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In the de Sitter approximation, a canonically rescaled scalar has superhorizon powers $(-\tau {)}^{1/2\pm \nu }$. The physical field has powers $(-\tau {)}^{3/2\pm \nu }$. For $H^2<m^2<2H^2$, both physical branches decay, while the slower rescaled branch grows. The generic late-time dimensionless power spectrum is proportional to $(k/(aH){)}^{3-2\nu }$.

Use $T={\xi }^0$ and ${\xi }^i={\partial }^i\lambda$, and define the physical background Hubble parameter by $H=\dot{a}/(\bar{N}a)$. At first order, the spacetime metric components are $\delta g_{00}=-2{\bar{N}}^2\Phi$, $\delta g_{0i}=a^2{B}_{,i}$ and $\delta g_{ij}=-2a^2(\Psi {\delta }_{ij}+E_{,ij})$. Apply the given passive cosmological gauge transformation component by component. The temporal component changes by $2\bar{N}\dot{\bar{N}}T+2{\bar{N}}^2\dot{T}$; the mixed component changes by ${\bar{N}}^2{T}_{,i}-a^2{\dot{\lambda }}_{,i}$; and the spatial component changes by $-2a\dot{a}T{\delta }_{ij}-2a^2{\lambda }_{,ij}$. HenceThese are the scalar gauge transformations with a background lapse in the paper's sign convention.

Starting in synchronous gauge, preserving both $\Phi =0$ and $B=0$ requiresThe first equation integrates to $\bar{N}T=C(\mathbf{x})$; substituting in the second and integrating givesThe arbitrary lower integration limit is absorbed into $D$. Thus specifying synchronous lapse and shift does not fix the origins of the freely falling clocks or all spatial coordinate labels. This is the residual gauge freedom in synchronous gauge.

During cosmic inflation, choosing these clocks remains arbitrary. For example, the $C$ mode shifts the spatial potential by $HC$ and the density perturbation by $3HC(\bar{\rho }+\bar{P})$. In the de Sitter approximation, $H$ varies little, so a nearly constant metric perturbation can contain a pure coordinate contribution. An inflaton fluctuation or density contrast in synchronous gauge therefore cannot be declared physical solely from its time behavior. Use a uniform-density curvature perturbation or an appropriate field-based gauge-invariant variable, or fix a physical initial slicing. The exact de Sitter case has no evolving homogeneous density clock, which makes density-defined slicing degenerate.

During the standard hot Big Bang, a cold-matter rest frame supplies a convenient additional condition: imposing zero cold-matter velocity removes the nontrivial spatially varying time-shift mode. Before that condition is imposed, the pure-gauge density contrast for constant $w$ is $3H(1+w)C$, which decays as $t^{-1}$ in either radiation or matter domination. In conformal time this is proportional to ${\tau }^{-2}$ during radiation domination and ${\tau }^{-3}$ during matter domination. It can be confused with an independent decaying physical mode unless gauge freedom is fixed. The $D$ mode is a time-independent spatial relabeling and must likewise be fixed by a coordinate or initial-metric convention.

Finally, a scalar density transforms by evaluating the homogeneous scalar at the shifted time:To reach Newtonian gauge from synchronous quantities, impose $\tilde{E}=\tilde{B}=0$. For the nonhomogeneous scalar modes, chooseUsing background stress-energy conservation, $\dot{\bar{\rho }}=-3\bar{N}H(\bar{\rho }+\bar{P})$, gives the synchronous-to-Newtonian density transformation with a lapse:The expression includes the lapse factors because dots refer to the arbitrary time coordinate, not automatically to proper time.

