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The baryon transfer function retains acoustic oscillations generated while baryons belonged to the photon-baryon fluid. After recombination, baryons fall into cold-dark-matter potential wells, reducing the relative perturbation while leaving a small oscillatory imprint.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 5 / b / Solution Created 2026-10-03 Updated 2026-10-14

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For small baryon fraction the CDM growing mode is

δ_{C} \propto a

, and its gravity forces

δ_{B}^{''} + H δ_{B}^{'} + c_{s}^{2} k^{2} δ_{B} ≃ 4 π G a^{2} \overset{ρ}{ˉ}_{C} δ_{C} .

(1)

For an adiabatic cosmological perturbation to follow the unsuppressed growing mode, pressure must be below the gravitational forcing frequency:

c_{s}^{2} k^{2} ≲ 4 π G a^{2} \overset{ρ}{ˉ}_{C}

. With physical wavelength

λ = 2 πa / k

, this becomes

λ ≳ λ_{J} = c_{s} \frac{π}{G ρ ˉ _{C}} .

(2)

This is Jeans support versus forced baryon response. The wording “only grow” must refer to unsuppressed gravitational structure growth; it is not a theorem that all shorter-wavelength baryon perturbations are absent or constant. In the pressure-dominated regime a forced response

δ_{B} ≃ (k_{J} / k)^{2} δ_{C}

remains. In particular, a falling sound speed can make this small response grow even before it catches up to CDM.

Before photon decoupling, the photon-baryon fluid has sound speed of order

c / 3

. Its Jeans scale is roughly a horizon scale, so smaller perturbations oscillate acoustically instead of freely following the growing CDM structures. After decoupling the baryon sound speed drops sharply. Above the new Jeans scale, pressure is negligible and subtraction of the CDM equation gives

(δ_{B} - δ_{C})^{''} + H (δ_{B} - δ_{C})^{'} = 0

. For

a \propto τ^{2}

, the difference is a constant plus a decaying

1/ τ

term; its ratio to the growing CDM contrast tends to zero. Baryons therefore catch up. Under the stipulated post-decoupling

c_{s} \propto T \propto a^{- 1}

, the physical Jeans length scales as

a^{1/2}

, the comoving scale as

a^{- 1/2}

and the corresponding Jeans mass as

a^{- 3/2}

For the requested instantaneous mass estimates, define the baryonic Jeans mass as the baryon content of a sphere of radius

λ_{J} /2

M_{J B} = \frac{π}{6} \overset{ρ}{ˉ}_{B} λ_{J}^{3}, \overset{ρ}{ˉ}_{B} ≃ 0.1 \overset{ρ}{ˉ}_{C} .

(3)

Using the supplied matter-era age gives

t_{dec} = t_{0} (1 + z_{dec})^{- 3/2} ≃ 9.5 \times 1 0^{13}

s and

\overset{ρ}{ˉ}_{C} ≃ 1/ (6 π G t_{dec}^{2}) ≃ 6 \times 1 0^{- 23} g c m^{- 3}

. Then

M_{J B}^{before} \sim 6 \times 1 0^{18} M_{⊙} (c_{s} = c / 3), M_{J B}^{after} \sim 3 \times 1 0^{4} M_{⊙} (c_{s} = 1 0^{- 5} c) .

(4)

The drop is by

(3 \times 1 0^{- 5})^{3}

. A definition using the total gravitating mass instead of baryon content gives values about ten times larger. The pre-decoupling number is an acoustic Jeans order estimate, not an accurate application of the nonrelativistic equations from part a when

c_{s} \sim c

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 75 / 1 / b / Solution Created 2026-10-03 Updated 2026-10-14

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For adiabatic initial conditions, the coupled photon-baryon fluid begins with a coherent displacement and approximately zero initial acoustic velocity. Neglecting slowly varying gravitational driving, its effective temperature satisfies an oscillator equation with phase

k r_{s}

, where

c_{s}^{2} = \frac{1}{3 ( 1 + R )}, R = \frac{3 ρ _{b}}{4 ρ _{γ}}, r_{s} (τ_{*}) = \int_{0}^{τ_{*}} c_{s} d τ .

(1)

Thus the Sachs-Wolfe combination has a dominant cosine phase, while the velocity has a sine phase. At last scattering, the first three temperature Cosmic microwave background acoustic peaks correspond approximately to

k r_{s} = π, 2 π, 3 π

: first compression, first rarefaction, and second compression of the initially displaced plasma in a potential well. Baryon inertia shifts the oscillator's equilibrium and enhances compressions relative to rarefactions, producing the familiar odd/even peak contrast. Potential decay and radiation driving alter the amplitudes, especially of the earliest peaks.

