Jeans Instability and Cosmological Perturbation Regimes

Every galaxy, cluster and filament of the cosmic web began as a whisper — a density contrast of one part in a hundred thousand — and grew because gravity is unstable. A region a shade denser than average pulls harder on its surroundings, drags in more mass, and pulls harder still. The only thing that can stop the runaway is pressure, which pushes outward and tries to smooth the lump away. Sir James Jeans, in 1902, turned this tug-of-war into a number: the Jeans length, the scale above which gravity outruns pressure and a perturbation collapses, and below which pressure wins and the perturbation merely rings as a sound wave. The classical calculation is disarmingly clean — linearise the fluid equations of a self-gravitating gas, look for plane waves, and read off a dispersion relation whose sign decides everything. But it rests on the notorious "Jeans swindle": a static, uniform, infinite background, which flatly contradicts Poisson's equation. The cosmological setting rescues it. The correct background is the expanding Friedmann universe, in which the mean density sources the Hubble flow rather than an unbalanced force. Perturbing about that self-consistent background reproduces Jeans' criterion locally while adding one decisive new ingredient — a Hubble-friction term $2H\dot\delta$ that drains the collapse of energy. The consequence is profound: unstable modes no longer blow up exponentially but crawl upward as power laws of the scale factor, $\delta\propto a$ in matter domination. This single change, together with the hierarchy of regimes it organizes — super-horizon versus sub-horizon, radiation versus matter era, pressure-supported baryons versus pressureless cold dark matter — is the microscopic engine behind the transfer function, the acoustic peaks of the microwave background, and the sudden freeing of baryons to make galaxies once recombination collapses their Jeans mass.

Key takeaways

The Jeans instability sets a critical length $\lambda_J$ : above it self-gravity overwhelms pressure and a perturbation collapses; below it pressure restores equilibrium and the perturbation oscillates as a sound wave.
In a static medium the dispersion relation $\omega^2=c_s^2k^2-4\pi G\rho_0$ makes modes with $k<k_J$ unstable, with $\lambda_J=c_s\sqrt{\pi/(G\rho_0)}$ set by the free-fall time racing the sound-crossing time.
The static derivation uses the "Jeans swindle" — a static uniform background — which the expanding-universe treatment cures by letting the mean density source the Hubble flow.
In an expanding universe a Hubble-drag term $2H\dot\delta$ damps growth: pressureless matter grows only as a power law, $\delta\propto a\propto t^{2/3}$ in Einstein–de Sitter, not exponentially.
The Jeans mass $M_J\sim\rho\lambda_J^3\propto c_s^3\rho^{-1/2}$ sets the smallest collapsing mass; it plummets from $\sim10^{17}M_\odot$ to $\sim10^{5}M_\odot$ at recombination when baryons shed photon pressure.
Different regimes — super-horizon vs sub-horizon, radiation vs matter era, baryons vs cold dark matter — each grow differently, and generalized Jeans analysis is the framework that organizes them.
Key idea: Gravity wins on large scales, pressure wins on small scales; the crossover is the Jeans length $\lambda_J=c_s\sqrt{\pi/(G\rho)}$ , set by the free-fall time $\sim(G\rho)^{-1/2}$ racing the sound-crossing time $\sim\lambda/c_s$ — and in an expanding universe Hubble drag turns exponential collapse into power-law growth.

The core insight

Gravity wins on large scales, pressure wins on small scales; the crossover — the Jeans length $\lambda_J=c_s\sqrt{\pi/(G\rho)}$ — is set by the free-fall time $\sim(G\rho)^{-1/2}$ racing the sound-crossing time $\sim\lambda/c_s$ . In an expanding universe a Hubble-drag term $2H\dot\delta$ turns exponential collapse into power-law growth, $\delta\propto a$ in matter domination.

Where this lives today

The cosmological Jeans framework is the working backbone of precision cosmology. The scale-dependent growth it predicts — logarithmic for modes entering during radiation domination, $\delta\propto a$ for those entering later — shapes the transfer function and imprints the matter-power-spectrum turnover at the equality scale, now measured to percent precision by DESI and Euclid. The pressure-supported oscillations of the sub-Jeans photon–baryon fluid became the acoustic peaks of the microwave background mapped by Planck, and their imprint in the baryons is the baryon acoustic oscillation feature — a $\sim150$ Mpc standard ruler that constrains dark energy. On smaller scales Silk damping sets the cutoff of surviving baryonic perturbations. The same Jeans physics runs the ALMA-era study of star-forming cores and constrains warm and fuzzy dark-matter models, whose nonzero effective Jeans length would suppress the smallest galaxies — tested against Milky Way satellite counts and the Lyman- $\alpha$ forest.

