= Solution
At cubic order, the expansion of $P(X,\phi)$ produces the schematic operators
$$
\varphi'^3,quad
\varphi'(\nabla\varphi)^2,quad
\varphi\varphi'^2,quad
\varphi(\nabla\varphi)^2,quad
\varphi^2\varphi',quad
\varphi^3.
$$
The derivative self-interactions involving $P_{XX}$ and $P_{XXX}$ are generally largest when $c_s\ll1$, while coefficients containing explicit $\phi$ derivatives of a slowly varying $P$ are commonly slow-variation suppressed. The former therefore tend to dominate <primordial non-Gaussianity>.
The terms contributing to $\varphi\varphi'^2$ are
$$
a^4P_{X\phi}\varphi\,\delta_2X
+\frac{a^4}{2}P_{XX\phi}\varphi(\delta_1X)^2.
$$
Thus
$$
S_3\supset\int d\tau\,d^3x\,a^2\lambda_0
\varphi\varphi'^2,
\qquad
\boxed{\lambda_0=\frac12(P_{X\phi}+2\bar XP_{XX\phi})}.
$$
Equivalently, if the complete time-dependent coefficient is called $\lambda$, then $\lambda(\tau)=a^2\lambda_0$.
Treat $\lambda_0$ as constant at leading slow variation. To cubic order the corresponding <interaction Hamiltonian> is
$$
H_I=-a^2\lambda_0
\int_{\mathbf p_1\mathbf p_2\mathbf p_3}
(2\pi)^3\delta^{(3)}(\mathbf p_1+\mathbf p_2+\mathbf p_3)
\varphi_{\mathbf p_1}\varphi'_{\mathbf p_2}\varphi'_{\mathbf p_3}.
$$
For $q_i=c_sk_i$, the mode function above satisfies
$$
\varphi_k(0)=A_k,
\qquad
\varphi_k^{*\prime}(\tau)=A_kq_k^2\tau e^{iq_k\tau},
\qquad
A_k=\frac{H}{\sqrt{2c_sk^3}}.
$$
Writing $K=k_1+k_2+k_3$, the needed regulated integral is
$$
\int_{-\infty(1-i\epsilon)}^0
(1-iq_i\tau)e^{ic_sK\tau}\,d\tau
=-\frac{i}{c_s}\left(\frac1K+\frac{k_i}{K^2}\right).
$$
There are three choices for the undifferentiated leg and two contractions interchanging the differentiated legs. The stated <in-in formalism> formula therefore gives
$$
\langle\varphi_{\mathbf k_1}\varphi_{\mathbf k_2}\varphi_{\mathbf k_3}\rangle
=(2\pi)^3\delta^{(3)}(\mathbf k_1+\mathbf k_2+\mathbf k_3)
B_\lambda(k_1,k_2,k_3),
$$
with
$$
\boxed{
B_\lambda
=\frac{\lambda_0H^4}{2(k_1k_2k_3)^3}
\sum_{\mathrm{cyc}}
k_j^2k_l^2
\left(\frac1K+\frac{k_i}{K^2}\right)}.
$$
The powers of $c_s$ cancel for this vertex with the normalization specified in part (i). Reversing the convention for the sign of $H_I$ reverses the displayed overall sign but not the momentum shape of the <primordial bispectrum>.
Solved by gpt-5.6-sol high.
Back to article page
