Linear cosmological perturbation theory

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.

A difference between the two scalar metric potentials in a specified cosmological convention. Nonrelativistic motions probe the potential in the time-time metric component, while weak gravitational lensing probes their sum. Joint tests can constrain modified gravity, but ordinary anisotropic stress can also generate slip and observational bias must be controlled.

Mass conservation in a Lagrangian dust displacement gives $1+\delta =\det (I+{\nabla }_q\psi {)}^{-1}$, hence $\delta =-{\nabla }_q\cdot \psi$ to first order. The perturbing gravitational potential determines the longitudinal part of the force, so the displayed equation holds for the curl-free displacement under the usual homogeneous boundary convention. A divergence-free transverse displacement instead satisfies ${\ddot{\psi }}_T+2H{\dot{\psi }}_T=0$. Taking the divergence gives $\ddot{\delta }+2H\dot{\delta }-4\pi G\bar{\rho }\delta =0$ without assuming the full displacement is longitudinal. The background Poisson equation with positive source requires $\nabla {\Phi }_0=+4\pi G\bar{\rho }r/3$.

For $h_{ij}=\mathrm{diag}(L,0,0)$, the determinant is zero for every $L$, even when one perturbation component is arbitrarily large. Linearized metric theory instead requires the eigenvalues of $h$ relative to the background spatial metric to be small, equivalently all its components in a background orthonormal frame to be small. The determinant of $1+h$ then expands as $1+\mathrm{tr}h+O(h^2)$, and the volume element as $1+\mathrm{tr}h/2+O(h^2)$.

A cosmological gauge transformation changes the identification of a perturbed spacetime with its homogeneous background. Individual field perturbations can change without changing physical observables. At first order the metric change is the linear metric gauge-transformation law; matter scalars obey the corresponding background-shift rule.

For a nonzero Fourier mode in synchronous gauge, write $h_{ij}=h{\delta }_{ij}/3+h_s({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)$. Under the passive displacement ${\xi }^{\mu }=(T,i{\hat{k}}_{i}L)e^{i\mathbf{k}\cdot \mathbf{x}}$, cancellation of spatial scalar shear requires $h_s+2kL=0$, and cancellation of the shift requires $kT-L^{\prime }=0$. The resulting cosmological Newtonian potential is $\Phi =(h_s^{\prime \prime }+\mathcal{H}{h}_s^{\prime })/(2k^2)$ and the spatial potential is $\Psi =-(h-h_s)/6-\mathcal{H}{h}_s^{\prime }/(2k^2)$. Both remain unchanged under residual synchronous-gauge freedom, for which $T^{\prime }+\mathcal{H}T=0$ and $L^{\prime }=kT$.

For the conformal metric with $g_{0i}=-a^2{\partial }_{i}B$ and trace-free scalar spatial shear $E_{ij}=({\partial }_i{\partial }_j-{\delta }_{ij}{\nabla }^2/3)E$, the passive coordinate change $\tilde{\eta }=\eta +T$, ${\tilde{x}}^i=x^i+{\partial }^{i}L$ gives the displayed rules and $\tilde{\varphi }=\varphi +\mathcal{H}T+{\nabla }^{2}L/3$. The combination $C=\varphi +{\nabla }^{2}E/3$ changes by $\mathcal{H}T$, while $B+v$ changes by $T$. Hence $\mathcal{R}=-C+\mathcal{H}(B+v)$ is the comoving curvature perturbation in this sign convention.

Density is a scalar field. Under a first-order time shift $\tilde{\tau }=\tau +T$, its background changes by ${\bar{\rho }}^{\prime }T$, so its perturbation changes by the negative of that amount. Spatial shifts act only at second order on a homogeneous background density.

For the passive convention $\tilde{x}=x+\xi$, expanding the tensor coordinate-transformation law around a fixed background gives this Lie derivative formula. The background time shift and Jacobian terms must both be retained. On an FLRW metric, the scalar lapse, shift and spatial potentials transform under two scalar coordinate functions.

A vector cosmological perturbation is the transverse-vector sector of linear metric and matter perturbations. For a Fourier mode with nonzero wavevector $\mathbf{k}$, transversality removes the component parallel to $\mathbf{k}$, leaving two helicity components. In a gauge with $d{s}^2=a^2[-d{\eta }^2+2B_{i}d\eta d{x}^i+{\delta }_{ij}d{x}^{i}d{x}^j]$, ${\partial }_i{B}_i=0$ describes the vector metric perturbation.