The physical picture

Picture a clump of gas a little denser than its surroundings, and start two clocks. The free-fall time $t_{\rm ff}\sim1/\sqrt{G\rho}$ is how long gravity needs to pull the region together; the sound-crossing time $t_{\rm sc}\sim\lambda/c_s$ is how long a pressure wave needs to cross the clump and push back. If gravity finishes first ( $t_{\rm ff}<t_{\rm sc}$ , i.e. the clump is larger than $\lambda_J$ ), the interior never learns it is being squeezed until too late, and it collapses. If the pressure wave finishes first, it communicates the compression outward, the clump bounces, and it rings as a standing sound wave. The Jeans length is just the distance a sound wave travels in one free-fall time, $\lambda_J\sim c_s t_{\rm ff}\sim c_s/\sqrt{G\rho}$ : larger clumps are "too big" for pressure to hold together. The first figure makes this quantitative through the dispersion relation itself — the parabola $\omega^2=c_s^2k^2-4\pi G\rho_0$ , whose single zero-crossing at $k_J$ is the boundary between the two fates.

Figure

The static Jeans dispersion relation ω² = c_s²k² − 4πGρ₀ as a parabola in k: it dips below zero for k < k_J where ω² < 0 gives runaway collapse δ ∝ e^(±t/τ), and rises above zero for k > k_J where ω² > 0 gives stable sound waves δ ∝ e^(±iωt); the crossing is the Jeans wavenumber k_J = √(4πGρ₀)/c_s

Now let the universe expand, and a third player enters: the Hubble flow, which stretches every perturbation and drains the kinetic energy of collapse. It appears as a friction term $2H\dot\delta$ — Hubble drag — and it is decisive. Even a genuinely unstable mode can no longer grow exponentially, because space pulls apart nearly as fast as gravity pulls together; the best the growing mode manages in matter domination is the power law $\delta\propto a$ . This one fact separates cosmological structure formation from ordinary star formation, where the static exponential picture is a fine approximation. The two extremes of matter behave very differently. Cold dark matter is pressureless, $c_s\approx0$ , so its Jeans length is essentially zero and it clusters on all scales, limited only by Hubble drag and the horizon. Baryons before recombination are the opposite: locked to photons, they inherit an enormous sound speed $c_s\to c/\sqrt3$ and a colossal Jeans mass, so they cannot collapse — they oscillate. The most consequential event in the story is recombination, when the baryons decouple, their sound speed crashes to the thermal speed of neutral hydrogen, and their Jeans mass falls by roughly twelve orders of magnitude. The second figure tracks that plunge.

Figure

The baryonic Jeans mass M_J ∝ c_s³ρ^(−1/2) versus cosmic time: while baryons are locked to photons (c_s ≈ c/√3) it sits near 10^17 M_⊙, larger than any galaxy, so nothing collapses; at recombination near z ≈ 1100 the baryons go neutral, c_s drops to the ≈ few km/s thermal speed of hydrogen, and M_J plunges about twelve orders of magnitude to ≈ 10^5 M_⊙, freeing baryons to fall into dark-matter wells

The core equations

The whole subject flows from one dispersion relation. Linearise the equations of a self-gravitating ideal fluid about a uniform background, feed in a plane wave $\delta\rho\propto e^{i(\mathbf k\cdot\mathbf r-\omega t)}$ , and the pressure restoring force ( $c_s^2k^2$ ) competes head-on with self-gravity ( $4\pi G\rho_0$ ). The sign of $\omega^2$ decides everything.

\omega^2 = c_s^2 k^2 - 4\pi G\rho_0 \qquad (1)

where: $\omega$ — mode frequency; $c_s$ — adiabatic sound speed, $c_s^2=(\partial p/\partial\rho)_S$ ; $k=2\pi/\lambda$ — wavenumber; $G$ — Newton's constant; $\rho_0$ — unperturbed density. Reading it: large $k$ (small $\lambda$ ) makes $\omega^2>0$ and the mode oscillates; small $k$ makes $\omega^2<0$ , $\omega$ imaginary, and the amplitude grows or decays exponentially. Setting $\omega^2=0$ isolates the marginal scale where pressure and gravity exactly balance — the Jeans length, the single most important number in the theory.

