---
title: Structure Formation and the Growth of Perturbations
draft: false
module: The Hot Big Bang
moduleNumber: 12
lessonNumber: 6
order: 1206
summary: >
  The near-uniform early universe grew its galaxies and clusters by gravitational
  instability acting on the tiny inflationary perturbations. In an expanding
  background the growth is slowed to a power law rather than the exponential of a
  static medium; perturbations stall during radiation domination and grow with the
  scale factor once matter dominates. The transfer function turns the primordial
  spectrum into the processed matter power spectrum, and cold dark matter builds
  structure from the bottom up.
topics: [The Hot Big Bang]
sources:
  - book: Ryden
    ref: "Ch. 11 — Structure Formation: Gravitational Instability; Ch. 12"
  - book: Carroll & Ostlie
    ref: "Ch. 30 — The Early Universe; §30.5"
  - book: Maoz
    ref: "Ch. 10 — Tests and Probes of Big Bang Cosmology"
---

The CMB shows the universe at last scattering to be uniform to one part in
$10^5$; the present universe is clumped into stars, galaxies, clusters, and a
cosmic web of filaments and voids, contrasts of order unity and far larger. The
bridge between the two is gravitational instability: an overdense region pulls in
its surroundings and grows denser, an underdense region empties. This lesson
derives the growth of small perturbations in an expanding universe, the reason
growth is suppressed until matter dominates, the transfer function that shapes the
matter power spectrum, and the hierarchical assembly of structure in cold dark
matter.

## The Jeans instability in a static medium

The classical result, before adding expansion, is the **Jeans instability**. A
uniform self-gravitating fluid of density $\rho$ and sound speed $c_s$ supports
small perturbations $\delta = \delta\rho/\rho$ that obey a wave equation with a
gravitational source,

$$
\ddot\delta = 4\pi G\rho\,\delta - c_s^2 k^2\,\delta ,
$$

for a Fourier mode of wavenumber $k$. Two terms compete: self-gravity, which
amplifies the perturbation, and pressure, which resists compression through the
$c_s^2 k^2$ restoring force. The sign of the right side switches at the **Jeans
wavenumber**

$$
k_J = \frac{\sqrt{4\pi G\rho}}{c_s},
\qquad
\lambda_J = \frac{2\pi}{k_J} = c_s\sqrt{\frac{\pi}{G\rho}} .
$$

- **Small scales ($k > k_J$, $\lambda < \lambda_J$):** pressure wins, the right
  side is negative, and the perturbation oscillates as a sound wave.
- **Large scales ($k < k_J$, $\lambda > \lambda_J$):** gravity wins, the right
  side is positive, and the perturbation grows.

In the static case the growing mode is exponential, $\delta \propto e^{t/\tau}$
with $\tau = (4\pi G\rho)^{-1/2}$ of order the free-fall time. Gravity converts a
tiny seed into collapse on the free-fall timescale.

The scale $\lambda_J$ corresponds to a mass, the **Jeans mass**, the minimum mass
that can collapse against its own pressure,

$$
M_J = \frac{4}{3}\pi\rho\left(\frac{\lambda_J}{2}\right)^3
\propto \frac{c_s^3}{G^{3/2}\rho^{1/2}} .
$$

Only regions with $M > M_J$ collapse; smaller ones oscillate as sound waves. The
Jeans mass is the same criterion that governs star formation in molecular clouds,
applied here to the mean cosmological density.

> **Worked example.** After recombination the baryonic gas is neutral hydrogen at
> $T \approx 3000\ \text{K}$ with sound speed $c_s \approx 5\ \text{km s}^{-1}$
> and mean density $\rho \approx 4 \times 10^{-19}\ \text{kg m}^{-3}$. The Jeans
> mass is $M_J \sim c_s^3/(G^{3/2}\rho^{1/2}) \approx 10^5\ M_\odot$, the mass
> scale of a globular cluster. Before recombination, when the baryons were coupled
> to photons, the effective sound speed was $c_s \approx c/\sqrt{3}$, roughly
> $10^4$ times larger, and the Jeans mass exceeded $10^{16}\ M_\odot$ — larger
> than any cluster. The collapse of galaxy-scale baryonic structures could
> therefore not begin until recombination dropped the sound speed and the Jeans
> mass with it.

