How small fluctuations become structure

Compare density on a specified time slice

The density contrast $\delta=(\rho-\bar\rho)/\bar\rho$ measures fractional excess above the mean. Comparing densities at the same time requires a convention. Changing coordinates changes which events share a time label, and the mean density evolves. A large-scale density perturbation therefore depends on the gauge.

For relativistic perturbations we use conformal Newtonian gauge, with $dt=a\,d\eta$ defining conformal time and light speed set to one. The time potential $\Psi$ affects gravitational redshift and acceleration; $\Phi$ perturbs spatial geometry. They coincide only when anisotropic stress can be neglected.

ds^2=a^2(\eta)\left[-(1+2\Psi)d\eta^2+(1-2\Phi)d\mathbf{x}^2\right]

The background is spatially flat FLRW. Both potentials are dimensionless. Initial conditions and evolution must use compatible gauge conventions.

Reading: Enrico Pajer · David Tong

Derive growth from conservation

Restrict attention to slowly moving, pressureless matter on scales inside the Hubble radius. Use cosmic time $t$ , comoving position $\mathbf{x}$ and peculiar velocity $\mathbf v=a\dot{\mathbf{x}}$ . Continuity relates density growth to inflow. Euler’s equation adds gravitational acceleration and the dilution of peculiar velocity by expansion.

Assume matter supplies the fluctuating gravitational source, with other smooth components affecting only the expansion. Write $\bar\rho_m$ for mean mass density, in mass per physical volume. The velocity divergence $\theta=\nabla\cdot\mathbf v$ measures local outflow or convergence. Spatial derivatives act on comoving coordinates.

Differentiating continuity also differentiates the scale factor in its denominator: $\ddot\delta=-\dot\theta/a+H\theta/a$ . Substitution from Euler supplies a second expansion term. Together they produce the factor of two in the growth equation. Poisson’s equation then replaces the potential by the density perturbation.

\dot\delta=-\theta/a,\quad\dot\theta+H\theta=-\nabla^2\phi/a,\quad\nabla^2\phi=4\pi G a^2\bar\rho_m\delta

The Newtonian potential

\phi

has units of speed squared and approaches

\phi=c^2\Psi

in the subhorizon approximation.

\ddot\delta+2H\dot\delta-4\pi G\bar\rho_m\delta=0

Gravity promotes growth and expansion damps peculiar motion. Radiation or other substantially clustering components require coupled equations.

Solve a matter-dominated example completely

For a flat matter-only background, $a\propto t^{2/3}$ , $H=2/(3t)$ and $4\pi G\bar\rho_m=2/(3t^2)$ . Try $\delta\propto t^p$ . Every term has the same time dependence, leaving a quadratic equation for the exponent.

Its roots are two-thirds and minus one. The growing mode follows the scale factor; the decaying mode falls as inverse time. A pure growing contrast of one part in a thousand becomes two parts in a thousand when the scale factor doubles, provided the matter-only and linear approximations still hold. Near unit contrast, nonlinear evolution becomes important.

p(p-1)+\frac43p-\frac23=0\quad\Rightarrow\quad\delta_+\propto a,\quad\delta_-\propto t^{-1}

Substitution into the differential equation cancels the terms exactly. These solutions independently test a numerical growth solver.

Reading: David Tong

Pressure makes a compressed wave rebound

A Fourier wave with comoving wavenumber $k$ acquires a pressure-restoring term proportional to $c_s^2k^2/a^2$ . Shorter waves have steeper pressure gradients and more readily oscillate. Longer waves are easier for gravity to amplify. This competition underlies the Jeans criterion.

Before recombination, photons supply pressure to a tightly coupled photon-baryon fluid, and baryons add inertia. Cold dark matter does not share that photon pressure and can support gravitational potential wells. Different cosmic ingredients therefore evolve differently from the same initial perturbation. The next chapter follows the resulting acoustic oscillator.

Reading: David Tong · Hu & Sugiyama

How small fluctuations become structure

Compare density on a specified time slice

Derive growth from conservation

Solve a matter-dominated example completely

Pressure makes a compressed wave rebound

Try the prediction

Use peak heights to think about pressure and inertia

Exercises and explanations

Sources and calculation scope