A flat $(k=0)$ homogeneous and isotropic universe with scale factor $a(t)$ is filled with a scalar field $\phi(t)$ with potential $V(\phi)$. Its evolution satisfies the Friedmann and scalar field equations,

$$H^{2}=\frac{1}{3 M_{\mathrm{Pl}}^{2}}\left(\frac{1}{2} \dot{\phi}^{2}+c^{2} V(\phi)\right), \quad \ddot{\phi}+3 H \dot{\phi}+c^{2} \frac{d V}{d \phi}=0$$

where $H(t)=\frac{\dot{a}}{a}$ is the Hubble parameter, $M_{\mathrm{Pl}}$ is the reduced Planck mass, and dots denote derivatives with respect to cosmic time $t$, e.g. $\dot{\phi} \equiv d \phi / d t$.

(a) Use these equations to derive the Raychaudhuri equation, expressed in the form:

$$\dot{H}=-\frac{1}{2 M_{\mathrm{Pl}}^{2}} \dot{\phi}^{2}$$

(b) Consider the following ansatz for the scalar field evolution,

$$\phi(t)=\phi_{0} \ln \tanh (\lambda t)$$

where $\lambda, \phi_{0}$ are constants. Find the specific cosmological solution,

$$\begin{aligned} H(t) &=\lambda \frac{\phi_{0}^{2}}{M_{\mathrm{Pl}}^{2}} \operatorname{coth}(2 \lambda t) \\ a(t) &=a_{0}[\sinh (2 \lambda t)]^{\phi_{0}^{2} / 2 M_{\mathrm{Pl}}^{2}}, \quad a_{0} \text { constant. } \end{aligned}$$

(c) Hence, show that the Hubble parameter can be expressed in terms of $\phi$ as

$$H(\phi)=\lambda \frac{\phi_{0}^{2}}{M_{\mathrm{P} 1}^{2}} \cosh \left(\frac{\phi}{\phi_{0}}\right),$$

and that the scalar field ansatz solution ( $\dagger$ ) requires the following form for the potential:

$$V(\phi)=\frac{2 \lambda^{2} \phi_{0}^{2}}{c^{2}}\left[\left(\frac{3 \phi_{0}^{2}}{2 M_{\mathrm{Pl}}^{2}}-1\right) \cosh ^{2}\left(\frac{\phi}{\phi_{0}}\right)+1\right]$$

(d) Assume that the given parameters in $V(\phi)$ are such that $2 / 3<\phi_{0}^{2} / M_{\mathrm{Pl}}^{2}<2$. Show that the asymptotic limit for the cosmological solution as $t \rightarrow 0$ exhibits decelerating power law evolution and that there is an accelerating solution as $t \rightarrow \infty$, that is,

$$\begin{array}{lll} t \rightarrow 0, & \phi \rightarrow-\infty, & a(t) \sim t^{\phi_{0}^{2} / 2 M_{\mathrm{Pl}}^{2}} \\ t \rightarrow \infty, & \phi \rightarrow 0, & a(t) \sim \exp \left(\lambda \phi_{0}^{2} t / M_{\mathrm{Pl}}^{2}\right) . \end{array}$$

Find the time $t_{\mathrm{acc}}$ at which the solution transitions from deceleration to acceleration.