Linearized Einstein and Conservation Equations

Scalar perturbations obey constraints as well as evolution equations. In Newtonian gauge the lapse and curvature potentials are tied to density, momentum, pressure, and anisotropic stress by four scalar projections of the Einstein equation. Matter conservation then evolves the sources, while the linearized Bianchi identity ensures that part of this combined system is redundant rather than overdetermined.

Helpful background. Einstein Equations introduce the field equation and Bianchi identity; relativistic matter supplies its stress–energy sources. Formulations of general relativity distinguishes constraints from evolution, the cosmology overview locates the background dynamics, and PDEs and Green functions introduces Fourier modes. Bardeen Potentials and Curvature Perturbations explains the invariant variables entering this system.

Stress–energy and metric variables

Work to first order about a spatially flat FLRW solution. In Newtonian gauge,

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2 +(1-2\Psi)\delta_{ij}dx^i dx^j\right], \qquad \mathcal H=\frac{a'}a.$$

Use $e^{i\mathbf k\cdot\mathbf x}$, so $\nabla^2\to-k^2$, and define the mixed stress–energy perturbations by

$$\begin{aligned} \delta T^0{}_0&=-\delta\rho, &\delta T^0{}_i&=\partial_i\delta q,\\ \delta T^i{}_j&=\delta p\,\delta^i{}_j +\left(\partial^i\partial_j -\frac13\delta^i{}_j\nabla^2\right)\Pi. \end{aligned}$$

These equations define every sign and normalization used below. The momentum potential $V\equiv\delta q/(\rho+p)$ is convenient when $\rho+p\ne0$, and $c_a^2=p'/\rho'$ is the adiabatic background sound speed when $\rho'\ne0$. Neither quotient should be used for a vacuum-like component or at a singular background point.

The background equations needed to check the reduction are

$$\mathcal H^2=\frac{8\pi G}{3}a^2\rho, \qquad \mathcal H'-\mathcal H^2=-4\pi Ga^2(\rho+p), \qquad \rho'=-3\mathcal H(\rho+p),$$

with any cosmological constant included once in the total $\rho,p$ in this section.

Einstein constraints and evolution

A direct first-order expansion gives the mixed Einstein-tensor projections

$$\delta G^0{}_0=\frac{2}{a^2} \left[3\mathcal H(\Psi'+\mathcal H\Phi)-\nabla^2\Psi\right],$$ $$\delta G^0{}_i=-\frac{2}{a^2}\partial_i (\Psi'+\mathcal H\Phi),$$

and

$$\begin{aligned} \delta G^i{}_j={}&\frac{2}{a^2} \left[L_0+\frac13\nabla^2(\Phi-\Psi)\right]\delta^i{}_j\\ &+\frac{1}{a^2} \left(\partial^i\partial_j-\frac13\delta^i{}_j\nabla^2\right) (\Psi-\Phi), \end{aligned}$$

where

$$L_0=\Psi''+2\mathcal H\Psi'+\mathcal H\Phi' +(2\mathcal H'+\mathcal H^2)\Phi.$$

These expressions follow by inserting the perturbed Christoffel symbols into $G^\mu{}_\nu$ and using the background equations before collecting the scalar trace and trace-free derivative operator. The $00$ and longitudinal $0i$ equations are therefore the constraints

$$\boxed{ 3\mathcal H(\Psi'+\mathcal H\Phi)-\nabla^2\Psi =-4\pi Ga^2\delta\rho },$$ $$\boxed{ \Psi'+\mathcal H\Phi=-4\pi Ga^2\delta q }.$$

They restrict admissible perturbations on each time slice. In particular, density, momentum, and the two metric potentials cannot be initialized independently and then expected to converge to a solution.

The raw spatial trace and the traceless scalar equation are

$$L_0+\frac13\nabla^2(\Phi-\Psi)=4\pi Ga^2\delta p,$$ $$\boxed{ \Psi-\Phi=8\pi Ga^2\Pi }.$$

Using this slip equation inside the raw trace gives the often more convenient reduced trace equation

$$\begin{aligned} \Psi''+2\mathcal H\Psi'+\mathcal H\Phi' +(2\mathcal H'+\mathcal H^2)\Phi =4\pi Ga^2\left(\delta p+\frac23\nabla^2\Pi\right), \end{aligned}$$

The traceless slip equation is algebraic in the two potentials once $\Pi$ is specified; the reduced trace remains a differential evolution equation. Scalar anisotropic stress therefore changes both their difference and their evolution. These equations, including the arbitrary-gauge form from which they follow, are given in Malik and Wands 2009, §8.2.1, printed pp. 46–48, eqs. (8.16)–(8.31) (Open PDF).

