Linear Growth of Density Perturbations

Derivation

Linear Growth of Density Perturbations

D-348 Home PU-308 Threads matter · force · chance Depends on The Jeans Instability Criterion, The Friedmann Equations from Einstein's Equations

Statement

Linearising the Newtonian cosmological fluid equations (continuity, Euler, Poisson) about a homogeneous expanding background, the density contrast \( \delta \equiv \delta\rho/\bar\rho \) of a single Fourier mode of comoving wavenumber \( \mathbf{k} \) obeys \[ \ddot\delta + 2H\dot\delta + \left(\frac{c_s^2 k^2}{a^2} - 4\pi G\bar\rho\right)\delta = 0, \] whose pressureless limit admits, in the Einstein–de Sitter matter era, a growing mode \( \delta_+ \propto a \propto t^{2/3} \) and a decaying mode \( \delta_- \propto t^{-1} \), while during radiation domination sub-horizon matter perturbations grow only logarithmically, \( \delta_m \approx C_1 + C_2\ln a \) (the Mészáros effect).

Why it matters

All of large-scale structure — galaxies, clusters, the cosmic web — descends from tiny primordial perturbations amplified by gravity. The linear growth equation is the engine that turns the \( \delta \sim 10^{-5} \) fluctuations imprinted on the cosmic microwave background into the order-unity contrasts that later collapse into bound objects. Because the equation is linear, each Fourier mode evolves independently and the primordial power spectrum is simply rescaled by a single scale-independent growth factor \( D(a) \) on sub-horizon, pressureless scales.

The rate of growth is a sharp probe of cosmology: matter clusters as \( \delta\propto a \) only while matter dominates, is frozen during radiation domination, and is quenched again once dark energy takes over. Measuring \( D(a) \) and its logarithmic derivative \( f=d\ln D/d\ln a \) through redshift-space distortions and weak lensing therefore tests gravity and the energy budget of the Universe.

Assumptions

Newtonian, sub-horizon dynamics.If dropped (modes near or above the Hubble radius), spatial curvature and relativistic gauge effects enter and one must use the full general-relativistic perturbation theory; the Poisson source and even the meaning of "\( \delta \)" become gauge-dependent.

Single self-gravitating fluid with barotropic pressure \( p=p(\rho) \).If dropped (multiple components: baryons + cold dark matter + radiation), \( \delta \) becomes a vector obeying coupled equations, each species sourcing the common potential with its own pressure and velocity.

Linearity, \( |\delta|\ll 1 \) and small peculiar velocities.If dropped, quadratic terms \( (\mathbf v\cdot\nabla)\mathbf v \) and \( \nabla\!\cdot(\delta\mathbf v) \) survive, mode coupling appears, and the evolution must be followed by perturbation theory or N-body simulation.

The unperturbed background solves the Friedmann/Hubble-flow equations exactly, with \( \mathbf u_0=H\mathbf r \) and \( \bar\rho\propto a^{-3} \).If dropped, the cancellation of the zeroth-order terms that isolates the perturbation equation fails; the background flow must itself be treated as a solution before perturbing.

Irrotational flow (no vorticity) and negligible anisotropic stress.If dropped, the velocity field carries a decaying rotational (vector) mode that does not couple to \( \delta \), and shear viscosity modifies the damping term.

Derivation

\[ \frac{\partial\rho}{\partial t}+\nabla_r\!\cdot(\rho\,\mathbf u)=0,\qquad \frac{\partial\mathbf u}{\partial t}+(\mathbf u\!\cdot\!\nabla_r)\mathbf u=-\nabla_r\Phi-\frac1\rho\nabla_r p,\qquad \nabla_r^2\Phi=4\pi G\rho. \]

Newtonian self-gravitating fluid in physical coordinates \( \mathbf r \): mass conservation, momentum conservation, and the field equation for the gravitational potential. A

\[ \mathbf r=a(t)\,\mathbf x,\qquad \mathbf u=\dot a\,\mathbf x+\mathbf v,\qquad \mathbf v=a\dot{\mathbf x},\qquad \nabla_r=\frac1a\nabla_x. \]

