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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.