Radiation acoustic transfer function

During matter domination with nearly constant gravitational potential, a radiation density perturbation obeys a forced acoustic equation. The observable Sachs-Wolfe combination is the homogeneous oscillatory part.

Pressure-supported radiation perturbations well inside the horizon oscillate rather than exhibit secular gravitational growth. In the ideal relativistic fluid, $c_s^2=1/3$ and the leading conformal frequency is $k/\sqrt{3}$. Their rapidly changing source has a small late-time response in the slowly varying cold-matter density mode.

Let ${\delta }_c=C_g{\tau }^2+C_d{\tau }^{-3}$ be the matter-era growing and decaying density modes, with radiation neglected in the background gravity. The radiation density contrast isThe formula follows by the Green's function of the forced harmonic oscillator. For $k\tau \gg 1$, the slowly varying particular response to the decaying mode begins at $16C_d{\tau }^{-5}/{\omega }^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while cold dark matter continues growing as ${\tau }^2$.

In deep radiation domination, an oscillatory radiation density source in the matter equation has order ${\Delta }_r/{\tau }^2$ and frequency of order $k$. Its rapidly varying particular response is therefore of order ${\Delta }_r/(k\tau {)}^2$. For $k\tau \gg 1$ it does not control the secular matter growth, though forcing near horizon entry sets the constant and logarithmic homogeneous amplitudes.