Conformal density of a canonical scalar field ($T_{00}=a^2\rho$)

For $d{s}^2=a^2(d{\tau }^2-d{\mathbf{x}}^2)$ and a homogeneous canonical scalar, $\rho ={\varphi }^{\prime 2}/(2a^2)+V$ is the physical density, whereas $T_{00}=a^2\rho$. At $w=-1/3$, the kinetic and potential energies are $K=\rho /3$ and $V=2\rho /3$, so ${\varphi }^{\prime 2}=2a^2\rho /3=2T_{00}/3$. A single power of ${\varphi }^{\prime }$ cannot replace the square in this energy identity.

// Target: cosmology.bigb