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In the limit where scalar field fluctuations dominate metric fluctuations, put $\zeta =-H\delta \varphi /\dot{\bar{\varphi }}$ and neglect derivatives of slowly varying background coefficients. Splitting $\delta X=\delta X_1+\delta X_2$ givesThe cubic terms $P_{XX}\delta X_1\delta X_2+P_{XXX}\delta X_1^3/6$ therefore yieldThe expression before factoring remains well-defined when $P_{XX}=0$. These interactions can generate substantial primordial non-Gaussianity when the Sound speed of a P(X, phi) scalar perturbation is small.

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.

Write $\varphi =\bar{\varphi }+\phi$ and evaluate derivatives of $P$ on the homogeneous backgroundThe first- and second-order changes in $X$ areAfter using the background equation to remove the linear action, the quadratic action of the P(X, phi) scalar field theory iswhereVarying gives

The Sound speed of a P(X, phi) scalar perturbation isAt leading slow variation, take $H$, $c_s$, and the kinetic coefficients as nearly constant and neglect the effective mass and their logarithmic derivatives. The Fourier equation then becomesFor $u_k=a{\phi }_k$ and de Sitter spacetime $a^{\prime \prime }/a=2/{\tau }^2$, this isTwo independent solutions areThe Bunch-Davies vacuum selects the positive-frequency behavior $e^{-i{c}_{s}k\tau }$ as $-c_{s}k\tau \to \infty$. Canonical normalization of $v=a\sqrt{A}\phi$ gives the general amplitude; with the field normalization $P_X\simeq 1$, so $A\simeq c_s^{-2}$, it reduces, up to an overall phase, to