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Under the stated linear field expansion, the Bunch-Davies vacuum is a centered Gaussian state. By Wick theorem, vacuum correlators reduce to pair contractions; an odd product has no complete pairing. In particular,This holds for arbitrary spatial arguments as a regulated correlator or distribution. One can also use number parity: $\mathcal{P}=(-1{)}^{\hat{N}}$ leaves the vacuum unchanged and sends every creation or annihilation operator, hence the linear field $\hat{f}$, to its negative. An odd product consequently has expectation equal to its own negative. Thus vanishing odd correlators of a free Gaussian field givesFor a coincident product, impose a momentum regulator or smear the field before taking the expectation; the zero result holds for every such symmetry-preserving regulator. It is not a claim that the unregulated local power is an ordinary finite random variable. Interactions, a non-Gaussian state, or inclusion of a nonzero background mean can invalidate the odd-moment conclusion. Within this free perturbation calculation, the primordial bispectrum vanishes.

Use $⟨{\phi }_G(\mathbf{k}){\phi }_G({\mathbf{k}}^{\prime })⟩=(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}+{\mathbf{k}}^{\prime })P_{\phi }(k)$. For nonzero external momenta, the quadratic term in local-type primordial non-Gaussianity isInsert it in one of the three external fields. Wick theorem supplies two connected Wick contractions at each insertion, whereas the mean subtraction removes the disconnected zero-mode contribution. Thus the leading primordial bispectrum isLinear propagation by the cosmological Poisson equation gives $P_m(k)=\alpha (k{)}^2{P}_{\phi }(k)$ and multiplies the primordial bispectrum by ${\alpha }_1{\alpha }_2{\alpha }_3$. Defining the matter three-point function by its momentum-conserving Dirac delta function times $B_m$, the primordial contribution to the matter bispectrum isThe nonlinear gravitational contribution from standard perturbation theory in cosmology is excluded here.

In a squeezed bispectrum configuration, $k_1=k_L\ll k_2\simeq k_3=k_S$, soFor a nearly scale-invariant potential, $P_{\phi }(k_L)\gg P_{\phi }(k_S)$, so the second term is subleading andRelative to the product of matter power spectra, the squeezed response is enhanced by $k_L^{-2}$. Its sign is the sign of $f_{\mathrm{NL}}$. This is the three-point counterpart of the conditional short-scale variance modulation by local non-Gaussianity.

For an attractor with one adiabatic clock and a regular Bunch-Davies vacuum, a long comoving curvature perturbation ${\zeta }_L$ acts at leading order as a spatial dilation. The single-field inflation squeezed-limit consistency relation therefore fixesFor the local convention $\zeta ={\zeta }_G+(3/5)f_{\mathrm{NL}}({\zeta }_G^2-⟨{\zeta }_G^2⟩)$, comparison with its squeezed primordial bispectrum givesThe matter-era negative gravitational potential is $\phi =3\zeta /5$, so this is also the conventional $f_{\mathrm{NL}}$ of part (i). In single-field slow-roll inflation, the spectral tilt is small and the effective local coupling is of slow-roll size, typically of order $10^{-2}$ for a tilt of a few percent. This is a statement about the normalized coupling: the squeezed primordial bispectrum still contains the large long-mode power spectrum.

Formally putting this small coupling into the local template of part (ii) predicts only a tiny correction on accessible scales, rather than an order-unity local signal. For a physical halo abundance there is a stronger qualification. The consistency-relation modulation at fixed coordinate wavelength is a dilation; the coordinate size of a fixed physical halo must be dilated too. Those responses cancel. This is the absence of primordial scale-dependent halo bias in single-clock inflation: the leading consistency-relation term produces no physical $k^{-2}$ halo bias. The first physical long-mode response contains two spatial derivatives, so dividing it by ${\delta }_L\propto k_L^2{\phi }_L$ does not produce the local-template enhancement. Thus the leading primordial large-scale galaxy power spectrum remains the Gaussian-bias result, apart from ordinary transfer, bias, and projection effects.

The cancellation can be seen with a top-hat filter. At fixed local time, a constant long curvature rescales physical lengths by $e^{{\zeta }_L}$, so the coordinate radius of a fixed physical halo is $R_c=e^{-{\zeta }_L}{R}_0$, and the coordinate short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. Changing variables $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ givesThus the smoothed field and its variance at fixed physical mass are unchanged by the constant long mode.

The assumptions matter: a non-attractor such as ultra-slow-roll inflation, additional fluctuating fields, or an excited initial state need not obey this squeezed-limit conclusion. A sizable physical local modulation would test the stated single-clock attractor assumptions, rather than every possible one-field model.

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.

The operator ${\dot{\varphi }}^2\varphi$ is not invariant under a scalar-field shift symmetry because one field is undifferentiated. An arbitrary large coefficient would therefore radiatively generate equally unsuppressed nonderivative operators, including a mass and a steep scalar potential, and would spoil the approximate shift symmetry and flat potential needed for single-field slow-roll inflation. In a technically natural slow-roll model its coefficient must consequently be small, so it does not produce parametrically large primordial non-Gaussianity. Treated merely as an effective spectator-field interaction, it does generate the tree-level primordial bispectrum computed below, whose size is controlled by the dimensionless interaction strength at the Hubble scale; taking that strength large abandons the controlled slow-roll premise.

