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radians; retaining both cosmic components near equality avoids an inaccurate pure-matter extrapolation there. The scale factor cancels between physical size and angular diameter distance, and

H_{0}

cancels from the ratio.

If cosmological recombination occurs only a modest expansion after equality, its causal patch still subtends a small part of the sky. Widely separated regions of the observed Cosmic microwave background have nearly the same temperature even though their past particle horizons did not overlap in the ordinary decelerating history. Local thermalization after the Big Bang therefore cannot explain this large-angle uniformity. This is the Horizon problem; inflation supplies an earlier connected region that can grow to encompass the observed sky.

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Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 9C / b / Solution Created 2026-09-24 Updated 2026-10-05

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The baryon-to-photon ratio is extremely small, of order

1 0^{- 9}

. Thus even when the typical photon energy is below

I

, the high-energy tail of the blackbody radiation contains enough photons above the ionization energy to reionize almost every newly formed hydrogen atom. Cosmological recombination becomes effective only when this exponentially small tail is also small compared with the baryon-to-photon ratio. Consequently recombination occurs at a temperature much lower than $I / k$ , consistent with

k T ≃ 0.03 I

. The exact freeze-out history also depends on reaction rates and expansion; the equilibrium estimate explains the large suppression of the temperature.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 4 / i / Solution Created 2026-10-03 Updated 2026-10-06

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Put

Γ (η) = a \overset{n}{ˉ}_{e} σ_{T} \geq 0

, the Thomson scattering rate per unit conformal time. The cosmological optical depth is the integrated scattering rate from emission time to observation, so a Poisson scattering process has survival probability

E (η) = e^{- τ (η)} = Pr (no further scattering between η and η_{0}) .

(1)

Since

\overset{τ}{˙} = - Γ

, the cosmological visibility function is

g (η) = Γ (η) e^{- τ (η)} = \frac{d E}{d η} .

(2)

A photon scatters in an interval

d η

with rate factor

Γ d η

and then reaches us without another scattering with probability

E

, so

g d η

is its probability of last scattering in that interval. For an optically thick initial epoch,

\int_{η_{i}}^{η_{0}} g (η) d η = 1 - e^{- τ (η_{i})} ≃ 1.

(3)

If the initial optical depth is finite, the missing weight represents photons that have already stopped scattering before

η_{i}

Before cosmological recombination, the optical depth to the present is enormous:

E ≃ 0

, and

g

is also small because almost every scattering is followed by another. Through photon decoupling,

E

rises rapidly and

g

has its dominant positive peak. After decoupling, without reionization,

E

is close to one and reaches exactly one at

η_{0}

, while

g

is small because the remaining electron density and scattering rate are small. Residual ionization can give a weak tail; there is no second reionization peak. The peak marks the Cosmic microwave background last-scattering surface.

Figure 1.
The photon visibility function during recombination
. Normalized schematic without reionization. The right curve is the derivative of the left curve and has unit area; the horizontal scale and width are illustrative, not a numerical recombination calculation.

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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 / 2021 / iii / Paper 310 / 3 / e / Solution 2026-09-28

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The Cosmic neutrino background began free streaming at neutrino decoupling, long before cosmological recombination, so its directional flux can retain information about density fluctuations from epochs inaccessible to the Cosmic microwave background. Earlier decoupling does not by itself guarantee a larger comoving radial distance. A sufficiently massive neutrino eventually becomes nonrelativistic, and its travelled distance is

χ_{ν} = \int_{a_{dec}}^{1} \frac{v ( a )}{a ^{2} H ( a )} d a,

(1)

which may be smaller than the photon distance because

v (a) < c

. Hence the cosmic neutrino background last-scattering surface need not lie beyond the Cosmic microwave background last-scattering surface; it generally does for neutrinos that remain sufficiently relativistic for sufficiently long.

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 9B / b / Solution Created 2026-09-24 Updated 2026-09-25

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Write

n_{b} = n_{e} + n_{H} ≃ η n_{γ}

. Charge neutrality gives

n_{p} = n_{e}

, and therefore

n_{e} = X_{e} n_{b}, n_{H} = (1 - X_{e}) n_{b} .

(1)

Substitution into the Saha ionization equation yields

\frac{1 - X _{e}}{X _{e}^{2}} = n_{b} (\frac{2 π ℏ ^{2}}{m _{e} k _{B} T})^{3/2} e^{I / (k_{B} T)} .

(2)

Using the given photon density,

\frac{1 - X _{e}}{X _{e}^{2}} = η \frac{2 ζ ( 3 ) ( 2 π ) ^{3/2}}{π ^{2}} (\frac{k _{B} T}{m _{e} c ^{2}})^{3/2} e^{I / (k_{B} T)} .

(3)

The baryon-to-photon ratio is extremely small. Even when

k_{B} T ≪ I

, the high-energy tail of the enormous photon population contains enough ionizing photons per baryon to delay cosmological recombination. The exponential factor must become large enough to overcome both

η \sim 1 0^{- 9}

and the phase-space factor, which occurs near

k_{B} T ≃ 0.3 eV

rather than near

I

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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 / 2023 / iii / Paper 310 / 3 / b / Solution 2026-09-28

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After a spatial Fourier transform, the preceding equation is

δ_{r}^{''} + \frac{k ^{2}}{3} δ_{r} = - \frac{4}{3} k^{2} Φ.

