Spherical Collapse Model

A sphere that falls behind, then falls in

The universe expands. Almost everything goes along for the ride — but not quite everything. A patch that starts even slightly denser than its surroundings has slightly more gravity, so it expands a little more slowly than the cosmic average. Over time that lag compounds: the patch keeps decelerating, reaches a maximum size, halts, and then begins to fall in on itself. When it has fallen far enough, the infall stops, the structure settles, and a bound object is born — a galaxy, a group, a cluster, a dark matter halo. The spherical collapse model is the cleanest possible cartoon of this process: take one perfectly uniform overdense sphere, follow its radius through cosmic time, and see what falls out.

What falls out is remarkable. Despite assuming a geometry that never occurs in nature — real overdensities are lumpy and triaxial — the model produces two pure numbers that turn out to govern the formation of structure in the real universe. The first is the linear collapse threshold δ_c ≈ 1.686: the density contrast that linear perturbation theory would predict at the moment a region has actually collapsed. The second is the virial overdensity Δ_vir = 18π² ≈ 178: how much denser a freshly virialized halo is than the surrounding universe. These two numbers are not decorative. δ_c is the threshold in the Press–Schechter halo mass function; Δ_vir defines the edge of a halo. Computational cosmology runs on them daily.

How it works: a tiny closed universe

The trick that makes the problem solvable is Birkhoff's theorem: in general relativity, the gravity inside a spherically symmetric shell depends only on the mass it encloses, and a uniform overdense sphere behaves exactly like its own miniature closed (positively curved) universe, independent of what the rest of the cosmos is doing. So we can borrow the machinery of cosmological dynamics and apply it to a single ball of matter.

Consider a sphere of physical radius R(t) enclosing a fixed mass M. Its equation of motion is just Newtonian gravity (which agrees with the GR result for pressureless matter):

d²R/dt² = − G M / R²

Integrating once gives an energy equation, ½(dR/dt)² − GM/R = E. If the sphere is overdense enough, the total energy E is negative — it is bound — and the sphere is destined to recollapse. The motion is a cycloid, the same curve traced by a point on a rolling wheel, written parametrically with a development angle θ:

R(θ) = A (1 − cos θ)
t(θ) = B (θ − sin θ)        with   A³ = G M B²

Three milestones live on this cycloid. At θ = 0 the sphere is born with the background (R = 0, t = 0). At θ = π the radius is maximal, R = 2A — this is turnaround, the instant expansion stops and collapse begins. At θ = 2π the radius returns to zero — formal collapse to a point. Because t(2π) = 2·t(π), full collapse takes exactly twice as long as reaching turnaround. Everything quantitative in the model is read off these three points.

Where 1.686 comes from

The collapse, taken literally, ends in a singularity of infinite density — useless as a threshold. The clever move, due to the way the model is used, is to ask a counterfactual question: what density contrast would the lazy, always-wrong linear theory have predicted at the moment of full nonlinear collapse?

Linear perturbation theory in an Einstein–de Sitter (matter-only, flat) universe says the density contrast grows in proportion to the scale factor, δ_lin ∝ a ∝ t^(2/3). Expand the cycloid solution for small θ to extract the leading overdensity, then evolve that linear growth forward to the collapse time t(2π). The arithmetic delivers a clean closed form:

δ_c = (3/20) (12π)^(2/3) = 1.68647…

This is the headline number. It is not the real overdensity at collapse — that is formally infinite. It is a bookkeeping device: a region whose linearly-extrapolated density contrast has reached 1.686 is, in the honest nonlinear picture, a region that has already collapsed into a bound object. Because linear theory is trivial to compute on a whole density field while nonlinear collapse is not, swapping the hard question ("has this patch collapsed?") for the easy one ("has its linear δ crossed 1.686?") is the foundational shortcut of analytic structure formation.

Where 178 comes from

The second number describes how dense the collapsed object actually is. Three facts combine:

• Turnaround density. At θ = π the sphere's mean density relative to the background is exactly (3π/4)² ≈ 5.55. It has fallen behind the cosmic expansion by this factor.
• Collapse to half the turnaround radius. A self-gravitating system cannot reach a point; it relaxes to virial equilibrium, where the energy theorem 2T + U = 0 holds. Starting from rest at turnaround (all potential energy, no kinetic), conservation of energy forces the virial radius to be exactly half the turnaround radius, R_vir = R_ta/2. Halving the radius makes the sphere 2³ = 8 times denser in itself.
• Background keeps diluting. Between turnaround and collapse the cosmic time doubles, so in an Einstein–de Sitter universe the background density drops by (t_ta/t_coll)² = 1/4.

Multiply: relative to the background at the collapse epoch, the virialized sphere is denser by 5.55 × 8 × 4 = 18π² ≈ 178. That is Δ_vir, the virial overdensity. It is the reason a "halo" has a well-defined edge: walk outward until the enclosed mean density drops to ~178 times the background (or, in common practice, a round 200 times), and you have hit the boundary, R_vir or R_200, with enclosed mass M_vir or M_200.

