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is the physical adiabatic sound speed, gradients refer to comoving coordinates, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating

u

and

ϕ

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - 4 π G \overset{ρ}{ˉ} δ = 0.

(2)

A Fourier mode with comoving wavenumber

k

therefore obeys

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

(3)

The Jeans wavenumber occurs where the coefficient in parentheses vanishes. Since a comoving wavelength is

2 π / k

, the result is

k_{J} = \frac{a 4 π G ρ ˉ}{c _{s}}, λ_{J}^{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

Below this comoving Jeans length, pressure restores a displaced fluid element: there are acoustic waves rather than growing Jeans instability. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination photon-baryon fluid, Silk damping supplies a separate erasure mechanism.

For collisionless matter, replace the sound speed by a characteristic velocity dispersion. For an isotropic Maxwell-Boltzmann velocity distribution, defining

σ^{2} = ⟨ u_{x}^{2} ⟩ = ⟨ ∣ u ∣^{2} ⟩ /3

, the static collisionless marginal-stability calculation gives

λ_{J, collisionless}^{com} = \frac{σ}{a} \frac{π}{G ρ ˉ} = \frac{v _{rms}}{a} \frac{π}{3 G ρ ˉ} .

(5)

The coefficient depends on the distribution and the convention for velocity dispersion. Small-scale suppression is now collisionless free streaming and phase mixing, rather than collisional acoustic waves. An instantaneous collisionless Jeans length must be distinguished from the accumulated comoving distance

\int v d t / a

traveled since decoupling.

For the requested thermal histories, an Einstein-de Sitter universe describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible cosmological constant, passing from radiation domination,

a \propto t^{1/2}

, to matter domination,

a \propto t^{2/3}

. For the usual schematic dynamical Jeans scale use

\overset{ρ}{ˉ}_{tot}

in the estimate, so

λ_{dyn}^{com} \propto {c_{s} a, c_{s} a^{1/2}, t ≪ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 4}, t ≫ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 3} .

(6)

Near the particle horizon a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.

Before cosmological recombination, adiabatic initial conditions keep the entropy per baryon fixed in the tightly coupled photon-baryon fluid. With

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

, its photon-baryon sound speed is

c_{s}^{2} = \frac{c ^{2}}{3 ( 1 + R _{b} )} .

(7)

For example, this follows from

δ ρ_{γ} / \overset{ρ}{ˉ}_{γ} = (4/3) δ ρ_{b} / \overset{ρ}{ˉ}_{b}

and

δ P = c^{2} δ ρ_{γ} /3

. In the radiation-dominated limit

R_{b} ≪ 1

c_{s} ≃ c / 3

; in the strongly baryon-loaded matter-era limit

R_{b} ≫ 1

c_{s} \propto a^{- 1/2}

. Thus the baryon Jeans length across recombination has the asymptotic history

λ_{J, b}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, approximately constant, a^{- 1/2} \propto t^{- 1/3}, t ≪ t_{eq}, t_{eq} ≪ t < t_{rec}, t > t_{rec} with adiabatic gas cooling .

(8)

The transition to the plateau is smooth: retaining

1 + R_{b}

gives

λ_{J, b}^{com} \propto a^{1/2} / 1 + R_{b}

in matter domination. If baryon loading is still small, this intermediate segment instead rises as

a^{1/2}

until loading becomes important.

At photon decoupling, radiation ceases to provide pressure support to the baryons, so the sound speed drops from the coupled-fluid value to that of a nonrelativistic ideal gas,

c_{s, b}^{2} = γ k_{B} T_{b} / (μ m_{p})

, with

γ = 5/3

. The drop in comoving Jeans length at fixed density is the same ratio of sound speeds, typically several orders of magnitude. Subsequently an isentropic process gives