With the Fourier transform convention $\varphi (\mathbf{k})=\int d^{3}x{e}^{-i\mathbf{k}\cdot \mathbf{x}}\varphi (\mathbf{x})$, scale invariance of the position-space three-point function givesBecause ${\delta }^{(3)}(\mu {\sum }_i{\mathbf{k}}_i)={\mu }^{-3}{\delta }^{(3)}({\sum }_i{\mathbf{k}}_i)$, the reduced primordial bispectrum must satisfyIn the result of part b, the bracket has momentum degree five, while $k_1^3{k}_2^3{k}_3^3{K}^2$ has degree eleven. Its total degree is therefore $-6$, exactly as required.

Statistical homogeneity forces a scalar two-point function to have momenta $(\mathbf{k},-\mathbf{k})$. Statistical isotropy then makes it a function only of $|\mathbf{k}|$, so it is parity even without using perturbation theory.

For a scalar three-point function, momentum conservation gives ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$, so all three vectors lie in one plane. A rotation by $\pi$ around the normal to that plane sends every ${\mathbf{k}}_a$ to $-{\mathbf{k}}_a$. Rotational invariance therefore identifies a triangle with its parity reverse, proving nonperturbatively that the scalar primordial bispectrum is parity even.

A rotationally invariant local parity-odd three-scalar vertex must contain a Levi-Civita symbol contracted with three spatial momenta. Its momentum-space factor is proportional tobecause momentum conservation makes the momenta linearly dependent. The equivalent position-space expression is a total divergence; its spatial integral vanishes under the stated vanishing boundary condition. Thus a parity-odd interaction of three scalar fields contributes nothing to the action.

Substitution of $\delta N=C\phi$ and conversion to conformal time turn the interaction intoTreat $C$ and $H$ as constant at leading slow-roll order and use the massless de Sitter modeFor $K=k_1+k_2+k_3$, the in-in formalism time integral at a vertex with leg $i$ undifferentiated is proportional toSumming the three choices of undifferentiated leg gives the primordial bispectrumThe sign follows from the interaction sign displayed in the question and $H_I=-L_{\mathrm{int}}$ at this order.

At cubic order, the expansion of $P(X,\varphi )$ produces the schematic operatorsThe derivative self-interactions involving $P_{XX}$ and $P_{XXX}$ are generally largest when $c_s\ll 1$, while coefficients containing explicit $\varphi$ derivatives of a slowly varying $P$ are commonly slow-variation suppressed. The former therefore tend to dominate primordial non-Gaussianity.

The terms contributing to $\phi {\phi }^{\prime 2}$ areThusEquivalently, if the complete time-dependent coefficient is called $\lambda$, then $\lambda (\tau )=a^2{\lambda }_0$.

Treat ${\lambda }_0$ as constant at leading slow variation. To cubic order the corresponding interaction Hamiltonian isFor $q_i=c_s{k}_i$, the mode function above satisfiesWriting $K=k_1+k_2+k_3$, the needed regulated integral isThere are three choices for the undifferentiated leg and two contractions interchanging the differentiated legs. The stated in-in formalism formula therefore giveswithThe powers of $c_s$ cancel for this vertex with the normalization specified in part (i). Reversing the convention for the sign of $H_I$ reverses the displayed overall sign but not the momentum shape of the primordial bispectrum.

The comoving curvature perturbation is dimensionless. With spatial coordinates and time carrying inverse mass dimensions, the cubic interaction $\int dt{d}^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ needs a mass-squared coefficient. Its standard gravitational normalization is $M_{\mathrm{Pl}}^2$. Six De Sitter curvature mode functions supply $M_{\mathrm{Pl}}^{-6}$; this vertex then supplies $M_{\mathrm{Pl}}^2$, leaving the physical $M_{\mathrm{Pl}}^{-4}$ curvature primordial bispectrum. The Wick-pairing multiplicity for a zeta zeta-prime-squared vertex is independently six: three choices of undifferentiated external leg times two differentiated-leg assignments.

For a scale-invariant primordial bispectrum, the displayed shape function is invariant under common rescaling of all three wavenumbers. It isolates triangle dependence from the overall power spectrum amplitude. It indicates intrinsic shape preference but alone does not determine observational signal-to-noise, which also depends on transfer functions, covariance and survey weights.

Linear cosmological transfer maps the primordial bispectrum to the reduced CMB bispectrum. A Fourier representation of the momentum delta function and three Rayleigh plane-wave expansions separate the radial transfer integrals from a Gaunt integral. The six angular factors and three Fourier measures give $(4\pi {)}^6/(2\pi {)}^9=(2/\pi {)}^3$. The observer-position phase cancels by momentum conservation; the remaining radial position is an auxiliary integration variable.

A primordial bispectrum is squeezed when one external Fourier mode has much smaller wavenumber than the other two. Momentum conservation then makes the two short wavevectors almost opposite. This configuration measures the response of short-scale correlations to a long perturbation.