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 Incoming links: Jeans wavenumber

Jeans length 2026-10-14

2 π / k

, the Jeans length is

2 π / k_{J}

, where the Jeans wavenumber is

4 π Gρ / c_{s}

. Linear pressure support dominates gravity below this length, whereas longer wavelengths can grow. In an expanding background the growth also depends on the Hubble damping term, so the static threshold does not by itself give the time dependence.

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Jeans modes of a four-thirds polytropic cosmological fluid 2026-10-14

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For a nonrelativistic polytropic equation of state

P = K ρ^{4/3}

during matter domination,

c_{s}^{2} \propto a^{- 1}

and the comoving Jeans wavenumber is constant. The leading density equation becomes

\ddot{δ} + 4 \dot{δ} / (3 t) + [2 (q - 1) / (3 t^{2})] δ = 0

, giving the displayed powers. A growing mode exists for

q < 1

; at

q = 1

one mode is constant. At

q = 25/24

the repeated root has the second solution

t^{- 1/6} ln t

. For

q > 25/24

both real solutions oscillate in

ln t

with envelope

t^{- 1/6}

. Thus the instability threshold and the onset of oscillatory solutions are distinct in an expanding background.

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Past exam of the astrophysics course of the University of Oxford / 2018 / 2 / a / Solution 2026-10-25

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Expand the density contrast in spatial Fourier modes using fixed comoving coordinates:

δ (x, t) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} δ_{k} (t) e^{i k \cdot x} .

(1)

The spatial Laplacian multiplies a mode by

- k^{2}

, while its comoving wavenumber

k = ∣ k ∣

is time-independent. Writing

H = \overset{a}{˙} / a

, the linear cosmological density perturbation equation becomes

\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.

(2)

The proper wavenumber is

k^{(p)} = k / a

, since the proper wavelength is

2 πa / k

. The proper Jeans wavenumber balances the pressure and gravity terms:

k_{J}^{(p)} = \frac{4 π G ρ ˉ}{c _{s}}, \ddot{δ}_{k} + 2 H \dot{δ}_{k} + c_{s}^{2} [(k^{(p)})^{2} - (k_{J}^{(p)})^{2}] δ_{k} = 0.

(3)

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 64 / 2 / ii / Solution Created 2026-10-03 Updated 2026-10-14

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Let

w = c_{s}^{2}

be constant and small, and write

θ = i k \cdot v

. The supplied synchronous gauge equations imply

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

(1)

Differentiating the first and eliminating

θ

gives

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

(2)

For the dominant single component, the Einstein equation gives

h^{''} + H h^{'} = - 3 H^{2} (1 + 3 w) δ

. Therefore the exact algebraic intermediate relation is

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

(3)

The requested nonrelativistic density equation in synchronous gauge follows to leading nonrelativistic order: discard pressure corrections to the background, Hubble damping, and metric tensor terms while retaining

c_{s}^{2} k^{2} δ

, which can be comparable to gravity at the Jeans wavenumber. Since

w ≪ 1

, that scale lies well inside the Hubble radius, so keeping its enhanced gradient term is consistent. The resulting equation is

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

(4)

where the Friedmann equation gave

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

. Now

δ^{'} = a \dot{δ}

and

δ^{''} = a^{2} (\ddot{δ} + H \dot{δ})

, so

\ddot{δ} + 2 H \dot{δ} - (4 π G \overset{ρ}{ˉ}_{m} - \frac{c _{s}^{2} k ^{2}}{a ^{2}}) δ = 0.

(5)

This is the specified approximation, not an exact finite-

w

relativistic identity. The pressure-gradient term stabilizes short wavelengths, while gravity drives the long-wavelength Jeans instability.

Now apply the stipulated equation to

P = K ρ^{4/3}

with

K > 0

. In the pressure-negligible matter background,

a \propto t^{2/3}, \overset{ρ}{ˉ}_{m} = \frac{1}{6 π G t ^{2}}, c_{s}^{2} = \frac{4}{3} K \overset{ρ}{ˉ}_{m}^{1/3} \propto t^{- 2/3} \propto a^{- 1} .

