= Solution
When $H\ll m_\sigma$ over an oscillation, the background equation reduces to
$$
\ddot{\bar\sigma}+m_\sigma^2\bar\sigma=0,
\qquad
\bar\sigma=\Sigma\cos(m_\sigma t+\varphi).
$$
Time averaging gives $\langle\dot{\bar\sigma}^2\rangle=m_\sigma^2\langle\bar\sigma^2\rangle$, hence
$$
\langle P_\sigma\rangle=0,
\qquad
\langle\rho_\sigma\rangle=\frac12m_\sigma^2\Sigma^2.
$$
The oscillating quadratic scalar is therefore <pressureless matter>. Its continuity equation gives
$$
\boxed{\rho_\sigma\propto a^{-3}},
$$
with the slowly varying amplitude obeying $\Sigma\propto a^{-3/2}$.
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