Linear growth factor ($D(t)$)

The linear growth factor multiplies the growing mode of a linear cosmological density perturbation, so $\delta (\mathbf{x},t)=D(t)\delta (\mathbf{x},t_0)/D(t_0)$ when scale-independent growth applies.

For pressureless linear cosmological density perturbations in an Einstein-de Sitter universe, $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. The growth equation is an Euler differential equation whose indicial equation is $(p-2/3)(p+1)=0$. Hence the independent modes are ${\delta }_{+}\propto t^{2/3}\propto a$ and ${\delta }_{-}\propto t^{-1}\propto a^{-3/2}$. The first is the growing linear growth factor, while the second is a decaying transient.

With physical matter and radiation densities, primordial amplitudes and spatial flatness fixed, extra smooth late dark energy increases $H_0$ but keeps ${\Omega }_m{H}_0^2$ fixed. Provided the dark energy is negligible at equality, the early cosmological transfer function and equality wavenumber in physical units are unchanged. Earlier accelerated expansion suppresses the linear growth factor, lowering present dimensional matter power by its squared growth ratio. Units involving $h$ can additionally relabel the horizontal and vertical axes.

For pressureless sub-Hubble perturbations, combine the linearized cosmological continuity equation, cosmological Euler equation and gravitational Poisson equation with matter as its source. The expansion contributes a $2H$ friction term and matter self-gravity contributes the negative coefficient of $\delta$. The background expansion can include radiation or dark energy; treating their perturbations as additional sources requires a more complete system.

Use a monotone linear growth factor $b(t)$ as time, $\mathbf{u}=\dot{\mathbf{x}}/\dot{b}$, and $\psi =a\phi /(Cb)$ with $C=3H_0^2{\Omega }_{m0}/2$ and $a_0=1$. The cosmological Poisson equation becomes ${\nabla }^2\psi =\delta /b$; the continuity equation becomes ${\partial }_b\delta +\nabla \cdot [(1+\delta )\mathbf{u}]=0$. The material derivative is $D_b={\partial }_b+\mathbf{u}\cdot \nabla$. Substitution into the pressureless cosmological Euler equation, followed by the growth equation and $\dot{b}=Hbf$, proves the displayed motion equation, with $f=d\ln b/d\ln a$. The density derivative in the conservative continuity equation is Eulerian, not material.

For a pure scalar growing density contrast with time-independent boundary conditions, ${\nabla }^2\psi ={\delta }_{\ast }$ fixes the scaled peculiar gravitational potential up to an irrelevant constant. The linearized continuity equation fixes the longitudinal scaled velocity to $-\nabla \psi$. The remaining homogeneous velocity is ${\mathbf{u}}_h=\mathbf{C}(\mathbf{x})/(a^2\dot{b})$; a divergence-free such field does not change the growing density at linear order. Excluding this decaying velocity component is necessary for time-independent $\mathbf{u}$. In an Einstein-de Sitter universe, $b=a$ and ${\mathbf{u}}_h\propto a^{-3/2}$, furnishing a counterexample if the pure growing-velocity hypothesis is omitted.

For dust, curvature and a constant cosmological constant, write $H^2=A{a}^{-3}+B{a}^{-2}+C$. Then $\dot{H}=-3A{a}^{-3}/2-B{a}^{-2}$ and $\ddot{H}=H(9A{a}^{-3}/2+2B{a}^{-2})$, which imply $\ddot{H}+2H\dot{H}-3A{a}^{-3}H/2=0$. Thus $H$ is a solution of the pressureless linear matter perturbation growth equation. In an Einstein-de Sitter universe, $H\propto t^{-1}$ is the decaying mode and the independent growing mode is $t^{2/3}$. Additional dynamical components need not preserve this identity.

For pressureless matter on subhorizon scales, the linear growth factor obeysThe second term is dilution by cosmic expansion, while the final term drives Jeans instability.

Replacing the self-gravity term in the matter growth equation by $(1-f)4\pi G{\bar{\rho }}_{m}D$, with $0<1-f<1$, weakens clustering while leaving the background expansion unchanged.

During matter domination, the growing mode under a reduced clustering strength $1-f$ isIt grows more slowly than the standard $t^{2/3}$ mode whenever $0<f<1$.

In a matter-dominated universe, the long-wavelength pressureless growth equation has modes

When matter self-gravity is negligible during radiation domination, $a\propto t^{1/2}$ and $\ddot{\delta }+2H\dot{\delta }=0$ gives $\delta =C_1+C_2\log a$.

In a radiation-dominated background with matter self-gravity retained, ${\delta }_{aa}+{\delta }_a/a-3\delta /(2a{a}_{\mathrm{eq}})=0$. A change to $x=\sqrt{6a/a_{\mathrm{eq}}}$ gives the Modified Bessel differential equation of order zero. The Modified Bessel function of the first kind increases as $1+3a/(2a_{\mathrm{eq}})+\cdots$; the Modified Bessel function of the second kind decreases with a leading negative logarithm. The early solution space has the constant/logarithmic behaviors, not the $a$ growth of matter domination. This background approximation must not be extrapolated through equality.

During curvature domination $a\propto t$, so a pressureless perturbation without a self-gravity term has modes $1$ and $a^{-1}$. The surviving mode is constant.