[You may work in units of the speed of light, so that $c=1$.]

(a) Combining the Friedmann and continuity equations

$$H^{2}=\frac{8 \pi G}{3} \rho, \quad \dot{\rho}+3 H(\rho+P)=0$$

derive the Raychaudhuri equation (also known as the acceleration equation) which expresses $\ddot{a} / a$ in terms of the energy density $\rho$ and the pressure $P$.

(b) Assuming an equation of state $P=w \rho$ with constant $w$, for what $w$ is the expansion of the universe accelerated or decelerated?

(c) Consider an expanding, spatially-flat FLRW universe with both a cosmological constant and non-relativistic matter (also known as dust) with energy densities $\rho_{c c}$ and $\rho_{d u s t}$ respectively. At some time corresponding to $a_{e q}$, the energy densities of these two components are equal $\rho_{c c}\left(a_{e q}\right)=\rho_{d u s t}\left(a_{e q}\right)$. Is the expansion of the universe accelerated or decelerated at this time?

(d) For what numerical value of $a / a_{e q}$ does the universe transition from deceleration to acceleration?