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. This discussion ignores baryonic acoustic structure and nonlinear clustering. The Mészáros equation solution basis includes both a constant and a logarithmic branch during radiation domination; the physical logarithmic growth is selected by matching the earlier radiation forcing, so not every mathematical solution grows logarithmically.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 310 / 3 / c / Solution Created 2026-10-03 Updated 2026-10-05

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Before decay, the nonrelativistic

χ

density redshifts as

a^{- 3}

, while the photon density redshifts as

a^{- 4}

. Therefore their ratio grows as

a

. Let a minus sign denote the instant just before decay and define

r_{d} = \frac{ρ _{χ} ( t _{d}^{-} )}{ρ _{γ} ( t _{d}^{-} )} = \frac{a _{d}}{a _{ref}} \frac{ρ _{χ} ( t _{ref} )}{ρ _{γ} ( t _{ref} )} .

(1)

Using the assumed radiation domination law

a \propto t^{1/2}

gives

r_{d} = (t_{d} / t_{ref})^{1/2} ρ_{χ} (t_{ref}) / ρ_{γ} (t_{ref})

Across an instantaneous decay the scale factor does not change. Energy conservation and complete thermalization into a zero-chemical-potential photon bath give

ρ_{γ} (t_{d}^{+}) = ρ_{γ} (t_{d}^{-}) + ρ_{χ} (t_{d}^{-}) = ρ_{γ} (t_{d}^{-}) (1 + r_{d}) .

(2)

Since blackbody radiation has

ρ_{γ} \propto T_{γ}^{4}

, instantaneous photon heating by a decaying relic gives

\frac{T _{γ}^{+}}{T _{γ}^{-}} = (1 + r_{d})^{1/4} .

(3)

The already decoupled Cosmic neutrino background is not heated. Dividing its unchanged temperature by the new photon temperature therefore yields

\frac{T _{ν}}{T _{γ}} = (\frac{4}{11})^{1/3} [1 + (\frac{t _{d}}{t _{ref}})^{1/2} \frac{ρ _{χ} ( t _{ref} )}{ρ _{γ} ( t _{ref} )}]^{- 1/4} .

(4)

The relation written with

a_{d} / a_{ref}

is independent of the expansion approximation. Replacing it by the time ratio uses the radiation-dominated background assumed in the question; an appreciable matter contribution would require solving for that background instead. This photon entropy injection is not adiabatic, so the pre-decay photon entropy need not be conserved across the decay.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 3 / iii / Solution Created 2026-10-03 Updated 2026-10-05

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Write

ψ_{0} = ψ (0, k)

. For regular adiabatic initial conditions during radiation domination, use

Θ_{0} = - ψ_{0} /2 + O (k η)

Θ_{2} = O ((k η)^{2})

, and constant potentials. The dipole equation in the neutrino Boltzmann hierarchy gives

\dot{Θ}_{1} = k (Θ_{0} + ψ_{0}) - \frac{2 k}{5} Θ_{2} = \frac{1}{2} k ψ_{0} + O (k^{2} η) .

(1)

Regularity removes a constant dipole, and integration yields

Θ_{1} = \frac{1}{2} ψ_{0} k η + O ((k η)^{2}) .

(2)

Then the quadrupole equation gives

\dot{Θ}_{2} = \frac{2 k}{3} Θ_{1} - \frac{3 k}{7} Θ_{3} = \frac{1}{3} ψ_{0} k^{2} η + O (k^{3} η^{2}) .

(3)

Its regular initial value is zero, so

Θ_{2} = \frac{1}{6} ψ_{0} (k η)^{2} + O ((k η)^{3}) .

(4)

For the usual regular growing solution, the monopole has no linear correction and parity improves the dipole and quadrupole errors to

O ((k η)^{3})

and

O ((k η)^{4})

, respectively. The weaker bounds above already follow from the assumptions explicitly given. Although the quadrupole is small on superhorizon scales, the trace-free Einstein field equations divide its stress by

k^{2}

, allowing a finite leading potential difference.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 3 / iv / Solution Created 2026-10-03 Updated 2026-10-05

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For one Fourier mode, the trace-free differential operator in the Einstein field equations is

- k^{2} Q_{ij}

, with

Q_{ij} = \hat{k}_{i} \hat{k}_{j} - δ_{ij} /3

. Part (ii) gives

Π_{ij} = (4/5) \overset{ρ}{ˉ}_{ν} Θ_{2} Q_{ij}

. The two minus signs in the field equation therefore imply

k^{2} (ϕ - ψ) = \frac{32 π G}{5} a^{2} \overset{ρ}{ˉ}_{ν} Θ_{2} .

(1)

Insert

Θ_{2} = ψ_{0} k^{2} η^{2} /6

. During radiation domination,

a \propto η

H = 1/ η

, and the Friedmann equation gives

8 π G a^{2} \overset{ρ}{ˉ} = 3 H^{2}

. Thus

ϕ_{0} - ψ_{0} = \frac{16 π G}{15} a^{2} \overset{ρ}{ˉ}_{ν} η^{2} ψ_{0} = \frac{2}{5} f_{ν} ψ_{0}, ϕ_{0} = (1 + \frac{2}{5} f_{ν}) ψ_{0} .

