Consider the equation of motion of a scalar field $\phi(t,x^i)$, $$\nabla^\mu\nabla_\mu\phi=\frac{dV(\phi)}{d\phi}$$ where $V(\phi)$ is the potential. Specialise to a massless field ($V(\phi)=0$) in the FRW/FLRW metric (coordinates $\{t,r,\varphi,\theta\}$, with $\phi=\phi(t)$. I need to show that $\phi(t)-\phi(t_0)=k\int_{t_0}^tdt'a(t')^{-3}$ for some constant of integration $k$.
My attempt is as follows:
$$\nabla^\mu\nabla_\mu\phi=g^{\alpha\mu}(\partial_\alpha\nabla_\mu\phi-\Gamma^\lambda_{\alpha\mu}\nabla_\lambda\phi)=g^{tt}(\partial_t\partial_t\phi-\Gamma^\lambda_{tt}\partial_t\phi)+g^{ij}(\partial_i\partial_j\phi-\Gamma^\lambda_{ij}\partial_\lambda\phi)$$ where I used that $\nabla=\partial$ for scalars, and I carried out the implicit (Einstein) summation. Next, I use the following information
- the only relevant non-zero Christoffel symbol is $\Gamma^t_{ij}=a\dot{a}\delta_{ij}$ (where $\dot{}$ denotes differentiation wrt $t$)
- $\phi$ is a function of $t$ only, so $\partial_i\phi=0$
- $g^{tt}=-1$ and $g^{ij}=\delta^{ij}/a^2$
to find
$$\nabla^\mu\nabla_\mu\phi=-\partial^2_t\phi-\frac{\delta^{ij}\delta_{ij}\dot{a}}{a}\partial_t\phi=-\partial^2_t\phi-\frac{3\dot{a}}{a}\partial_t\phi$$
Therefore, I am left with
$$-\partial^2_t\phi-\frac{3\dot{a}}{a}\partial_t\phi=0$$
No clue how to proceed. Any hints? Thanks.