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In a measuring observer's orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor. Using the positive Euclidean spatial metric ${\delta }_{ij}$, defineThe trace here is the spatial Euclidean trace. Writing a Lorentzian mixed trace $T^{\hat{k}}{}_{\hat{k}}$ without its signature convention can reverse its sign in signature $(+---)$.

An isotropic perfect fluid in general relativity has $T^{\hat{i}\hat{j}}=P{\delta }^{ij}$ in its rest frame and therefore zero anisotropic stress. Different directional pressures or an angular quadrupole of particle momenta can give nonzero ${\pi }^{ij}$. In particular, the photon quadrupole determines the linear photon contribution.

Some linear cosmological perturbation theory conventions use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{\hat{i}\hat{j}}=P{\delta }^{ij}-{\Pi }^{ij}$. This sign must accompany a photon Euler equation; trace removal and the choice of sign are separate conventions.

Cosmological perturbation theory studies departures of the metric tensor and matter fields from a homogeneous and isotropic FRW metric, order by order. Linear cosmological perturbation theory keeps only first-order terms. Higher orders describe nonlinear coupling between perturbations. Nonlinear cosmological perturbation theory also uses expansions that allow finite amplitudes, such as the long-wavelength approximation in cosmology.

David Wands and collaborators gave a conservation-based account of long-wavelength fluctuations in linear cosmological perturbation theory. Their 2000 work treated slowly varying regions as locally homogeneous expanding universes and identified the conditions for conserved uniform-density curvature perturbations, without selecting particular gravitational field equations.^[ref]

The important qualifications are negligible spatial gradients and negligible non-adiabatic pressure perturbations; large wavelength alone is insufficient.

Linear cosmological perturbation theory describes small departures from a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric. Introduce a bookkeeping parameter $ϵ$ and expand the metric tensor and stress-energy tensor asSubstituting into the Einstein field equations separates the background equations from linear equations for the perturbations; products of two first-order perturbations are discarded. Smallness of $h_{ab}$ is assessed relative to the background metric tensor, not by its determinant alone.

Spatial homogeneity lets one analyze each Fourier mode separately. The fields and particle species at a given wavevector can remain coupled, but different wavevectors do not exchange amplitude through the linear equations. Products retained in nonlinear cosmological perturbation theory instead generate convolutions between modes. The linear density equations help connect initially small fluctuations to the growth of cosmic structure.^[ref]

A cosmological gauge transformation changes how the perturbed and background spacetimes are identified. For example, under $\tilde{\eta }=\eta +T$, a density perturbation transforms aswhere the prime is a conformal time derivative. A nonzero coordinate-dependent density perturbation can therefore represent a time relabelling of unchanged matter. One fixes a gauge or forms invariant combinations before relating perturbations to observations.

Tensor cosmological perturbations describe gravitational-wave degrees of freedom, while density and velocity perturbations describe other responses. A weak gravitational potential does not guarantee a small density contrast; collapsed structures can require a nonlinear matter treatment even when their metric potentials remain small.

Define the momentum density $q_i=(\bar{\rho }+\bar{P})v_{\perp i}$, lowering its spatial label with ${\delta }_{ij}$, so the mixed stress-energy tensor obeys $T^i{}_0=q_i$ and $T^0{}_i=-q_i$. All products of $J_{ij}$ or its derivatives with $q_i$ are second order and must be dropped in linear cosmological perturbation theory.

For $\nu =0$, the three terms in stress-energy conservation areHere the trace $J_{ii}^{\prime }$ vanishes, ${\Gamma }_{00}^i={\Gamma }_{i0}^0=0$, and ${\partial }_i{q}_i=(\bar{\rho }+\bar{P}){\partial }_i{v}_{\perp }^i=0$. ThusFor $\nu =i$, spatial pressure gradients vanish because this is a purely vector cosmological perturbation. The first two terms areIn the remaining contraction, the two background-connection times momentum density terms cancel:The other terms are ${\Gamma }_{0i}^0\bar{\rho }=0$ and $-\bar{P}{\Gamma }_{ji}^j=0$. ConsequentlyExpanding this conservation law and using the background cosmological continuity equation gives the equivalent peculiar velocity equation, when $\bar{\rho }+\bar{P}\ne 0$,

At linear order a coordinate displacement ${\xi }^{\mu }$ changes a metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. Work in conformal time with ${\bar{g}}_{\mu \nu }=a^2\mathrm{diag}(-1,1,1,1)$, write ${\xi }^0=\alpha$ and lower spatial indices with ${\delta }_{ij}$. Directly evaluating the Lie derivative of a covariant tensor field givesHere primes on $\alpha ,{\xi }_i$ mean conformal-time derivatives, while primes on $q_{0\mu }^{\prime }$ label transformed components. Setting the transformed lapse and shift perturbations to zero requiresThese are first-order time equations. Their explicit local solutions areThus four coordinate functions can impose the four synchronous conditions in any regular perturbative patch. This is synchronous gauge fixing by coordinate displacement. The arbitrary spatial functions $C,D_i$ remain as residual gauge freedom; the conditions alone do not fix the coordinate system completely.

Geometrically, synchronous coordinates follow a congruence of freely falling observers launched normally from a spatial slice, using their proper time. This gives unit lapse and zero shift in cosmic time; conversion to background conformal time gives the displayed convention. “Always possible” is local: geodesic caustics or singularities can prevent a global synchronous chart. Such global obstructions do not invalidate local linear cosmological perturbation theory.

In linear cosmological perturbation theory, an unsourced vector cosmological perturbation of a perfect fluid in synchronous gauge obeys stress-energy conservation in the form ${\mathbf{q}}^{\prime }+4\mathcal{H}\mathbf{q}=0$, where $\mathbf{q}=(\bar{\rho }+\bar{P}){\mathbf{v}}_{\perp }$ is the momentum density, $\mathcal{H}$ the conformal Hubble parameter, and the prime a conformal time derivative. Its conserved quantity is $a^4\mathbf{q}$. For constant equation-of-state parameter $w\ne -1$, the background cosmological continuity equation gives ${\mathbf{v}}_{\perp }\propto a^{3w-1}$. Thus a pressureless matter vector velocity falls as $a^{-1}$, while a radiation vector velocity is constant. This statement assumes vanishing vector anisotropic stress and no momentum exchange with another component.