= Solution
Now impose the stipulated negligible shear and work where $\Pi\ne0$. The <Hamiltonian constraint>, <momentum constraint> and local expansion evolution give
$$
H^2=\frac{8\pi G}{3}(\tfrac12\Pi^2+V),\qquad \partial_iH=-4\pi G\Pi\partial_i\phi,\qquad \dot H=-4\pi GN\Pi^2.
$$
Time differentiation of the first equation, followed by $\dot\phi=N\Pi$, gives $\dot\Pi=-N(3H\Pi+V_{,\phi})$. Spatial differentiation of the same equation and use of the <momentum constraint> gives
$$
\Pi\partial_i\Pi+V_{,\phi}\partial_i\phi=-3H\Pi\partial_i\phi,\qquad
\partial_i\Pi=-(3H+V_{,\phi}/\Pi)\partial_i\phi.
$$
These are the temporal and spatial identities needed for the <nonlinear curvature covector>. Differentiate its definition, keeping the spatially varying <lapse function>:
$$
\dot\zeta_i=-\partial_i(NH)+\left[-4\pi GN\Pi+\frac{NH(3H\Pi+V_{,\phi})}{\Pi^2}\right]\partial_i\phi+\frac H\Pi\partial_i(N\Pi).
$$
The terms proportional to $\partial_iN$ cancel. The terms $-N\partial_iH$ and $-4\pi GN\Pi\partial_i\phi$ cancel by the <momentum constraint>, and the remaining two terms cancel by the spatial momentum identity. Therefore
$$
\boxed{\dot\zeta_i\simeq0}
$$
to the retained gradient order. No homogeneous-lapse assumption was needed.
For linear <adiabatic cosmological perturbations>, expansion of the local scale and <scalar field> gives $\zeta_i=\partial_i[\Psi+\bar H\delta\phi/\bar\Pi]$. For nonzero spatial modes, linearizing the <Hamiltonian constraint> and the <momentum constraint> gives
$$
\delta\rho=\frac{3\bar H}{4\pi G}\delta H=-3\bar H\bar\Pi\delta\phi,\qquad \bar\rho+\bar P=\bar\Pi^2.
$$
Thus
$$
\boxed{\zeta_i\simeq\partial_i\left[\Psi-\frac{\delta\rho}{3(\bar\rho+\bar P)}\right]=\partial_i\zeta.}
$$
This uses the paper's overall sign for the <uniform-density curvature perturbation>; the opposite overall sign is also common. For an adiabatic single-field attractor, <superhorizon conservation of single-field comoving curvature> allows the primordial amplitude to be transported through later epochs without following every short-scale process. This proof specifically discarded the shear contribution to the <momentum constraint> and assumed a usable field clock. It is not a claim that every single-field background conserves every curvature mode: <growing curvature perturbation in ultra-slow-roll inflation> is a non-attractor counterexample, and $\Pi=0$ makes this particular covector undefined.
Back to article page
