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For a spatially homogeneous minimally coupled scalar field with zero scalar potential, the stress-energy tensor has $\rho =P={\dot{\varphi }}^2/2$. Its field equation is $\ddot{\varphi }+3H\dot{\varphi }=0$, so $a^3\dot{\varphi }$ is constant and its energy density decreases as $a^{-6}$. Thus its homogeneous stress is precisely that of a stiff fluid. Spatial gradients would introduce directional stresses, so this equivalence requires homogeneity.

With negligible shear, a minimally coupled scalar field serving as a clock, and $\Pi \ne 0$, the Hamiltonian constraint and momentum constraint imply $\dot{H}=-4\pi GN{\Pi }^2$ and ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$. Differentiating the local Friedmann relation yields $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$ and ${\partial }_i\Pi =-(3H+V_{,\varphi }/\Pi ){\partial }_i\varphi$. Inserting these identities into ${\dot{\zeta }}_i$ cancels all terms, including gradients of the lapse function. The conserved covector becomes the gradient of a uniform-density curvature perturbation at linear adiabatic order. Dropping the shear constraint is essential; this is not a proof of conservation in an arbitrary non-attractor phase.

For a minimally coupled scalar field with spatial induced metric $h_{ij}$, the field's derivative along the future unit normal is $\Pi$. If the shift vector enters as $d{x}^i-N^{i}dt$, then $\Pi =(\dot{\varphi }+N^i{\partial }_i\varphi )/N$. The canonical momentum of the density is ${\pi }_{\varphi }=\sqrt{h}\Pi$. Confusing this density with the normal derivative loses a volume factor.

For the minimally coupled scalar field, keep the metric tensor fixed and vary $\varphi$. A scalar's first covariant derivative equals its ordinary derivative, soApply integration by parts for tensor fields. The boundary contribution is proportional to $\delta \varphi$ and vanishes under the given boundary condition, leavingThe interior variation is arbitrary. Thus the scalar equation of motion isHere the d'Alembert operator contains the Levi-Civita connection in the second covariant derivative, even though the first derivative of the scalar field is ordinary.

The metric dependence of the minimally coupled scalar field Lagrangian is $\partial L/\partial g^{\alpha \beta }=-\frac{1}{2}{\nabla }_{\alpha }\varphi {\nabla }_{\beta }\varphi$. Part (ii) therefore gives the stress-energy tensorUse metric compatibility and the symmetric Hessian of a scalar field to computeThis is the scalar stress-energy divergence identity, an instance of the diffeomorphism Noether identity for a scalar field. On the scalar equation of motion it gives stress-energy conservation, $\boxed{{\nabla }_{\mu }{T}^{\mu }{}_{\alpha }=0}$.

For a minimally coupled scalar field, put $q^2=h^{ij}{D}_i\varphi D_j\varphi$ using the spatial covariant derivative. Spatial and normal projections of its stress-energy tensor give $\rho ={\Pi }^2/2+q^2/2+V$, ${\mathcal{J}}_i=-\Pi D_i\varphi$, and $S_{ij}=D_i\varphi D_j\varphi +h_{ij}({\Pi }^2/2-q^2/2-V)$. The trace-free stress is $D_i\varphi D_j\varphi -h_{ij}{q}^2/3$, with coefficient one. This makes the scalar anisotropic stress second order in a spatial gradient expansion.

For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.