Consider the de Sitter metric

$$d s^{2}=-d t^{2}+e^{2 H t}\left(d x^{2}+d y^{2}+d z^{2}\right)$$

where $H>0$ is a constant.

(a) Write down the Lagrangian governing the geodesics of this metric. Use the Euler-Lagrange equations to determine all non-vanishing Christoffel symbols.

(b) Let $\mathcal{C}$ be a timelike geodesic parametrized by proper time $\tau$ with initial conditions at $\tau=0$,

$$t=0, \quad x=y=z=0, \quad \dot{x}=v_{0}>0, \quad \dot{y}=\dot{z}=0,$$

where the dot denotes differentiation with respect to $\tau$ and $v_{0}$ is a constant. Assuming both $t$ and $\tau$ to be future oriented, show that at $\tau=0$,

$$\dot{t}=\sqrt{1+v_{0}^{2}}$$

(c) Find a relation between $\tau$ and $t$ along the geodesic of part (b) and show that $t \rightarrow-\infty$ for a finite value of $\tau$. [You may use without proof that

$$\left.\int \frac{1}{\sqrt{1+a e^{-b u}}} d u=\frac{1}{b} \ln \frac{\sqrt{1+a e^{-b u}}+1}{\sqrt{1+a e^{-b u}}-1}+\text { constant }, \quad a, b>0 .\right]$$

(d) Briefly interpret this result.