T_{b} \propto \overset{ρ}{ˉ}_{b}^{γ - 1} \propto a^{- 2}

, so

c_{s, b} \propto a^{- 1}

and

λ_{J, b}^{com} \propto a^{- 1/2}

. Residual energy exchange through Compton scattering can initially keep

T_{b} \propto a^{- 1}

, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:

log(comoving baryon Jeans scale)
  ^
  |                   __________________
  |                 /                   |
  |               /                     |  loss of photon pressure
  |             /                       |
  |           /                         +---\
  |         /                                \
  +-------------------|-----------------|----------> log(time)
                    t_eq              t_rec
        slope 1/2          slope ~0        slope -1/3

For the cold-dark-matter kinetic Jeans scale, take the stated toy history to mean that collisions maintain a particle temperature equal to the photon temperature until kinetic decoupling. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering

t_{NR} < t_{dec} < t_{eq} < t_{rec}

. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled,

σ \propto T \propto a^{- 1/2}

; after kinetic decoupling, particle momentum redshifts as

a^{- 1}

and

σ \propto a^{- 1}

. Combining these with the background dynamical density gives

λ_{dyn, CDM}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, a^{1/2} \propto t^{1/4}, constant, a^{- 1/2} \propto t^{- 1/3}, t < t_{NR}, t_{NR} < t < t_{dec}, t_{dec} < t < t_{eq}, t > t_{eq} .

(9)

There is no sudden change at cosmological recombination: cold dark matter has already decoupled. Its small initial velocity dispersion makes its suppression scale much smaller than the pre-recombination baryon scale.

log(comoving CDM dynamical Jeans scale)
  ^
  |                        _______________
  |                    ___/               \
  |                ___/                    \       no jump
  |              /                          \      at t_rec
  |            /                             \
  +-----------|------------|--------------|----|------> log(time)
            t_NR         t_dec          t_eq t_rec
   slope 1/2    slope 1/4       slope 0       slope -1/3

The density convention matters. If one instead defines a species' self-gravitating collisionless Jeans length using only its own nonrelativistic density

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, its thermally coupled segment is constant and its decoupled segment decreases as

a^{- 1/2}

even during radiation domination. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined cosmological Poisson equation.

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 346 / 1 / a / Solution Created 2026-09-28 Updated 2026-10-05

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For an adiabatic cosmological perturbation, the baryon and photon fractional density perturbations obey

δ_{b} = 3 δ_{γ} /4

. Since the pressure of the photon-baryon fluid is supplied by the photons,

d P = \frac{1}{3} d \overset{ρ}{ˉ}_{γ}, d \overset{ρ}{ˉ} = d \overset{ρ}{ˉ}_{γ} + d \overset{ρ}{ˉ}_{b} = (1 + \frac{3 ρ ˉ _{b}}{4 ρ ˉ _{γ}}) d \overset{ρ}{ˉ}_{γ} .

(1)

The photon-baryon sound speed is therefore

c_{s} = c \frac{d P}{d ρ ˉ} = \frac{c}{3} (1 + \frac{3 ρ ˉ _{b}}{4 ρ ˉ _{γ}})^{- 1/2} .

(2)

Use

\overset{ρ}{ˉ}_{b} \propto a^{- 3}

and

\overset{ρ}{ˉ}_{γ} \propto a^{- 4}

in the comoving Jeans length

λ_{J, com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ _{b}} .

(3)

Well before matter-radiation equality, photon inertia dominates,

c_{s} ≃ c / 3

, and

λ_{J, com} \propto a^{1/2} \propto t^{1/4} .

(4)

Once baryon loading dominates while tight coupling still holds,

c_{s} \propto a^{- 1/2}

, so

λ_{J, com}

is approximately constant. At cosmological recombination, photon pressure support disappears and the baryonic Jeans scale drops sharply. For a subsequently adiabatic monatomic gas,

T_{b} \propto a^{- 2}

and

c_{s} \propto a^{- 1}

, giving

λ_{J, com} \propto a^{- 1/2} \propto t^{- 1/3}

(5)

in an Einstein-de Sitter universe. The requested graph therefore rises as

t^{1/4}

, flattens before recombination, jumps downward there, and then decreases as

t^{- 1/3}

For collisionless matter, the same instantaneous estimate uses its one-dimensional velocity dispersion

σ_{v}

instead of

c_{s}

; more precisely, suppression is described by collisionless free streaming. Thus

λ_{J, com} \sim \frac{σ _{v}}{a} \frac{π}{G ρ ˉ _{cdm}} .