Move to comoving coordinates \( \mathbf x \) and split the velocity into the Hubble flow \( \dot a\,\mathbf x = H\mathbf r \) plus a peculiar velocity \( \mathbf v \); the background is \( \rho=\bar\rho(t) \), \( \mathbf v=0 \). B

\[ \rho=\bar\rho(t)\big[1+\delta(\mathbf x,t)\big],\qquad \Phi=\Phi_0+\phi,\qquad \nabla_r^2\Phi_0=4\pi G\bar\rho, \]

Perturb about the homogeneous background. The background potential \( \Phi_0=\tfrac{2}{3}\pi G\bar\rho\, r^2 \) accounts for the Hubble flow (it reproduces \( \ddot a/a=-\tfrac{4}{3}\pi G\bar\rho \)); \( \phi \) is the peculiar potential. B

\[ \frac{\partial\delta}{\partial t}+\frac1a\nabla_x\!\cdot\mathbf v=0. \]

Insert into continuity and keep only first order in \( \delta,\mathbf v \). The zeroth-order part reproduces \( \dot{\bar\rho}+3H\bar\rho=0 \), i.e. \( \bar\rho\propto a^{-3} \); the linear part is this equation. A

\[ \frac{\partial\mathbf v}{\partial t}+H\mathbf v=-\frac1a\nabla_x\phi-\frac{c_s^2}{a}\nabla_x\delta,\qquad c_s^2\equiv\left(\frac{\partial p}{\partial\rho}\right)_{\!S}. \]

Linearised Euler equation. The \( H\mathbf v \) term is the "Hubble drag" arising from the expansion of the background; the pressure gradient is written via the adiabatic sound speed with \( \delta p=c_s^2\,\bar\rho\,\delta \). B

\[ \nabla_x^2\phi=4\pi G\,a^2\,\bar\rho\,\delta. \]

Linearised Poisson equation in comoving coordinates (the factor \( a^2 \) comes from \( \nabla_r^2=a^{-2}\nabla_x^2 \)); the background piece has already been absorbed into \( \Phi_0 \). A

\[ \frac{\partial^2\delta}{\partial t^2}+\frac1a\nabla_x\!\cdot\frac{\partial\mathbf v}{\partial t}-\frac{H}{a}\nabla_x\!\cdot\mathbf v=0. \]

Take \( \partial/\partial t \) of the continuity equation (step 4), using \( \partial(1/a)/\partial t=-H/a \). This isolates \( \nabla\!\cdot\dot{\mathbf v} \), which step 5 supplies. B

\[ \frac1a\nabla_x\!\cdot\frac{\partial\mathbf v}{\partial t}=-\frac{H}{a}\nabla_x\!\cdot\mathbf v-\frac{1}{a^2}\nabla_x^2\phi-\frac{c_s^2}{a^2}\nabla_x^2\delta. \]

Take the divergence of the Euler equation (step 5) and divide by \( a \). Irrotationality lets \( \nabla\!\cdot\mathbf v \) fully capture the velocity field. A

\[ \frac{\partial^2\delta}{\partial t^2}+2H\,\frac{\partial\delta}{\partial t}-\frac{1}{a^2}\nabla_x^2\phi-\frac{c_s^2}{a^2}\nabla_x^2\delta=0. \]

Substitute step 8 into step 7 and use \( \nabla_x\!\cdot\mathbf v=-a\,\dot\delta \) (from step 4). The two \( \nabla\!\cdot\mathbf v \) terms combine to give the friction \( 2H\dot\delta \). B

\[ \delta(\mathbf x,t)=\sum_{\mathbf k}\delta_{\mathbf k}(t)\,e^{i\mathbf k\cdot\mathbf x}\ \Rightarrow\ \nabla_x^2\to -k^2,\qquad \frac{1}{a^2}\nabla_x^2\phi=4\pi G\bar\rho\,\delta_{\mathbf k}. \]