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.

For the metric in vector cosmological perturbation, an orthonormal tetrad is $E_0=a^{-1}{\partial }_{\eta }$, $E_i=a^{-1}(B_i{\partial }_{\eta }+{\partial }_i)$ to first order. A photon has $p^0=(ϵ/a^2)(1+B_i{e}^i)$ and $p^i=(ϵ/a^2)e^i$, so the covariant component is $p_0=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$. The derivative of the scale factor cancels by the null vector condition, proving the redshift formula.

An adiabatic mode is a long-wavelength physical perturbation whose leading local field configuration is generated by a residual gauge transformation. Its action on shorter modes produces a cosmological soft theorem or consistency relation.

An adiabatic perturbation obeys ${\Delta }_r=(4/3){\Delta }_c$. In a radiation background this makes the cold-matter density equation ${\Delta }_c^{\prime \prime }+{\Delta }_c^{\prime }/\tau =4{\Delta }_c/{\tau }^2$, whose growing solution is ${\tau }^2$. In a matter background it instead becomes ${\Delta }_c^{\prime \prime }+2{\Delta }_c^{\prime }/\tau =6{\Delta }_c/{\tau }^2$, with the same growing conformal-time exponent and a different scale-factor dependence.

A tensor cosmological perturbation is the transverse, trace-free part of a metric perturbation about a Friedmann-Lemaître-Robertson-Walker metric. In conformal coordinates, its spatial metric has the form $g_{ij}=a^2({\delta }_{ij}+h_{ij})$, with ${\partial }_i{h}_{ij}=0$ and $h_{ii}=0$. Its two independent polarizations describe a gravitational wave. Neutrino tensor anisotropic stress can damp its evolution through neutrino free streaming.

If the tensor metric perturbation is $h_{ij}=2E_{ij}$ and the polarization tensors have contraction $M_{ij}^{(p)}{M}^{(q)\ast ij}={\delta }_{pq}$, the quadratic action for each Fourier amplitude is $(M_{\mathrm{Pl}}^2/2)\int a^2(|{\psi }_{(p)}^{\prime }{|}^2-k^2|{\psi }_{(p)}{|}^2)$. Thus each helicity is normalized like a real massless scalar by the displayed rescaling. Its power per polarization is $(H_k/(2\pi M_{\mathrm{Pl}}){)}^2$. Factors of two in other tensor spectra depend on the definitions of the metric amplitude, the polarization norm and whether both helicities are summed.

A complex helicity-$+2$ plane wave along the $z$ axis has polarization contraction ${\sin }^2\theta e^{2i\phi }$. Its remaining tensor CMB line-of-sight source depends only on $\cos \theta$. Orthogonality of the azimuthal factors of spherical harmonics therefore restricts its temperature coefficients to $m=2$. A real perturbation also includes the conjugate mode.

For a matter-era superhorizon tensor with $h=h_{\mathrm{prim}}[1-(k\eta {)}^2/10+\cdots ]$, the tensor CMB line-of-sight source gives ${\Theta }_{22}\propto h_{\mathrm{prim}}(k{\eta }_0{)}^2$ and ${\Theta }_{32}/{\Theta }_{22}=-ik{\eta }_0/(3\sqrt{7})$ when ${\eta }_{\ast }\ll {\eta }_0$. The extra spatial phase makes the octupole one order smaller.

For a tensor cosmological perturbation $h_{ij}$, the time geodesic equation in a local orthonormal tetrad gives $d\log ϵ/d\eta =-{\dot{h}}_{ij}{e}^i{e}^j/2$. The photon comoving energy removes the background redshift; the dot is a partial time derivative, whereas the energy derivative follows the ray.

For massless neutrinos with a temperature perturbation $\Theta$ and background energy density ${\bar{\rho }}_{\nu }$, the physical trace-free spatial stress-energy tensor isSome conventions define ${\Pi }^{ij}=-{\pi }_{\nu }^{ij}$ and correspondingly reverse the sign of the stress source in the gravitational-wave equation.

Projecting free-streaming neutrinos onto the tensor spherical harmonic givesHere $j_2$ is a Spherical Bessel function. The continuous value at zero follows by integrating $(1-{\mu }^2{)}^2$, whose integral is $16/15$.

For either tensor helicity, neutrino free streaming produces the memory equationThe neutrino tensor free-streaming kernel encodes the angular propagation of the perturbation. Energy transferred to neutrino directional anisotropy damps the oscillation amplitude of a gravitational wave after horizon entry. This effect was calculated by Weinberg.