## Growth in an expanding universe

Expansion changes the growth law. The perturbation equation acquires a Hubble
friction term, exactly as the inflaton did, from the expanding background,

$$
\ddot\delta + 2H\dot\delta = 4\pi G\rho_m\,\delta - \frac{c_s^2 k^2}{a^2}\,\delta ,
$$

where $\rho_m$ is the matter density and the pressure term is negligible for cold
dark matter ($c_s \approx 0$). On scales well above the Jeans length the pressure
term drops and

$$
\ddot\delta + 2H\dot\delta - 4\pi G\rho_m\,\delta = 0 .
$$

The $2H\dot\delta$ friction slows the growth from exponential to a power law. In a
matter-dominated (Einstein-de Sitter) universe, $H = 2/3t$ and $4\pi G\rho_m =
2/3t^2$, and substituting a trial $\delta \propto t^n$ gives $n = 2/3$ or $n =
-1$. The two solutions are

$$
\delta_+ \propto t^{2/3} \propto a,
\qquad
\delta_- \propto t^{-1} ,
$$

a growing mode that increases linearly with the scale factor and a decaying mode.
The linear growth $\delta \propto a$ is far gentler than the exponential of a
static medium: expansion competes with collapse, and structure builds slowly.[^ryden-structure]

$$
% caption: Hubble friction slows perturbation growth. In a static medium a
% super-Jeans mode grows exponentially; in a matter-dominated expanding universe
% it grows only linearly with the scale factor.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {time};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {amplitude (log)};
% exponential (static) using bezier to avoid overflow
\draw[black, very thick, dashed] (0.4,0.4) .. controls (3.0,0.8) and (4.5,2.0) .. (6.0,4.1);
\node[black, anchor=south east] at (5.9,3.9) {static: exponential};
% linear growth in a: straight line
\draw[acc, very thick] (0.4,0.5) -- (8.8,3.0);
\node[acc, anchor=north] at (6.6,2.4) {expanding: grows as scale factor};
\end{tikzpicture}
$$

The linear $\delta \propto a$ result is exact only for a matter-dominated,
spatially flat universe. In the concordance cosmology dark energy comes to
dominate at late times, and its accelerated expansion increases the Hubble
friction, so growth slows below $\delta \propto a$ for $z \lesssim 1$. The full
result is written as a **growth factor** $D(a)$, the solution of the growth
equation normalized to $D = a$ deep in the matter era. A useful approximation is
$D(a) \propto a\,g(a)$ with a suppression $g < 1$ that scales roughly as
$\Omega_m(a)^{0.55}$, so that structure has effectively stopped growing today:
the onset of dark-energy domination froze the large-scale structure into its
present pattern.

## Suppression before equality and the role of dark matter

Growth depends on which component drives the expansion. During radiation
domination the expansion rate is set by the radiation, not the matter, and the
matter perturbations grow only logarithmically — effectively frozen. The physical
reason is the **Meszaros effect**: the radiation-driven expansion is too fast for
the weak self-gravity of the sub-dominant matter to act; a dark-matter
perturbation cannot grow faster than the background expands. Only after
matter-radiation equality, when matter dominates the expansion, does $\delta
\propto a$ begin.

Baryons face an additional delay. Before recombination the baryons are locked to
the photons in the acoustic oscillations, so baryon perturbations oscillate rather
than grow, and photon pressure prevents their collapse. Only after decoupling are
the baryons free to fall.

