Linear Growth of Matter Perturbations

Matter perturbations grow when self-gravity overcomes pressure or free-streaming support, but expansion supplies Hubble drag and smooth components dilute the gravitational source. The familiar scale-independent growth factor is therefore a controlled limit, not the general answer. This page derives that limit, recovers its exact Einstein–de Sitter modes, and then shows where radiation, neutrinos, and relativistic metric evolution restore scale dependence.

Helpful background. The Einstein Equation supplies Einstein Equations and their constraints; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects fluid and kinetic closures; Relativistic Cosmology supplies the Relativistic Cosmology background; Computational Relativity supplies convergence and independent numerical checks; and Probability and Statistical Inference supplies covariance and uncertainty concepts used when growth is inferred.

The relativistic starting point

Use conformal time $\eta$, cosmic time $t$, $\mathcal H=aH$, and the Fourier convention $e^{i\mathbf k\cdot\mathbf x}$. Primes and dots denote $d/d\eta$ and $d/dt$, respectively. For pressureless matter in Newtonian gauge, the linear conservation equations give

$$\delta_m'=-\theta_m+3\Psi', \qquad \theta_m'=-\mathcal H\theta_m+k^2\Phi.$$

Eliminating $\theta_m$ gives the exact linear relation

$$\delta_m''+\mathcal H\delta_m' =3(\Psi''+\mathcal H\Psi')-k^2\Phi.$$

It already exposes the domain of the usual growth equation. On scales $k\gg\mathcal H$, for slowly varying potentials, negligible anisotropic stress, and clustering pressureless matter, the comoving Poisson equation reduces the right side to $4\pi Ga^2\rho_m\delta_m$. In cosmic time,

$$\boxed{ \ddot\delta_m+2H\dot\delta_m-4\pi G\rho_m\delta_m=0 }.$$

The density contrast is gauge dependent on horizon scales. There one must retain the metric terms and either state the slicing or use a gauge-invariant comoving density perturbation. The subhorizon reduction must not be continued through $k/\mathcal H=O(1)$ merely by changing a numerical coefficient.

For a nonrelativistic fluid with rest-frame sound speed $c_s$, the same continuity–Euler–Poisson calculation yields

$$\ddot\delta+2H\dot\delta +\left(\frac{c_s^2k^2}{a^2}-4\pi G\rho\right)\delta=0.$$

The instantaneous comoving Jeans wavenumber is $k_J=a\sqrt{4\pi G\rho}/c_s$ when $c_s\gt0$. Modes with $k\gg k_J$ have pressure-supported oscillations when their coefficients vary slowly over a period; expansion still changes the oscillation amplitude. Collisionless species require a hierarchy rather than a single $c_s$: their momentum distribution and anisotropic stress produce scale-dependent free streaming. Bernardeau and collaborators derive the pressureless moment hierarchy, single-stream closure, and linear equation in Bernardeau et al. 2002, §§2.1–2.3, printed pp. 13–18, eqs. (9)–(33) (Open PDF).

Growth factor and Einstein–de Sitter check

When every relevant matter species follows one pressureless growing mode, write

$$\delta_m(\mathbf k,a) =\frac{D_+(a)}{D_+(a_{\rm ref})}\, \delta_m(\mathbf k,a_{\rm ref}).$$

The ratio makes the arbitrary normalization of $D_+$ explicit. One may set $D_+(a_{\rm ref})=1$, or use the early-matter normalization below, provided the same choice is retained in every transfer and spectrum.

Using $N=\ln a$, the scale-independent equation becomes

$$\frac{d^2D}{dN^2} +\left(2+\frac{d\ln H}{dN}\right)\frac{dD}{dN} -\frac32\Omega_m(a)D=0.$$

In Einstein–de Sitter, $H\propto a^{-3/2}$ and $\Omega_m=1$. Trying $D\propto a^p$ gives

$$p^2+\frac12p-\frac32=0,$$

so

$$D_+\propto a, \qquad D_-\propto a^{-3/2}.$$

The growing solution and decaying solution are both needed to specify general initial data. Selecting only $D_+$ is an initial-condition statement, not a consequence of the second-order equation.

For spatially flat pressureless matter plus a smooth cosmological constant, an exact integral representation of the growing solution is

$$D_+(a)=\frac{5\Omega_{m0}}{2}E(a) \int_0^a\frac{da'}{a'^3E^3(a')}, \qquad E^2(a)=\Omega_{m0}a^{-3}+\Omega_{\Lambda0}.$$

This normalization makes $D_+\to a$ in the early matter era; divide by $D_+(1)$ when using the common present-day normalization $D(1)=1$. Bernardeau et al. give this quadrature and its decaying partner in Bernardeau et al. 2002, §2.3, printed pp. 17–18, eqs. (29)–(33) (Open PDF).