\lambda_J = \frac{2\pi}{k_J} = c_s\sqrt{\frac{\pi}{G\rho_0}} \qquad (2)

where: $\lambda_J$ — critical wavelength separating collapse from oscillation; $k_J=\sqrt{4\pi G\rho_0}/c_s$ — Jeans wavenumber. Reading it: $\lambda_J$ grows with sound speed (more pressure support) and shrinks as density rises (stronger gravity) — exactly a sound wave's travel distance in one free-fall time. Enclosing the marginal scale in a sphere converts length into mass — the smallest mass that can overcome its own pressure.

M_J = \frac{4}{3}\pi\rho_0\left(\frac{\lambda_J}{2}\right)^3 \;\sim\; \frac{c_s^3}{G^{3/2}\rho_0^{1/2}} \qquad (3)

where: $M_J$ — Jeans mass, the mass in a sphere of radius $\lambda_J/2$ . Reading it: the steep $c_s^3$ dependence is why a factor-of-ten change in sound speed swings $M_J$ by a thousand — and why recombination, which drops $c_s$ by $\sim10^4$ , drops $M_J$ by $\sim10^{12}$ . The static analysis is a scaffold. Perturbing the fluid equations on an expanding Friedmann background in comoving coordinates yields the same pressure-versus-gravity bracket plus one new term.

\ddot{\delta}_k + 2H\dot{\delta}_k + \left(\frac{c_s^2 k^2}{a^2} - 4\pi G\bar\rho\right)\delta_k = 0 \qquad (4)

where: $\delta_k$ — Fourier amplitude of the density contrast at comoving wavenumber $k$ ; $H=\dot a/a$ — Hubble parameter; $a$ — scale factor, so $k/a$ is the physical wavenumber; $\bar\rho$ — mean clustering density. Reading it: the middle term $2H\dot\delta_k$ is Hubble drag — friction from expansion — and the bracket vanishes at the physical Jeans wavenumber, recovering the static criterion locally. For pressureless matter ( $c_s\to0$ ) in an Einstein–de Sitter universe, equation (4) has two power-law solutions — a growing mode that builds structure and a decaying mode that fades.

\delta_+ \propto a \propto t^{2/3}, \qquad \delta_- \propto a^{-3/2} \propto t^{-1} \qquad (5)

where: $\delta_+$ — growing mode, dominates at late times; $\delta_-$ — decaying mode, negligible at late times. Reading it: the growing mode is a power law, not an exponential — the signature of gravity fighting Hubble drag to a near-standstill. Before recombination the baryons are far from pressureless: photon pressure gives the coupled photon–baryon fluid a relativistic sound speed, reduced only by the inertia the baryons add.

c_s = \frac{c}{\sqrt{3}}\sqrt{\frac{1}{1+R}}, \qquad R = \frac{3\rho_b}{4\rho_\gamma} \qquad (6)

where: $R$ — baryon-to-photon momentum ratio; $\rho_b,\rho_\gamma$ — baryon and photon energy densities. Reading it: with $R\to0$ (photon-dominated) the sound speed is $c/\sqrt3$ ; loading in baryons ( $R$ up) lowers $c_s$ and shrinks the acoustic scale. In the radiation era a further subtlety appears — a sub-horizon CDM mode barely grows, $\delta\propto\ln a$ (the Meszaros effect), because the fast radiation-driven expansion overwhelms its feeble self-gravity, freezing small-scale power and carving the turnover into the transfer function.

Derivation AStatic Jeans criterion from the fluid equations

The goal is to derive the dispersion relation $\omega^2=c_s^2k^2-4\pi G\rho_0$ and the Jeans length $\lambda_J=c_s\sqrt{\pi/(G\rho_0)}$ from the continuity, Euler and Poisson equations of a self-gravitating ideal gas, while being honest about the "Jeans swindle" — the inconsistent static uniform background the calculation assumes. We linearise about rest, combine the three equations into a single wave equation for the density perturbation, and insert a plane wave to read off stability.

