= Solution
Small primordial fluctuations seed later structure. For <adiabatic cosmological perturbations> on scales outside the horizon, the <uniform-density curvature perturbation> is conserved when the <non-adiabatic pressure perturbation> and the decaying mode can be neglected. A <density contrast> outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.
For subhorizon, pressureless matter, the linearized <continuity equation>, <Euler equation> and <Poisson equation> combine into
$$
\boxed{\ddot\delta+2H\dot\delta-4\pi G\bar\rho_m\delta=0}.
$$
Hubble expansion damps peculiar motion; <self-gravity> amplifies overdensities. During <radiation domination>, <radiation> <pressure> drives acoustic oscillations and <cold dark matter> growth after horizon entry is only logarithmic. During <matter domination> the growing and decaying solutions are $\delta\propto a$ and $\delta\propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.
For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2k^2\delta/a^2$, giving a <Jeans length> separating acoustic support from gravitational growth. Before <recombination>, <baryons> are tightly coupled to <photons> and oscillate rather than freely following <dark matter>. <Silk damping> from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, <baryons> can fall into dark-matter potential wells. <free streaming> smooths species-dependent scales and distinguishes cold from warm or hot <dark matter>.
Schematically, the linear <matter power spectrum> can be written $P_\delta(k,a)=A k^{n_s}T^2(k)D^2(a)$ under the usual primordial-curvature normalization. The <cosmological transfer function> records horizon entry, matter-radiation equality, acoustic physics and damping; the <linear growth factor> records later amplification. <density contrasts> approaching unity require nonlinear collapse and halo formation beyond this approximation. <cosmic microwave background> anisotropies and <galaxy clustering> test the same initial perturbations through different transfer histories; <galaxy bias>, <redshift-space distortions> and observational selection enter the galaxy comparison.
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