Fourier-expand in comoving wavenumber \( \mathbf k \); each mode decouples because the equation is linear with spatially constant coefficients, and Poisson (step 6) fixes the potential term. A

\[ \boxed{\ \ddot\delta_{\mathbf k}+2H\dot\delta_{\mathbf k}+\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta_{\mathbf k}=0\ } \]

Collect terms: the growth equation. Gravity (\( -4\pi G\bar\rho \)) drives collapse; pressure (\( +c_s^2k^2/a^2 \)) resists it; the sign of the bracket at \( k=k_J \), where \( c_s^2k_J^2/a^2=4\pi G\bar\rho \), reproduces the Jeans criterion. B

\[ \ddot\delta+2H\dot\delta-4\pi G\bar\rho\,\delta=0\quad(k\ll k_J). \]

On scales far below the Jeans wavenumber pressure is negligible; the equation becomes scale-free, so \( \delta_{\mathbf k}(t)=D(t)\,\delta_{\mathbf k}(t_i) \) with a single growth factor \( D \). A

\[ a\propto t^{2/3},\ H=\frac{2}{3t},\ 4\pi G\bar\rho=\frac32H^2=\frac{2}{3t^2}\ \Rightarrow\ \ddot\delta+\frac{4}{3t}\dot\delta-\frac{2}{3t^2}\delta=0. \]

Einstein–de Sitter matter era (\( \Omega_m=1 \)): use the Friedmann solution and \( 4\pi G\bar\rho=\tfrac32\Omega_m H^2 \). The equation is now equidimensional (Euler type). B

\[ \delta\propto t^{n}:\quad 3n^2+n-2=0\ \Rightarrow\ n=\frac23\ \text{or}\ n=-1. \]

Substitute the power-law ansatz into step 13 and solve the indicial equation. Two independent roots give the general solution. A

\[ \delta(t)=A\,t^{2/3}+B\,t^{-1}=A'\,a+B'\,H,\qquad \delta_+\propto a,\quad \delta_-\propto t^{-1}\propto H. \]

Growing mode \( \delta_+\propto a\propto t^{2/3} \); decaying mode \( \delta_-\propto t^{-1} \), which equals the Hubble rate — the generic decaying solution of the growth equation in any era. A

\[ \text{Radiation era: } a\propto t^{1/2},\ H=\frac{1}{2t},\ 4\pi G\bar\rho_m\ll H^2 \ \Rightarrow\ \ddot\delta_m+\frac1t\dot\delta_m\approx0. \]

During radiation domination the expansion is driven by radiation, so the matter self-gravity term is negligible; radiation pressure prevents \( \delta_\gamma \) from clustering sub-horizon, leaving only Hubble drag on \( \delta_m \). C

\[ \dot\delta_m\propto t^{-1}\ \Rightarrow\ \delta_m(t)=C_1+C_2\ln t=C_1+2C_2\ln a. \]

Integrate step 16 (set \( u=\dot\delta_m \), then \( \dot u+u/t=0 \)). Matter perturbations are nearly frozen, growing only logarithmically — the Mészáros effect, which suppresses small-scale power on scales entering the horizon before equality. C

Result

\[ \ddot\delta+2H\dot\delta+\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta=0,\qquad \begin{cases}\delta_+\propto a,\ \delta_-\propto t^{-1} & \text{matter era}\\[4pt]\delta_m\simeq C_1+C_2\ln a & \text{radiation era}\end{cases} \]

Reading. The density contrast behaves like a damped, anti-restoring oscillator. The middle term \( 2H\dot\delta \) is Hubble friction: expansion continually drains the peculiar motion and slows collapse, which is why gravitational growth is a power law in time rather than the exponential runaway of a static medium. The bracket sets the competition between self-gravity, which wants to grow \( \delta \), and pressure, which wants to oscillate it; the crossover is the Jeans scale. In the matter era gravity wins and the dominant solution grows exactly in step with the scale factor. In the radiation era the matter term is overwhelmed by the radiation-driven Hubble rate, so matter perturbations merely limp along logarithmically until matter–radiation equality releases them.

