Consider a homogeneous free scalar field $\phi$ of mass m which has a potential
$$V(\phi) = \frac{1}{2}m^2\phi^2$$
Show that, for $m ≫ H$, the scalar field undergoes oscillations with frequency given by $m$ and that its energy density dilutes as $a^{−3}$.
This is from Modern Cosmology, Scott Dodelson, Chapter 6.
For the part "Show that its energy density dilutes as $a^{−3}$", following is my attempt:
In the equation $\frac{\partial \rho}{\partial t} = -3H(P+\rho)$, put $P = \frac{1}{2} \dot{\phi}^2-V(\phi)$ and $\rho=\frac{1}{2} \dot{\phi}^2+V(\phi)$ to get
$$\frac{\partial \rho}{\partial t} = -3\frac{\dot{a}}{a}\dot{\phi}^2,$$
where $H = \dot{a}/a$. I am not sure how to proceed or proceed or whether this is the correct approach. Should I use Friedmann's equation instead? But it involves densities of other species as well, and there is no assumption here whether one species dominates. Or Should I convert $d\rho/dt$ to $d \rho/ da$?
Kindly provide only a hint as to how to proceed.
