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.

b

 0  0

Solution

 0  0

The shell equation is conservative because its radial acceleration is the negative derivative of the specific potential energy

U (r) = - \frac{GM}{r} - \frac{Λ r ^{2}}{6} .

(1)

Multiplying the equation of motion by

\overset{r}{˙}

gives

\frac{d}{d t} (\frac{1}{2} \overset{r}{˙}^{2} - \frac{GM}{r} - \frac{Λ r ^{2}}{6}) = \overset{r}{˙} (\overset{r}{¨} + \frac{GM}{r ^{2}} - \frac{Λ r}{3}) = 0.

(2)

Hence

E = \frac{1}{2} \overset{r}{˙}^{2} - \frac{GM}{r} - \frac{Λ r ^{2}}{6} = constant .

(3)

This is energy per unit shell mass. The total energy of a uniform sphere requires integrating the energies of its shells.

For a sphere of physical radius

R

, the shell theorem gives

M (r) = M (r / R)^{3}

and

d M = 3 M r^{2} d r / R^{3}

. Assemble the sphere shell by shell to obtain its Newtonian gravitational potential energy:

W_{G} (R) = - \int_{0}^{R} \frac{GM ( r )}{r} d M = - \frac{3 G M ^{2}}{R ^{6}} \int_{0}^{R} r^{4} d r = - \frac{3}{5} \frac{G M ^{2}}{R} .

(4)

The cosmological constant contributes a prescribed one-body potential

- Λ r^{2} /6

, rather than a pair interaction, so there is no additional factor

1/2

W_{Λ} (R) = - \frac{Λ}{6} \int_{0}^{R} r^{2} d M = - \frac{Λ M}{2 R ^{3}} \int_{0}^{R} r^{4} d r = - \frac{1}{10} Λ M R^{2} .

(5)

Evaluating at the turnaround radius gives

W_{G, ta} = - \frac{3}{5} \frac{G M ^{2}}{r _{ta}}, W_{Λ, ta} = - \frac{1}{10} Λ M r_{ta}^{2} .

(6)

As in the question,

Λ

here includes the factor

c^{2}

that would multiply the geometrical cosmological constant in SI units.

For the final equilibrium apply the scalar virial theorem. The gravitational virial is

W_{G}

, whereas the cosmological constant force contributes

\sum_{α} r_{α} \cdot F_{Λ, α} = \frac{Λ}{3} \int r^{2} d M = - 2 W_{Λ} .

(7)

Thus, neglecting a surface-pressure term,

2 T_{f} + W_{G, f} - 2 W_{Λ, f} = 0.

(8)

At fixed

R

, a positive cosmological constant reduces the kinetic energy needed for virial equilibrium because its outward acceleration partially offsets self-gravity.

For the uniform-sphere collapse with a cosmological constant, assume a uniform final sphere of the same mass, with no energy loss or escaping matter. Put

x = \frac{r _{f}}{r _{ta}}, A = \frac{3 G M ^{2}}{5 r _{ta}}, B = \frac{Λ M r _{ta}^{2}}{10} .

(9)

The sphere has no bulk kinetic energy at turnaround, so

E_{ta} = - A - B

. The final virial theorem gives

T_{f} = \frac{A}{2 x} - B x^{2}, E_{f} = T_{f} - \frac{A}{x} - B x^{2} = - \frac{A}{2 x} - 2 B x^{2} .

(10)

Since

ρ_{ta} = 3 M / (4 π r_{ta}^{3})

\frac{B}{A} = \frac{Λ r _{ta}^{3}}{6 GM} = \frac{η}{2}, η = \frac{Λ}{4 π G ρ _{ta}} .

(11)

Conservation of energy therefore gives

1 + \frac{η}{2} = \frac{1}{2 x} + η x^{2}, 2 η x^{3} - (2 + η) x + 1 = 0.

(12)

Select the collapsing branch continuous from

x = 1/2

η = 0

, rather than an unrelated root of the cubic equation.

For

∣ η ∣ ≪ 1

, the root lies close to

1/2

. Linearize

x^{3}

there:

x^{3} = \frac{1}{8} + \frac{3}{4} (x - \frac{1}{2}) + O ((x - \frac{1}{2})^{2}) = \frac{3}{4} x - \frac{1}{4} + O ((x - \frac{1}{2})^{2}) .

(13)

Substitution into the cubic equation gives

(η /2 - 2) x + 1 - η /2 ≃ 0

, whence

\frac{r _{f}}{r _{ta}} ≃ \frac{1 - η /2}{2 - η /2} = \frac{1}{2} - \frac{η}{8} + O (η^{2}) .

(14)

The rational expression is an approximation, not an exact solution of the cubic equation. In fact both it and the exact collapsing root agree through order

η^{2}

; their first difference is of order

η^{3}

With zero cosmological constant, spherical collapse gives

r_{f} = r_{ta} /2

. For small positive

Λ

, the same sphere with the same turnaround radius reaches a smaller final radius. This does not mean that repulsion promotes collapse: as the sphere contracts,

W_{Λ}

becomes less negative, so some released gravitational binding energy pays the energetic cost of moving inward against the repulsive force. Combined with the modified virial theorem, energy conservation requires a more compact endpoint. Positive

Λ

can prevent turnaround altogether; an inward acceleration at

r_{ta}

requires

- \frac{GM}{r _{ta}^{2}} + \frac{Λ}{3} r_{ta} < 0, η < 1.

(15)

The perturbative radius formula applies well inside this limit, for spheres that actually turn around. A small negative cosmological constant instead gives