Comparing the stated curvature <power spectrum> with $P_{\mathcal R}=2\pi^2A_s/k^3$ at the pivot scale gives the <primordial scalar amplitude>
$$
A_s=\frac{H^2}{8\pi^2\epsilon_VM_{\rm Pl}^2}
\simeq
\boxed{\frac{\Lambda^{12}}
{12\pi^2M_{\rm Pl}^6m^4\phi^2}}.
$$
The <scalar spectral index> is
$$
n_s-1\simeq-6\epsilon_V+2\eta_V
\simeq
\boxed{-\frac{3M_{\rm Pl}^2m^4\phi^2}{\Lambda^8}
-\frac{2M_{\rm Pl}^2m^2}{\Lambda^4}}.
$$
Sufficiently close to the hilltop, $\epsilon_V$ is suppressed by $\phi^2$ and the $\eta_V$ term controls the <spectral tilt>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 2 / e / Solution

Expand $\delta=a\delta_1+a^2\delta_2+a^3\delta_3+\cdots$. For a Gaussian linear density field, odd linear correlators vanish and <Wick theorem> reduces the fourth-order terms to products of the linear <power spectrum>. Hence
$$
P_{\rm 1-loop}(k)
=a^2P(k)+a^4[P_{22}(k)+2P_{13}(k)].
$$
The two contractions of the two quadratic fields give
$$
\boxed{P_{22}(k)=2\int\frac{d^3q}{(2\pi)^3}
F_2(\mathbf q,\mathbf k-\mathbf q)^2
P(q)P(|\mathbf k-\mathbf q|)}.
$$
The three choices for which argument of $F_3$ carries the external momentum give
$$
\boxed{P_{13}(k)=3P(k)\int\frac{d^3q}{(2\pi)^3}
F_3(\mathbf k,\mathbf q,-\mathbf q)P(q)}.
$$
Thus the contribution conventionally called $P_{13}+P_{31}$ is $2P_{13}=6P(k)\int F_3P$. These are the two one-loop diagrams of the <one-loop matter power spectrum>.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 2 / ii / Solution

{wiki}

The power spectrum describes how the variance of a random field or stationary process is distributed over spatial wavenumber or temporal frequency.


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