Step 1. Write the governing equations: continuity $\partial_t\rho+\nabla\cdot(\rho\mathbf v)=0$ , Euler $\rho(\partial_t\mathbf v+\mathbf v\cdot\nabla\mathbf v)=-\nabla p-\rho\nabla\Phi$ , and Poisson $\nabla^2\Phi=4\pi G\rho$ . Take the unperturbed state uniform and at rest, $(\rho_0,\mathbf v_0=0,p_0,\Phi_0)$ . Why: this triad is the complete Newtonian description of a gas that feels its own gravity — pressure through $\nabla p$ , self-gravity through $\nabla\Phi$ .

Step 2. Note the swindle explicitly: a uniform $\rho_0\ne0$ forces $\nabla\Phi_0\ne0$ through Poisson, so a truly static uniform background cannot satisfy Euler with $\mathbf v_0=0$ . We proceed anyway, setting the background force to zero and deferring the honest fix to the cosmological treatment. Why: the perturbation dynamics depend on gradients of the perturbed field, not the ill-defined background force, so the crossover scale still comes out correct — but the step is formally inconsistent.

Step 3. Linearise with $\rho=\rho_0+\delta\rho$ , $\mathbf v=\mathbf v_1$ , $\delta p=c_s^2\delta\rho$ , keeping only first-order terms: $\partial_t\delta\rho+\rho_0\nabla\cdot\mathbf v_1=0$ , $\rho_0\partial_t\mathbf v_1=-c_s^2\nabla\delta\rho-\rho_0\nabla\Phi_1$ , $\nabla^2\Phi_1=4\pi G\delta\rho$ . Why: products of small quantities are negligible; the equation of state enters only through the adiabatic sound speed $c_s^2=(\partial p/\partial\rho)_S$ .

Step 4. Eliminate $\mathbf v_1$ and $\Phi_1$ : take $\partial_t$ of continuity, substitute the divergence of Euler, and use Poisson for $\nabla^2\Phi_1$ . The system collapses to $\partial_t^2\delta\rho-c_s^2\nabla^2\delta\rho-4\pi G\rho_0\,\delta\rho=0$ . Why: a single wave equation exposes the two competing effects — the $c_s^2\nabla^2$ term is pressure, the $4\pi G\rho_0$ term is self-gravity.

Step 5. Insert the plane wave $\delta\rho\propto e^{i(\mathbf k\cdot\mathbf r-\omega t)}$ to get $\omega^2=c_s^2k^2-4\pi G\rho_0$ . Instability ( $\omega^2<0$ ) requires $k<k_J=\sqrt{4\pi G\rho_0}/c_s$ , giving $\lambda_J=2\pi/k_J=c_s\sqrt{\pi/(G\rho_0)}$ . Why: the sign of $\omega^2$ is the whole physics — real $\omega$ is bounded oscillation, imaginary $\omega$ is unbounded growth, i.e. collapse.

Go deeper · Derivation B · 1 minPerturbation growth in an expanding universe

Derivation BPerturbation growth in an expanding universe

The aim is to show how the static wave equation changes once the background is the expanding Friedmann universe, producing the Hubble-drag term $2H\dot\delta$ absent from Jeans' original analysis. Working in comoving coordinates removes the bulk Hubble flow and isolates the peculiar dynamics; the resulting equation both cures the swindle and generalizes the Jeans criterion to any cosmological epoch.

Step 1. Split physical position into the Hubble flow plus a peculiar displacement, $\mathbf r=a(t)\mathbf x$ , so the physical velocity is $\dot{\mathbf r}=H\mathbf r+\mathbf u$ with peculiar velocity $\mathbf u=a\dot{\mathbf x}$ . Rewriting continuity and Euler in the comoving coordinate $\mathbf x$ turns $\nabla_r$ into $a^{-1}\nabla_x$ and injects explicit factors of $H$ . Why: the mean expansion is not an instability and must be factored out, so that $\delta$ and $\mathbf u$ describe only departures from a smooth Friedmann background.

Step 2. The linearised comoving equations are $\dot\delta=-a^{-1}\nabla_x\cdot\mathbf u$ and $\dot{\mathbf u}+H\mathbf u=-a^{-1}c_s^2\nabla_x\delta-a^{-1}\nabla_x\phi$ , with $\nabla_x^2\phi=4\pi G\bar\rho a^2\delta$ . The new piece is $H\mathbf u$ in the Euler equation — a drag on peculiar velocity produced purely by expansion. Why: as space stretches, a peculiar velocity is redshifted away ( $\mathbf u\propto a^{-1}$ in free streaming) — the microscopic origin of Hubble friction.

