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Cosmological perturbation theory studies departures of the metric tensor and matter fields from a homogeneous and isotropic FRW metric, order by order. Linear cosmological perturbation theory keeps only first-order terms. Higher orders describe nonlinear coupling between perturbations. Nonlinear cosmological perturbation theory also uses expansions that allow finite amplitudes, such as the long-wavelength approximation in cosmology.

Nonlinear cosmological perturbations allow finite fluctuations of the spatial scale, density and other fields. Expanding in spatial gradients can remain useful even when a linear expansion in their amplitudes is invalid. The long-wavelength approximation in cosmology is one such organization of the equations.

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.

The long-wavelength approximation in cosmology organizes equations in powers of $ϵ\sim k/(aH)\ll 1$, while allowing finite perturbation amplitudes. For fields and lapse varying on the same large spatial scale, a spatial derivative is of order $ϵ$ relative to a temporal expansion scale. Intrinsic spatial curvature, scalar gradient energy, and second spatial derivatives of the lapse are therefore second order and can be neglected at leading order. First-gradient constraints must still be retained: they relate neighboring locally homogeneous patches. During cosmic inflation, physical wavelengths stretch while the Hubble scale changes slowly, improving this expansion. Sharp spatial features or independently large anisotropic expansion require additional care.

Set the shift to zero, let $S^i{}_j={\tilde{K}}^i{}_j$, $S^2=S^i{}_j{S}^j{}_i$, and keep the lapse explicit. The normal scalar momentum is $\Pi =\dot{\varphi }/N$, and the shear-inclusive long-wavelength scalar equations obtained from the given Einstein and scalar equations areTerms with two spatial gradients have been omitted in the time equations; the last equation is the retained first-gradient momentum constraint. The Friedmann-type equation is the Hamiltonian constraint. To check the Hubble evolution, the trace equation gives $\dot{H}=-3N{H}^2+8\pi GNV$; inserting the Hamiltonian constraint gives the displayed kinetic and cosmological shear terms. Shear energy is not a spatial-gradient correction and cannot be discarded merely because the wavelength is large.

The trace-free evolution integrates immediately:The arbitrary spatial integration tensor is trace free and must obey the initial constraints. This inflationary shear damping suppresses anisotropic expansion exponentially during sustained inflation; its contribution $S^2$ falls as $a^{-6}$. It does not force the spatial conformal metric to be exactly flat: a frozen anisotropic shape or tensor perturbation can survive with negligible time-dependent cosmological shear.

After this decaying cosmological shear mode and its first-gradient contribution become negligible, the equations simplify to $3H^2=8\pi G({\Pi }^2/2+V)$ and $D_{i}H=-4\pi G\Pi D_i\varphi$. Spatially differentiating the first relation givesSubstitute the momentum constraint and divide by $\Pi \ne 0$ to obtain $D_i\Pi =-(3H+V_{,\varphi }/\Pi )D_i\varphi$. The time scalar equation, together with $\dot{\varphi }=N\Pi$, similarly becomes $\dot{\Pi }=-(3H+V_{,\varphi }/\Pi )\dot{\varphi }$. Finally $\dot{H}=-4\pi G\Pi \dot{\varphi }$. Thus the requested single-clock relations areThese retain arbitrary lapse and reduce to proper-time expressions when $N=1$. The division assumes the scalar is a valid local clock; turning points with $\Pi =0$ need another regular variable or a limiting treatment. The shear-neglect step, rather than the long-wavelength expansion alone, is what permits these single-clock spatial relations.

Retain zeroth and first spatial-gradient orders in the long-wavelength approximation in cosmology. Spatial Ricci curvature, $D_i{D}_{j}N$, $q^2$ and scalar anisotropic stress are of second gradient order. Put $A^i{}_j={\tilde{K}}^i{}_j$ and $A^2=A^i{}_j{A}^j{}_i$. The supplied Einstein field equations then reduce toThe scalar-field stress projections show that the momentum constraint is a first-gradient equation and must be retained. The square $A^2$ is not a spatial derivative and cannot yet be dropped merely by invoking the long-wavelength approximation in cosmology. The time-derivative dot on the trace-free evolution equation is visible in the PDF and missing from the TeX aid.

Using $NH=\dot{a}/a$, the last equation becomes ${\partial }_t(a^3{A}^i{}_j)\simeq 0$. HenceThe time-independent spatial tensor must also satisfy the momentum constraint and the symmetry inherited from $K_{ij}$. This is inflationary shear damping: expanding regions lose anisotropic expansion, without requiring the frozen spatial metric shape to vanish. The matter evolution, useful below, is $\dot{\varphi }=N\Pi$ and $\dot{\Pi }\simeq -N(3H\Pi +V_{,\varphi })$, the leading Klein-Gordon equation in each local region.

Decompose the mixed extrinsic curvature of a spatial hypersurface into $K^i{}_j=\frac{1}{3}K{\delta }^i{}_j+{\tilde{K}}^i{}_j$. In the long-wavelength approximation in cosmology, the inflationary shear damping law makes ${\tilde{K}}^i{}_j$ fall as the inverse local volume. During cosmic inflation, many e-folds therefore suppress the anisotropic expansion, while $K$ still measures the local isotropic expansion through the local volume Hubble parameter.

Neglecting spatial gradients at leading order gives approximately independent expanding patches. Restrict also to scalar cosmological perturbations, and neglect the independent frozen tensor cosmological perturbation. The remaining spatial scale can be written as a background scale times a local fluctuation:This is the nonlinear scalar metric in a cosmological gradient expansion. The exponential keeps the metric positive and permits finite $\zeta$; no expansion in its amplitude is needed. Scalar lapse and shift perturbations describe the remaining freedom in time slicing and spatial threading. Choose $N=1+\Psi$ and $N_i=-a_{\mathrm{bg}}^2{\partial }_{i}B$. The negative shift vector convention then produces the positive mixed term $2a_{\mathrm{bg}}^2{\partial }_{i}Bdtd{x}^i$.

The exact completed-square form hasThe quadratic shift expression must be understood as this norm. If the raised derivative in the displayed ansatz is intended as a flat derivative, the corresponding factors of the spatial scale are needed. In either convention the shift-square term is of second gradient order, and the leading long-wavelength approximation in cosmology is unaffected.

Damping of shear motivates locally isotropic scalar expansion, not the disappearance of every tensor perturbation. A time-independent anisotropic shape can survive; the conformally flat scalar ansatz additionally neglects that tensor sector.