(6)

While the particles are relativistic,

σ_{v} ≃ c

and

λ_{J, com} \propto a^{1/2}

. Once nonrelativistic but thermally coupled to radiation,

T_{cdm} \propto a^{- 1}

, so

σ_{v} \propto a^{- 1/2}

and the scale is constant. After kinetic decoupling, momentum redshifts as

a^{- 1}

, so

σ_{v} \propto a^{- 1}

and

λ_{J, com} \propto a^{- 1/2}

. The second graph joins these three power laws at

t_{NR}

and

t_{dec}

; unlike the baryonic graph, its final decline begins at dark-matter kinetic decoupling rather than recombination.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 3 / a / Solution Created 2026-09-24 Updated 2026-10-05

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Before recombination, tight coupling makes photons and baryons share an adiabatic perturbation. Since

δ ρ_{b} / ρ_{b} = (3/4) δ ρ_{r} / ρ_{r}

and only radiation supplies appreciable pressure,

δ P = \frac{c ^{2}}{3} δ ρ_{r}, δ ρ = δ ρ_{r} (1 + \frac{3 ρ _{b}}{4 ρ _{r}}) .

(1)

Thus the photon-baryon sound speed is

c_{s} = \frac{c}{3} (1 + \frac{3 ρ _{b}}{4 ρ _{r}})^{- 1/2} .

(2)

Because

ρ_{r} \propto a^{- 4}

and

ρ_{b} \propto a^{- 3}

c_{s} \propto {a^{0}, a^{- 1/2}, t < t_{eq}, t_{eq} < t < t_{rec} .

(3)

After cosmological recombination, baryons decouple from radiation; for adiabatically cooling nonrelativistic gas,

T_{b} \propto a^{- 2}

and

c_{s} \propto a^{- 1}

The comoving Jeans length is

λ_{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

During radiation domination

\overset{ρ}{ˉ} \propto a^{- 4}

, so

λ_{com} \propto a

. During matter domination before recombination,

\overset{ρ}{ˉ} \propto a^{- 3}

and

c_{s} \propto a^{- 1/2}

, so

λ_{com}

is approximately constant. Recombination causes a sharp downward jump in sound speed, after which

λ_{com} \propto a^{- 1/2}

. The requested log-log sketch therefore rises with slope one, reaches a plateau after equality, drops at recombination, and then declines with slope

- 1/2

For collisionless particles, replace

c_{s}

by their one-dimensional velocity dispersion

v_{rms}

λ_{com}^{coll} = \frac{v _{rms}}{a} \frac{π}{G ρ ˉ} .

(5)

For a thermally produced cold relic, identify four epochs:

while coupled and relativistic, $v_{rms} ≃ c$ and $λ_{com} \propto a$ in radiation domination;
after decoupling but while still relativistic, momentum redshifts but speed remains near $c$ , so the same $λ_{com} \propto a$ scaling continues with collisionless free streaming;
after becoming nonrelativistic but before equality, $v_{rms} \propto a^{- 1}$ and $\overset{ρ}{ˉ} \propto a^{- 4}$ , giving $λ_{com} \propto a^{0}$ ;
after equality, $v_{rms} \propto a^{- 1}$ and $\overset{ρ}{ˉ} \propto a^{- 3}$ , giving $λ_{com} \propto a^{- 1/2}$ .

Its plot rises through the two relativistic epochs, turns onto a plateau at the nonrelativistic transition, and falls after equality. Recombination does not dynamically affect collisionless dark matter.

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