Units check. \( \delta \) is dimensionless. \( [\,4\pi G\bar\rho\,]=(\mathrm{m^3\,kg^{-1}\,s^{-2}})(\mathrm{kg\,m^{-3}})=\mathrm{s^{-2}} \); \( [H^2]=\mathrm{s^{-2}} \); \( [c_s^2k^2/a^2]=(\mathrm{m^2\,s^{-2}})(\mathrm{m^{-2}})=\mathrm{s^{-2}} \); and \( [\ddot\delta]=\mathrm{s^{-2}} \). Every term carries \( \mathrm{s^{-2}}\times\delta \), so the equation is dimensionally homogeneous.

Limiting cases

Static medium \( (H\to0) \): the friction term vanishes and \( \ddot\delta=4\pi G\bar\rho\,\delta \) gives exponential Jeans growth \( \delta\propto e^{t/\tau} \) with \( \tau=(4\pi G\bar\rho)^{-1/2} \) — the free-fall time.
Pressure-dominated, \( k\gg k_J \): the bracket is positive and \( \delta \) executes damped acoustic oscillations \( \delta\propto a^{-1/2}e^{\pm i\!\int c_s k\,dt/a} \); no gravitational growth.
Einstein–de Sitter, sub-Jeans: \( \delta_+\propto a\propto t^{2/3} \), \( f\equiv d\ln D/d\ln a=1 \).
\( \Lambda \)-dominated future \( (H\to\text{const}) \): \( \ddot\delta+2H\dot\delta=0 \Rightarrow \delta\to \) const; growth freezes as dark energy takes over, \( f\to0 \).
Open/low-density matter era: \( D_+\propto H(a)\int_0^a da'/(a'H(a'))^3 \) grows more slowly than \( a \); the general growing mode is this integral.

Breaks when

Perturbations reach \( \delta\sim1 \). Linearisation fails; the neglected \( (\mathbf v\cdot\nabla)\mathbf v \) and \( \nabla\!\cdot(\delta\mathbf v) \) terms drive mode coupling, shell crossing and virialisation — the regime of the spherical-collapse model and N-body simulation.
Scales approach or exceed the Hubble radius, \( k\lesssim aH \). The Newtonian treatment and the notion of an absolute \( \delta \) break down; one must use relativistic gauge-invariant perturbation theory, where super-horizon growth and gauge artefacts appear.
Multiple coupled species (baryons + CDM + photons) with different pressures. A single-fluid \( c_s \) is inadequate; baryon–photon acoustic oscillations, Silk damping and CDM decoupling require the full Boltzmann hierarchy.
Relativistic or free-streaming components (neutrinos, hot dark matter). Pressure and free-streaming erase perturbations below the free-streaming length, which no barotropic \( c_s^2 \) captures.

Failure modes

Dropping the Hubble-drag term. Writing \( \ddot\delta=4\pi G\bar\rho\,\delta \) in an expanding universe and finding spurious exponential growth; the \( 2H\dot\delta \) term is what converts it to a power law.
Confusing physical and comoving wavenumbers. The pressure term is \( c_s^2k^2/a^2 \) with comoving \( k \); forgetting the \( a^{-2} \) (or using physical \( k \) inconsistently) misplaces the Jeans scale by factors of \( a \).
Using \( 4\pi G\bar\rho=\tfrac32 H^2 \) outside Einstein–de Sitter. That identity assumes \( \Omega_m=1 \); in \( \Lambda \)CDM one must keep \( 4\pi G\bar\rho=\tfrac32\Omega_m(a)H^2 \).
Assuming \( \delta\propto a \) always. The linear \( \delta\propto a \) law holds only in the matter-dominated, sub-Jeans regime; students misapply it through radiation domination (where growth is logarithmic) and into the \( \Lambda \) era (where growth freezes).
Keeping the decaying mode as physically important. After a few e-folds \( \delta_-\propto t^{-1} \) is utterly negligible; retaining it in structure-formation estimates is a sign of not having applied initial conditions.
Treating radiation as clustering sub-horizon. Setting \( \bar\rho \) to the total (matter+radiation) density in the driving term during radiation domination; radiation pressure keeps \( \delta_\gamma \) oscillating, so only matter self-gravity (negligible then) sources growth.