Step 3. Differentiate the continuity equation in time, substitute the Euler and Poisson relations, and Fourier-transform in $\mathbf x$ to eliminate $\mathbf u$ and $\phi$ . The result is $\ddot\delta_k+2H\dot\delta_k+(c_s^2k^2/a^2-4\pi G\bar\rho)\delta_k=0$ . Why: the same elimination that gave a clean wave equation in the static case now leaves behind the $2H\dot\delta_k$ term, because the background is time-dependent — this is equation (4).

Step 4. Read the physics off the bracket: it vanishes at the physical Jeans wavenumber $k_J/a=\sqrt{4\pi G\bar\rho}/c_s$ , recovering Jeans' criterion locally, while $2H\dot\delta_k$ ensures even super-Jeans modes grow only as a power of $a$ . Why: cosmology reproduces Jeans' crossover scale exactly but replaces exponential collapse with power-law growth — on a self-consistent background where $\bar\rho$ sources the Hubble flow, not an unbalanced force.

Go deeper · Derivation C · 1 minGrowing and decaying modes in an Einstein–de Sitter universe

Derivation CGrowing and decaying modes in an Einstein–de Sitter universe

The goal is to solve equation (4) for pressureless matter in a flat matter-dominated (Einstein–de Sitter) background and obtain the canonical solutions $\delta_+\propto a$ and $\delta_-\propto a^{-3/2}$ , showing concretely that gravitational instability in an expanding universe yields power-law rather than exponential growth. We drop pressure, insert the EdS background relations, and solve the resulting Euler equation with a power-law ansatz.

Step 1. Set $c_s=0$ to drop the pressure term — exact for cold dark matter and valid for baryons far above their Jeans length — leaving $\ddot\delta+2H\dot\delta-4\pi G\bar\rho\,\delta=0$ . Why: where pressure is irrelevant, the only competition left is self-gravity versus Hubble drag — the regime that builds cosmic structure.

Step 2. Insert the Einstein–de Sitter background: $a\propto t^{2/3}$ , so $H=2/(3t)$ and $4\pi G\bar\rho=\tfrac32 H^2=2/(3t^2)$ . The equation becomes $\ddot\delta+\tfrac{4}{3t}\dot\delta-\tfrac{2}{3t^2}\delta=0$ . Why: in EdS every background quantity is a pure power of $t$ , so the equation reduces to an equidimensional (Euler) ODE.

Step 3. Try $\delta\propto t^{p}$ . Cancelling the common $t^{p-2}$ gives $p(p-1)+\tfrac43 p-\tfrac23=0$ , i.e. $3p^2+p-2=0$ . Why: a power-law ansatz is exact for an equidimensional equation and already signals a power-law answer, not an exponential.

Step 4. Solve: $p=\dfrac{-1\pm5}{6}$ , giving $p=\tfrac23$ (growing) and $p=-1$ (decaying). Via $a\propto t^{2/3}$ this is $\delta_+\propto t^{2/3}\propto a$ and $\delta_-\propto t^{-1}\propto a^{-3/2}$ . Why: the growing root is only $\delta\propto a$ — Hubble drag has throttled exponential collapse down to linear growth in the scale factor, the central result of linear structure formation.

Limiting cases

Which behaviour a perturbation shows depends entirely on which competing scale or timescale dominates:

Case	Result	Meaning
Small scale, static medium ( $\lambda\ll\lambda_J$ )	$\omega^2\approx c_s^2k^2>0$	Pressure dominates; $\delta\propto e^{\pm i\omega t}$ are stable acoustic oscillations.
Large scale, static medium ( $\lambda\gg\lambda_J$ )	$\omega^2\approx-4\pi G\rho_0<0$	Gravity dominates; $\delta\propto e^{\pm t/\tau}$ with $\tau=(4\pi G\rho_0)^{-1/2}$ : exponential collapse.
Pressureless matter, matter era	$\delta_+\propto a,\ \delta_-\propto a^{-3/2}$	In an EdS background the growing mode is only $\delta\propto a\propto t^{2/3}$ — power-law, not exponential, due to Hubble drag.
Sub-horizon, radiation era	$\delta\propto\ln a$ (Meszaros effect)	CDM entering the horizon before equality grows only logarithmically, freezing the small-scale suppression of the transfer function.
Baryons, before vs after recombination	$M_J:\ \sim10^{17}M_\odot\to\sim10^{5}M_\odot$	The baryonic Jeans mass plunges $\sim12$ orders as $c_s$ falls from $\sim c/\sqrt3$ to the thermal speed of neutral hydrogen.
Cold dark matter, all epochs	$\lambda_J\to0$ ( $c_s\approx0$ )	Pressureless CDM has no Jeans scale and clusters on every scale, checked only by Hubble drag and the horizon.

