= Solution
{c}
Write the <equation of state> as $P=w\rho c^2$. Integrating the <cosmological perfect-fluid continuity equation> for constant $w$ gives $\rho\propto R^{-3(1+w)}$. With spatial flatness and the geometric <cosmological constant> set to zero, the <Friedmann equation> is $H^2=8\pi G\rho/3$.
In S1, $w=-1$ describes <vacuum energy>: $T_{\mu\nu}=-\rho c^2g_{\mu\nu}$ and $\rho$ is constant. Its negative <pressure> drives accelerated expansion, $\ddot R/R=8\pi G\rho/3=H_1^2$, and the expanding solution is $R\propto e^{H_1t}$. This is the ideal <de Sitter spacetime> approximation to <cosmic inflation>. An exactly constant vacuum component cannot end its phase unaided; a phase transition or <reheating> mechanism is required for the assumed change to S2.
In S2, $w=1/3$ describes an isotropic bath of ultrarelativistic <particles>, such as <photons>, with negligible <rest mass> relative to their <energy>. The number per physical <volume> falls as $R^{-3}$, while the <energy> per particle redshifts as $R^{-1}$, so $\rho\propto R^{-4}$. This <radiation-dominated universe> has $R\propto(t-t_*)^{1/2}$ on its expanding branch.
In S3, $w=0$ describes <pressureless matter>, or <relativistic dust>, with negligible random <kinetic energy>. Conserved particle number and approximately fixed <rest energy> give $\rho\propto R^{-3}$. A flat universe containing only this component is the <Einstein-de Sitter universe>, with $R\propto(t-t_*)^{2/3}$. The constants $t_*$ are integration offsets in the individual phases; they need not represent one common start time when phases are matched.
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