For the homogeneous background, $X=\dot{\bar\phi}^{,2}/2$. The second <Friedmann equation> gives
$$
\frac12P_{,X}\dot{\bar\phi}^{,2}
=-\dot H M_{\rm Pl}^2
=\epsilon H^2M_{\rm Pl}^2.
$$
Dividing by $H M_{\rm Pl}^2\dot{\bar\phi}$ shows that the coefficient found above is
$$
\boxed{C=\frac{H\epsilon}{\dot{\bar\phi}}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 2 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-312/2/iv/solution