= Solution
In <cosmic time>, all three <Friedmann-Lemaître-Robertson-Walker metrics> can be written
$$
ds^2=-dt^2+a^2(t)\left[d\chi^2+S_K^2(\chi)d\Omega_2^2\right],
$$
where
$$
S_K(\chi)=
\begin{cases}
\sin\chi,&K=+1\quad\hbox{closed},\\
\chi,&K=0\quad\hbox{flat},\\
\sinh\chi,&K=-1\quad\hbox{open}.
\end{cases}
$$
The closed spatial slice is a <three-sphere> of radius $a(t)$, so
$$
\boxed{V_{K=+1}=2\pi^2a^3}.
$$
The complete flat and open spatial slices are noncompact and have
$$
\boxed{V_{K=0}=V_{K=-1}=\infty}.
$$
For a homogeneous fluid in the <Spatially flat FLRW metric>, the equations in part (a) reduce to the <cosmological perfect-fluid continuity equation>
$$
\boxed{\dot\rho+3H(\rho+P)=0},
$$
while homogeneity makes the spatial <relativistic Euler equation> automatic. The <Einstein field equations> give the flat <Friedmann equation> and <Friedmann acceleration equation>,
$$
\boxed{H^2=\frac{8\pi G}{3}\rho,
\qquad
\dot H=-4\pi G(\rho+P)},
$$
equivalently $\ddot a/a=-4\pi G(\rho+3P)/3$.
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