This is where cold dark matter is essential. Because it does not couple to
photons, dark matter begins growing at equality, well before recombination. By the
time the baryons decouple, the dark-matter perturbations have already deepened
their potential wells; the baryons then fall into ready-made wells and quickly
catch up, $\delta_b \to \delta_c$. Without dark matter, structure would have to
grow from the $10^{-5}$ perturbations only after recombination, and the linear
$\delta \propto a$ growth over a factor of $\sim 1100$ in scale factor would reach
only $\delta \sim 10^{-2}$ today — far short of the collapsed structures observed.
The head start dark matter provides is what makes the observed universe possible.

$$
% caption: Growth histories of the perturbations. Dark matter begins growing at
% matter-radiation equality; baryons oscillate with the photons until decoupling,
% then fall into the dark-matter wells and catch up.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {scale factor (log)};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {amplitude (log)};
% dark matter: flat then rising at equality
\draw[acc, very thick] (0.4,1.2) -- (2.6,1.35) -- (8.8,3.9);
\node[acc, anchor=south east] at (6.6,3.3) {dark matter};
% equality marker
\draw[black, densely dotted] (2.6,0) node[below, black!70] {equality} -- (2.6,1.35);
% baryons: oscillate (wavy) until decoupling then rise to meet dark matter
\draw[black, very thick, dashed] (0.4,1.0)
  .. controls (1.0,1.3) and (1.4,0.7) .. (1.8,1.0)
  .. controls (2.2,1.3) and (2.6,0.7) .. (3.0,1.0)
  .. controls (3.4,1.3) and (3.8,0.7) .. (4.2,1.0)
  -- (5.2,1.15) .. controls (6.2,1.6) and (7.4,2.9) .. (8.6,3.6);
\draw[black, densely dotted] (4.2,0) node[below, black!70] {decoupling} -- (4.2,1.0);
\node[black, anchor=north] at (5.8,1.6) {baryons};
\end{tikzpicture}
$$

## The transfer function and the matter power spectrum

The statistical description of the density field is the **power spectrum** $P(k)
= \langle |\delta_k|^2\rangle$, the variance of the Fourier amplitudes as a
function of wavenumber. Inflation supplies a near-scale-invariant primordial
spectrum, $P_{\text{prim}}(k) \propto k^{n_s}$ with $n_s \approx 1$. Processing by
the growth physics multiplies it by the square of a **transfer function** $T(k)$,

$$
P(k) = A\,k^{n_s}\,T^2(k)\,D^2(a),
$$

with $D(a)$ the linear growth factor. The transfer function encodes the
scale-dependent suppression from the epoch when each mode entered the horizon:

- **Large scales ($k$ small, $\lambda \gg$ horizon at equality):** these modes
  entered the horizon after equality, during matter domination, and were never
  suppressed. Here $T(k) \to 1$ and $P(k) \propto k^{n_s} \approx k$.
- **Small scales ($k$ large):** these modes entered the horizon during radiation
  domination and stalled by the Meszaros effect until equality, losing amplitude
  relative to the large-scale modes. Here $T(k) \propto k^{-2}$ and $P(k) \propto
  k^{n_s - 4} \approx k^{-3}$.

The two regimes meet at a **turnover** near the horizon scale at equality,
$k_{\text{eq}} \sim 0.01\ \text{Mpc}^{-1}$. The power spectrum rises as $k$ on
large scales, peaks at $k_{\text{eq}}$, and falls on small scales. The location of
the turnover measures $z_{\text{eq}}$ and hence the matter density, an independent
probe of $\Omega_m$.

$$
% caption: The processed matter power spectrum. Large-scale modes retain the
% primordial slope; small-scale modes are suppressed because they entered the
% horizon during radiation domination and stalled, producing a turnover at the
% horizon scale of matter-radiation equality.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {wavenumber (log)};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {power (log)};
% rising then turnover then falling
\draw[acc, very thick] (0.5,0.6) -- (3.4,3.6)
  .. controls (4.0,4.0) and (4.6,3.9) .. (5.2,3.4) -- (8.8,0.8);
\draw[black, densely dotted] (4.0,0) node[below, black!70] {turnover} -- (4.0,3.7);
\node[acc, anchor=north west] at (0.7,2.2) {rises as k};
\node[acc, anchor=south west] at (6.4,2.1) {falls on small scales};
\node[black, anchor=south] at (4.0,3.9) {horizon at equality};
\end{tikzpicture}
$$

