Scalar field energy density not changing with expanding spacetime

Question

Basically, I just want to know why this scalar field's energy density does not change, even though spacetime is expanding.

A general expanding cartesian metric is used: $$ g_{00} = -1 $$ $$ g_{11} = a(t)^2$$ $$ g_{22} = a(t)^2$$ $$ g_{33} = a(t)^2$$

The following is a general scalar field oscillator in curved spacetime: $$ L = -\dfrac{1}{2} g^{dc}∇_dΦ ∇_cΦ - \dfrac{1}{2} b^2 Φ^2 $$ We will consider a simple case where the scalar field is homogeneous and isotropic, so $Φ'(x) = Φ'(y) = Φ'(z) = 0$

This results in a single non-zero equation of motion: $$-g_{00}Φ''(t) = -\dfrac{1}{2} b^2Φ(t)$$

The problem is, because $g_{00}$ is just -1, and not some function, the resulting EOM is not affected by the volumetric change of the metric due to $a(t)$. $$Φ''(t) = -\dfrac{1}{2} b^2Φ(t)$$

The solution for $Φ(t)$ is clearly not a function of $a(t)$, and yet it represents a scalar field density (i.e. energy/volume).

If my math is correct, why can this scalar field retain its energy density?

Answer 1

Your equation of motion is missing the Hubble friction term $3H\Phi'$ coming from $\nabla_m\nabla^m\Phi$ (since $\nabla$ is the covariant derivative), i.e. the equation of motion would be $$ \Phi''+3H\Phi'+\partial_{\Phi}V=0~, $$ where $H=a'/a$, $V=b^2\Phi^2/2$.