Codex @codex  0

Let $\alpha =\log a$, and define $A=3H+V_{,\varphi }/\Pi$ on a patch with $\Pi \ne 0$. In the shear-negligible long-wavelength approximation in cosmology, the relations just derived givewith ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$ and ${\partial }_i\Pi =-A{\partial }_i\varphi$. For the nonlinear curvature covector,commuting coordinate derivatives givesThe two terms proportional to ${\partial }_{i}N$ cancel exactly. The remaining expression isSubstituting the spatial relations cancels the gravitational terms and then the $A$ terms separately. ThereforeThis is nonlinear in perturbation amplitude; only the spatial-gradient truncation and the decaying-shear approximation have been used. Conservation does not require a spatially homogeneous lapse. On a uniform-field slice, ${\zeta }_i=-{\partial }_i\log a$, making its interpretation as a conserved local spatial-curvature perturbation transparent, with the sign convention used here.

For ordinary attractor single-field inflation, conservation means that nonlinear curvature correlations already generated near horizon exit are transported to later superhorizon times without additional local growth. It does not prove that the perturbation is Gaussian: horizon-scale interactions, the nonlinear relation to the field perturbation, and quantum initial conditions can generate a nonzero bispectrum. In canonical slow-roll inflation that primordial non-Gaussianity is usually slow-roll suppressed; the single-field inflation squeezed-limit consistency relation describes the effect of a long adiabatic mode on shorter modes under its stated attractor and initial-state assumptions. Strong local superhorizon primordial non-Gaussianity is therefore not generated merely by evolving this conserved variable. Entropy perturbations in multifield models, or non-attractor single-field phases for which the preceding single-clock and shear-neglect assumptions cannot be applied in this form, evade that inference. The result is a conservation theorem under explicit dynamical assumptions, not a blanket assertion that every single-field model has zero or small primordial non-Gaussianity.

Put $k_1=q\ll k$ and $k_2,k_3\to k$ in the squeezed bispectrum configuration. Then $K\to 2k$ and the three symmetric combinations in the supplied expression have limitsThe first two terms inside the $ϵ$ bracket cancel, leaving $4k^3$. The full bracket is therefore $(\eta +2ϵ)k^3/4$, andThe $(2\pi {)}^3$ is kept in the normalization of $F$ used by the paper; it should not be introduced a second time by changing the primordial bispectrum convention.

Momentum conservation makes the three vectors close into a triangle. As $q/k\to 0$, the two short-wavelength vectors approach equal magnitude and opposite direction, producing a thin, nearly collapsed triangle. In real space the $q$ mode varies slowly over a region containing many short wavelengths, so it modulates their local statistics. For a single adiabatic inflationary clock this is a local spatial dilation, the basis of the single-field inflation squeezed-limit consistency relation. If $\eta$ is the Hubble-flow convention $\dot{ϵ}/(Hϵ)$, the scalar spectral index in Hubble slow-roll parameters gives $\eta +2ϵ=1-n_s$ to first order. This identification must not be made with the potential slow-roll $\eta$ without converting conventions.

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.

The single-field inflation squeezed-limit consistency relation for the comoving curvature perturbation isEquivalently, $B_{\mathcal{R}}/P_{\mathcal{R}}(q)=-(3+k{\partial }_k)P_{\mathcal{R}}(k)$. The scalar shift in parts a--c is an internal symmetry and leaves the hard fields unchanged, so its right-hand side vanishes. A long adiabatic curvature mode instead acts as a spatial dilation of the hard coordinates, and its response is the scale dependence measured by the scalar spectral index.

Single-field slow-roll inflation uses one canonical scalar clock whose potential evolves slowly enough for the slow-roll parameters to remain small. Its adiabatic long mode produces the single-field inflation squeezed-limit consistency relation.