Common trapCosmological perturbations grow exponentially, and the Jeans swindle makes the whole thing invalid

Both halves of this common belief are wrong. First, the static result $\delta\propto e^{t/\tau}$ does not carry over to cosmology: the Hubble-drag term $2H\dot\delta$ throttles growth to a power law, $\delta\propto a\propto t^{2/3}$ in matter domination, so structure builds up linearly in the scale factor, not explosively — expecting exponential growth badly overestimates how fast perturbations reach nonlinearity. Second, the "Jeans swindle" — assuming a static uniform background with no net gravitational force, which contradicts Poisson's equation — is a flaw of the static derivation only, not of the physics. The cosmological treatment cures it completely: the mean density sources the Hubble expansion through the Friedmann equations rather than an unbalanced force, and perturbing about that self-consistent background reproduces Jeans' crossover scale with no swindle. The static calculation is a mnemonic for the right answer, not a valid derivation of it.

Worked examples

1 · Jeans mass of a cold molecular-cloud core (fully worked)

A dense core in a molecular cloud has temperature $T=10$ K, mean molecular weight $\mu=2.33$ (molecular hydrogen with helium), and number density $n=10^4\ {\rm cm}^{-3}$ . Find its sound speed, Jeans length, and Jeans mass, and say what it implies for star formation.

Step 1. Sound speed. Use the isothermal sound speed $c_s=\sqrt{k_BT/(\mu m_H)}$ appropriate for cold, efficiently-cooled gas. With $k_B=1.38\times10^{-23}$ J/K and $\mu m_H=2.33\times1.67\times10^{-27}=3.90\times10^{-27}$ kg, $c_s=\sqrt{(1.38\times10^{-23}\cdot10)/(3.90\times10^{-27})}=\mathbf{188\ m/s\approx0.19\ km/s}$ .

Step 2. Mass density. Convert the number density: $n=10^4\ {\rm cm}^{-3}=10^{10}\ {\rm m}^{-3}$ , so $\rho_0=n\mu m_H=10^{10}\times3.90\times10^{-27}=\mathbf{3.90\times10^{-17}\ kg/m^3}$ .

Step 3. Jeans length. $\lambda_J=c_s\sqrt{\pi/(G\rho_0)}$ . With $G\rho_0=2.60\times10^{-27}\ {\rm s}^{-2}$ , the free-fall factor is $\sqrt{\pi/(G\rho_0)}=3.48\times10^{13}$ s, so $\lambda_J=188\times3.48\times10^{13}=6.5\times10^{15}\ {\rm m}=\mathbf{0.21\ pc}$ .

Step 4. Jeans mass. $M_J=\tfrac43\pi\rho_0(\lambda_J/2)^3$ . With $\lambda_J/2=3.28\times10^{15}$ m, its cube is $3.52\times10^{46}\ {\rm m}^3$ , giving $M_J=5.7\times10^{30}\ {\rm kg}$ . Dividing by $M_\odot=1.99\times10^{30}$ kg, $M_J=\mathbf{2.9\ M_\odot}$ .

Step 5. Interpretation. A cold core a few tenths of a parsec across holds several Jeans masses of gas, so it is unstable and fragments into clumps of order a solar mass — exactly the observed characteristic scale of stars. The Jeans mass of $\approx3\ M_\odot$ explains why molecular clouds fragment into stellar-mass, not galactic-mass, objects.

2 · Linear growth from last scattering to today

A cold-dark-matter perturbation has density contrast $\delta=2\times10^{-3}$ at last scattering ( $z=1100$ ). Treating the intervening universe as Einstein–de Sitter, by what factor does it grow, and does it reach nonlinearity ( $\delta\sim1$ )?

