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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 310

At <cosmological horizon exit>, $d\log k/dN=1-\epsilon\simeq1$. Differentiating the <slow-roll curvature power spectrum> and using the <Hubble slow-roll parameters> gives
$$
n_s-1=\frac{d\log\Delta_{\mathcal R}^2}{d\log k}
=-2\epsilon-\eta.
$$
The supplied relations imply
$$
\epsilon=\frac{M_{\rm Pl}^2}{2}\left(\frac{V'}V\right)^2,
\qquad
\eta=4\epsilon-2M_{\rm Pl}^2\frac{V''}V.
$$
Consequently the <scalar spectral index> is
$$
\boxed{n_s-1
=-3M_{\rm Pl}^2\left(\frac{V'}V\right)^2
+2M_{\rm Pl}^2\frac{V''}V}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 4 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 4 / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 4 / c / Solution

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