Linear Growth of Structure

Linear growth is not a historical relic — it is the working engine of precision cosmology's central tension. The amplitude of the linear power spectrum is summarized by $\sigma_8$, and the growth factor is what evolves it from the CMB epoch to now. Planck, measuring fluctuations at $z\approx1100$ and evolving them forward with $\Lambda$CDM, predicts $S_8\equiv\sigma_8\sqrt{\Omega_m/0.3}\approx0.83$; weak-lensing surveys (KiDS, DES, HSC) that measure the low-redshift field directly find $S_8\approx0.76$–$0.78$. This few-percent "$S_8$ tension" is a discrepancy in how much structure grew, and if it survives it points at physics — massive neutrinos, dark-energy dynamics, or modified gravity — that alters $D(a)$. On the velocity side, redshift-space distortions in BOSS, eBOSS, and DESI measure $f\sigma_8(z)$ through the Kaiser anisotropy of clustering, testing the GR prediction $\gamma\approx0.55$ across cosmic time; Euclid and Rubin/LSST will pin $\gamma$ to the percent level. And every cosmological N-body simulation begins by displacing particles off a grid with the Zel'dovich approximation, where displacements scale as $D(a)$ and velocities as $\dot D = fHD$ — so an accurate growth factor is the literal first line of code in modelling the nonlinear cosmic web.

Picture an overdense blob in an expanding, raisin-bread universe. Its excess gravity pulls surrounding matter inward, but the whole loaf is stretching, so the collapse runs uphill against the expansion. In the linearized equations that expansion enters as a velocity-proportional drag, $2H\dot\delta$ — mathematically identical to a mass on a spring immersed in molasses, with the Hubble rate $H$ as the viscosity. Large $H$ means strong drag and sluggish growth; the self-gravity that would explode exponentially in a static medium is throttled to a power law. The first figure makes the tug-of-war concrete and pins down the matter-era balance that gives $D\propto a$.

The growth factor is scale-independent for cold matter because Newtonian self-gravity carries no built-in length. A region twice as large collapses by the same fractional amount as a small one — gravitational and inertial forces scale the same way with size — so there is no preferred wavenumber and every super-Jeans mode grows by the identical factor $D(a)$. A length scale can only enter through pressure (a sound speed, hence a Jeans length) or a finite horizon, and for cold dark matter above the Jeans length after equality neither is at work. During radiation domination the story flips: the breakneck expansion is powered by photons that do not cluster, so matter feels a drag it cannot itself source, and the growing mode nearly freezes — the Mészáros effect — which is why structure formation waits for matter–radiation equality to switch on. Late-time dark energy does gravity's opposite, piling on drag with no collapsing source, so $D(a)$ saturates. The second figure quantifies the late-time story through the growth rate $f(a)$.

Linear theory expands the cosmic-fluid equations to first order in $\delta$ and the peculiar velocity $\mathbf v$; eliminating $\mathbf v$ and the peculiar potential $\Phi$ collapses the three conservation laws into one second-order equation per Fourier mode. The full mode equation keeps a pressure term, and everything specific to linear growth follows from switching that term off.

where: $\delta_k$ — Fourier amplitude at comoving wavenumber $k$; $H=\dot a/a$ — Hubble rate; $c_s$ — sound speed; $c_s^2 k^2/a^2$ — pressure restoring term; $4\pi G\bar\rho_m$ — gravitational drive; $\bar\rho_m$ — mean matter density. The bracket is a competition: pressure resists collapse on small scales, gravity drives it on large ones, balancing at the Jeans wavenumber $k_J$. For cold dark matter on scales far above the Jeans length ($k\ll k_J$) the sound speed is negligible and the pressure term drops out entirely. What remains has no explicit $k$, which is the whole origin of scale-independent growth.

