The Friedmann equations and the conservation of energy-momentum for a spatially homogeneous and isotropic universe are given by:

$$3 \frac{\dot{a}^{2}+k}{a^{2}}-\Lambda=8 \pi \rho, \quad \frac{2 a \ddot{a}+\dot{a}^{2}+k}{a^{2}}-\Lambda=-8 \pi P, \quad \dot{\rho}=-3 \frac{\dot{a}}{a}(P+\rho),$$

where $a$ is the scale factor, $\rho$ the energy density, $P$ the pressure, $\Lambda$ the cosmological constant and $k=+1,0,-1$.

(a) Show that for an equation of state $P=w \rho, w=$ constant, the energy density obeys $\rho=\frac{3 \mu}{8 \pi} a^{-3(1+w)}$, for some constant $\mu$.

(b) Consider the case of a matter dominated universe, $w=0$, with $\Lambda=0$. Write the equation of motion for the scale factor $a$ in the form of an effective potential equation,

$$\dot{a}^{2}+V(a)=C$$

where you should determine the constant $C$ and the potential $V(a)$. Sketch the potential $V(a)$ together with the possible values of $C$ and qualitatively discuss the long-term dynamics of an initially small and expanding universe for the cases $k=+1,0,-1$.

(c) Repeat the analysis of part (b), again assuming $w=0$, for the cases:

(i) $\Lambda>0, k=-1$,

(ii) $\Lambda<0, k=0$,

(iii) $\Lambda>0, k=1$.

Discuss all qualitatively different possibilities for the dynamics of the universe in each case.