Eqs. (4.13) → (4.14): squeezed limit of $\langle\zeta_{\vec k_1}\gamma^{s_2}_{\vec k_2}\gamma^{s_3}_{\vec k_3}\rangle$

Let $\dot\rho_*>0$, $\dot\phi_*\ne0$, $k>0$, and let $E$ be real (the polarization contraction $\epsilon^{s_2}_{ij}\epsilon^{s_3}_{ij}$). As $(k_1,k_3)\to(0,k)$ through pairs for which $(k_1,k,k_3)$ is a momentum triangle with positive sides,

where $B_{\zeta\gamma\gamma}$ is the correlator (4.13), $P_\zeta$ and $P_\gamma$ are the power spectra (2.20) and (2.29), and $n_t$ is the tensor tilt (2.30).

For equal polarizations, $\epsilon^{s}_{ij}\epsilon^{s}_{ij} = 2$, and this reads $\langle\zeta\gamma\gamma\rangle\sim -n_t\langle\zeta\zeta\rangle'\langle\gamma\gamma\rangle'$. This is the agreement of (4.13) with (4.14).

Formalization Note $M_{pl}=1$, and the horizon-crossing times $*$, $*'$ are identified at leading order. The polarization contraction enters as the parameter $E$.