Eqs. (4.15) → (4.17): squeezed limit of $\langle\gamma^{s_1}\gamma^{s_2}\gamma^{s_3}\rangle$

Let $\hat n\in\mathbb R^3$ be a unit vector and $\vec k_2\ne0$, with $k_2=|\vec k_2|$. Let $\epsilon^1,\epsilon^2,\epsilon^3$ be real symmetric traceless $3\times3$ matrices with $\epsilon^1\hat n=0$, $\epsilon^2\vec k_2=0$ and $\epsilon^3\vec k_2=0$. These are the polarizations of the soft graviton and of the two hard gravitons, the latter in the limiting configuration $\vec k_3\to-\vec k_2$. Let $\dot\rho_*$ be real. Put $\vec k_1=t\hat n$ and $\vec k_3=-\vec k_1-\vec k_2$. Then, with $B_{\gamma\gamma\gamma}$ the correlator (4.15),

For equal hard polarizations, $\epsilon^2_{ij}\epsilon^3_{ij}=2\delta_{s_2s_3}$, and the right-hand side is the coefficient in (4.17). The statement is the agreement of (4.15) with (4.17) in the overlapping region.

Formalization Note $M_{pl}=1$, and the horizon-crossing times $*$, $*'$ are identified at leading order. The polarizations are held fixed along the limit, and the hard polarizations are taken transverse to $\vec k_2$, the limiting direction of $\pm\vec k_3$.