 Solution

For the flat FRW metric, $g^{00}=-1$, $g^{0i}=0$, and $g^{ij}=a^{-2}{\delta }^{ij}$. Consequently the kinetic contraction of the scalar field isInserting this in $T^{\mu \nu }={\partial }^{\mu }\varphi {\partial }^{\nu }\varphi +g^{\mu \nu }\mathcal{L}$, with ${\partial }^0\varphi =-\dot{\varphi }$ and ${\partial }^i\varphi =a^{-2}{\partial }_i\varphi$, givesThese are contravariant coordinate components of the stress-energy tensor; spatial components in an orthonormal frame carry different powers of $a$. In particular $T^{00}$ is the energy density seen by the comoving observer.