Adjacent displacement extrema differ by

π

, so

Δ k = \frac{π}{r _{s}}, Δ ℓ ≃ \frac{π D _{M} ( τ _{*} )}{r _{s} ( τ _{*} )} = \frac{π}{θ _{s}}, θ_{s} = \frac{r _{s}}{D _{M}} .

(2)

Here

D_{M}

is the transverse comoving distance to last scattering; in a flat model it is

τ_{0} - τ_{*}

. If

c_{s}

is treated as constant, this reduces to

Δ ℓ ≃ π (τ_{0} - τ_{*}) / (c_{s} τ_{*})

. No numerical spacing is determined without a cosmological distance and sound horizon. Gravitational phase shifts, baryon loading and projection displace the exact peak locations from integer multiples of this spacing. These are primarily coherent density-plus-potential acoustic peaks; the label “Doppler peaks” does not mean the velocity term alone creates their sequence.

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 64 / 3 / ii / Solution Created 2026-10-03 Updated 2026-10-14

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Before decoupling, rapid Thomson scattering makes the photon distribution nearly isotropic in the local photon-fluid rest frame. A local blackbody temperature perturbation changes its energy density by

δ_{γ} = 4 δ T / T

. A first-order boost by the fluid velocity adds a temperature photon dipole

n \cdot v

; it does not generate a photon quadrupole until second order in that boost. Thus the photon brightness perturbation at instantaneous decoupling is

Δ_{*} = δ_{γ *} + 4 n \cdot v_{*} .

(1)

Higher angular moments are suppressed by the photon mean-free-path to perturbation-length ratio in the tight-coupling approximation. The idealized equilibrium assumption neglects those finite-mean-free-path corrections, polarization effects and finite thickness of the last-scattering surface. It is not an assertion that collisionless photons remain isotropic after decoupling. The distinction between a fluid treatment and the subsequent hierarchy is discussed in the primary synchronous and Newtonian perturbation treatment.

Let

μ = \hat{k} \cdot n

and

W (τ) = e^{ik μ (τ - τ_{0})}

. Multiplication of the synchronous photon brightness equation by its integrating factor gives

(e^{ik μτ} Δ)^{'} = - 2 e^{ik μτ} h_{ij}^{'} n^{i} n^{j} .

(2)

Integration from

τ_{*} = τ_{dec}

τ_{0}

and division by four yield

Θ (k, μ, τ_{0}) = (\frac{δ _{γ *}}{4} + n \cdot v_{*}) e^{- ik μ (τ_{0} - τ_{*})} - \frac{1}{2} \int_{τ_{*}}^{τ_{0}} W (τ) h_{ij}^{'} n^{i} n^{j} d τ, Θ = \frac{Δ T}{T} .

(3)

For the real-space observation point

x_{0}

, the corresponding background ray is

x (τ) = x_{0} - n (τ_{0} - τ)

. Fourier inversion gives the synchronous Sachs-Wolfe line-of-sight formula

Θ (x_{0}, n, τ_{0}) = \frac{1}{4} δ_{γ} (x_{*}, τ_{*}) + n \cdot v (x_{*}, τ_{*}) - \frac{1}{2} \int_{τ_{*}}^{τ_{0}} h_{ij}^{'} (x (τ), τ) n^{i} n^{j} d τ, x_{*} = x (τ_{*}) .

(4)

Every source is evaluated at its appropriate point on the ray, rather than at the observation position. Here

n

is photon propagation direction, opposite to the outward observer-to-source sky direction; changing that convention changes the written sign of the Doppler projection consistently.

The first term is the intrinsic photon temperature at last scattering. On wavelengths larger than the sound horizon, it retains primordial adiabatic information; inside that horizon the photon-baryon fluid has acoustic energy density oscillations, producing the acoustic temperature peaks. The second term is the Doppler effect from the fluid's bulk motion at emission. A regular long-wavelength velocity is gradient-suppressed; the Doppler contribution is important near and below the acoustic horizon, with its acoustic phase displaced from that of the energy density oscillation.

The integral is the gravitational energy shift produced by the changing spatial metric tensor along the photon path. In synchronous gauge it includes endpoint gravitational redshifts as well as evolving-potential effects. It must not be identified solely with the Integrated Sachs-Wolfe effect: in Newtonian gauge the same total separates into the ordinary last-scattering Sachs-Wolfe effect, local observer terms, and an integral of time-varying potentials. The ordinary contribution is important on scales outside the sound horizon at decoupling. A late integrated contribution is strongest on large angular scales when dark energy or curvature changes the potentials; evolution around matter-radiation equality also contributes near the first acoustic scales.