Discussion

The growth equation is the linear heart of gravitational instability. Its structure — inertia, friction, and a restoring/anti-restoring term — is that of a parametric oscillator whose "spring constant" \( c_s^2k^2/a^2-4\pi G\bar\rho \) changes sign at the Jeans scale. Above the Jeans length gravity wins and perturbations grow; below it pressure wins and they ring as sound waves. The single new ingredient relative to the static Jeans analysis is the expansion, entering both as explicit Hubble friction and implicitly through the time dependence of \( H \), \( a \) and \( \bar\rho \). That friction is decisive: it is the reason cosmic structure grows algebraically, giving the Universe just enough time to build galaxies without collapsing everything at once.

The result \( \delta_+\propto a \) in the matter era is remarkably clean and has deep consequences. Because growth is scale-independent on sub-horizon pressureless scales, the shape of the matter power spectrum is preserved and only its amplitude scales as \( D^2(a) \). The transition epochs are imprinted permanently: modes that entered the horizon during radiation domination were held back by the Mészáros effect, bending the primordial spectrum at the equality scale \( k_{\rm eq} \) — a feature directly visible in galaxy surveys and the CMB. Thus the humble growth factor connects the physics of the first minutes (radiation domination, equality) to the distribution of galaxies today.

Beyond Einstein–de Sitter the growing mode is the integral \( D_+(a)\propto H(a)\int_0^a da'/[a'H(a')]^3 \), obtained by reduction of order using the decaying solution \( \delta_-\propto H \) (a fact that follows because \( H \) itself satisfies the homogeneous growth equation, as one can verify from the Friedmann equations). Its logarithmic derivative \( f(a)=d\ln D/d\ln a\approx\Omega_m(a)^\gamma \) with growth index \( \gamma\approx0.55 \) in general relativity is a precision test of gravity: modified-gravity theories predict different \( \gamma \), so redshift-space-distortion measurements of \( f\sigma_8 \) probe whether cosmic acceleration is dark energy or a breakdown of Einstein gravity on large scales.

Common misconceptions. (i) Expansion does not "help" perturbations grow — it opposes them through Hubble friction; gravity grows structure despite the expansion, converting would-be exponential collapse into slow power-law growth. (ii) "Matter perturbations don't grow during radiation domination" is only approximately true: they grow logarithmically, not not-at-all. (iii) The growth factor \( D(a) \) is not the scale factor in general — the equality \( D\propto a \) is a special property of the matter-dominated Einstein–de Sitter phase.

Worked examples

Example 1 — Growing/decaying decomposition of a static initial perturbation (the 3/5 factor).

\[ \delta(t)=A\,t^{2/3}+B\,t^{-1},\qquad \dot\delta(t)=\tfrac23 A\,t^{-1/3}-B\,t^{-2}. \]

General EdS solution and its derivative; impose initial data \( \delta(t_i)=\delta_i,\ \dot\delta(t_i)=0 \) (a perturbation released from rest). A

\[ 0=\tfrac23 A\,t_i^{-1/3}-B\,t_i^{-2}\ \Rightarrow\ B=\tfrac23 A\,t_i^{5/3}. \]

Solve the velocity condition for the decaying amplitude \( B \). A

\[ \delta_i=A\,t_i^{2/3}+\tfrac23 A\,t_i^{5/3}t_i^{-1}=\tfrac53 A\,t_i^{2/3}\ \Rightarrow\ A=\tfrac35\,\delta_i\,t_i^{-2/3},\ \ B=\tfrac25\,\delta_i\,t_i. \]

Substitute into the position condition and solve for both amplitudes: 3/5 lands in the growing mode, 2/5 in the decaying mode. B

\[ \delta(t)=\tfrac35\delta_i\left(\frac{t}{t_i}\right)^{2/3}+\tfrac25\delta_i\left(\frac{t}{t_i}\right)^{-1}=\tfrac35\delta_i\,\frac{a}{a_i}+\tfrac25\delta_i\,\frac{H}{H_i}. \]