where: $\delta$ — density contrast of a super-Jeans cold-matter mode; the two independent solutions are a growing mode $D_+(a)$ and a decaying mode $D_-(a)$. With no length scale, all such modes share the time dependence $\delta\propto D(a)$. The driving term comes from the perturbed Poisson equation, written in comoving coordinates so only the perturbation, not the smooth background, appears on the right.

where: $\Phi$ — peculiar gravitational potential; $a$ — scale factor; the factor $a^2$ comes from converting the physical Laplacian $\nabla_r=a^{-1}\nabla_x$ to comoving coordinates. Subtracting the background leaves the overdensity $\delta\rho=\bar\rho_m\delta$ as the sole source. Given the two solutions of equation (2), the growing mode has a closed form as a quadrature over the expansion history — the most useful formula for any $\Lambda$CDM background, since it needs only $H(a)$.

where: $D_+$ — growing-mode growth factor; $\Omega_m$ — present matter density parameter; $H_0$ — Hubble constant; $H(a)$ — Hubble rate versus scale factor. The prefactor $\tfrac52\Omega_m H_0^2$ is chosen so $D_+\to a$ deep in matter domination, fixing the normalization. The logarithmic derivative of $D$ is the growth rate $f$, which sets peculiar-velocity amplitudes and is what surveys measure. In general relativity it is captured to sub-percent accuracy by a single power law in the matter density parameter.

where: $f$ — growth rate; $\Omega_m(a)=\Omega_m a^{-3}/[H(a)/H_0]^2$ — matter fraction at scale factor $a$; $\gamma$ — growth index, $\approx0.55$ in GR. A measured $\gamma\ne0.55$ would flag modified gravity or dynamical dark energy. Finally, because equation (2) has no explicit spatial dependence, the field separates into a fixed spatial pattern times a growing amplitude — which is why the linear power spectrum simply scales as $D^2$.

where: $a_i$ — any initial epoch in the linear regime; $P(k,a)$ — matter power spectrum. The spatial map is frozen; only its contrast deepens, uniformly across all scales.

The pressure-free growth equation $\ddot\delta + 2H\dot\delta = 4\pi G\bar\rho_m\delta$ takes a different character in each cosmological era, according to what dominates the expansion and whether that component clusters.

1 · Late-time growth suppression and the growth rate today (fully worked)

For a flat $\Lambda$CDM universe with $\Omega_m=0.31$, $\Omega_\Lambda=0.69$, compute the present growth rate $f_0$ and the present growing-mode amplitude $D_0$ relative to the Einstein–de Sitter extrapolation $D_+=a$. Interpret both numbers physically.

Step 1. Growth rate from the fitting form. Equation (5) at $a=1$ gives $f_0 = \Omega_m(1)^{0.55} = 0.31^{0.55}$. Taking logs, $\ln f_0 = 0.55\ln 0.31 = -0.644$, so $f_0 = e^{-0.644} = \mathbf{0.53}$. A perfectly matter-dominated universe would have $f=1$; today's $0.53$ is depressed because dark energy has begun to freeze growth.

Step 2. Growth suppression from the Carroll–Press–Turner fit. The quadrature (4) at $a=1$ is captured to $\sim1\%$ by $g_0\equiv D_0/a = \tfrac52\Omega_m\big[\Omega_m^{4/7}-\Omega_\Lambda+(1+\tfrac12\Omega_m)(1+\tfrac{1}{70}\Omega_\Lambda)\big]^{-1}$. With $\Omega_m^{4/7}=0.512$, the bracket is $0.512 - 0.69 + (1.155)(1.00986) = 0.988$, so $g_0 = 2.5\times0.31/0.988 = \mathbf{0.78}$ — the growing mode has reached only 78% of the EdS amplitude, a direct fingerprint of $\Lambda$.

Step 3. Physical reading. The two numbers describe the same freezing from two angles: $D_0/a<1$ says the accumulated growth is below the matter-era track, while $f_0<1$ says the instantaneous growth is slowing. Because peculiar velocities scale as $v\propto fHD$, the sub-unity $f_0$ directly lowers the velocity field that redshift-space distortions measure. Answer: $f_0\approx 0.53$ and $D_0/a\approx0.78$ — late-time growth is suppressed roughly 20% by dark energy.

