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For a single barotropic perfect fluid in general relativity, the perturbations obey $\delta P=w\delta \rho =w\bar{\rho }\delta$. This adiabatic closure is needed: constant background $P/\rho$ alone would not eliminate an independent entropy perturbation. The absence of scalar anisotropic stress allows the common potential used in Newtonian gauge in cosmology.

Substituting the density constraint into the pressure equation gives the gravitational potential evolution of a barotropic fluidSince $a\propto {\tau }^{2/(1+3w)}$, the conformal Hubble parameter is $\mathcal{H}=2/[(1+3w)\tau ]$. The bracket cancels identically, leavingFor a Fourier transform mode, ${\nabla }^2$ becomes $-k^2$.

During radiation domination, $w=1/3$ and ${\Phi }_k^{\prime \prime }+4{\Phi }_k^{\prime }/\tau +(k^2/3){\Phi }_k=0$. Set $x=k\tau /\sqrt{3}$ and write $\Phi =u/x$. The resulting equation is $u_{xx}+2u_x/x+(1-2/x^2)u=0$, so the two Spherical Bessel functions in the hint giveAt $x\ll 1$, the two solutions approach a constant and a mode proportional to ${\tau }^{-3}$. The regular adiabatic mode, normalized to its primordial potential, isAfter entry into the sound horizon, $x\gg 1$, the potential oscillates at cosmological sound speed $1/\sqrt{3}$ with envelope $x^{-2}\propto a^{-2}$. The Hubble radius and sound horizon differ by the sound-speed factor; outside the Hubble radius the regular potential is constant, while well inside it radiation supports acoustic oscillations.

During matter domination, $w=0$ gives ${\Phi }_k^{\prime \prime }+6{\Phi }_k^{\prime }/\tau =0$ at every wavenumber. HenceThe growing density mode has a constant potential both outside and inside the Hubble radius; the other potential mode decays. Pressureless matter has zero cosmological sound speed, so horizon entry does not produce the radiation acoustic decay. These formulas cover both independent solutions, while the subsequent sketches select the regular adiabatic growing mode.

Once a mode is well inside the horizon, radiation pressure opposes gravitational collapse. For an ideal radiation fluid its cosmological sound speed is $c_s=1/\sqrt{3}$, and its density perturbation undergoes subhorizon radiation acoustic oscillations with frequency approximately $k/\sqrt{3}$. Unlike the superhorizon growing mode, its amplitude does not acquire sustained gravitational growth. Free-streaming relativistic components can additionally phase-mix the perturbation; the acoustic-fluid description is the appropriate idealization for the stated system.

During deep radiation domination, the radiation source in the dark-matter equation oscillates rapidly compared with a Hubble time. Its accumulated late-time effect averages away. More quantitatively, an oscillatory source of magnitude ${\Delta }_r/{\tau }^2$ gives a rapidly varying particular solution of order ${\Delta }_r/(k\tau {)}^2$ when $k\tau \gg 1$. It can therefore be neglected in computing the slowly varying homogeneous dark-matter growth, although its forcing near horizon entry fixes the integration constants.

The dark-matter self-gravity term is also subleading while $a/a_{\mathrm{eq}}\ll 1$. Keeping the leading expansion drag and averaging the radiation source yieldsConsequentlyThis logarithmic growth of matter perturbations during radiation domination is the Mészáros effect. It is the leading behavior well after horizon entry but before equality; close to equality the neglected self-gravity matters. The radiation perturbation is not small simply because it is radiation: its rapid oscillations, rather than a small radiation background density, justify dropping its source for this slow mode.