The relevant comoving sound scale is

r_{s} (τ_{*}) = \int^{τ_{*}} c_{s} (τ) d τ

, while the projection distance is

D_{*} ≃ τ_{0} - τ_{*}

. Modes project roughly to photon temperature multipole

ℓ \sim k D_{*}

, and the acoustic scale is

ℓ \sim D_{*} / r_{s}

, with successive peak spacing of order

π D_{*} / r_{s}

. At much shorter wavelengths Silk damping and the finite last-scattering thickness suppress energy density and velocity anisotropies; the instantaneous-decoupling approximation alone does not model that cutoff. These scale distinctions explain why the three terms cannot be assigned one common range of important wavelengths.

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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 / 2008 / iii / Paper 64 / 2 / ii / Solution Created 2026-10-03 Updated 2026-10-14

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In the tight-coupling approximation, repeated Thomson scattering drives the photon distribution toward isotropy in the local baryon rest frame. At linear order a temperature perturbation gives a monopole and a small boost gives only a dipole. A photon propagating along

n

has rest-frame energy

p (1 - n \cdot v)

, so

f_{1} = - q f_{0}^{'} (q) (Θ_{rest} + n \cdot v) .

(1)

Since the rest-frame density contrast is

δ_{γ} = 4 Θ_{rest}

, the initial photon brightness perturbation is

Δ_{*} = δ_{γ *} + 4 n \cdot v_{*} .

(2)

The quadrupole and higher photon temperature multipoles are small when the conformal collision rate

κ = a n_{e} σ_{T}

greatly exceeds both

k

and

H

. For example the quadrupole is suppressed by a factor of order

k / κ

relative to the dipole. Instantaneous decoupling idealizes this strongly coupled state as the initial data for collisionless propagation; finite mean free path and polarization corrections are omitted from that initial truncation.

Multiply the synchronous photon brightness equation by

e^{ik μτ}

and integrate from

τ_{*}

τ_{0}

. The line-of-sight solution for free-streaming photons is

Δ (k, n, τ_{0}) = e^{- ik μ (τ_{0} - τ_{*})} Δ_{*} (k, n) - 2 \int_{τ_{*}}^{τ_{0}} e^{- ik μ (τ_{0} - τ)} h_{ij}^{'} (k, τ) n^{i} n^{j} d τ .

(3)

The phase is essential: it specifies the emission point and the points traversed by the ray. In real space define

x (τ) = x_{0} - n (τ_{0} - τ)

and

x_{*} = x (τ_{*})

. Dividing by four gives the synchronous Sachs-Wolfe line-of-sight formula,

\frac{Δ T}{T} (x_{0}, n, τ_{0}) = \frac{1}{4} δ_{γ} (x_{*}, τ_{*}) + n \cdot v (x_{*}, τ_{*}) - \frac{1}{2} \int_{τ_{*}}^{τ_{0}} h_{ij}^{'} (x (τ), τ) n^{i} n^{j} d τ .

(4)

The spatial arguments suppressed in a shorthand version must be understood in this retarded sense; evaluating the emission terms at

x_{0}

instead would discard free streaming. Here

n

is propagation direction. The observer-to-source direction on the sky is

- n

, explaining the opposite Doppler sign often used with that convention. The formula describes the synchronous-frame observer; an additional observer peculiar velocity contributes a dipole.

The first term is the intrinsic temperature perturbation at emission, since

δ_{γ} = 4Θ

. The second is the Doppler CMB anisotropy from the plasma bulk velocity. The third is gravitational frequency shifting by the time-dependent spatial metric along the ray, and can include scalar and tensor metric contributions. In scalar Newtonian gauge, rearranging that metric contribution supplies the emission gravitational potential as well as the Integrated Sachs-Wolfe effect, up to the observer terms. It should not be identified solely with the integrated Sachs-Wolfe term: a constant Newtonian potential need not give a vanishing synchronous metric integral.

On large angles, modes are outside or comparable to the sound horizon at emission. For the growing adiabatic mode during matter domination, the intrinsic and emission-potential contributions give the ordinary Sachs-Wolfe effect,

Θ_{0} + Ψ = Ψ/3

in Newtonian variables. Velocity is a gradient effect, of order

k τ_{*} Ψ

on superhorizon scales, so its Doppler contribution is suppressed there. A scale-invariant primordial curvature spectrum produces approximately the familiar large-angle plateau in

ℓ (ℓ + 1) C_{ℓ}

. Evolving potentials during radiation-to-matter transition or late acceleration can add integrated gravitational anisotropy; constant matter-era scalar potentials do not generate a genuine scalar integrated Sachs-Wolfe source.