Step 1. Growth law. In EdS the growing mode is $\delta\propto a$ , and $a=1/(1+z)$ , so the growth factor from $z=1100$ to $z=0$ is $a_0/a_{\rm ls}=1+z=\mathbf{1101}$ .

Step 2. Extrapolate. $\delta_0\approx1101\times2\times10^{-3}=\mathbf{2.2}$ . This exceeds unity, so the perturbation enters the nonlinear regime and collapses into a bound halo before today — how a $10^{-3}$ seed becomes a galaxy.

Step 3. Reality check. The true universe is $\Lambda$ CDM: dark energy suppresses late-time growth, cutting the effective factor to $\sim800$ – $900$ , so $\delta_0\approx1.6$ – $1.9$ . Either way a $\sim10^{-3}$ perturbation at recombination reaches $\delta\gtrsim1$ today — linear theory's own signal that structure has gone nonlinear.

Physics of the portrait

The Jeans instability is where a single, almost throwaway calculation about interstellar gas turned out to govern the largest structures in existence — a criterion written down for star formation that, generalized to an expanding universe, seeds the entire cosmic web.

Origins · 1 minJames Jeans and how this was discovered

James Jeans

1877–1946 · self-gravitating gas and gravitational instability

Sir James Hopwood Jeans was an English physicist, astronomer and mathematician who worked at the intersection of thermodynamics, radiation and gravitation. In 1902, analysing the stability of a self-gravitating gaseous nebula, he derived the criterion that now bears his name: a uniform gas is unstable to collapse on scales larger than $\lambda_J=c_s\sqrt{\pi/(G\rho_0)}$ , where self-gravity outruns pressure support. His derivation famously assumed a static uniform background — the "Jeans swindle," so named because a uniform density cannot produce a zero net gravitational field consistent with Poisson's equation — yet it captured the correct physical scale, and it remains the starting point of every textbook treatment of gravitational collapse. Jeans also contributed the low-frequency Rayleigh–Jeans law of blackbody radiation, whose "ultraviolet catastrophe" helped force the quantum revolution, and later became one of the great popularizers of science through books such as The Mysterious Universe. The modern resolution of his swindle — replacing the static background with the expanding Friedmann universe, so the mean density sources the Hubble flow — came decades later, with Lifshitz's relativistic perturbation theory and Bonnor's Newtonian expanding-universe analysis, which together turned Jeans' criterion into the foundation of cosmological structure formation.

History

1902

— Jeans publishes his gravitational-instability criterion for a self-gravitating gas

1946

— Lifshitz derives the relativistic theory of perturbations in an expanding universe

1957

— Bonnor treats the Jeans instability in an expanding Newtonian cosmology

1968

— Silk computes the photon-diffusion damping of small-scale baryon perturbations

1974

— Meszaros shows dark-matter growth stalls during radiation domination

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §12.2 (Jeans criterion) and Ch. 30 (structure formation) · Binney & Tremaine, Galactic Dynamics, 2nd ed., §5.2 (Jeans instability and the swindle) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 12 · Dodelson, Modern Cosmology, 2nd ed., Ch. 7 (growth of structure) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 4 · Peebles, The Large-Scale Structure of the Universe (1980). Key results to keep in mind: the dispersion relation $\omega^2=c_s^2k^2-4\pi G\rho_0$ and Jeans length $\lambda_J=c_s\sqrt{\pi/(G\rho_0)}$ ; the EdS indicial roots $p=2/3,-1$ from $3p^2+p-2=0$ with $H=2/(3t)$ and $4\pi G\bar\rho=\tfrac32H^2$ ; the molecular-core Jeans mass $M_J\approx2.9\,M_\odot$ from $T=10$ K, $n=10^4\ {\rm cm}^{-3}$ ; and the order-of-magnitude drop of the baryonic Jeans mass ( $\sim10^{17}\to10^{5}M_\odot$ ) from the $c_s^3$ scaling.

Prerequisites. Density Perturbations and the Density Contrast · Linear Growth of Structure · The ΛCDM Concordance Model. Leads to. Transfer Function and the Matter Power Spectrum · Acoustic Oscillations in the Photon–Baryon Fluid · Baryon Acoustic Oscillations · Silk Damping.