Rewrite in terms of \( a\propto t^{2/3} \) and \( H\propto t^{-1} \). Now put numbers: \( \delta_i=10^{-3} \) at \( z_i=1000 \) (so \( a_i/a_0=1/1001 \)), evaluate today (\( z=0 \)). A

\[ \delta_0\approx\tfrac35(10^{-3})(1001)+\tfrac25(10^{-3})(10^{-3})\approx 0.60+4\times10^{-7}. \]

The decaying term has shrunk by \( (a_i/a_0)^{-1}\times\dots \) to parts in \( 10^{6} \) and is utterly negligible; only the growing mode survives. A

\[ \delta_0\approx 0.60 \quad(\text{growing mode }\tfrac35\delta_i\,a_0/a_i=0.60,\ \text{decaying mode }\sim4\times10^{-7}) \]

Reading. A matter perturbation of \( 10^{-3} \) at \( z=1000 \), released from rest, grows by \( \sim600\times \) in pure EdS to reach order unity today — the onset of nonlinearity. The famous factor \( 3/5 \) is why "how much has structure grown" depends on the initial velocity, not just amplitude.

Example 2 — Comoving Jeans length of the baryon gas after recombination.

\[ \frac{c_s^2k_J^2}{a^2}=4\pi G\bar\rho\ \Rightarrow\ k_J=\frac{a}{c_s}\sqrt{4\pi G\bar\rho},\qquad \lambda_J^{\rm phys}=\frac{2\pi a}{k_J}=c_s\sqrt{\frac{\pi}{G\bar\rho}}. \]

Set the bracket in the growth equation to zero to define the Jeans wavenumber, then convert to a physical wavelength. Symbols first. A

\[ c_s=\sqrt{\frac{5k_BT}{3m_p}}\ \text{(monatomic H)},\qquad \bar\rho=\bar\rho_{b,0}(1+z)^3. \]

Just after recombination the gas is neutral atomic hydrogen; use the adiabatic sound speed and scale the baryon density to \( z=200 \) with \( \bar\rho_{b,0}\approx4.2\times10^{-28}\,\mathrm{kg\,m^{-3}} \). B

\[ T\approx200\,\mathrm{K}\Rightarrow c_s=\sqrt{\frac{5(1.38\times10^{-23})(200)}{3(1.67\times10^{-27})}}\approx 1.66\times10^{3}\,\mathrm{m\,s^{-1}}. \]

Evaluate the sound speed at \( z\approx200 \), where the gas temperature has dropped adiabatically to roughly \( 200\,\mathrm K \). A

\[ \bar\rho=(4.2\times10^{-28})(201)^3\approx 3.4\times10^{-21}\,\mathrm{kg\,m^{-3}}. \]

Physical baryon density at \( z=200 \). A

\[ \lambda_J^{\rm phys}=c_s\sqrt{\frac{\pi}{G\bar\rho}}=(1.66\times10^{3})\sqrt{\frac{\pi}{(6.67\times10^{-11})(3.4\times10^{-21})}}\approx 6.2\times10^{18}\,\mathrm{m}. \]

Insert numbers. Convert: \( 6.2\times10^{18}\,\mathrm m\approx200\,\mathrm{pc} \); the enclosed Jeans mass is \( M_J=\tfrac{4}{3}\pi(\lambda_J/2)^3\bar\rho\sim10^{5}\,M_\odot \). B

\[ \lambda_J^{\rm phys}\approx6\times10^{18}\,\mathrm{m}\approx200\ \mathrm{pc},\qquad M_J\sim10^{5}\,M_\odot \]

Reading. After recombination the baryon Jeans mass plummets (the sound speed drops from relativistic to thermal), from \( \sim10^{16}M_\odot \) before decoupling to \( \sim10^{5}M_\odot \) after — of order a globular-cluster mass. Perturbations above this scale grow; below it they are pressure-supported. This collapse of the Jeans mass at recombination is what lets the first bound baryonic structures begin to form.