On smaller angles, the photon-baryon fluid has undergone acoustic oscillations. Its density and velocity are approximately in cosine and sine phases, producing acoustic structure in the intrinsic and Doppler sources and their projection onto angular multipoles. Silk damping suppresses sufficiently short wavelengths before decoupling, while the finite width of last scattering also smooths the real small-angle signal. These corrections refine the instantaneous-decoupling model rather than follow from a collisionless equation alone. The separate intrinsic, velocity and metric pieces depend on gauge convention; the complete observed anisotropy, including the appropriate observer terms and removal of the unobservable mean, is the physical quantity.

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 53 / 4 Created 2026-10-03 Updated 2026-10-07

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The broad turnover near

k \sim 1 0^{- 2} Mpc^{- 1}

marks the matter-radiation equality scale. Modes entering the Hubble radius after matter-radiation equality grow efficiently during matter domination. Modes entering earlier have their cold-matter growth delayed by radiation domination: their subhorizon growth is logarithmic rather than proportional to

a

. This is the Mészáros effect. The smaller undulations on the declining side are baryon acoustic oscillations inherited from the photon-baryon fluid, with an oscillatory wavenumber spacing set approximately by

π

divided by the sound horizon. They are damped at sufficiently short scales by photon diffusion and the transfer of the baryon perturbation into the total matter field. The plotted spectrum is a linear prediction; nonlinear evolution changes the actual sufficiently small-scale spectrum at redshift zero.

For an adiabatic growing mode, the comoving curvature perturbation transfers to the matter overdensity as

Δ_{m} (k, a) = \frac{2}{5} \frac{k ^{2}}{Ω _{m 0} H _{0}^{2}} T (k) D (a) R (k),

(1)

up to an irrelevant overall sign convention for

R

. Normalize the cosmological transfer function by

T \to 1

at small

k

and the linear growth factor by

D \to a

in matter domination. The relation distinguishes a dimensional matter power spectrum from a dimensionless cosmological power spectrum: the dimensional curvature spectrum is proportional to

k^{- 3} P_{R} (k)

. Therefore

P_{m} (k, a) \propto D (a)^{2} k T (k)^{2} P_{R} (k) .

(2)

For locally constant scalar spectral index

n_{s}

, the asymptotic cold-dark-matter power spectrum is

P_{m} (k) \propto {k^{n_{s}}, k^{n_{s} - 4} ln^{2} (k / k_{eq}), k ≪ k_{eq}, k ≫ k_{eq},

(3)

because the cold-dark-matter transfer function tends to one at small

k

and scales as

ln (k / k_{eq}) / (k / k_{eq})^{2}

at large

k

. The asymptotes refer to the smooth envelope, not the individual baryonic wiggles. The large-

k

local logarithmic slope is

n_{s} - 4 + 2/ ln (k / k_{eq})

. For the quadratic potential at

N ≃ 60

, derived below,

n_{s} ≃ 0.967

, so the limiting powers are approximately

k^{0.967}

and

k^{- 3.033} ln^{2} k

. Multiplying by

k^{3}

to form dimensionless matter power would change both exponents by three; that is not the plotted quantity.

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Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 53 / 2 / iii / Solution Created 2026-10-03 Updated 2026-10-06

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All dots in this solution denote differentiation with respect to conformal time

η

. This is required by comoving

k

, the streaming term

k Θ_{ℓ}

, and

\overset{τ}{˙} = - a \overset{n}{ˉ}_{e} σ_{T}

. The PDF's reference to proper-time derivatives is inconsistent with these coefficients. Proper-time equations follow by dividing the conformal-time equations by

a

, replacing streaming

k

k / a

and the collision rate by

- \overset{n}{ˉ}_{e} σ_{T}

The monopole equation of the photon Boltzmann hierarchy is

\dot{Θ}_{0} + k Θ_{1} /3 = \dot{ϕ}

; its collision term vanishes. Consequently the photon continuity equation is

\dot{δ}_{γ} - \frac{4}{3} k v_{γ} - 4 \dot{ϕ} = 0.

(1)

For the dipole the hierarchy gives

\dot{Θ}_{1} + k (\frac{2}{5} Θ_{2} - Θ_{0}) = \overset{τ}{˙} (Θ_{1} + v_{b}) + k ψ .

(2)

Using the relations from the preceding part, the photon Euler equation becomes

\overset{v}{˙}_{γ} + \frac{k}{4} δ_{γ} + k ψ + \frac{2}{3} k Π_{γ} = \overset{τ}{˙} (v_{γ} - v_{b}) .

