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

Table of contents
- Solution d

Solution

The expressions in part c imply

η = \frac{4}{α} ϵ, n_{s} - 1 = - 2 ϵ (1 + \frac{2}{α}) .

Since the tensor-to-scalar ratio is

r = 16 ϵ

,

n_{s} - 1 = - \frac{α + 2}{8 α} r .

For positive

α

, this class predicts a red scalar tilt and

0 < r < 8 (1 - n_{s}) .

A blue tilt with positive

r

, or

r \geq 8 (1 - n_{s})

, is incompatible. An allowed pair fixes

α = 2 r / [8 (1 - n_{s}) - r]

, subject to slow roll and observational bounds.

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