= Solution
Write the <dimensionless> <scale factor> as $a=R/R_0$ and the <density contrast> as $\delta=(\rho-\bar\rho)/\bar\rho$. Gravity amplifies overdensities while expansion and pressure oppose growth. In a sub-Hubble Newtonian single-fluid approximation, a comoving Fourier mode obeys
$$
\ddot\delta_k+2H\dot\delta_k+
\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta_k=0.
$$
For <pressureless matter> in an Einstein-de Sitter background, the growing solution is $\delta\propto t^{2/3}\propto a$ and the decaying solution is $t^{-1}$. Jeans support suppresses growth when the pressure term dominates. The equation is not a substitute for relativistic perturbation theory on horizon-sized scales.
<cold dark matter> clusters without baryonic pressure support. Sub-horizon growth is much weaker during <radiation domination>, and <baryons> are initially tied to the <photon> fluid, producing acoustic oscillations rather than unrestricted collapse. Horizon entry, radiation-matter equality, acoustic evolution and <diffusion> therefore give a scale-dependent <transfer function>. The primordial curvature spectrum and the later matter-density spectrum are not the same function: evolution and the gravitational constraints relate them. Late dark-energy domination reduces the growth rate.
When $|\delta|$ approaches unity, linear theory fails. Collapse, <virialization>, mergers and tidal interactions form a hierarchy of haloes. Gas must cool and lose <angular momentum> to make dense stellar systems; feedback then changes its supply and <distribution>. Simulations combine collisionless gravitational dynamics with increasingly complex gas physics rather than extending the linear growing mode indefinitely. Galaxy clustering measures biased tracers of matter; <gravitational lensing> and CMB constraints provide complementary tests. Approximately adiabatic, nearly scale-invariant initial perturbations explain the coherent acoustic pattern and supply seeds for this later growth.<Planck 2013 results. XVI. Cosmological parameters by Planck Collaboration>[abstract and standard-model temperature/lensing analysis]{ref}
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