How small ripples grow — linear perturbation theory

The Lecture 13 notes open mid-way through an overview of cosmic history, with a public-domain NASA/WMAP timeline: the earliest stages are opaque to photons, the later ones transparent and potentially observable. At early times the universe holds matter — dark matter and baryons — and radiation; the cosmological constant and curvature don't matter yet.

Non-relativistic matter obeys the equations of a self-gravitating fluid:

— continuity, momentum (Euler) and Poisson. [The notes write $+\nabla\phi$ in the momentum equation, here and in eq. (377); with $\nabla^2\phi = 4\pi G\rho$ the gravitational force per unit mass is $-\nabla\phi$, as here.] For radiation the pressure joins the density: $\rho$ becomes $\rho + p/c^2$ in the continuity and momentum equations and $\rho + 3p/c^2$ in Poisson's equation, with $p = \rho c^2/3$ for pure radiation (eq. 377).

Some cautions from the notes. Dark matter is collisionless: it really obeys the collisionless Boltzmann equation, and the fluid equations hold only for its averages (the Jeans equations); there is no single velocity at each point, since streams of particles can cross. Still, the fluid description gives the correct growth on large scales, where pressure (or velocity dispersion) can be ignored.

Write the density as a smooth background $\bar\rho$ plus a small density contrast

Linearising the fluid equations in an expanding background, with comoving coordinates $\vec x$ (physical $\vec r = a\vec x$), gives

where $v_s$ is the sound speed $c_s$ for baryons, the velocity dispersion $\sigma$ for dark matter, and $c/\sqrt3$ for radiation (eq. 380). Each term has a meaning: gravity ($4\pi G\bar\rho\,\delta$) makes overdensities grow; pressure ($v_s^2\nabla^2\delta$) resists; and Hubble drag ($2H\dot\delta$) — the expansion — slows everything down.

Splitting $\delta$ into Fourier modes $\hat\delta(\vec k, t)$ of comoving wavenumber $k$ (eq. 381) turns $\nabla^2$ into $-k^2$:

with $\frac{32\pi}{3}G\bar\rho$ in place of $4\pi G\bar\rho$ for radiation. [Eq. (381) of the notes writes $\hat\delta(\vec k, \delta)$; $\hat\delta(\vec k, t)$ is meant.]

In a static background ($H = 0$) the equation is $\ddot{\hat\delta} + \omega_0^2\hat\delta = 0$ with $\omega_0^2 = v_s^2k^2/a^2 - 4\pi G\bar\rho$ (eqs. 383–384). If $\omega_0^2 > 0$ the mode oscillates — a sound wave; if $\omega_0^2 < 0$ it grows. With the physical wavenumber $\tilde k = k/a$, the dividing line is

(for radiation, $\sqrt{32/3}\,\sqrt{\pi G\bar\rho}/v_s$). Perturbations larger than the Jeans length $\lambda_J$ collapse under gravity; smaller ones are held up by pressure and oscillate. (The explicit form $\pi v_s/\sqrt{\pi G\bar\rho}$ is this page's own step; the notes give the proportionality.) The corresponding mass is the Jeans mass, $M_J = \frac{4}{3}\pi\bar\rho\lambda_J^3$ (notes, p. 85).

On scales well above $\lambda_J$ the pressure term can be dropped. For a flat universe, $\bar\rho$ is the critical density $3H^2/8\pi G$ (eq. 388), so

with $H = 2/(3t)$ in the matter era and $H = 1/(2t)$ in the radiation era (eq. 390). Try $\hat\delta \propto t^n$:

The $n = -1$ solutions are decaying modes that die away and don't matter. The growing modes give

since $a \propto t^{2/3}$ and $a \propto t^{1/2}$ respectively. In general one writes $\delta = D\,\delta_0$ with the growth factor $D$, normalised to $D = 1$ today. Note the striking contrast with a static universe: there, gravitational instability grows exponentially; in an expanding one, Hubble drag tames it to a power law.

• A perturbation is described by its density contrast $\delta = (\rho - \bar\rho)/\bar\rho$; linearising the fluid equations gives $\ddot\delta + 2H\dot\delta = 4\pi G\bar\rho\,\delta + v_s^2\nabla^2\delta/a^2$ (with $\frac{32\pi}{3}G\bar\rho$ for radiation).
• Gravity drives growth, pressure resists, and Hubble drag slows it. Modes larger than the Jeans length $\lambda_J \propto v_s/\sqrt{\pi G\bar\rho}$ grow; smaller ones oscillate.
• Ignoring pressure, the growing mode is $\delta \propto a$ in the matter era and $\delta \propto a^2$ in the radiation era — power laws, not exponentials. One writes $\delta = D\delta_0$ with the growth factor $D$.