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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 1 / d / Solution Created 2026-10-03 Updated 2026-10-14

The prescribed cosmic string network scaling gives

\overset{ρ}{˙} = - 2 H ρ

. Comparing with the cosmological perfect-fluid continuity equation gives

- 2 = - 3 (1 + ω), ω = - \frac{1}{3} .

The averaged pressure is therefore

P = - ρ /3

, provided the network is treated as the stipulated isotropic homogeneous component. The Friedmann acceleration equation gives

\overset{a}{¨} = 0

. On an expanding branch let

ρ = C / a^{2}

and

B^{2} = 8 π GC /3 - K > 0

; then

a (t) = B (t - t_{B}), H = B, Ω = \frac{8 π GC /3}{B ^{2}} = constant .

This coasting cosmic-string universe has no growing flatness instability, but it does not drive an initially nonflat universe toward flatness. The comoving Hubble radius remains constant, so there is also no shrinking-radius mechanism of the usual cosmic inflation type.

The global causal issue is more precise than saying that a constant comoving Hubble radius solves every horizon problem. If this ideal coasting solution extends to its zero-size time

t_{B}

, then

\int_{t_{B}}^{t} d t^{'} / [B (t^{'} - t_{B})]

diverges logarithmically, so it has no finite particle horizon. If string domination starts at a finite earlier time after another cosmological phase, its causal history must instead be evaluated from that earlier phase. The string era alone does not establish that previously disconnected regions had time to equilibrate. A static branch with

B = 0

is a separate possibility; its Hubble-normalized

Ω

is undefined and is not covered by the expanding formulas.

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 9B / b / Solution Created 2026-09-24 Updated 2026-10-14

The physical particle horizon is

d_{H} (t) = a (t) \int_{0}^{t} \frac{c d t ^{'}}{a ( t ^{'} )} .

For an Einstein-de Sitter universe,

a (t) \propto t^{2/3}

, so

d_{H} (t_{0}) = t_{0}^{2/3} c \int_{0}^{t_{0}} t^{' - 2/3} d t^{'} = 3 c t_{0} .

Since

H_{0} = 2/ (3 t_{0})

d_{H} (t_{0}) = \frac{2 c}{H _{0}} .

(3)

The horizon problem is that widely separated regions of the observed universe have nearly identical conditions even though, in a decelerating standard Big Bang history, their past light cones do not overlap early enough to establish causal thermal equilibrium.

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

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 .

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.

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.

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 15B / e / Solution Created 2026-09-24 Updated 2026-10-14

In the decelerating hot Big Bang model, the particle horizon at recombination encloses only a finite comoving region. Widely separated parts of the cosmic microwave background therefore had no overlapping past light cones in that model, although they have almost the same temperature. This is the horizon problem. During cosmic inflation, accelerated expansion makes the comoving Hubble radius

(a H)^{- 1}

shrink. A small region that was initially in causal contact can then expand to contain the whole presently observable universe.

The flatness problem follows from

∣Ω - 1∣ = \frac{∣ k ∣}{a ^{2} H ^{2}} .

During ordinary radiation- or matter-dominated decelerating expansion this quantity grows, so the observed closeness of the cosmological density parameter

Ω

to one requires extreme tuning at early times. Inflation makes

a H

grow nearly exponentially and therefore drives

∣Ω - 1∣

rapidly toward zero. A sufficiently large number of e-folds makes spatial flatness a dynamical outcome rather than a finely tuned initial condition.

Past exam of the relativistic Astrophysics and Cosmology course of the University of Cambridge / 2014 / Paper 1 / 2 / Solution 2026-10-20

The metric uses signature

(+ - - -)

and a time-valued interval

d s

: a comoving clock has

d s = d t

. Here

t

is cosmic proper time,

R (t)

a scale factor with dimensions of length,

c

the speed of light,

χ

a dimensionless comoving radial coordinate, and

θ, ϕ

the usual polar and azimuthal angles. With curvature normalized to

k = + 1, 0, - 1

S (χ) = sin χ, χ, sinh χ

, respectively. Homogeneity and isotropy apply to each constant-

t

spatial slice. A comoving object's fixed coordinates do not imply a constant proper separation: spatial lengths scale with

R

Radial null propagation obeys

d χ = c d t / R (t)

in magnitude. Two neighbouring wave crests between the same comoving emitter and observer have the same radial integral, giving

δ t_{0} / R_{0} = δ t_{e} / R_{e}

. The wavelength redshift is consequently

1 + z = \frac{λ _{0}}{λ _{e}} = \frac{δ t _{0}}{δ t _{e}} = \frac{R _{0}}{R _{e}} .

