Maldacena's consistency relation (eqs. 4.5–4.7): $\langle\zeta\zeta\zeta\rangle/(P_\zeta(k_1)P_\zeta(k_3))\to-n_s$ as $k_3\to0$

Let $\dot\rho_*>0$, $\dot\phi_*\ne0$, $\ddot\phi_*$ real, and $k>0$. Let $B_\zeta(k_1,k_2,k_3)$ be the leading slow-roll three-scalar correlator of eqs. (4.5)–(4.6), $P_\zeta$ the scalar power spectrum (2.20), and $n_s$ the scalar tilt (2.22). Then, as $(k_2,k_3)\to(k,0)$ through pairs for which $(k,k_2,k_3)$ is a momentum triangle with positive sides,

In words: when one momentum is much smaller than the other two, the three-point function of $\zeta$ equals $-n_s$ times the product of the two-point functions, $\langle\zeta_{\vec k_1}\zeta_{\vec k_2}\zeta_{\vec k_3}\rangle\sim-n_s\langle\zeta_{\vec k_3}\zeta_{-\vec k_3}\rangle'\langle\zeta_{\vec k_1}\zeta_{\vec k_2}\rangle'$. This is the single-field consistency relation, eq. (4.7), recovered from the general formula (4.5)–(4.6).

Formalization Note $M_{pl}=1$, $H_*=\dot\rho_*$, the factor $(2\pi)^3\delta^3(\sum\vec k_i)$ is dropped, and the horizon-crossing times $*$, $*'$ are identified at leading order in slow roll. The limit is taken in $\mathbb R^2$ within the momentum-triangle set; momentum conservation forces $k_2\to k$.