2 · The Mészáros effect: how much a mode grows across radiation domination

A sub-horizon cold-dark-matter mode enters the horizon deep in the radiation era. Using the Mészáros growing solution $\delta_m\propto(1+\tfrac32 y)$ with $y\equiv a/a_{\rm eq}$, find the factor by which it grows between deep radiation domination ($y\ll1$) and equality ($y=1$), and contrast this with the growth an equal span would produce in the matter era.

Step 1. Evaluate the Mészáros mode at the endpoints. At $y\to0$ the growing solution is $\delta_m\propto1$; at $y=1$ it is $\delta_m\propto(1+\tfrac32)=\tfrac52$. The growth factor across the entire radiation era is therefore $\delta_m(1)/\delta_m(0) = \mathbf{5/2}$. However long radiation dominates, a sub-horizon matter mode grows by at most this fixed factor of 2.5 — nearly frozen.

Step 2. Contrast with matter-era growth. In the matter era $\delta\propto a\propto y$, so a mode spanning the same range of scale factor — say $y=10^{-3}$ (early horizon entry) to $y=1$ — would grow by a factor $10^3$. The Mészáros freezing costs roughly a factor $10^3/2.5\approx\mathbf{400}$ of "missing" growth. This is why small-scale modes that entered the horizon early are strongly suppressed in the transfer function relative to large-scale modes entering after equality.

Step 3. Physical reading. The freeze happens because radiation, not matter, drives the expansion, so the drag $2H\dot\delta$ is huge while the source $4\pi G\bar\rho_m\delta$ is small; matter cannot gather fast enough against a Hubble rate it does not control. Only after equality does $\delta\propto a$ resume. Answer: total growth across radiation domination is a fixed factor of 5/2, a few hundred short of unimpeded $\delta\propto a$ growth.

The theory of how a nearly smooth universe grew its structure is, more than any other body of cosmological work, the achievement of one person who built it into a calculable discipline over half a century.

Texts. Dodelson, Modern Cosmology, 2nd ed., ch. 7 (growth of structure) · Ryden, Introduction to Cosmology, 2nd ed., ch. 12 · Mo, van den Bosch & White, Galaxy Formation and Evolution, §4 (linear perturbation theory) · Binney & Tremaine, Galactic Dynamics, 2nd ed., §5.2 (Jeans equations in expanding backgrounds) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30 · Peebles, The Large-Scale Structure of the Universe (1980). The quadrature (4) is the Heath (1977)–Peebles form; the growth index $\gamma\approx0.55$ is the Linder (2005) GR value. Key numerical anchors: the EdS indicial roots are $n=2/3,-1$ (roots of $3n^2+n-2=0$), giving $D_+\propto a$ and $D_-\propto a^{-3/2}$; the Wronskian scales as $W\propto a^{-2}$, yielding the integrand $(a'H')^{-3}$; the Mészáros growth factor across equality is $5/2$; and for $\Omega_m=0.31$ one obtains $f_0=0.31^{0.55}=0.53$ and the Carroll–Press–Turner suppression $D_0/a=0.78$. All cosmological parameters used throughout ($\Omega_m=0.31$, $\Omega_\Lambda=0.69$, $z_{\rm eq}\approx3400$, $S_8\approx0.83$ vs $0.76$) are standard concordance values.

Prerequisites. Density Perturbations and the Density Contrast · Jeans Instability and Cosmological Perturbation Regimes · Cold Dark Matter. Leads to. Transfer Function and the Matter Power Spectrum · The Primordial Power Spectrum · Baryon Acoustic Oscillations.

Ask the tutor: Ask me to integrate the growth quadrature for your cosmology, or to show how $f\sigma_8(z)$ turns a survey into a direct test of gravity.