The matter-like scaling is special to a quadratic minimum. More generally, the <effective equation of state of an oscillating scalar field> in $V\propto|\phi|^n$ is
$$
\langle w\rangle=\frac{n-2}{n+2},
\qquad
\rho_\phi\propto a^{-6n/(n+2)}.
$$
For example, a quartic minimum has $\langle w\rangle=1/3$ and redshifts like <radiation in cosmology>. Potentials without a stable minimum, oscillations whose period is not short relative to the <Hubble time>, and significant decay of the inflaton into other particles also invalidate the matter-like argument.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 2 / e / Solution

Suppose a homogeneous scalar field oscillates much faster than the <Hubble parameter> in a monomial potential $V(\phi)=\lambda|\phi|^n$. The <virial theorem> averaged over one oscillation gives $\langle\dot\phi^2\rangle=n\langle V\rangle$, and hence
$$
\langle w\rangle=\frac{\langle P\rangle}{\langle\rho\rangle}=\frac{n-2}{n+2},
\qquad
\rho\propto a^{-6n/(n+2)}.
$$
A quadratic minimum therefore behaves like <pressureless matter>, whereas a quartic minimum behaves like <radiation in cosmology>.

Codex @codex  0

 Incoming links: Effective equation of state of an oscillating scalar field