## Hierarchical assembly and the halo mass function

Cold dark matter has negligible thermal velocity, so it retains power on small
scales; small perturbations survive and collapse first. Structure therefore
assembles **bottom-up**: low-mass dark-matter halos form earliest, then merge into
progressively larger halos, which host galaxies and then clusters. This
hierarchical picture is the defining prediction of cold dark matter, in contrast
with the top-down fragmentation that hot dark matter (fast-moving, free-streaming
neutrinos) would produce. The observed early appearance of small galaxies confirms
the bottom-up sequence.

When a perturbation grows to $\delta \sim 1$, linear theory breaks down. A useful
analytic model is **spherical collapse**: an overdense sphere expands with the
universe, decouples, turns around, and collapses. The linear-theory extrapolation
predicts collapse when the linearly evolved overdensity reaches the critical value
$\delta_c \approx 1.686$. Counting the fraction of the density field above this
threshold, smoothed on a mass scale $M$, gives the **Press-Schechter** halo mass
function,

$$
\frac{\d n}{\d M} = \sqrt{\frac{2}{\pi}}\,\frac{\bar\rho}{M}\,
\frac{\delta_c}{\sigma^2}\left|\frac{\d\sigma}{\d M}\right|
\exp\!\left(-\frac{\delta_c^2}{2\sigma^2(M)}\right),
$$

where $\sigma(M)$ is the rms density fluctuation smoothed on scale $M$, computed
from $P(k)$. The mass function is a power law at low mass and cut off exponentially
above the mass scale that has just collapsed, and it grows toward higher masses as
the universe ages — the quantitative statement of hierarchical growth. N-body
simulations, which integrate the gravitational dynamics of billions of dark-matter
particles, reproduce this mass function and the cosmic web of filaments, walls,
and voids in detail.

$$
% caption: Hierarchical assembly in cold dark matter. Small halos collapse first
% and merge into larger halos over cosmic time, building the mass hierarchy from
% the bottom up.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,-0.2) -- (9.0,-0.2) node[right, black!70] {time};
% early: many small halos
\foreach \x/\y in {0.6/1.6,1.0/2.4,0.9/0.9,1.5/1.9,1.3/2.9,0.7/2.9}
  \fill[acc!60] (\x,\y) circle (2pt);
\node[black, anchor=north] at (1.0,0.4) {many small halos};
% middle: merged into medium
\foreach \x/\y in {4.0/1.4,4.4/2.4,4.2/2.9}
  \fill[acc] (\x,\y) circle (4pt);
\node[black, anchor=north] at (4.2,0.4) {merged};
% late: one large halo
\fill[acc] (7.4,2.0) circle (9pt);
\node[black, anchor=north] at (7.4,0.4) {large halo};
% merger arrows
\draw[->, black] (1.9,1.9) -- (3.4,2.0);
\draw[->, black] (5.0,2.0) -- (6.4,2.0);
\end{tikzpicture}
$$

The perturbations tracked here are the same $10^{-5}$ fluctuations imaged in the
[CMB anisotropies](/astrophysics-cosmology/the-hot-big-bang/cmb-anisotropies-and-cosmological-parameters),
grown by gravity into the halos that host galaxies. The dark matter whose head
start makes the growth work is the subject of the [concluding
lesson](/astrophysics-cosmology/the-hot-big-bang/dark-matter-dark-energy-and-open-questions),
which collects the full evidence chain and the open questions of the concordance
model.

[^ryden-structure]: Ryden, Ch. 11–12 — the Jeans instability, growth of perturbations in an expanding universe, the Meszaros suppression during radiation domination, the transfer function and matter power spectrum, and the Press-Schechter mass function.