A scalar cosmological perturbation transforms as a scalar under spatial rotations and can be represented using scalar metric potentials, density and pressure perturbations, and the longitudinal part of the fluid velocity.

The average of the two conformal Newtonian gauge potentials controls scalar gravitational lensing. A conformal rescaling makes the spatial optical metric flat and its time component $1+4\Psi$ to first order. Vanishing scalar anisotropic stress makes $\varphi =\psi$ in general relativity, but lensing depends on their sum even when they differ.

For the Newtonian gauge in cosmology with potentials $\psi ,\varphi$, photon comoving momentum $ϵ$ obeys $d\ln ϵ/d\eta ={\varphi }^{\prime }-\mathbf{e}\cdot \nabla \psi =-d\psi /d\eta +{\varphi }^{\prime }+{\psi }^{\prime }$. The homogeneous physical-energy redshift is separated by $E=ϵ/a$. The total derivative follows the unperturbed photon path at linear order.

Let $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$ and $\bar{\Pi }=\mathcal{D}\bar{\varphi }$. For negative-expansion extrinsic curvature of a spatial hypersurface, its trace perturbation is $\kappa$. With $\delta \Pi =\mathcal{D}\delta \varphi -\bar{\Pi }\Psi$, scalar evolution is $\mathcal{D}\delta \Pi -\Psi \mathcal{D}\bar{\Pi }+3H\delta \Pi -\kappa \bar{\Pi }-\Delta \delta \varphi +V_{,\varphi \varphi }\delta \varphi =0$. Combining background acceleration terms produces the standard lapse source $-\bar{\Pi }(\kappa +\mathcal{D}\Psi -3H\Psi )+2\Psi V_{,\varphi }$.

A comoving gauge in cosmology chooses hypersurfaces with zero scalar momentum density for the specified fluid or total system. For cold pressureless matter it can be implemented by taking the matter velocity potential to vanish. The choice of component must be stated in a multi-fluid system; density contrasts on this slicing can differ from those in a Newtonian or spatially flat gauge.

The uniform-density curvature perturbation is a dimensionless spatial-curvature amplitude evaluated on hypersurfaces where the density perturbation vanishes. Fix the scalar spatial metric conventionHere primes denote conformal time, $\mathcal{H}=a^{\prime }/a$ and ${\bar{\rho }}^{\prime }\ne 0$. Under the passive cosmological gauge transformation $\tilde{\eta }=\eta +T$, $\tilde{\Phi }=\Phi +\mathcal{H}T$ and $\delta \tilde{\rho }=\delta \rho -{\bar{\rho }}^{\prime }T$, so the two changes cancel in $\zeta$. Choosing $T=\delta \rho /{\bar{\rho }}^{\prime }$ reaches uniform-density slicing and gives $\zeta =-{\Phi }_{\delta \rho =0}$.

Using background energy conservation, the same convention isIt agrees with the uniform-density curvature sign test. For a spatially flat background, the intrinsic three-dimensional Ricci perturbation on these slices is $\delta {}^{(3)}R=-4a^{-2}{\nabla }^2\zeta$: $\zeta$ is its scalar amplitude, rather than the Ricci scalar itself. The full-Hessian $E$ term is a spatial-coordinate contribution and cancels from that Ricci scalar.

An overall reversal of the definition of $\zeta$ is also used, including in the explicitly signed superhorizon conservation of uniform-density curvature equation. Signs must follow the declared metric and curvature definition together. Uniform-density slicing also differs from comoving gauge in cosmology; their regular adiabatic large-scale relation needs additional dynamical assumptions.^[ref]

For spatial metric $a^2(1-2\Phi ){\delta }_{ij}$, the combination $\zeta =-\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is invariant under a time displacement: $\Phi$ shifts by $\bar{N}HT$ and $\delta \rho$ by $3\bar{N}H(\bar{\rho }+\bar{P})T$. The opposite overall sign is also valid. The sum $\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is not invariant. This is a useful convention check before applying superhorizon conservation of uniform-density curvature.

With the convention $\zeta =\Phi -\delta \rho /[3(\bar{\rho }+\bar{P})]$, energy conservation gives ${\zeta }^{\prime }=\mathcal{H}\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})+(\nabla \cdot \mathbf{v})/3$. The non-adiabatic pressure perturbation vanishes for adiabatic cosmological perturbations, and regular velocity gradients are suppressed on superhorizon scales. Curvature is then conserved to leading gradient order, allowing primordial initial conditions to be propagated between cosmological eras.

