---
title: "How small fluctuations become structure"
description: "Define density perturbations and gauge, derive pressureless growth, and separate gravitational attraction from expansion and pressure support."
canonical_url: "https://sandboxphysics.com/stories/cosmology/linear-perturbations/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/cosmology/linear-perturbations/"
last_modified: "2026-10-28"
---

[Cosmology from expansion to inference](https://sandboxphysics.com/stories/cosmology/)

L2 · Chapter 2

# How small fluctuations become structure

Sandbox Physics · 2026-10-28

A slightly denser region attracts more matter and grows denser. Expansion slows the inflow; pressure can push the material back out. Subtracting the mean expansion lets us follow these small departures and identify what controls their evolution.

## Before you begin

Read chapter one. You need derivatives and a sinusoidal wave. A second derivative describes how a rate changes. Linear theory keeps only first-order departures and applies while the density contrast is small.

[Review the preceding chapter](https://sandboxphysics.com/stories/cosmology/expansion-distances/)

- Explain why a density perturbation needs a time-slicing convention.
- Derive growth from continuity, motion and gravity.
- Find growing and decaying solutions in a matter-only universe.

## 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

## Try the prediction

### [Use peak heights to think about pressure and inertia](https://sandboxphysics.com/labs/cmb-acoustic-peaks/)

1. Predict how increasing baryon density changes sound speed and relative peak heights.
2. Change only baryon density and compare the spectrum with the fixed reference.
3. Connect the result to photon pressure and baryon inertia, remembering that the full model changes other background quantities too.

## Exercises and explanations

1. Why does the growing solution usually dominate a late matter-only perturbation that started with both modes?

   Read the explanation

   Its amplitude increases as time to the power two-thirds, while the decaying amplitude falls as inverse time. Any nonzero growing component eventually dominates their ratio.

2. Can the pressureless subhorizon equation describe photon perturbations during radiation domination?

   Read the explanation

   Use coupled relativistic perturbation equations with pressure instead. Photon pressure, changing gravitational potentials and horizon-scale effects can all change the evolution.

[Previous: Following light through an expanding universe](https://sandboxphysics.com/stories/cosmology/expansion-distances/) [Next: Why primordial sound leaves a series of peaks](https://sandboxphysics.com/stories/cosmology/acoustic-oscillations/)

## Sources and calculation scope

1. [David Tong, Cosmology, Chapter 3](https://davidtong.org/teaching/cosmology/cosmohtml/S3)

   Structure Formation: linear growth, pressure support, transfer functions and the microwave sky. The pressureless growth example applies inside the Hubble scale, during matter domination.

2. [Enrico Pajer, Cosmology, Part III course (2018–2019)](https://www.damtp.cam.ac.uk/user/ep551/cosmo.html)

   Course reading entry for background geometry and cosmological perturbation theory. Our Newtonian-gauge convention is defined in chapter two; this link is further study, not a claim of a numerical reproduction.

3. [Hu & Sugiyama, Small-Scale Cosmological Perturbations: An Analytic Approach, arXiv:astro-ph/9510117v2 (1996)](https://arxiv.org/abs/astro-ph/9510117v2)

   Tight-coupling acoustic physics and baryon loading. The constant-potential, constant-baryon-loading oscillator in chapter three is a restricted analytic example; the interactive spectra come from the frozen CAMB calculation.