(3)

Because

\overset{τ}{˙} < 0

, the collision term drives the photon velocity towards the baryon velocity, as Thomson scattering should.

For pressureless baryons, stress-energy conservation without energy exchange gives

\dot{δ}_{b} - k v_{b} - 3 \dot{ϕ} = 0

. At this order Thomson scattering changes momentum but not the local photon energy: energy transfer by the relative bulk motion is second order, and recoil/thermal energy exchange is neglected. Total momentum conservation fixes the baryon force to be the negative of the photon force. The momentum densities are proportional to their enthalpy densities,

4 \overset{ρ}{ˉ}_{γ} /3

and

\overset{ρ}{ˉ}_{b}

. Thus

\frac{4}{3} \overset{ρ}{ˉ}_{γ} \overset{τ}{˙} (v_{γ} - v_{b}) + \overset{ρ}{ˉ}_{b} F_{b} = 0, F_{b} = - \frac{τ ˙}{R} (v_{γ} - v_{b}), R = \frac{3 ρ ˉ _{b}}{4 ρ ˉ _{γ}} .

(4)

Including the pressureless expansion and gravitational terms yields the baryon Euler equation with Thomson drag

\overset{v}{˙}_{b} + \frac{a ˙}{a} v_{b} + k ψ = - \frac{τ ˙}{R} (v_{γ} - v_{b}) .

(5)

The expansion coefficient for photons cancels against the changing background pressure, while pressureless baryons retain the Hubble drag term. The opposite drag signs and the enthalpy ratio ensure that the combined photon-baryon fluid conserves momentum.

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 312 / 2 / i / Solution Created 2026-10-03 Updated 2026-10-06

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Throughout this question dots denote conformal time derivatives, and

H = \overset{a}{˙} / a

is the conformal Hubble parameter. The monopole of the photon Boltzmann hierarchy gives

\dot{Θ}_{0} + k Θ_{1} /3 = \dot{ϕ}

. With the specified velocity convention, the photon continuity equation is therefore

\dot{δ}_{γ} - \frac{4}{3} k v_{γ} = 4 \dot{ϕ} .

(1)

The dipole equation is

\dot{Θ}_{1} + k (2 Θ_{2} /5 - Θ_{0}) = \overset{τ}{˙} (Θ_{1} + v_{b}) + k ψ

. Thus the photon Euler equation is

\overset{v}{˙}_{γ} + \frac{k}{4} δ_{γ} - \frac{2 k}{5} Θ_{2} + k ψ = \overset{τ}{˙} (v_{γ} - v_{b}) .

(2)

In particular,

\overset{τ}{˙} < 0

makes the scattering term damp the photon-baryon velocity slip; it does not amplify it.

The linear momentum density of the photon-baryon fluid is weighted by enthalpy, not merely energy density:

q_{tot} = \frac{4}{3} \overset{ρ}{ˉ}_{γ} v_{γ} + \overset{ρ}{ˉ}_{b} v_{b} .

(3)

Elastic Thomson scattering exchanges momentum internally. Its photon force density is

(4 \overset{ρ}{ˉ}_{γ} /3) \overset{τ}{˙} (v_{γ} - v_{b})

, so the baryon force density is its negative. Dividing by

\overset{ρ}{ˉ}_{b}

gives the baryon Euler equation with Thomson drag

\overset{v}{˙}_{b} + H v_{b} + k ψ = \frac{τ ˙}{R} (v_{b} - v_{γ}), R = \frac{3 ρ ˉ _{b}}{4 ρ ˉ _{γ}} .

(4)

This

R

is the baryon loading parameter, the baryon-to-photon inertia ratio. The two collision terms cancel exactly when the equations are weighted by their enthalpies. Background conservation gives

\overset{ρ}{ˉ}_{b} \propto a^{- 3}

and

\overset{ρ}{ˉ}_{γ} \propto a^{- 4}

, so

\dot{R} = H R

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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 / 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 / iii / Paper 310 / 3 / e / Solution Created 2026-09-28 Updated 2026-10-03

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Yes. The baryons carry the oscillatory baryon transfer function inherited from the pre-recombination photon-baryon fluid. Their later infall toward cold-dark-matter overdensities suppresses the relative mode and drives

δ_{b} / δ_{c} \to 1

, but gravitational evolution does not remove every scale-dependent acoustic phase. The total cosmological density power spectrum therefore contains the small wiggles known as a baryon acoustic oscillation in the matter power spectrum.

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