Use exactly the stipulated unit-sound-speed mode functions for this integral. Their late-time values are real and $u_k^{\prime }{}^{\ast }(\tau )=H{k}^2\tau e^{ik\tau }/\sqrt{4ϵk^3}$. ConsequentlyThe vacuum contour lies above the negative real time axis, so $e^{iK\tau }$ decays at its past endpoint. With $s=iK+0^{+}$, repeated differentiation of ${\int }_{-\infty }^0{e}^{s\tau }d\tau =1/s$ givesSubstitution gives the bispectrum with consistent operator ordering and the stated Hamiltonian:The sign is positive in front of $\mathcal{A}$, contrary to the target's negative sign. The printed negative result is obtained algebraically if one uses the unstarred derivatives, $e^{-iK\tau }$, and the conjugate past contour, which gives $-2i/K^3$, while retaining $-2i$ outside. That combination is not the stated vacuum contour or the stated external-before-vertex contraction. A consistently conjugated representation must also conjugate the external prefactor and has the positive result above. The unit-speed cubic time-derivative bispectrum records these conventions explicitly.

The approximation treats $H$, $ϵ$ and the interaction coefficients as constant, uses the de Sitter approximation, retains only the stipulated vertex at tree level, and takes $k_i|{\tau }_{\mathrm{final}}|\to 0$ after doing the convergent early-time integral. The $i0$ prescription must be kept while imposing the past boundary condition; an arbitrary early-time cutoff leaves spurious oscillatory boundary terms. For a physical noncanonical quadratic action with $c_s\ne 1$, the mode functions also change to $H(1+i{c}_{s}k\tau )e^{-i{c}_{s}k\tau }/\sqrt{4ϵc_s{k}^3}$. They were not supplied here; the displayed calculation is conditional on the paper's specified modes, rather than a full self-consistent noncanonical power-spectrum calculation.

With the stipulated power spectrum $P_{\zeta }(k)=H^2/(4ϵk^3)$, an equilateral bispectrum configuration gives $B(k,k,k)/P(k{)}^2=2\mathcal{F}/9$. For the local-style normalization evaluated only at that triangle, $B(k,k,k)=(18/5)f_{\mathrm{NL}}^{\mathrm{effective}}P(k{)}^2$, this meansThis is a normalization convention for an equilateral amplitude, not a claim that the shape is local. In a squeezed bispectrum configuration, the ratio to the long-short power product is suppressed by $(k_L/k_S{)}^2$. Small sound speed or large higher-kinetic derivatives can make $|\mathcal{F}|$ large enough for potentially observable primordial non-Gaussianity, whereas $c_s\to 1$ with finite $\mathcal{A}$ suppresses this vertex. Detectability is conditional on amplitudes, the other cubic interactions, and perturbative control; it cannot be asserted from the given constants alone.

In the constant-$H$ de Sitter approximation, $a=-1/(H\tau )$ with $\tau <0$, so $a^{\prime \prime }/a=2/{\tau }^2$. The massive rescaled mode obeysFor a fixed mass in the specified open interval, at sufficiently late superhorizon scales with $|k\tau |\ll 1$, the $k^2$ term is negligible compared with the inverse-square term in the limiting equation. Try $v\propto (-\tau {)}^p$. The indicial equation is $p(p-1)+m_{\chi }^2/H^2-2=0$, givingThus both independent branches of the massive de Sitter superhorizon scalar mode are present inThe coefficients are constant in time and fixed by initial conditions. Writing $-k\tau >0$ avoids ambiguous fractional powers of the negative number $k\tau$; a chosen complex phase in the printed notation can be absorbed into the coefficients.

For the specified mass range, $1/2<\nu <\sqrt{5}/2<3/2$. The physical fluctuation is $\delta {\chi }_{\mathbf{k}}=v_{\mathbf{k}}/a=-H\tau v_{\mathbf{k}}$, so its two branches behave as $(-\tau {)}^{3/2-\nu }$ and $(-\tau {)}^{3/2+\nu }$. Both exponents are strictly positive. Therefore both physical-field branches decay as $\tau \to 0^{-}$:The slower branch of $v$ itself grows because $1/2-\nu <0$, but the expanding scale factor more than cancels that growth. A spatially homogeneous background field of this mass satisfies the corresponding $k=0$ equation and also has these decaying powers.