The expressions in part c imply
$$
\eta=\frac4\alpha\epsilon,
\qquad
n_s-1=-2\epsilon\left(1+\frac2\alpha\right).
$$
Since the <tensor-to-scalar ratio> is $r=16\epsilon$,
$$
\boxed{n_s-1=-\frac{\alpha+2}{8\alpha}r}.
$$
For positive $\alpha$, this class predicts a red scalar tilt and
$$
0<r<8(1-n_s).
$$
A blue tilt with positive $r$, or $r\geq8(1-n_s)$, is incompatible. An allowed pair fixes $\alpha=2r/[8(1-n_s)-r]$, subject to slow roll and observational bounds.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 4 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-310/4/d/solution