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The creation and annihilation operators obeyThe two-point correlation function in their vacuum has Fourier amplitude $|f_k{|}^2/a^2$. After cosmological horizon exit, $|k\tau |\ll 1$, and the stated Bunch-Davies vacuum mode and $a=-1/(H\tau )$ giveThe dimensionless power spectrum is consequently

Since $dt=ad\tau$ and $\dot{\varphi }=a^{-1}{\varphi }^{\prime }$, the interaction isto first order in $\lambda$. The tree-level in-in formalism givesThere are two Wick contractions for each choice of the undifferentiated field at the vertex. With $K=k_1+k_2+k_3$, the stated Bunch-Davies vacuum mode obeysThe two powers of $\tau$ from the differentiated modes cancel $a^2=1/(H^2{\tau }^2)$, and the remaining integral isRemoving the momentum-conserving delta function, the bispectrum from a time-derivative cubic scalar interaction is therefore

In the far past, $|ck\tau |\gg 1$, the term $2/{\tau }^2$ is negligible and the rescaled mode is a quantum harmonic oscillator of angular frequency $ck$. The stated choice of its ground state is the Bunch-Davies vacuum.

At late times, after the mode is far outside the Hubble scale, $|ck\tau |\ll 1$. Its variance becomesSince $P_k$ has dimensions of length cubed, the dimensionless power is proportional to $k^3{P}_k$. More precisely, with the standard isotropic Fourier convention,which is independent of the wavenumber $k$. Thus equal logarithmic intervals in $k$ carry equal late-time power. This is the sense in which inflation naturally generates a scale-invariant inflationary power spectrum.

The positive-frequency solution for $u_k=a{f}_k$ is proportional to $e^{-ick\tau }$. Since $a=-1/(H\tau )$, its scalar-field mode function has the stated form $f_k=\mathcal{N}\tau e^{-ick\tau }$. The canonical momentum in conformal time is $\pi =a^2{\phi }^{\prime }$, so the canonical commutation relation requires the Wronskian normalizationSubstitution gives $2ick{a}^2|\mathcal{N}{|}^2{\tau }^2=i$. Hence, up to an irrelevant constant phase,The choice $e^{-ick\tau }$ is the Bunch-Davies vacuum condition at early conformal time.

In one standard Schwinger-Keldysh propagator convention, suppressing the momentum delta function,The two bulk-to-boundary propagators to the late-time insertion areInterchanging every $L$ and $R$ gives the equivalent opposite contour-label convention. Direct differentiation of the supplied Bunch-Davies vacuum mode gives

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

The Gaussian Bunch-Davies vacuum is selected by tilting the early endpoint to $-\infty (1-i\epsilon )$ for integrands $e^{iK\tau }$ with $K>0$. They then decay at the early boundary. The complex-conjugate integrand requires the opposite tilt. Ordered contractions and their contour must be conjugated together.