Cubic time-derivative curvature bispectrum ($H_{\mathrm{int}}\propto {\dot{\zeta }}^3$)

Put $C=ϵ+1-c_s^2$. For constant $H,ϵ,c_s>0$ and cubic interaction $H_{\mathrm{int}}=-M_{\mathrm{Pl}}^2\int d^{3}x{a}^3ϵC{\dot{\zeta }}^3/(H{c}_s^2)$, the Bunch-Davies vacuum modes $u_k=H(1+ik{c}_s\tau )e^{-ik{c}_s\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{c}_s{k}^3}$ giveTo derive this primordial bispectrum, transform $dt=ad\tau$, $\dot{\zeta }={\zeta }^{\prime }/a$, use the six connected Wick contractions, and integrate ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{i{c}_s(k_1+k_2+k_3)\tau }d\tau =2i/[c_s(k_1+k_2+k_3){]}^3$ in the in-in formalism. The normalized equilateral amplitude is $B(k,k,k)/P(k{)}^2=2C/9$; raw inverse powers of $c_s$ must not be confused with normalized primordial non-Gaussianity.

For $H_{\mathrm{int}}=-\int d^{3}x{a}^3ϵ\mathcal{F}{\dot{\zeta }}^3/H$, $a=-1/(H\tau )$ and stipulated modes $u_k=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵk^3}$, six connected Wick contractions and ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{iK\tau }d\tau =2i/K^3$ yield $B=3H^4\mathcal{F}/(8{ϵ}^2{k}_1{k}_2{k}_3{K}^3)$. Here $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3\delta (\sum {\mathbf{k}}_i)B$. Using unstarred vertex modes with the external-before-vertex prefactor incorrectly reverses the sign. Non-unit sound speed also changes the quadratic modes, so this result is conditional on the specified unit-speed modes.