(6)

Thus

c_{s}^{2} k^{2} / a^{2}

is proportional to

t^{- 2}

, exactly like the gravitational term. Define

k_{J}^{2} = \frac{4 π G ρ ˉ _{m} a ^{2}}{c _{s}^{2}}, q = \frac{k ^{2}}{k _{J}^{2}} .

(7)

The comoving Jeans wavenumber and

q

are constant in this era. The equation becomes the Euler-Cauchy equation

\ddot{δ} + \frac{4}{3 t} \dot{δ} + \frac{2 ( q - 1 )}{3 t ^{2}} δ = 0.

(8)

Setting

δ \propto t^{p}

gives

p^{2} + p /3 + 2 (q - 1) /3 = 0

. For distinct real roots the full pair of modes is

δ (t) = A (\frac{t}{t _{*}})^{p_{+}} + B (\frac{t}{t _{*}})^{p_{-}}, p_{\pm} = \frac{- 1 \pm 25 - 24 q}{6} .

(9)

These are the Jeans modes of a four-thirds polytropic cosmological fluid. The physical Jeans length is

λ_{J} = \frac{2 πa}{k _{J}} = c_{s} \frac{π}{G ρ ˉ _{m}}, q = (\frac{λ _{J}}{λ})^{2}, λ = \frac{2 πa}{k} .

(10)

Here

λ_{J} \propto a

, so its comoving value is constant. For

λ > λ_{J}

q < 1

and

p_{+} > 0

, giving genuine growth; the other solution decays. In the long-wavelength pressure-free limit,

p_{+} = 2/3

and

p_{-} = - 1

, the familiar matter-era modes. At

λ = λ_{J}

, the two powers are

0

and

- 1/3

, so the leading mode is constant rather than growing.

For

1 < q < 25/24

both powers are negative: pressure prevents growth, although the modes still decay without oscillation. At

q = 25/24

they coincide and the complete solution is

δ = t^{- 1/6} [A + B ln (t / t_{*})] .

(11)

For

q > 25/24

they become a complex-conjugate pair; a real basis is

δ = t^{- 1/6} [A cos (ω ln (t / t_{*})) + B sin (ω ln (t / t_{*}))], ω = \frac{24 q - 25}{6} .

(12)

These short-wavelength acoustic oscillations have a decaying envelope. The onset of instability is

q = 1

, while the threshold for oscillatory time dependence is slightly larger,

q = 25/24

; expansion damping explains why these two thresholds are not identical.

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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 / 2010 / ii / Paper 4 / 10D / Solution Created 2026-09-24 Updated 2026-10-14

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The Jeans wavenumber in physical coordinates is

k_{J} = 4 π Gρ / c_{s}

. With wavelength defined as

2 π / k

, the Jeans length is

λ_{J} = \frac{2 π}{k _{J}} = c_{s} \frac{π}{Gρ} .

(1)

Pressure opposes gravitational growth on physical wavelengths shorter than this; on longer wavelengths the gravitational term wins. Expansion also provides the damping term

2 H \dot{δ}

, so the actual growth is not generally the exponential growth of a static medium.

For the pressure-free, spatially flat matter solution,

H = 2/ (3 t)

and the Friedmann equation gives

ρ = 1/ (6 π G t^{2})

. The perturbation equation becomes

\ddot{δ}_{k} + \frac{4}{3 t} \dot{δ}_{k} - \frac{2}{3 t ^{2}} δ_{k} = 0.

(2)

Substituting

δ_{k} = t^{ν}

gives

3 ν^{2} + ν - 2 = (3 ν - 2) (ν + 1) = 0

. Thus

δ_{k} (t) = A_{k} t^{2/3} + B_{k} t^{- 1} .

(3)

The growing mode scales as

a (t)

and permits the formation of structure; the decaying mode becomes negligible at late times. This is a linear result, valid while the density contrast remains small.

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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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