 Solution

Write $\mathcal{H}=a^{\prime }/a$, with the prime denoting ${\partial }_0=c^{-1}{\partial }_{\eta }$. The metric and inverse metric are $g_{\mu \nu }=a^2{\eta }_{\mu \nu }$ and $g^{\mu \nu }=a^{-2}{\eta }^{\mu \nu }$. Since $a$ depends only on $x^0$, the Christoffel symbols reduce toConsequentlyFor the stated de Sitter spacetime scale factor, $\mathcal{H}=-1/(c\eta )$. The factor $c^{-1}$ is required because the prime is not differentiation with respect to $\eta$ itself.