(1)

The constant particular solution is

δ_{r} = - 4Φ

. Hence the Sachs-Wolfe combination

S = \frac{1}{4} δ_{r} + Φ

(2)

contains only the homogeneous harmonic oscillator modes and satisfies

S^{''} + \frac{k ^{2}}{3} S = 0.

(3)

For adiabatic initial conditions on large scales,

S = Φ/3 = R /5

and

S^{'} = 0

at the start of matter domination. Up to an irrelevant choice of the phase origin,

S (k, τ) = \frac{1}{5} R (k, 0) cos (\frac{k τ}{3}), T_{S} (k, τ) = \frac{1}{5} cos (\frac{k τ}{3}) .

(4)

Thus the radiation acoustic transfer function is flat at

1/5

k \to 0

and oscillates on smaller scales. Projection onto the sky produces a large-angle Sachs-Wolfe plateau and, after the full photon-baryon physics is included, the acoustic structure of the Cosmic microwave background power spectrum.

The baryon transfer function should also oscillate soon after cosmological recombination: before decoupling, Thomson scattering forced baryons to share the acoustic motion of the photon-baryon fluid, and recombination does not instantly erase the resulting density pattern.

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 9B / Solution Created 2026-09-23 Updated 2026-09-25

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In the nonrelativistic regime,

E (p) = m c^{2} + \frac{p ^{2}}{2 m} + O (\frac{p ^{4}}{m ^{3} c ^{2}}) .

(1)

The assumption

E - μ ≫ k_{B} T

puts the Fermi-Dirac distribution in its Maxwell-Boltzmann limit, so

n ≃ \frac{4 π g _{s}}{h ^{3}} e^{(μ - m c^{2}) / (k_{B} T)} \int_{0}^{\infty} p^{2} e^{- p^{2} / (2 m k_{B} T)} d p = \frac{4 π g _{s}}{h ^{3}} e^{(μ - m c^{2}) / (k_{B} T)} \frac{π}{4} (2 m k_{B} T)^{3/2} .

(2)

Therefore the nonrelativistic Maxwell--Boltzmann number density is

n = g_{s} (\frac{2 πm k _{B} T}{h ^{2}})^{3/2} exp (\frac{μ - m c ^{2}}{k _{B} T}) .

(3)

For

p + e^{-} \leftrightarrow H + γ

, chemical equilibrium and the vanishing photon chemical potential imply

μ_{p} + μ_{e} = μ_{H} .

(4)

Applying the number-density formula to each massive species and eliminating the chemical potentials gives

\frac{n _{e} n _{p}}{n _{H}} = \frac{g _{e} g _{p}}{g _{H}} (\frac{2 π k _{B} T}{h ^{2}})^{3/2} (\frac{m _{e} m _{p}}{m _{H}})^{3/2} exp (- \frac{( m _{p} + m _{e} - m _{H} ) c ^{2}}{k _{B} T}) .

(5)

Use charge neutrality,

n_{p} ≃ n_{e}

, the mass approximation

m_{H} ≃ m_{p}

, and the convention in the question that suppresses the order-one internal-degeneracy ratio

g_{e} g_{p} / g_{H}

. With the Hydrogen binding energy

E_{bind} = (m_{p} + m_{e} - m_{H}) c^{2},

(6)

we obtain the Saha ionization equation

\frac{n _{e}^{2}}{n _{H}} ≃ (\frac{2 π m _{e} k _{B} T}{h ^{2}})^{3/2} exp (- \frac{E _{bind}}{k _{B} T}) .

(7)

The assumptions are thermal equilibrium and chemical equilibrium before cosmological recombination, a nonrelativistic nondegenerate gas, zero photon chemical potential, charge neutrality, negligible proton-electron mass correction in the translational prefactor, and the stated degeneracy-factor approximation.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 2 / b / Solution Created 2026-09-24 Updated 2026-09-25

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Since

n_{e} = n_{p} = X_{e} n_{b}

and

n_{H} = (1 - X_{e}) n_{b}

\frac{1 - X _{e}}{X _{e}^{2}} = n_{b} \frac{n _{H}}{n _{e}^{2}} .

(1)

Using

n_{b} = η n_{γ}

and

n_{γ} = 2 ζ (3) T^{3} / π^{2}

yields

\frac{1 - X _{e}}{X _{e}^{2}} = \frac{2 ζ ( 3 )}{π ^{2}} η (\frac{2 π T}{m _{e}})^{3/2} e^{B_{H} / T} .

(2)

Cosmological recombination occurs far below

B_{H}

because the baryon-to-photon ratio is only about

1 0^{- 9}

. There are roughly a billion photons per baryon, so the high-energy tail of the blackbody distribution continues to photoionize hydrogen until the exponential Boltzmann factor overcomes this enormous entropy factor.

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