent, so ν climbs to about 4.8, exp(−ν²/2) drops by a further factor of ~1000, and 10¹⁵ M☉ clusters become genuinely scarce. That steep redshift dependence is exactly the signal cluster surveys exploit to measure the growth of structure and constrain dark energy.

Press-Schechter versus its successors

The original formula captures the right physics but the wrong numbers in the tails. Generations of fits, calibrated against ever-larger N-body simulations, refined it without discarding the σ–δ_c framework.

Model	Year	Collapse picture	Low-mass end	Cluster tail accuracy
Press-Schechter	1974	Spherical, fixed barrier	Overpredicts ~30–50%	Underpredicts by ×3–10 near 10¹⁵ M☉
Excursion set (BCEK)	1991	Random-walk first crossing	Same shape, factor-2 derived	Same tail as PS
Sheth-Tormen	1999	Ellipsoidal, moving barrier	~10% accurate	~10–20% — major improvement
Jenkins et al.	2001	Empirical fit to simulations	~10–20%	~10–20%
Warren et al.	2006	Empirical, friends-of-friends	~5%	~5–10%
Tinker et al.	2008	Empirical, spherical-overdensity	~5%	~5% — the cluster-cosmology standard

The Sheth-Tormen refinement is the most physically motivated upgrade. Real protohalos collapse ellipsoidally, not spherically: the most overdense regions collapse along their shortest axis first, and tidal shear means lower-mass objects need a slightly higher effective barrier. Replacing the fixed barrier with a mass-dependent "moving barrier" raises the cluster abundance and tames the low-mass overcount, bringing the analytic curve within ~10 percent of simulations. Modern survey pipelines — the Dark Energy Survey, eROSITA, SPT, and the upcoming Euclid and Rubin LSST cluster analyses — use the Tinker or similar emulator-calibrated mass functions, but every one of them is a child of the Press-Schechter idea.

History: a four-page paper that founded a field

William H. Press and Paul L. Schechter wrote "Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation" while at Caltech, publishing it in the Astrophysical Journal in 1974 (ApJ 187, 425). It predates the cold-dark-matter paradigm entirely — the authors were thinking about self-gravitating gas and the self-similar growth of bound objects. The Gaussian-field, threshold-crossing argument was so clean that it survived the conceptual revolution of the 1980s, when dark matter and inflation reshaped cosmology, essentially unchanged. The 1991 excursion-set rederivation by Bond, Cole, Efstathiou and Kaiser put it on a rigorous footing; Ravi Sheth and Giuseppe Tormen's 1999 ellipsoidal-collapse extension made it quantitatively competitive with simulations. Press went on to even broader fame as lead author of Numerical Recipes; Schechter is also the namesake of the Schechter luminosity function for galaxies. The formalism remains one of the most-cited results in extragalactic astrophysics half a century on.

Where it is used today

• Cluster cosmology. The number of galaxy clusters above a mass threshold as a function of redshift is one of the sharpest probes of σ₈, Ω_m, and the dark-energy equation of state. eROSITA, SPT, ACT, and DES all turn cluster counts into cosmological constraints using a calibrated mass function descended from Press-Schechter.
• Reionisation modelling. The collapsed fraction f_coll above the minimum halo mass that can host star formation (~10⁸ M☉ for atomic cooling) sets how many ionising sources exist at z > 6. Press-Schechter gives f_coll analytically, feeding 21 cm and CMB-optical-depth predictions.
• Semi-analytic galaxy formation. Extended Press-Schechter merger trees provide the halo assembly histories that codes like GALFORM and L-Galaxies populate with baryonic physics — far cheaper than running a full N-body simulation for every model variation.
• First stars and dark-matter constraints. The high-mass exponential tail at high redshift tells you when the first 10⁶–10⁸ M☉ halos appear, anchoring Population III star formation; the low-mass end is sensitive to warm dark matter, which would suppress σ(M) below a cutoff and erase small halos.
• Survey forecasting. Before a telescope is built, mass-function predictions tell you how many clusters or how many high-z galaxies a given survey volume and depth will yield — JWST's surprising abundance of bright z > 10 galaxies was framed precisely against Press-Schechter / Sheth-Tormen expectations.

Common misconceptions and subtleties

• "δ_c = 1.686 means the halo is 1.686 times denser than average." No. 1.686 is a linear-theory bookkeeping value. The actual virialised halo is ~178 times denser than the background. The whole point of using 1.686 is to stay in linear theory and avoid following nonlinear collapse.
• "The factor of two is arbitrary." It looked arbitrary in 1974, but excursion-set theory derives it from the cloud-in-cloud accounting. It is not a free parameter — it is forced once you require all mass to end up in halos.
• "Press-Schechter is obsolete because we have simulations." Simulations calibrate the mass function, but you cannot run a billion-particle simulation for every point in a cosmological parameter scan. Analytic and emulator mass functions — all built on the σ–δ_c skeleton — are what actually go into likelihood pipelines.
• "It works equally well at all masses." It does not. The original formula overpredicts dwarf-scale halos and badly underpredicts cluster-scale halos. Knowing which way it errs in each regime is essential before trusting a back-of-envelope estimate.
• "σ₈ and δ_c are interchangeable knobs." They play different roles. σ₈ sets the overall amplitude (and shape) of σ(M); δ_c is a near-universal collapse threshold from gravity. Changing σ₈ slides the whole mass function; δ_c barely moves with cosmology.
• "Halos are spheres because spherical collapse assumes a sphere." Real halos are triaxial. Spherical collapse is a simplifying assumption for the threshold; the Sheth-Tormen ellipsoidal correction exists precisely because real collapse is not spherical.