For $N=\bar{N}(1+\Psi )$, $g_{0i}=a^2{\partial }_{i}B$ and spatial scalar convention $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}-2{\partial }_i{\partial }_{j}E]$, a passive displacement $({\xi }^0,{\partial }^i\lambda )$ gives $\tilde{\Psi }=\Psi -{\dot{\xi }}^0-(\dot{\bar{N}}/\bar{N}){\xi }^0$, $\tilde{B}=B+{\bar{N}}^2{\xi }^0/a^2-\dot{\lambda }$, $\tilde{\Phi }=\Phi +(\dot{a}/a){\xi }^0$ and $\tilde{E}=E+\lambda$. These follow from subtracting the background metric Lie derivative.

For spatial scalar convention $g_{ij}=a^2[(1-2\Psi ){\delta }_{ij}-2E_{,ij}]$ and $g_{0i}=a^2{B}_{,i}$, a passive gauge displacement gives $\tilde{E}=E+\lambda$ and $\tilde{B}=B-\dot{\lambda }+{\bar{N}}^2{\xi }^0/a^2$. Moving from synchronous gauge to Newtonian gauge therefore requires $\lambda =-E_S$ and ${\xi }^0=-a^2{\dot{E}}_S/{\bar{N}}^2$. A scalar density transforms by $\delta \tilde{\rho }=\delta \rho -\dot{\bar{\rho }}{\xi }^0$, giving the displayed density contrast relation. Keeping the background lapse function explicit prevents cosmic-time and conformal-time factors from being confused.

Choose the ADM shift vector ${\beta }^i$ throughwith $N=\bar{N}(1+\Psi )$ and ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2{\partial }_i{\partial }_{j}E]$. For negative extrinsic curvature, $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{\beta }_j-D_j{\beta }_i)/(2N)$. The scalar shear potential isRaised spatial derivatives use $a^{-2}{\delta }^{ij}$, so $\Delta =a^{-2}{\nabla }^2$. The expression follows by keeping the trace-free part of ${\dot{\gamma }}_{ij}$ and the symmetrized shift gradient in the extrinsic-curvature formula. The sign of $B$ is therefore tied to the metric cross term; it is not determined by the name “shift” alone.

The positive-shift scalar shear convention instead writes ${\chi }_{+}=-\chi$ for this same explicit metric. Its trace-free and trace shear terms both reverse their displayed sign. Stating the line element prevents a mistaken combination of a minus-cross-term shift definition with a plus-cross-term extrinsic-curvature formula.

With negative extrinsic curvature convention, the background trace is $\bar{K}=-3H$, where $H=\dot{a}/(a\bar{N})$. The scalar expansion perturbation is $\kappa =K+3H$. Use the explicit ADM conventionwith ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2{\partial }_i{\partial }_{j}E]$. Substituting into $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{\beta }_j-D_j{\beta }_i)/(2N)$ giveswhere $\chi =a^2(\dot{E}-B)/\bar{N}$ is the scalar shear potential and $\Delta =a^{-2}{\nabla }^2$.

Thus the positive metric cross term and ${\beta }_i=+a^2{\partial }_{i}B$ accompany this choice of $\chi$; changing the shift sign requires changing its definition too. For the future unit-normal congruence, the expansion is $-K=3H-\kappa$, so positive $\kappa$ means reduced expansion. A homogeneous perturbation with $\Psi =0$, $B=E=0$ checks the sign: increasing $\Phi$ decreases the local spatial scale and gives $\kappa =3\dot{\Phi }/\bar{N}$.

For $N_i=a^2{B}_{,i}$, ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2E_{,ij}]$ and $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$, put ${\chi }_{+}=a^2(B-\dot{E})/\bar{N}$. Then $K^i{}_j=-H{\delta }^i{}_j+(\dot{\Phi }/\bar{N}+H\Psi ){\delta }^i{}_j+{\partial }^i{\partial }_j{\chi }_{+}$ and $\kappa =3(\dot{\Phi }/\bar{N}+H\Psi )+\Delta {\chi }_{+}$. Defining $\chi =-{\chi }_{+}$ instead makes the trace-free tensor coefficient negative and the trace gradient term $-\Delta \chi$. A sign change must affect both pieces.

Synchronous gauge sets the lapse and shift perturbations to zero, $h_{0\mu }=0$. Its coordinates follow freely falling observers, but residual coordinate freedom produces gauge modes that must be distinguished from physical perturbations.

