Density contrast ($\delta$)

The density contrast is the fractional perturbation $\delta =(\rho -\bar{\rho })/\bar{\rho }$.

In matter domination with the constant growing potential, the full Newtonian-gauge energy constraint gives the stated density contrast. The second term dominates inside the Hubble radius and grows as $a$, because ${\mathcal{H}}^2\propto a^{-1}$. Outside the Hubble radius, the constant $-2\Phi$ term dominates and is gauge dependent. The comoving density combination ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ cancels it for ${\theta }_c=2k^2\Phi /(3\mathcal{H})$, leaving the cosmological Poisson equation relation. Matter-spectrum power laws must specify whether they refer to this comoving density or to subhorizon Newtonian-gauge density.

The smoothed matter density variance measures the root-mean-square linear density contrast after applying a window corresponding to a Lagrangian mass scale. For a homogeneous isotropic field it is ${\sigma }^2=(1/(2{\pi }^2))\int k^{2}P(k)|W(kR){|}^{2}dk$. The variance convention, growth epoch and smoothing mass-density relation must agree with the collapse barrier used in a halo abundance calculation.

A spherical top-hat window function averages a field uniformly inside a sphere of radius $R$ and gives zero weight outside it. Its normalized real-space kernel is $W_R(\mathbf{x})=3/(4\pi R^3)$ for $|\mathbf{x}|\le R$, and its Fourier transform is

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.

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.

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.

For a pressureless fluid withthe first-order density contrast $\delta =\delta \rho /\bar{\rho }$ obeys