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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.

In the Spatially flat FLRW metric, neglect metric perturbations and write $\varphi =\bar{\varphi }+\delta \varphi$. The kinetic invariant is then exactly quadratic in field derivatives:Subtracting $\bar{X}={\dot{\bar{\varphi }}}^2/2$ gives the second-order expansionIn the derivative-dominated approximation, $\delta \varphi =-\dot{\bar{\varphi }}\zeta /H$ with $\dot{\bar{\varphi }}/H$ treated as slowly varying. Thus $\delta \dot{\varphi }\simeq -\dot{\bar{\varphi }}\dot{\zeta }/H$ andThis uses the linear flat-to-comoving slicing relation. Differentiating its slowly varying coefficient would add terms involving undifferentiated $\zeta$; it is not an exact nonlinear gauge transformation, and those neglected terms do not alter the leading pure ${\dot{\zeta }}^3$ coefficient computed here.

Only $P_{,XX}\delta X_1\delta X_2$ and $P_{,XXX}\delta X_1^3/6$ contribute to the requested three-time-derivative term. Therefore the directly evaluated cubic coefficient isThis is the derivative-cubic coefficient of a noncanonical scalar. For a physical $P(X,\varphi )$ scalar field theory, differentiating $\rho =2X{P}_{,X}-P$ at fixed $\varphi$ gives ${\rho }_{,X}=P_{,X}+2X{P}_{,XX}$. Its sound speed is consequentlyThe minus sign printed in the denominator of $c_s^2$ reverses the asserted coefficient identity; it must be a plus. The printed background equation also lacks a power: it is $3H^2=\rho$ in the stated reduced-Planck units. A stable propagating scalar has positive time-kinetic coefficient $P_{,X}+2X{P}_{,XX}$ and positive gradient coefficient $P_{,X}$, as summarized by noncanonical scalar kinetic stability.

The unfactored expression for $C$ stays meaningful if $P_{,XX}=0$, although $\mathcal{A}$ separately then need not exist. For a canonical kinetic term with no higher kinetic derivatives, $C=0$. The cubic curvature action for a P(X, phi) scalar field can include other vertices and gravitational terms; they are outside this requested derivative coefficient.

In cosmic time, all three Friedmann-Lemaitre-Robertson-Walker metrics can be writtenwhereThe closed spatial slice is a three-sphere of radius $a(t)$, soThe complete flat and open spatial slices are noncompact and have

For a homogeneous fluid in the Spatially flat FLRW metric, the equations in part (a) reduce to the cosmological perfect-fluid continuity equationwhile homogeneity makes the spatial relativistic Euler equation automatic. The Einstein field equations give the flat Friedmann equation and Friedmann acceleration equation,equivalently $\ddot{a}/a=-4\pi G(\rho +3P)/3$.

A suitable Spatially flat FLRW metric isBoth source and detector are comoving, so cosmological redshift stretches the wave period in proportion to the scale factor. If $f_e$ is the local source frequency at emission and $f_o$ is the frequency measured in Earth's local frame, thenThe same relation applies to gravitational waves propagating on the FLRW background in the geometric-optics regime.