 Solution

For a homogeneous scalar in conformal time, $(\partial \bar{\varphi }{)}^2=-{\bar{\varphi }}^{\prime 2}/a^2$. The time component of the stress-energy tensor is $T_{00}={\bar{\varphi }}^{\prime 2}/2+a^{2}V$, while $T_{ij}={\delta }_{ij}[{\bar{\varphi }}^{\prime 2}/2-a^{2}V]$. Raising one index and applying the prescribed definitions givesThese are the scalar kinetic energy plus/minus potential energy. In proper cosmological time $d{t}_{\mathrm{proper}}=adt$, the same kinetic term is ${\dot{\bar{\varphi }}}^2/2$. Potential domination gives negative pressure, while kinetic domination gives $P\simeq \rho$.