= Spatial-gradient cubic curvature bispectrum
{title2=$H_{\rm int}=g\int d^3x\,a\zeta(\partial\zeta)^2$}
With cosmic-time interaction coefficient $g$, constant $H,\epsilon,c_s$, and modes whose positive-frequency part is $H(1+ikc_s\tau)e^{-ikc_s\tau}/\sqrt{4\epsilon c_sk^3}$ in units $M_p=1$, two <Wick contractions> per differentiated pair and the <late-time scalar gradient vertex integral> give
$$
B=-\frac{gH^4}{16\epsilon^3c_s^2\prod_i k_i^3}\left(\sum_{i<j}\mathbf k_i\cdot\mathbf k_j\right)\left[-K+\frac{s_2}{K}+\frac{s_3}{K^2}\right].
$$
Here $\langle\zeta^3\rangle_c=(2\pi)^3\delta^3(\sum\mathbf k_i)B$. The extra factor $a$ in $dt=a\,d\tau$ must be included. A positive $g$ yields a negative bispectrum in an <equilateral bispectrum configuration>; changing the sign of the interaction changes the sign of $B$.
Back to article page
