Linear density modes with polytropic pressure

In a flat matter-dominated universe with $a\propto t^{2/3}$ and $P\propto {\rho }^{4/3}$, the squared sound speed scales as $t^{-2/3}$. Each comoving Fourier mode then satisfies an Euler-Cauchy equation, with gravitational growth at long wavelengths and logarithmic-time acoustic oscillations at short wavelengths.

When the characteristic exponents of the density-mode Euler-Cauchy equation coincide, the second independent solution gains a logarithm. In the polytropic matter-era example the complete solution at this threshold is $t^{-1/6}(A+B\log t)$.