4. The Friedmann-Robertson-Walker (FRW) universe is described by the
following metric of the spacetime:
2
2
dr2
2
ds² = -dt² + a²(t) \left[1 - \frac{kr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2)\right]
(5)
(a) (10 marks) Please classify the three cases: open, flat, and closed
universe, and plot the history of the Universe for these three cases.
(b) (10 marks) With the metric in hand, please compute the following
connection coefficients: $\Gamma^1_{11}$, $\Gamma^1_{22}$, $\Gamma^3_{33}$, and the rest of non-zero connec-
tion coefficients include:
$\Gamma^1_{01} = \Gamma^1_{10} = \Gamma^2_{02} = \Gamma^2_{20} = \Gamma^3_{03} = \Gamma^3_{30} = \frac{\dot{a}}{a}$
$\Gamma^1_{22} = -r(1 - kr^2)$
$\Gamma^1_{33} = -r(1 - kr^2)\sin^2\theta$
$\Gamma^2_{12} = \Gamma^2_{21} = \Gamma^3_{13} = \Gamma^3_{31} = \frac{1}{r}$
$\Gamma^3_{23} = \Gamma^3_{32} = \cot\theta$
(6)
Please calculate the curvature tensor, the Ricci tensor, and the Ricci
scalar.
(c) (10 marks) Recall that the energy-momentum tensor for a perfect
fluid can be written as
$\mathcal{T}_{\mu\nu} = (p + \rho)U_\mu U_\nu + pg_{\mu\nu}$,
(7)
where $\rho$ and $p$ are the energy density and pressure (respectively) as
measured in the rest frame, and $U^\mu$ is the four-velocity of the fluid.
Please derive the Friedmann equations:
$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3p)$, (8)
and
$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}$.
(9)
