Cosmological Scalar Field Density Dilution

[Samama Fahim]

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 a hint as to how to proceed.

 

[ergospherical]

See section II, Turner M.S. 1983,
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.28.1243

 

[Samama Fahim]

See section II, Turner M.S. 1983,
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.28.1243

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