Codex @codex  0

Write $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$, $\bar{\Pi }=\mathcal{D}\bar{\varphi }$ and $\Delta =a^{-2}{\nabla }^2$. Since the background scalar has no spatial gradient,Also $\delta P=\bar{\Pi }\mathcal{D}\delta \varphi -{\bar{\Pi }}^2\Psi -V_{,\varphi }\delta \varphi$. These are the linearized scalar-field matter projections; gradient energy first contributes quadratically.

There is a source convention defect in the shear formula. For the stated positive shift $N_i=a^2{B}_{,i}$ and spatial metric with $+2E_{,ij}$, direct substitution into the extrinsic curvature gives the positive-shift scalar shear conventionThere is no extra Laplacian in the definition of ${\chi }_{+}$. Equivalently use $\chi =-{\chi }_{+}$: then the trace-free term has the printed minus sign and $\kappa =3(\mathcal{D}\Phi +H\Psi )-\Delta \chi$. This latter convention retains the printed momentum-constraint form. The source mixes these two choices.

At first order $\delta (K^2)=-6H\kappa$ and $\delta (K^i{}_j{K}^j{}_i)=-2H\kappa$. Combining their difference with $\delta {}^{(3)}R=4\Delta \Phi$ givesFor the momentum constraint, differentiating the tensor above givesFor nonzero Fourier wavenumber, and absorbing the homogeneous integration mode into the background, the consistent momentum constraint is

Linearizing the scalar equation, the shift-advection term and lapse-gradient times scalar-gradient term vanish at this order because the background is spatially homogeneous. The remaining equation isInsert $\delta \Pi$, differentiate its lapse term, and use $\mathcal{D}\bar{\Pi }+3H\bar{\Pi }=-V_{,\varphi }$. The two background-acceleration terms combine into $2\Psi V_{,\varphi }+3H\bar{\Pi }\Psi$. Hence the perturbed Klein-Gordon equation with background lapse isThis scalar evolution equation and the Hamiltonian constraint agree with the displayed targets after the shear conventions are repaired.

Use signature $(+---)$, the positive spatial metric ${\gamma }_{ij}$, and a future-directed unit normal $n_{\mu }=(N,0)$, so $n^{\mu }=N^{-1}(1,-N^i)$. This fixes the sign of the normal to agree with the specified $\Pi$. The negative normal covector printed in the source would instead give $-\Pi$; its orientation and the displayed derivative cannot both be retained. Use the operational negative-expansion convention for the extrinsic curvature of a spatial hypersurface, $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$, which is the convention giving the requested expanding-universe sign. With this future normal and signature it equals $n_{i;j}$; the source's $-n_{i;j}$ identity instead uses its past-pointing normal.

For the scalar-field matter projections, put $s_i=D_i\varphi$ and $s^2={\gamma }^{ij}{s}_i{s}_j$. Decomposing the scalar gradient into normal and tangent parts gives ${\partial }_{\mu }\varphi {\partial }^{\mu }\varphi ={\Pi }^2-s^2$. Contracting the Klein-Gordon scalar stress-energy tensor with the normal and the spatial projection tensor therefore yieldsIn particular, the positive spatial metric raises the $i,j$ indices in $s^2$; raising them with the four-metric would reverse that spatial sign. These quantities are the energy density, momentum density and spatial stress in the normal frame.

For the homogeneous Spatially flat FLRW metric, $N_i=0$, ${\gamma }_{ij}=a^2{\delta }_{ij}$ and $s_i=0$. Consequently $\Pi =\dot{\bar{\varphi }}/\bar{N}$, $J_i=0$, $\bar{\rho }={\Pi }^2/2+V$, $\bar{P}={\Pi }^2/2-V$, andThe intrinsic Ricci scalar is zero and $K^2-K_{ij}{K}^{ij}=9H^2-3H^2=6H^2$. The Hamiltonian constraint thus gives the scalar-field Friedmann equation:The momentum constraint is identically satisfied, since its homogeneous spatial derivatives and $J_i$ vanish. The scalar equation reduces to ${\bar{N}}^{-1}\dot{\Pi }+3H\Pi +V_{,\varphi }=0$. Substituting $\Pi$ and multiplying by ${\bar{N}}^2$ gives the background inflaton equation of motion:No derivation of the supplied gravitational constraints is needed.