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In the Spatially flat FLRW metric, neglect metric perturbations and write $\varphi =\bar{\varphi }+\delta \varphi$. The kinetic invariant is then exactly quadratic in field derivatives:Subtracting $\bar{X}={\dot{\bar{\varphi }}}^2/2$ gives the second-order expansionIn the derivative-dominated approximation, $\delta \varphi =-\dot{\bar{\varphi }}\zeta /H$ with $\dot{\bar{\varphi }}/H$ treated as slowly varying. Thus $\delta \dot{\varphi }\simeq -\dot{\bar{\varphi }}\dot{\zeta }/H$ andThis uses the linear flat-to-comoving slicing relation. Differentiating its slowly varying coefficient would add terms involving undifferentiated $\zeta$; it is not an exact nonlinear gauge transformation, and those neglected terms do not alter the leading pure ${\dot{\zeta }}^3$ coefficient computed here.

Only $P_{,XX}\delta X_1\delta X_2$ and $P_{,XXX}\delta X_1^3/6$ contribute to the requested three-time-derivative term. Therefore the directly evaluated cubic coefficient isThis is the derivative-cubic coefficient of a noncanonical scalar. For a physical $P(X,\varphi )$ scalar field theory, differentiating $\rho =2X{P}_{,X}-P$ at fixed $\varphi$ gives ${\rho }_{,X}=P_{,X}+2X{P}_{,XX}$. Its sound speed is consequentlyThe minus sign printed in the denominator of $c_s^2$ reverses the asserted coefficient identity; it must be a plus. The printed background equation also lacks a power: it is $3H^2=\rho$ in the stated reduced-Planck units. A stable propagating scalar has positive time-kinetic coefficient $P_{,X}+2X{P}_{,XX}$ and positive gradient coefficient $P_{,X}$, as summarized by noncanonical scalar kinetic stability.

The unfactored expression for $C$ stays meaningful if $P_{,XX}=0$, although $\mathcal{A}$ separately then need not exist. For a canonical kinetic term with no higher kinetic derivatives, $C=0$. The cubic curvature action for a P(X, phi) scalar field can include other vertices and gravitational terms; they are outside this requested derivative coefficient.

For the P(X, phi) scalar field theory, the background pressure is $p=P$ and the energy density is $\rho =2X{P}_X-P$. Differentiating at fixed $\varphi$ gives $p_X=P_X$ and ${\rho }_X=P_X+2X{P}_{XX}$, so the Sound speed of a P(X, phi) scalar perturbation isThis is the propagation speed of the scalar fluctuation; the derivative is taken at fixed field, rather than along an arbitrary background trajectory.

On the fixed expanding metric,Write the comoving curvature perturbation in the paper's convention as $\zeta =-H\delta \varphi /\dot{\bar{\varphi }}$. Neglecting slow-roll derivatives of $H$ and $\dot{\bar{\varphi }}$ yieldsWith $\delta X_1=-2\bar{X}\dot{\zeta }/H$ and $\delta X_2=\bar{X}[{\dot{\zeta }}^2-a^{-2}({\partial }_i\zeta {)}^2]/H^2$, the only cubic terms areThe identities $ϵ=\bar{X}{P}_X/H^2$ and $(1-c_s^2)/c_s^2=2\bar{X}{P}_{XX}/P_X$ give the cubic curvature action for a P(X, phi) scalar fieldIf $P_{XX}=0$, the unfactored expression above supplies the regular limiting result.

For the requested primordial bispectrum, define $C=ϵ(1-c_s^2)/(H{c}_s^2)$ and ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$. The cubic Hamiltonian and $dt=ad\tau$ combine to giveHere the positive sign comes from the two spatial derivatives and $H_{\mathrm{int}}=-L_{\mathrm{int}}$ at cubic order. Wick theorem gives two contractions for each choice of which external leg meets ${\zeta }^{\prime }$. At the external time zero,Thus, after removing the momentum-conserving delta function, the Hamiltonian contraction isPut $A_k=H/\sqrt{4ϵc_s{k}^3}$. Then $u_k(0)=A_k$ and $u_k^{\ast \prime }=A_k{c}_s^2{k}^2\tau e^{ik{c}_s\tau }$. The $\tau$ cancels the $1/\tau$ in $a=-1/(H\tau )$; moreover,Inserting these expressions into the in-in formalism and conjugating the expression inside the real part giveswhere $K=k_1+k_2+k_3$. Conjugating the integrand also conjugates the contour: the original positive-frequency-product integral has lower limit $-\infty (1-i{0}^{+})$, whereas the negative exponential displayed here requires $-\infty (1+i{0}^{+})$. This supplies the convergent interpretation of the paper's integral.