For the scalar metric variable satisfying $\ddot{h}+(\dot{R}/R)\dot{h}=4\pi G{\bar{\rho }}_m{R}^{2}h$, the Einstein-de Sitter universe has $\dot{R}/R=2/\tau$ and $4\pi G{\bar{\rho }}_m{R}^2=6/{\tau }^2$. The indicial equation is $p(p-1)+2p=6$, giving exponents $2$ and $-3$. Growth of this metric variable is not growth of every gauge-invariant potential; the growing dust density contrast is compatible with constant scalar gravitational potentials.

Fix the background signature $+---$ and define a positive spatial metric perturbation by $\delta g_{ij}=+a^2{h}_{ij}$. Thus the synchronous line element isFor a nonzero Fourier mode, take the passive displacement $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$. The transformed scalar shear is $h_S-2kL$, so setting it to zero gives $L=h_S/(2k)$. Removing the mixed metric term gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$.

Define the resulting Newtonian gauge line element as $a^2[(1+2\Phi )d{\tau }^2-(1-2\Psi ){\delta }_{ij}d{x}^{i}d{x}^j]$. Its potentials areThese signs follow directly from the linear metric gauge-transformation law. Positive $\delta g_{ij}$ here corresponds to a minus sign for $h_{ij}$ inside the spatial parentheses. The synchronous-to-Newtonian scalar Fourier transformation uses a different explicit background/spatial-amplitude convention. Its displacement signs must be translated along with that convention. The present definitions also fix the signs in the photon redshift identity with positive spatial perturbations.

Let $q=\mathbf{k}\cdot \mathbf{n}$, $E=e^{iq(\tau -{\tau }_0)}$ and $T=h_S^{\prime }/(2k^2)$. The synchronous-to-Newtonian transformation with positive spatial perturbations gives $(h^{\prime }-h_S^{\prime })/6={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. Since $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$, the displayed identity follows by integration. The emission endpoint combines intrinsic and velocity shifts into the ordinary Sachs-Wolfe effect; the local observer endpoint is a monopole plus a dipole. Without anisotropic stress the remaining integral is the Integrated Sachs-Wolfe effect, $2\int E{\Phi }^{\prime }d\tau$. The angular term in the synchronous contraction is positive for this metric convention.

For the FRW metric in conformal time, a displacement ${\xi }^{\mu }=(\alpha ,{\xi }^i)$ changes a raw metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. The displayed first-order time equations set the lapse and shift perturbations to zero. They can be integrated locally for smooth perturbations, demonstrating accessibility of synchronous gauge in cosmology. Their homogeneous solutions leave residual spatial functions, so the gauge is not fully fixed. A globally regular synchronous chart need not survive caustics of its freely falling congruence.

For constant $P=w\rho$, the perfect fluid stress tensor and inverse perturbed spatial metric give $T^{ij}=a^{-2}w\bar{\rho }[(1+\delta ){\delta }^{ij}-h^{ij}]$ at first order. Normalizing the velocity makes $u^0=a^{-1}$ at this order. Stress-energy conservation yields the synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge; the pressure projection includes $v_i{\bar{P}}^{\prime }$, which changes the velocity damping from $\mathcal{H}$ to $(1-3w)\mathcal{H}$.

// Target: cosmology.bigb

For constant $P=w\rho$ and velocity potential $v_i=i{k}_i\theta$, stress-energy conservation gives the displayed density equation in synchronous gauge in cosmology. Here $h$ is the trace of the spatial metric perturbation, and primes denote conformal time. In particular cold baryons obey ${\delta }_b^{\prime }=k^2{\theta }_b-h^{\prime }/2$. A positive sign in front of $k^2{\theta }_b$ on the left contradicts this potential convention.

Preserving $\Psi =B=0$ under the scalar gauge transformations with a background lapse requires $\bar{N}{\xi }^0=C(\mathbf{x})$ and $\dot{\lambda }=C(\mathbf{x})\bar{N}/a^2$. Thus arbitrary spatial functions $C,D$ remain after imposing synchronous gauge in cosmology. This freedom can generate changes in the remaining scalar metric perturbations.

A residual synchronous time displacement ${\xi }^0=C(\mathbf{x})/a$ changes a fluid density contrast by $-{\bar{\rho }}^{\prime }{\xi }^0/\bar{\rho }$. For constant equation of state this gives the displayed mode: it scales as ${\tau }^{-2}$ during radiation domination and as ${\tau }^{-3}$ during matter domination. Such density changes alone do not indicate physical structure growth. Choosing the cold-matter rest frame fixes the residual time displacement, while a spatial relabeling fixes the remaining integration convention for the metric trace.

