= Solution
The flat matter-dominated <Friedmann equation> gives $\rho=1/(6\pi Gt^2)$ when $a=t^{2/3}$. With $P=C\rho^{4/3}$,
$$
c_s^2=\frac{dP}{d\rho}=\frac{4C}3\rho^{1/3}=\bar c_s^2t^{-2/3},\qquad\bar c_s^2=\frac{4C}3(6\pi G)^{-1/3}.
$$
For <linear density modes with polytropic pressure>, substitution gives the <Euler-Cauchy equation>
$$
\ddot\delta+\frac4{3t}\dot\delta+\frac{\bar c_s^2k^2-2/3}{t^2}\delta=0.
$$
A <power law> has exponent satisfying $n(n-1)+4n/3+\bar c_s^2k^2-2/3=0$. Thus, for distinct roots,
$$
\boxed{\delta=A_kt^{n_+}+B_kt^{n_-},\qquad n_\pm=-\frac16\pm\sqrt{\frac{25}{36}-\bar c_s^2k^2}.}
$$
For $k\ll k_J$, the two modes approach $t^{2/3}$ and $t^{-1}$: gravity produces a growing <linear cosmological density perturbation> and a decaying mode. For $k\gg k_J$, put $\nu=\sqrt{\bar c_s^2k^2-25/36}$; the real solution is
$$
\delta=t^{-1/6}\{C_k\cos(\nu\log t)+D_k\sin(\nu\log t)\}.
$$
These are pressure-supported oscillations with a decaying envelope. For the <repeated-root cosmological density mode> $k=k_J$, the complete solution is \b[$t^{-1/6}(A_k+B_k\log t)$]. The specified $k_J=5/(6\bar c_s)$ separates real exponents from logarithmic oscillations; the growing exponent actually crosses zero at $k=\sqrt{2/3}/\bar c_s$, a slightly smaller value.
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