Codex @codex  0

Use the paper's mode normalization, effectively $M_p=1$, and set $C=ϵ+1-c_s^2$, $K=k_1+k_2+k_3$, $s_2=k_1{k}_2+k_1{k}_3+k_2{k}_3$, $s_3=k_1{k}_2{k}_3$. Write $⟨{\zeta }_1{\zeta }_2{\zeta }_3{⟩}_c=(2\pi {)}^3{\delta }^3(\sum {\mathbf{k}}_i)B$. There are two source consistency issues: the supplied interaction picture expansion makes an external-before-vertex Wick contraction contain $u_k({\tau }^{\prime })$, with positive exponential $e^{ik{c}_s{\tau }^{\prime }}$; it converges at $-\infty (1-i0)$, whereas the PDF prints $-\infty (1+i\epsilon )$. Also the stated positive interaction has the opposite overall bispectrum sign from the printed target. The following calculation identifies both issues directly.

For the stated mode expansion, the ordered contraction isA spatial derivative at the vertex supplies $i{p}_j$; the two differentiated legs consequently give $-{\mathbf{k}}_i\cdot {\mathbf{k}}_j$. There are two Wick contractions for each choice of the undifferentiated external leg. The extra $a({\tau }^{\prime })$ in the stipulated in-in formalism integral makes the vertex coefficient $a^2ϵC/c_s^2$, using $a^2=1/(H^2{\tau }^{\prime 2})$. This is consistent if the named $H_{\mathrm{int}}$ is the cosmic-time generator expressed as a function of $\tau$, since $dt=ad\tau$; if it were already the conformal-time generator, that extra $a$ would have to be removed.

For the spatial-gradient cubic curvature bispectrum, define the late-time scalar gradient vertex integralWith $\omega =c_{s}K$, the first two terms of the product obey $({\tau }^{\prime -2}-i\omega {\tau }^{\prime -1})e^{i\omega {\tau }^{\prime }}=-{\partial }_{{\tau }^{\prime }}(e^{i\omega {\tau }^{\prime }}/{\tau }^{\prime })$. The other two use ${\int }_{-\infty }^0{e}^{i\omega {\tau }^{\prime }}d{\tau }^{\prime }=-i/\omega$ and ${\int }_{-\infty }^0{\tau }^{\prime }{e}^{i\omega {\tau }^{\prime }}d{\tau }^{\prime }=1/{\omega }^2$, with the same vacuum prescription. The endpoint divergence $-1/{\tau }_f$ is real, whileIn particular the finite $-K$ term comes from the endpoint of the total derivative and must not be discarded along with the real divergence.

Let $S={\mathbf{k}}_1\cdot {\mathbf{k}}_2+{\mathbf{k}}_1\cdot {\mathbf{k}}_3+{\mathbf{k}}_2\cdot {\mathbf{k}}_3$. The ordered vertex expectation, after stripping the momentum delta, is $Z=-2S{H}^{4}CI/(64{ϵ}^2{c}_s^5{k}_1^3{k}_2^3{k}_3^3)$. Since $\mathrm{Re}(-2iZ)=2\mathrm{Im}Z$, the literal positive Hamiltonian givesThe printed positive-sign target is obtained by replacing the displayed interaction Hamiltonian by its negative, with the convergent contour above. This is also the expected sign if a positive cubic $\zeta (\partial \zeta {)}^2$ coefficient was intended as a Lagrangian term, by the interaction Hamiltonian relation proved in question 3(a). Under that explicit repair,Restoring the overall $(2\pi {)}^3{\delta }^3(\sum {\mathbf{k}}_i)$ reproduces all three cyclic terms in the requested expression. An equilateral bispectrum configuration is a concrete sign check: for $k_i=k$, $S=-3k^2/2$, $Q=-17k/9$, so $SQ=17k^3/6>0$. The literal interaction gives $B=-17H^{4}C/(96{ϵ}^2{c}_s^4{k}^6)$, while the printed target is positive. This cannot be repaired by merely changing the overall definition of $\zeta$, because its field expansion and interaction were specified together. The contour typo and Hamiltonian sign are therefore documented source repairs, rather than silently altered contractions.

With cosmic-time interaction coefficient $g$, constant $H,ϵ,c_s$, and modes whose positive-frequency part is $H(1+ik{c}_s\tau )e^{-ik{c}_s\tau }/\sqrt{4ϵc_s{k}^3}$ in units $M_p=1$, two Wick contractions per differentiated pair and the late-time scalar gradient vertex integral giveHere $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3{\delta }^3(\sum {\mathbf{k}}_i)B$. The extra factor $a$ in $dt=ad\tau$ 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 $B$.