Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 57 / 3 / c

Put $k_1=q\ll k$ and $k_2,k_3\to k$ in the squeezed bispectrum configuration. Then $K\to 2k$ and the three symmetric combinations in the supplied expression have limitsThe first two terms inside the $ϵ$ bracket cancel, leaving $4k^3$. The full bracket is therefore $(\eta +2ϵ)k^3/4$, andThe $(2\pi {)}^3$ is kept in the normalization of $F$ used by the paper; it should not be introduced a second time by changing the primordial bispectrum convention.

Momentum conservation makes the three vectors close into a triangle. As $q/k\to 0$, the two short-wavelength vectors approach equal magnitude and opposite direction, producing a thin, nearly collapsed triangle. In real space the $q$ mode varies slowly over a region containing many short wavelengths, so it modulates their local statistics. For a single adiabatic inflationary clock this is a local spatial dilation, the basis of the single-field inflation squeezed-limit consistency relation. If $\eta$ is the Hubble-flow convention $\dot{ϵ}/(Hϵ)$, the scalar spectral index in Hubble slow-roll parameters gives $\eta +2ϵ=1-n_s$ to first order. This identification must not be made with the potential slow-roll $\eta$ without converting conventions.