Problems

Verify by direct substitution that \( \delta\propto a \) is an exact solution of \( \ddot\delta+2H\dot\delta-4\pi G\bar\rho\,\delta=0 \) in the Einstein–de Sitter era.
Solution
Take \( \delta=a=a_0(t/t_0)^{2/3} \). Then \( \dot\delta=\tfrac23 a/t \), \( \ddot\delta=-\tfrac29 a/t^2 \), and \( H=\tfrac{2}{3t} \), \( 4\pi G\bar\rho=\tfrac{2}{3t^2} \). Substitute: \( -\tfrac29\frac{a}{t^2}+2\cdot\tfrac{2}{3t}\cdot\tfrac23\frac{a}{t}-\tfrac{2}{3t^2}a=\frac{a}{t^2}\left(-\tfrac29+\tfrac89-\tfrac69\right)=\frac{a}{t^2}\cdot0=0. \) Confirmed.
Show that the second EdS mode is \( \delta_-\propto t^{-1} \) and that it equals the Hubble rate \( H(t) \). Explain in one sentence why \( \delta_-\propto H \) generically.
Solution
The indicial equation \( 3n^2+n-2=0 \) has roots \( n=2/3,\,-1 \); the second gives \( \delta_-\propto t^{-1} \). Since \( H=\dot a/a=\tfrac{2}{3t}\propto t^{-1} \), indeed \( \delta_-\propto H \). Generically \( H(t) \) solves the homogeneous growth equation because differentiating the Friedmann/acceleration equations shows \( \ddot H+2H\dot H-4\pi G\bar\rho\,H \) reduces to a background identity; hence the decaying mode is always \( \propto H \).
During radiation domination (\( a\propto t^{1/2} \)) the matter self-gravity term is negligible. Solve \( \ddot\delta_m+2H\dot\delta_m=0 \) and show growth is logarithmic in \( a \).
Solution
With \( H=\tfrac{1}{2t} \), the equation is \( \ddot\delta_m+\tfrac1t\dot\delta_m=0 \). Let \( u=\dot\delta_m \): \( \dot u+u/t=0\Rightarrow u=u_0 t_0/t \). Integrate: \( \delta_m=C_1+u_0t_0\ln t=C_1+C_2\ln t \). Since \( a\propto t^{1/2} \), \( \ln t=2\ln a+\text{const} \), so \( \delta_m=C_1'+C_2'\ln a \) — logarithmic (Mészáros) growth rather than the power-law \( a \) of the matter era.
A perturbation has \( \delta_i=2\times10^{-4} \) and \( \dot\delta_i=0 \) at \( z_i=3400 \) (matter–radiation equality). Using the EdS growing/decaying decomposition, estimate \( \delta \) at \( z=10 \). Ignore \( \Lambda \).
Solution
From the 3/5 result, the surviving amplitude is \( \delta(a)=\tfrac35\delta_i\,(a/a_i) \). With \( a/a_i=(1+z_i)/(1+z)=3401/11\approx309 \): \( \delta=\tfrac35(2\times10^{-4})(309)\approx3.7\times10^{-2} \). The decaying mode, \( \tfrac25\delta_i(a_i/a)\approx2\times10^{-4}\cdot0.4/309\approx3\times10^{-7} \), is negligible. So \( \delta(z=10)\approx0.037 \), still safely linear.
Below what comoving wavelength does a gas with sound speed \( c_s=10\,\mathrm{km\,s^{-1}} \) in a region of density \( \bar\rho=10^{-22}\,\mathrm{kg\,m^{-3}} \) at scale factor \( a=1 \) oscillate rather than grow? Give the physical Jeans length.
Solution
\( \lambda_J^{\rm phys}=c_s\sqrt{\pi/(G\bar\rho)}=(10^{4})\sqrt{\pi/[(6.67\times10^{-11})(10^{-22})]} \). Inside the root: \( \pi/(6.67\times10^{-33})=4.71\times10^{32} \); square root \( =2.17\times10^{16} \). Times \( 10^4 \): \( \lambda_J^{\rm phys}\approx2.2\times10^{20}\,\mathrm{m}\approx7\,\mathrm{kpc} \). Modes with wavelength below \( \sim7\,\mathrm{kpc} \) are pressure-supported and oscillate as sound waves; larger modes grow gravitationally.