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 Incoming links: Baryon Jeans length across recombination

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 64 / 1 / ii / Solution Created 2026-10-03 Updated 2026-10-14

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Put

θ = i k \cdot v

. The synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge give

δ^{'} = - (1 + w) (θ + h^{'} /2), θ^{'} + (1 - 3 w) H θ - \frac{w}{1 + w} k^{2} δ = 0.

(1)

Differentiate the first equation and use the second and the scalar trace metric equation. Before making a nonrelativistic approximation, the result is

δ^{''} + H δ^{'} + [w k^{2} - \frac{3}{2} (1 + w) (1 + 3 w) H^{2}] δ = - 3 w (1 + w) H θ .

(2)

For

w = w_{m} = c_{s}^{2} ≪ 1

, discard relative pressure corrections to the background gravitational and damping terms, while retaining

c_{s}^{2} k^{2}

because a large wavenumber can compensate a small sound speed. Matter domination and the flat Friedmann equation give

\frac{3}{2} H^{2} = 4 π G \overset{ρ}{ˉ}_{m} a^{2}

. Thus the leading nonrelativistic density equation in synchronous gauge is

δ_{m}^{''} + H δ_{m}^{'} + (c_{s}^{2} k^{2} - 4 π G \overset{ρ}{ˉ}_{m} a^{2}) δ_{m} ≃ 0.

(3)

The approximation sign records the small finite-

w

terms just displayed; the printed Newtonian-form equation is not an exact relativistic equation at nonzero

w

The comoving Jeans wavenumber balances pressure and gravity:

c_{s}^{2} k_{J}^{2} = 4 π G \overset{ρ}{ˉ}_{m} a^{2}

. Defining physical wavelength as

2 πa / k

, the physical Jeans length is

λ_{J} = \frac{2 πa}{k _{J}} = c_{s} \frac{π}{G ρ ˉ _{m}} .

(4)

For wavelengths longer than this scale, gravity can overcome pressure and density modes grow; shorter modes are pressure-supported acoustic oscillations, with expansion modifying both growth and damping. Before cosmological recombination, the tightly coupled photon-baryon fluid has substantial photon pressure,

c_{s, γb}^{2} = 1/ [3 (1 + R_{b})]

, where

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

. The baryonic Jeans scale is correspondingly large, so baryons oscillate instead of freely collapsing on small scales. Cold dark matter has a much smaller pressure scale and can supply gravitational potential wells. After photon decoupling, baryons lose photon pressure support and their sound speed falls to its thermal-gas value; the baryon Jeans length across recombination drops, allowing baryons to fall into those wells. The earlier photon-coupled fluid is not itself a constant-small-

w

fluid; its role is a physical comparison of pressure support, not an extension of the preceding approximation through recombination.

Define a normalized smoothing window on physical radius

R

, with comoving radius

r = R / a

. If

⟨ δ_{k} δ_{k^{'}}^{*} ⟩ = (2 π)^{3} δ_{D} (k - k^{'}) P (k, τ)

, the smoothed matter density variance is

σ_{R}^{2} (τ) = ⟨ δ_{R}^{2} ⟩ = \frac{1}{2 π ^{2}} \int_{0}^{\infty} k^{2} P (k, τ) ∣ W (k r) ∣^{2} d k .

(5)

Thus

σ_{R}

is the root-mean-square amplitude. For a Gaussian window,

W (u) = e^{- u^{2} /2}

; this choice makes the scale-free integral ultraviolet convergent.

On scales where pressure is negligible, a flat matter-dominated universe has

a \propto τ^{2}

H = 2/ τ

, and

4 π G \overset{ρ}{ˉ}_{m} a^{2} = 6/ τ^{2}

. The density equation has powers satisfying

s (s - 1) + 2 s - 6 = 0

, so

δ_{m} = C_{+} τ^{2} + C_{-} τ^{- 3} .

(6)

Select the growing mode and exclude residual synchronous gauge modes. Starting with

P (k, τ_{eq}) = A k

gives

P (k, τ) = A k (τ / τ_{eq})^{4}

. The dimensionless cosmological power spectrum is then

P_{δ} (k, τ) = \frac{k ^{3} P ( k , τ )}{2 π ^{2}} = \frac{A}{2 π ^{2} τ _{eq}^{4}} (k τ)^{4} .

(7)

At cosmological horizon crossing,

k \sim a H = H

, hence

k τ \sim 2

, and

P_{δ} (k, τ_{H}) = \frac{8 A}{π ^{2} τ _{eq}^{4}}, independent of k .

(8)

For the smoothed variance, change variables to

u = k r

σ_{R}^{2} = \frac{A}{2 π ^{2} r ^{4}} (\frac{τ}{τ _{eq}})^{4} \int_{0}^{\infty} u^{3} ∣ W (u) ∣^{2} d u .

(9)

The Gaussian integral equals

1/2

. At a physical Hubble-radius window,

R = H^{- 1}

and

r = τ /2

, giving the Gaussian horizon-crossing variance for a Harrison-Zeldovich spectrum

σ_{H^{- 1}}^{2} = \frac{4 A}{π ^{2} τ _{eq}^{4}}, constant at crossing .

(10)

The window-dependent numerical constant is secondary; cancellation of the scale dependence is the Harrison-Zeldovich spectrum property. This demonstration uses the growing pressureless matter-era idealization and therefore applies to modes crossing in that era. Modes which crossed before equality require radiation-era transfer and cannot be evolved through that period with the matter equation. A pure

P \propto k

spectrum with a real-space top-hat window also has a logarithmically divergent ultraviolet variance unless a physical high-

k

cutoff or transfer function is supplied; the Gaussian choice makes the stated idealized variance well defined.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-05

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Write

x

for a comoving coordinate,

u

for the peculiar velocity,

H = \overset{a}{˙} / a

for the Hubble parameter, and

δ = δ ρ / \overset{ρ}{ˉ}

for the density contrast. For a homogeneous self-gravitating fluid, the linearized continuity equation, Euler equations for an inviscid fluid, and cosmological Poisson equation are

\dot{δ} = - \frac{1}{a} \nabla \cdot u, \dot{u} + H u = - \frac{c _{s}^{2}}{a} \nabla δ - \frac{1}{a} \nabla ϕ, \nabla^{2} ϕ = 4 π G a^{2} \overset{ρ}{ˉ} δ .

(1)

Here

c_{s}

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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