Exact growth versus a growth-index approximation

Define the logarithmic growth rate

$$f(a)=\frac{d\ln D_+}{d\ln a}.$$

It obeys

$$\frac{df}{dN}+f^2 +\left(2+\frac{d\ln H}{dN}\right)f =\frac32\Omega_m(a).$$

For flat $\Lambda$CDM, expanding about $\Omega_m=1$ and trying $f=\Omega_m^\gamma$ gives $\gamma=6/11$ at first order. A normalized approximate growth factor follows from

$$D_\gamma(a) =\exp\left[-\int_a^1\Omega_m(a')^{6/11}\,d\ln a'\right].$$

As an explicit check, take $\Omega_{m0}=0.3$, $\Omega_{\Lambda0}=0.7$. Direct quadrature gives:

Scale factor	Exact $D/D(1)$	Growth-index $D_\gamma$	Relative difference
$0.25$	$0.3188404$	$0.3182967$	$-0.171\%$
$0.50$	$0.6118058$	$0.6108101$	$-0.163\%$
$2/3$	$0.7731812$	$0.7721195$	$-0.137\%$
$1$	$1$	$1$	$0$

At $a=0.5$, the exact rate is $f=0.8692851$ and $\Omega_m^{6/11}=0.8697060$. At $a=1$, the approximation is less accurate: $0.5185533$ versus $0.5127962$. These numbers check the approximation in one declared model; they are not universal fitting errors.

When growth becomes scale dependent

Radiation domination stalls subhorizon CDM growth to a logarithm while photon pressure drives acoustic oscillations. Baryon growth changes across drag and recombination. Massive neutrinos can cluster on wavelengths longer than their free-streaming length but are comparatively smooth on shorter wavelengths, so the gravitational source contains a scale-dependent weighted sum of species. Dark energy perturbations, modified gravity, or nongravitational dark-sector interactions can also change both the friction and source terms.

Consequently, a single $D(a)$ is adequate only after stating the species, scale range, gravity theory, and smooth-component approximation. A Boltzmann solver returns transfer functions $D_i(k,a)$ in the general linear problem. An executable treatment of the wider parameter space must retain its solver version, inputs, convergence evidence, and the analytic checks stated here.

Scope and related topics

This page derives matter evolution, not the direct observability of $\delta_m$. Galaxy surveys measure biased tracers in redshift space on the past light cone; extracting a growth rate requires a tracer model, velocities, window functions, selection effects, covariance, and a declared gravity model. Those steps are developed later in this chapter. The quoted $f=\Omega_m^\gamma$ comparison is therefore an approximation diagnostic, not a model-independent estimator to apply directly to a catalog.

Detailed gas cooling, feedback, star formation, and other baryonic astrophysics are explicitly excluded from this linear treatment. On sufficiently small scales they change the matter distribution and must be modeled or removed by scale cuts. Conversely, a measured departure from the smooth-$\Lambda$ growth equation is not by itself evidence for modified gravity: neutrino mass, clustering dark energy, interactions, nonlinear modeling, selection, or calibration can produce alternatives. Dedicated gravity-test volumes own the theory-specific field equations and consistency tests.

Exercises

Derive $\gamma=6/11$ for flat $\Lambda$CDM by expanding the growth-rate equation to first order in $x=1-\Omega_m$.

Solution

For matter plus a cosmological constant,

$$\frac{d\Omega_m}{dN}=-3\Omega_m(1-\Omega_m), \qquad \frac{d\ln H}{dN}=-\frac32\Omega_m.$$

With $f=\Omega_m^\gamma$ and $\Omega_m=1-x$,

$$\frac{df}{dN}=-3\gamma x+O(x^2), \qquad f=1-\gamma x+O(x^2).$$

The left side of the growth-rate equation is therefore

$$\frac32+\left(\frac32-\frac{11}{2}\gamma\right)x+O(x^2),$$

whereas its right side is $3/2-(3/2)x+O(x^2)$. Matching the coefficients gives

$$\gamma=\frac{6}{11}.$$

References

• Bernardeau, F., S. Colombi, E. Gaztañaga, and R. Scoccimarro, “Large-Scale Structure of the Universe and Cosmological Perturbation Theory,” Physics Reports 367, 1–248 (2002), DOI, Open PDF.
• Tong, D., Cosmology, University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019, §3.1, printed pp. 133–147, Official PDF.