Ask the tutor: Ask me to walk through why a $10^{-5}$ ripple at last scattering grows into a galaxy, or how the Jeans scale becomes the acoustic peaks of the CMB.

What exactly is the "Jeans swindle," and does it invalidate the result?

The swindle is the static derivation's assumption of a uniform background density that produces no net gravitational force. That is inconsistent: Poisson's equation says a uniform nonzero density must produce a nonzero potential gradient, so a truly static uniform self-gravitating medium cannot exist. It does not invalidate the answer, though — the perturbation dynamics depend on gradients of the perturbed quantities, not the ill-defined background force, so the crossover scale $\lambda_J$ comes out correct. And the cosmological treatment removes the swindle entirely: the uniform density sources the Hubble expansion via the Friedmann equations, and perturbing about that background gives the same $\lambda_J$ honestly.

Why does the Jeans length depend on the sound speed at all?

Because pressure resists collapse, and the sound speed measures how fast pressure information travels. The Jeans length is the distance a sound wave covers in one free-fall time, $\lambda_J\sim c_s/\sqrt{G\rho}$ . If a region is smaller than this, a pressure wave crosses it and equalizes the compression before gravity wins, so it bounces and oscillates; if larger, the interior collapses before pressure can react. A higher sound speed communicates faster and stabilizes larger regions, which is why $\lambda_J$ grows with $c_s$ — and why pressureless cold dark matter ( $c_s\approx0$ ) has essentially no Jeans length and collapses on all scales.

How can cold dark matter collapse if it has no pressure to define a Jeans scale?

It collapses freely — that is the point. With $c_s\approx0$ the Jeans length vanishes, so there is no small-scale pressure cutoff and dark matter clusters on every scale. What limits CDM growth is not pressure but the other two terms in the perturbation equation: Hubble drag, which slows growth to $\delta\propto a$ in matter domination, and the horizon, which caps causal physics. During radiation domination there is an extra brake — the Meszaros effect — where the fast radiation-driven expansion overwhelms the feeble self-gravity of sub-horizon CDM and freezes its growth to a slow logarithm. So CDM's clustering is set by expansion and causality, not by a Jeans scale.

Why does the baryonic Jeans mass drop so dramatically at recombination?

Because the Jeans mass scales as $M_J\propto c_s^3$ , and the baryons' sound speed changes by about four orders of magnitude at recombination. Before recombination the baryons are locked to photons with a nearly relativistic $c_s\approx c/\sqrt3$ ; the Jeans mass is then $\sim10^{17}M_\odot$ , larger than any galaxy, so baryons only oscillate as sound waves. Once hydrogen recombines and the photons stream free, $c_s$ crashes to the thermal speed of neutral gas, a few km/s or $\sim10^{-4}c$ . Cubing that factor of $10^4$ drops $M_J$ by $\sim10^{12}$ , to $\sim10^{5}M_\odot$ — small enough that baryons can finally fall into the dark-matter wells that had been growing all along and form the first bound structures.

If dark matter has been collapsing all along, why did baryons have to wait?

Because dark matter and baryons feel gravity identically but pressure very differently. Pressureless cold dark matter started clustering well before recombination — its perturbations grew (slowly, then as $\delta\propto a$ ) and built deepening potential wells through the radiation and early matter eras. The baryons could not join in because photon pressure held their Jeans mass enormous, so they oscillated instead. When recombination freed the baryons they found ready-made dark-matter wells to fall into, which is why baryonic structure formation accelerated so sharply. This head start is essential: without it, structure could not have formed as early as observed, since the baryons only began growing at recombination.

Why does the growing mode go exactly as δ ∝ a in matter domination?

It is a precise balance in the Einstein–de Sitter equation. Setting pressure to zero leaves $\ddot\delta+2H\dot\delta-4\pi G\bar\rho\,\delta=0$ ; inserting the EdS background ( $H=2/3t$ , $4\pi G\bar\rho=2/3t^2$ ) gives an equidimensional equation with exponents $p=2/3$ and $p=-1$ , and since $a\propto t^{2/3}$ the growing root $\delta\propto t^{2/3}$ is exactly $\delta\propto a$ . Physically, self-gravity tries to make the perturbation grow while Hubble drag tries to freeze it, and in matter domination these nearly cancel, leaving gentle linear-in- $a$ growth rather than the exponential runaway of the static case. This is why $a$ itself is often used as the growth factor in matter domination.