Worked example: a cluster-scale sphere

Take a region destined to become a galaxy cluster of mass M = 10¹⁵ M_⊙ that reaches full collapse at the present epoch, z = 0, in an Einstein–de Sitter universe with present mean matter density ρ̄_0 ≈ 4.1 × 10⁻³⁰ g/cm³ (Ω_m = 1 normalization for the toy calculation). We can read its history straight off the cycloid.

Mass enclosed         M        = 1.0 × 10¹⁵ M_⊙
Virial overdensity    Δ_vir    = 18π²  ≈ 178
Mean density at z=0   ρ̄_0      ≈ 4.1 × 10⁻³⁰ g/cm³
Halo mean density     ρ_halo   = 178 × ρ̄_0 ≈ 7.3 × 10⁻²⁸ g/cm³

Virial radius from  M = (4/3)π R_vir³ ρ_halo:
  R_vir = [ 3M / (4π · 178 · ρ̄_0) ]^(1/3) ≈ 1.9 Mpc
Turnaround radius:    R_ta   = 2 R_vir   ≈ 3.8 Mpc
Linear δ at z=0:      δ_lin(collapse) = 1.686   (by construction)
Linear δ at turnaround: δ_lin(θ=π)    ≈ 1.06

Every step is dictated by the two universal numbers. The cluster's virial radius of about 1.9 Mpc and its mean interior density of 178 times the cosmic average are not free parameters — they are forced by δ_c and Δ_vir. Run the same arithmetic for a Milky-Way-mass halo (M ≈ 10¹² M_⊙) and you get R_vir ≈ 0.2 Mpc, the familiar ~200 kpc halo edge that rotation-curve studies and N-body simulations both recover. The model with two numbers reproduces the sizes of objects spanning three decades in mass.

From one sphere to the whole halo population

The reason δ_c = 1.686 is famous is the Press–Schechter formalism (Press & Schechter 1974). Take the linear density field of the early universe — a Gaussian random field — and smooth it with a filter of comoving scale corresponding to mass M. The smoothed contrast δ_M is itself Gaussian with a variance σ²(M) that you can compute from the matter power spectrum. Press & Schechter then made an audacious ansatz: the fraction of mass that has collapsed into halos of mass ≥ M equals the fraction of space where δ_M exceeds δ_c = 1.686. That single threshold converts statistics into objects.

dn/dM = √(2/π) (ρ̄/M) (δ_c/σ²) |dσ/dM| exp[ − δ_c² / (2σ²) ]

The exponential cutoff at high mass — the reason massive clusters are exponentially rare — is set entirely by the ratio δ_c/σ(M). The excursion-set refinement (Bond, Cole, Efstathiou & Kaiser 1991) fixed Press & Schechter's notorious missing factor of two by reframing the threshold as a barrier crossed by a random walk as the smoothing scale shrinks. In every version, the barrier height is the spherical-collapse threshold 1.686. Modern fits (Sheth–Tormen, Tinker) add empirical corrections calibrated to N-body simulations, but they are perturbations around this analytic skeleton.

Variants and regimes

• Open and Λ-dominated universes. In an open (Ω_m < 1) or dark-energy universe the cycloid is modified and δ_c drifts mildly with redshift — typically dropping about one percent below 1.686 by z = 0 for ΛCDM. Δ_vir is more sensitive; the Bryan & Norman (1998) fit Δ_vir = 18π² + 82x − 39x² (x = Ω_m(z) − 1) gives ≈ 337 relative to the mean density at z = 0 for Ω_m = 0.3.
• Ellipsoidal collapse. Sheth, Mo & Tormen (2001) replaced the sphere with a triaxial ellipsoid that collapses along its shortest axis first. This makes the effective threshold mass-dependent (higher for low-mass, less-spherical peaks) and substantially improves the predicted mass function.
• Turnaround radius as a probe. The boundary where infall exactly cancels Hubble expansion — the turnaround radius — has been promoted to an observable; in dark-energy cosmologies it has a maximum allowed value, offering an independent test of Λ.
• Self-similar secondary infall. Bertschinger (1985) and Fillmore & Goldreich (1984) extended the single shell to a continuum of shells turning around in sequence, producing the self-similar density profiles that anticipate the NFW form.
• Splashback radius. A modern correction recognizes that infalling shells do not stop dead at R_vir but reach a first apocenter — the splashback radius at roughly 1.5 R_200 — a more physical halo edge now measured in cluster galaxy distributions.

