The <single-field inflation squeezed-limit consistency relation> for the <comoving curvature perturbation> is
$$
\boxed{\lim_{q\to0}
B_{\mathcal R}(q,k,|\mathbf k+\mathbf q|)
=-(n_s-1)P_{\mathcal R}(q)P_{\mathcal R}(k)}.
$$
Equivalently, $B_{\mathcal R}/P_{\mathcal R}(q)=-(3+k\partial_k)P_{\mathcal R}(k)$. The scalar shift in parts a--c is an internal symmetry and leaves the hard fields unchanged, so its right-hand side vanishes. A long adiabatic curvature mode instead acts as a spatial dilation of the hard coordinates, and its response is the scale dependence measured by the <scalar spectral index>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 2 / d / Solution

Single-field slow-roll inflation uses one canonical scalar clock whose potential evolves slowly enough for the slow-roll parameters to remain small. Its adiabatic long mode produces the <single-field inflation squeezed-limit consistency relation>.

Single-field slow-roll inflation

For a single-clock adiabatic inflationary background, a long <comoving curvature perturbation> acts on short modes as a spatial dilation. The squeezed bispectrum therefore obeys
$$
\lim_{q\to0}B_{\mathcal R}(q,k,|\mathbf k+\mathbf q|)
=-(n_s-1)P_{\mathcal R}(q)P_{\mathcal R}(k)
$$
at leading order in $q/k$.

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