(2)

These are the radiation-era adiabatic potentials with free-streaming neutrinos; photons are taken to have negligible scalar anisotropic stress.

To normalize the potentials, use

\overset{ˉ}{P} = \overset{ρ}{ˉ} /3

. The denominator in the comoving curvature perturbation becomes

4 π G a^{2} (\overset{ρ}{ˉ} + \overset{ˉ}{P}) = \frac{16 π G}{3} a^{2} \overset{ρ}{ˉ} = 2 H^{2} .

(3)

With

\dot{ϕ} = 0

, the conserved curvature therefore obeys

R = - ϕ_{0} - ψ_{0} /2

. Combining it with the potential ratio gives

ψ_{0} (k) = - \frac{10}{15 + 4 f _{ν}} R (k), ϕ_{0} (k) = - \frac{10 + 4 f _{ν}}{15 + 4 f _{ν}} R (k) .

(4)

As a sign and normalization check,

f_{ν} = 0

gives

ϕ_{0} = ψ_{0} = - 2 R /3

. Increasing the free-streaming radiation fraction increases

ϕ_{0} / ψ_{0}

while preserving the supplied curvature convention. The TeX's malformed definition of

H

is restored by the PDF:

H = \overset{a}{˙} / a

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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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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 14D / iii / Solution Created 2026-09-24 Updated 2026-09-29

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Under the stated approximation the perturbation equation is

\ddot{δ} + 2 H \dot{δ} = 0, \dot{δ} \propto a^{- 2} .

(1)

During radiation domination,

a \propto t^{1/2}

, so

\dot{δ} \propto t^{- 1}

and

δ = C_{1} + C_{2} lo g t = C_{1} + 2 C_{2} lo g a .

(2)

Matter perturbations therefore have only logarithmic growth of matter perturbations during radiation domination.

In a curvature-dominated universe,

a \propto t

, so

\dot{δ} \propto t^{- 2}

and

δ = C_{1} + C_{2} t^{- 1} = C_{1} + C_{2} a^{- 1} .

(3)

The nondecaying mode is constant: curvature domination gives freezing of matter perturbations during curvature domination, rather than continued growth.

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3 / a / Solution 2026-09-28

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During matter domination,

4 π G \overset{ρ}{ˉ}_{m} = 3 H^{2} /2

and

a \propto t^{2/3}

. Substitution of

δ_{m} \propto t^{p}

gives

p = 2/3, - 1

, hence

δ_{m} = C_{1} a + C_{2} a^{- 3/2} .

(1)

During radiation domination, neglecting the rapidly oscillating radiation perturbation and the subdominant matter source leaves

\ddot{δ}_{m} + 2 H \dot{δ}_{m} = 0, a \propto t^{1/2} .

(2)

Therefore

δ_{m} = C_{3} + C_{4} lo g a .

(3)

This logarithmic behavior is the Mészáros effect for subhorizon cold dark matter.

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 15A / d / Solution Created 2026-09-24 Updated 2026-09-25

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Neglecting the pressure and self-gravity bracket leaves

\ddot{δ} + 2 H \dot{δ} = 0.

(1)

During radiation domination,

a (t) \propto t^{1/2}

and

H = 1/ (2 t)

, so

\ddot{δ} + \frac{1}{t} \dot{δ} = 0.

(2)

Consequently

δ (t) = A + B lo g t = A + 2 B lo g a .

(3)

Matter perturbations therefore grow only logarithmically while radiation dominates.

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Radiation-era adiabatic potentials with free-streaming neutrinos 2026-10-05

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For regular adiabatic initial conditions on superhorizon scales during radiation domination, the unweighted neutrino Boltzmann hierarchy gives

Θ_{1} = ψ k η /2

and

Θ_{2} = ψ (k η)^{2} /6

. With

f_{ν} = \overset{ρ}{ˉ}_{ν} / (\overset{ρ}{ˉ}_{ν} + \overset{ρ}{ˉ}_{γ})

and neutrinos the only source of scalar anisotropic stress, the trace-free Einstein field equations gives

ϕ = (1 + 2 f_{ν} /5) ψ

. For the sign convention

R = - ϕ - ψ /2

ψ = - \frac{10}{15 + 4 f _{ν}} R, ϕ = - \frac{10 + 4 f _{ν}}{15 + 4 f _{ν}} R .

(1)

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Radiation-matter scale factor in conformal time Created 2026-10-06 Updated 2026-10-07

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In a flat universe containing separately conserved radiation in cosmology and pressureless matter, the Friedmann equation becomes

(a^{'})^{2} = H_{0}^{2} (Ω_{r, 0} + Ω_{m, 0} a)

in conformal time. Setting

a = 0

τ = 0

gives the stated quadratic-plus-linear scale factor. The linear term controls radiation domination and the quadratic term controls matter domination. The formula presumes