Common pitfalls and misconceptions

• Thinking δ_c = 1.686 is the real overdensity at collapse. It is not. The true nonlinear overdensity at the formal collapse instant is infinite; 1.686 is the value linear theory would have (wrongly) predicted there. It is a threshold for the linear field, not a physical density.
• Confusing Δ_vir ≈ 178 with δ_c ≈ 1.686. They answer different questions. δ_c is a dimensionless threshold on the linear contrast (a number near 1.7). Δ_vir is a nonlinear density ratio (a number near 178). They are linked by the same cycloid but are not interchangeable.
• Forgetting which density Δ is measured against. Δ_vir ≈ 178 is relative to the mean matter density. Quoted relative to the critical density in ΛCDM at z = 0 the same halo is only ~100× overdense, because Ω_m ≈ 0.3. M_200c and M_200m differ — always check the reference density.
• Believing the sphere actually collapses to a point. No real system does; violent relaxation virializes it at half the turnaround radius. The singular endpoint of the idealized cycloid is an artifact of perfect symmetry and zero velocity dispersion.
• Assuming spherical symmetry is realistic. Zel'dovich showed collapse goes pancake → filament → knot, never a clean implosion. The spherical model is accurate for the scalar outputs (threshold, overdensity, timing) precisely because those depend on energy and mass, not on shape.

Reference numbers from the cycloid

Quantity	Symbol	Value (EdS)	Cycloid stage	Role
Linear collapse threshold	δ_c	(3/20)(12π)^(2/3) ≈ 1.686	θ = 2π	Press–Schechter barrier
Virial overdensity (vs. mean)	Δ_vir	18π² ≈ 178	post-virialization	Halo boundary definition
Turnaround overdensity	ρ/ρ̄	(3π/4)² ≈ 5.55	θ = π	Energy reference state
Linear δ at turnaround	δ_lin(π)	≈ 1.062	θ = π	Onset of collapse marker
Collapse / turnaround time ratio	t_coll/t_ta	2	θ: π → 2π	Sets collapse timescale
Virial / turnaround radius	R_vir/R_ta	1/2	virialization	Fixes final halo size
Density vs. turnaround	ρ_vir/ρ_ta	2³ = 8	virialization	Collapse compression
Δ_vir at z=0 (ΛCDM)	Δ_vir	≈ 337 (vs. mean)	Bryan & Norman fit	Why catalogs use Δ=200

Observational status and applications

You cannot watch a single sphere collapse — the timescales are billions of years — but the model's outputs are tested constantly and indirectly. Cosmological N-body simulations grow structure from Gaussian initial conditions under gravity alone and find that halos do form with mean interior densities near Δ ≈ 178–200 times the background, vindicating the virial-overdensity argument despite the absence of any literal sphere. Halo finders such as friends-of-friends and spherical-overdensity codes use these thresholds to define which particles belong to which halo.

On the threshold side, the abundance of massive galaxy clusters as a function of redshift — measured through the Sunyaev–Zel'dovich effect, X-ray surveys, and weak-lensing mass calibration — is one of the sharpest cosmological probes, and its interpretation runs through a mass function whose exponential tail is fixed by δ_c = 1.686. The amplitude of matter clustering σ_8 is in essence a statement about how σ(M) compares to that threshold at cluster scales. JWST's discovery of surprisingly massive galaxies at z > 10 is partly a stress test of the same machinery: too many massive halos too early would mean the field crossed 1.686 sooner than ΛCDM allows. The humble overdense sphere, expanding, turning around, and collapsing, is therefore embedded in the front-line cosmological tension of the moment.

Quantitative analysis: assembling the threshold

Here is the threshold derivation in compact form. Parameterize the bound sphere with the cycloid R = A(1 − cos θ), t = B(θ − sin θ). Match to an Einstein–de Sitter background whose scale factor obeys a ∝ t^(2/3) and whose mean density is ρ̄ = 1/(6πGt²). The sphere's true (nonlinear) overdensity is

1 + δ_nl(θ) = (9/2) (θ − sin θ)² / (1 − cos θ)³

Evaluate at turnaround (θ = π): 1 + δ_nl = (9/2)(π²)/(2³) = 9π²/16 ≈ 5.55, the turnaround overdensity. To get the linear value, expand for small θ. Using cos θ ≈ 1 − θ²/2 + θ⁴/24 and sin θ ≈ θ − θ³/6 + θ⁵/120, the leading nonzero term of δ_nl is

δ_lin(θ) = (3/20) θ²       (small-θ growing mode)

Linear theory grows as δ ∝ a ∝ t^(2/3) ∝ (θ − sin θ)^(2/3); since δ_lin ∝ θ² in the small-θ limit and θ² ∝ [t(θ)]^(2/3) at leading order, the linear contrast simply tracks (3/20)θ² for all θ. At collapse, θ = 2π, so

δ_c = δ_lin(2π) = (3/20)(2π)²·[correction] = (3/20)(12π)^(2/3) = 1.6865

and at turnaround δ_lin(π) = (3/20)(6π)^(2/3) ≈ 1.062. The same energy argument that fixes R_vir = R_ta/2 then converts the 5.55 turnaround overdensity into Δ_vir = 18π² ≈ 178 at collapse. Two short expansions of one cycloid, and the entire quantitative backbone of halo formation — threshold, overdensity, timing, and size — is in hand.