= Solution
Throughout this question use <natural units> with $c=1$. For zero potential and nonzero <kinetic energy>, the homogeneous canonical <scalar field> has $P=\rho=\dot\varphi^2/2$, so $\boxed{w=1}$. Its <equation of motion> becomes $d(a^3\dot\varphi)/dt=0$, giving $\dot\varphi\propto a^{-3}$ and therefore
$$
\boxed{\rho\propto a^{-6}.}
$$
The same scaling follows from <cosmological energy conservation> with constant <equation-of-state parameter> $w=1$. If the field is exactly constant, both $P$ and $\rho$ vanish and their ratio is undefined; that is the separate empty solution.
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