For scalar perturbations in cosmic time with $d{s}^2=-d{t}^2+a^2({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j$, define the covariant fluid velocity potential $u$ by $\delta u_i={\partial }_{i}u$. The linear momentum density is $T^0{}_i=(\bar{\rho }+\bar{P}){\partial }_{i}u$. Normalize scalar anisotropic stress bywhere derivatives inside these parentheses are Cartesian comoving derivatives. Then ${\partial }_j{\pi }^j{}_i=2{\partial }_i{\nabla }^2\Pi /(3a^2)$. Spatial stress-energy conservation gives, for nonzero scalar Fourier modes,Writing ${\pi }^S=2\Pi /(3a^2)$ reproduces the form with ${\nabla }^2{\pi }^S$; the coefficient depends on that stress normalization. The time derivative acts on the entire momentum-density product, not only on $u$.

For pressureless matter with zero scalar stress, background $\dot{\bar{\rho }}+3H\bar{\rho }=0$ reduces the equation to $\dot{u}=0$. The physical peculiar velocity is $\mathbf{v}=\nabla u/a$ and therefore decays as $a^{-1}$. This checks the difference between covariant velocity potential and physical velocity and the expansion factor in the conservative equation.

Newtonian gauge removes scalar shear and writes scalar metric perturbations in terms of gravitational potentials. The fluid Euler equation then contains an explicit gradient of the Newtonian potential.

For one constant-$w$ perfect fluid without anisotropic stress, Newtonian gauge continuity gives ${\Delta }^{\prime }=-(1+w)\theta$. Combining it with the Euler equation and the Einstein constraint yields a closed equation for the shifted density perturbation. For matter domination, the large-scale limit is ${\Delta }^{\prime \prime }+4{\Delta }^{\prime }/\eta =0$, with leading modes constant and ${\eta }^{-3}$. This shifted variable should not be conflated with the comoving-gauge density contrast, which includes a velocity term instead.

The potential $\Phi$ describes the lapse perturbation in conformal Newtonian gauge. The spatial curvature perturbation is conventionally denoted $\Psi$; the two are equal when the linear scalar anisotropic stress vanishes. On scales where the Newtonian limit applies, $\Phi$ agrees with the peculiar gravitational potential. Its Fourier-mode variance and the power spectrum of a comoving curvature perturbation are different observables.

Rescale the perturbed cosmological metric by $a^{-2}(1+2\varphi )$. To first order, its spatial part is Euclidean and only its lapse is perturbed. The unparameterized null geodesics are unchanged. The nonzero linear connection coefficients are ${\hat{\Gamma }}_{00}^0=2{\partial }_{\eta }\Psi$, ${\hat{\Gamma }}_{0i}^0=2{\partial }_i\Psi$ and ${\hat{\Gamma }}_{00}^i=2{\partial }^i\Psi$. The affine parameter is that of the rescaled metric, not necessarily that of the original spacetime.

For reception at the origin with unit propagation direction $e$, the null geodesic condition fixes $v({\eta }_0)=(1+2{\Psi }_0)e$. The perturbative spatial acceleration is $x^{i\prime \prime }=2e^{i}d\Psi /d\eta -2({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi$. Its integrated trajectory is $x^i=e^i(\eta -{\eta }_0)+2e^i{\int }_{{\eta }_0}^{\eta }\Psi d{\eta }^{\prime }-2{\int }_{{\eta }_0}^{\eta }(\eta -{\eta }^{\prime })({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi d{\eta }^{\prime }$. Potentials in these already first-order integrals can be evaluated on the straight background ray.

For a flat background dominated by one constant-$w$ barotropic equation of state, assume $\delta P=w\delta \rho$ and no scalar anisotropic stress. The Einstein field equations then reduce the common Newtonian-gauge potential to the stated Fourier transform equation, with $\mathcal{H}=2/[(1+3w)\tau ]$. For radiation the regular solution is $3{\Phi }_0(\sin x-x\cos x)/x^3$, $x=k\tau /\sqrt{3}$, and oscillates with a decaying envelope after sound-horizon entry. For pressureless matter the two solutions are a constant and ${\tau }^{-5}$. A non-adiabatic pressure perturbation would supply an additional source, so constant background $w$ alone does not justify the homogeneous equation.