Structure Formation and the Growth of Perturbations

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,

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

• 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,

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.

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,

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

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

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.¹

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 $D(a)\propto ag(a)$ 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.

The transfer function and the matter power spectrum

The statistical description of the density field is the power spectrum $P(k)=⟨|{\delta }_k{|}^2⟩$, 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 $T(k)$,

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 ${\Omega }_m$.

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,

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.

The perturbations tracked here are the same $10^{-5}$ fluctuations imaged in the CMB anisotropies, 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, which collects the full evidence chain and the open questions of the concordance model.