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 Incoming links: Comoving Jeans length

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 3 / a / Solution Created 2026-09-24 Updated 2026-09-25

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}}) .

Thus the photon-baryon sound speed is

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

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} .

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} .

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} .

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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