Planck normalization of a cubic curvature interaction ($H_{\mathrm{int}}\propto M_{\mathrm{Pl}}^2{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$)

The comoving curvature perturbation is dimensionless. With spatial coordinates and time carrying inverse mass dimensions, the cubic interaction $\int dt{d}^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ needs a mass-squared coefficient. Its standard gravitational normalization is $M_{\mathrm{Pl}}^2$. Six De Sitter curvature mode functions supply $M_{\mathrm{Pl}}^{-6}$; this vertex then supplies $M_{\mathrm{Pl}}^2$, leaving the physical $M_{\mathrm{Pl}}^{-4}$ curvature primordial bispectrum. The Wick-pairing multiplicity for a zeta zeta-prime-squared vertex is independently six: three choices of undifferentiated external leg times two differentiated-leg assignments.