Peculiar Doppler shifts are excluded by the comoving assumption. Define angular diameter distance by

d_{θ} = ℓ_{⊥} / δ θ

, where

ℓ_{⊥}

is proper transverse size at emission. The transverse metric gives

ℓ_{⊥} = R_{e} S (χ) δ θ

, hence

d_{θ} = R_{e} S (χ) = \frac{R _{0} S ( χ )}{1 + z} .

This distance is neither the photon travel time times

c

nor the object's present proper distance.

The comoving particle horizon at

t_{1}

, with the initial boundary at the Big Bang, is

χ_{p} (t_{1}) = \int_{0}^{t_{1}} c d t / R (t)

. Its proper radial distance is

D_{p} = R (t_{1}) χ_{p}

; for a power-law scale factor

R = A t^{p}

0 < p < 1

χ_{p} = \frac{c t _{1}^{1 - p}}{A ( 1 - p )}, D_{p} = \frac{c t _{1}}{1 - p} .

(3)

Thus a matter-only flat Einstein-de Sitter universe,

p = 2/3

, has

D_{p} = 3 c t_{1}

, and the flat radiation-only case,

p = 1/2

, has

D_{p} = 2 c t_{1}

. Strictly, Einstein-de Sitter denotes the pressureless matter model; the radiation calculation uses the flat zero-

Λ

radiation analogue requested here.

For the source seen at time

t_{1} = t_{rec}

, the radial coordinate is the integral from recombination to today, not the earlier horizon integral. In these flat models,

χ_{s} = \frac{c}{A ( 1 - p )} (t_{0}^{1 - p} - t_{1}^{1 - p}), d_{θ} = \frac{c t _{1}^{p}}{1 - p} (t_{0}^{1 - p} - t_{1}^{1 - p}) .

(4)

Taking the linear horizon scale in the question to be

D_{p}

, its small angular extent is

θ_{p} = \frac{D _{p}}{d _{θ}} = \frac{( t _{1} / t _{0} ) ^{1 - p}}{1 - ( t _{1} / t _{0} ) ^{1 - p}} .

(5)

Therefore

θ_{p, m} = \frac{( t _{rec} / t _{0} ) ^{1/3}}{1 - ( t _{rec} / t _{0} ) ^{1/3}}, θ_{p, r} = \frac{( t _{rec} / t _{0} ) ^{1/2}}{1 - ( t _{rec} / t _{0} ) ^{1/2}} .

(6)

Using the supplied times gives

θ_{p, m} = 0.02684 rad = 1.5 4^{\circ}

and

θ_{p, r} = 0.004244 rad = 0.24 3^{\circ}

. There is a genuine radius/diameter convention to state: the supplied

3 c t_{1}

is a radial horizon distance. If angular diameter means the full diameter of a spherical horizon patch of radius

D_{p}

, the two answers are instead $2 θ_{p, m} = 3.0 8^{\circ}$ and $2 θ_{p, r} = 0.48 6^{\circ}$ . Both conventions follow from the same distance calculation; a factor of two should not be hidden in the word horizon.

Matter-only evolution is the better of these two approximations because the universe is already matter dominated around recombination and evolves mostly through a subsequent matter era, rather than remaining radiation dominated to the present. Neither is exact: pre-recombination radiation changes the particle horizon, and late-time dark energy changes the distance to the last-scattering surface. No supplied ages have been replaced by later best-fit values.

The cosmic microwave background originates near last scattering and samples many horizon-sized regions. In an always-decelerating hot-Big-Bang model, widely separated last-scattering patches lack overlapping early causal histories, motivating the horizon problem. The measured acoustic angular scale uses the smaller sound horizon, not automatically this light-travel particle horizon; baryon-photon pressure changes the propagation speed. Inflation supplies an earlier accelerated expansion during which the comoving Hubble radius shrinks, allowing a patch with a common earlier causal history to expand to the size of the observable universe. Quantum perturbations can then seed later acoustic and density fluctuations. This does not require local signals to propagate faster than light, and the Hubble radius must not be equated indiscriminately with the particle horizon. CMB temperature and lensing spectra test the resulting perturbation and expansion model.^[ref]

Figure 1.
Original horizon-length angular extents in the two stated power-law cosmologies; full patch diameters are twice the plotted values
. **Details**.