Several checks are immediate. With $\Pi=0$, the two potentials agree. Combining the two constraints gives the gauge-invariant Poisson equation

$$\nabla^2\Psi =4\pi Ga^2\delta\rho_{\rm com}, \qquad \delta\rho_{\rm com}=\delta\rho-3\mathcal H\delta q.$$

For $k\gg\mathcal H$ and slowly varying potentials this becomes $k^2\Psi\simeq-4\pi Ga^2\delta\rho$, the expected Newtonian sign: an overdensity sits in a negative potential well with this metric convention.

Energy and momentum conservation

For a mixed tensor, the linearized divergence follows from

$$\begin{aligned} \delta(\nabla_\mu T^\mu{}_{\nu})={}& \partial_\mu\delta T^\mu{}_{\nu} +\bar\Gamma^\mu{}_{\mu\alpha}\delta T^\alpha{}_{\nu} -\bar\Gamma^\alpha{}_{\mu\nu}\delta T^\mu{}_{\alpha}\\ &+\delta\Gamma^\mu{}_{\mu\alpha}\bar T^\alpha{}_{\nu} -\delta\Gamma^\alpha{}_{\mu\nu}\bar T^\mu{}_{\alpha}. \end{aligned}$$

Substituting the scalar stress tensor and separating the time projection from the longitudinal spatial projection leaves the two residual equations

$$\boxed{ \delta\rho'+3\mathcal H(\delta\rho+\delta p) -3(\rho+p)\Psi' +(\rho+p)\nabla^2V=0 },$$ $$\boxed{ V'+(1-3c_a^2)\mathcal HV+\Phi +\frac{1}{\rho+p} \left(\delta p+\frac23\nabla^2\Pi\right)=0 }.$$

Setting those residuals to zero gives the scalar projections of $\nabla_\mu T^\mu{}_\nu=0$. The first equation accounts for volume dilation, pressure work, perturbed expansion, and momentum divergence. The second is the relativistic Euler equation. In Fourier space the anisotropic-stress force carries the declared $-k^2$ from $\nabla^2$. Malik and Wands derive these equations in Malik and Wands 2009, §8.2.2, printed pp. 48–49, eqs. (8.32)–(8.37) (Open PDF).

For separately interacting species, write $\nabla_\mu T_A^{\mu}{}_\nu=Q_{A\nu}$ with $\sum_AQ_{A\nu}=0$. Each component equation then receives energy- and momentum-transfer sources, while their sum returns the total equations above. One must not apply the source-free component equations to the tightly coupled photon–baryon plasma.

Ma and Bertschinger provide the same system in synchronous and conformal Newtonian gauges, together with their velocity-divergence and shear normalizations, in Ma and Bertschinger 1995, §4, printed pp. 10–13, eqs. (19)–(30) (Open PDF). Their symbols $\phi$ and $\psi$ attach to the opposite metric roles from $\Phi$ and $\Psi$ here, so equations must be translated through the metrics.

Constraint propagation

The four Einstein projections and two conservation equations are not six independent equations. This can be checked without an appeal to counting. Define residuals

$$\begin{aligned} C_{00}&=3\mathcal H(\Psi'+\mathcal H\Phi)-\nabla^2\Psi +4\pi Ga^2\delta\rho,\\ C_{0i}&=\Psi'+\mathcal H\Phi+4\pi Ga^2\delta q,\\ C_{\rm tr}&=L_0+\frac13\nabla^2(\Phi-\Psi) -4\pi Ga^2\delta p, \end{aligned}$$

and let $E_\rho$ denote the left side of the energy-conservation equation. Using only the background equations, direct differentiation gives

$$C_{00}'+\mathcal HC_{00}-4\pi Ga^2E_\rho +\nabla^2C_{0i}=3\mathcal HC_{\rm tr}.$$

Thus the $00$ constraint, energy equation, and $0i$ constraint propagate the raw spatial trace wherever $\mathcal H\ne0$. The reduced trace printed above also uses the traceless slip equation. At a bounce or turnaround this identity remains true, but division by $\mathcal H$ cannot be used to infer the trace; one must impose or evolve an independent regular equation there. This is the linearized contracted Bianchi identity. A numerical solver may evolve a nonredundant subset and monitor the omitted constraint as an error measure; evolving inconsistent initial data does not make the extra equation an independent repair mechanism.

For a single perfect barotropic fluid, set $\Pi=0$, $\Phi=\Psi$, and $\delta p=c_a^2\delta\rho$. Eliminating the density with the $00$ constraint yields

$$\Psi''+3(1+c_a^2)\mathcal H\Psi' +\left[2\mathcal H'+(1+3c_a^2)\mathcal H^2 -c_a^2\nabla^2\right]\Psi=0.$$

For matter domination, $c_a^2=0$ and $2\mathcal H'+\mathcal H^2=0$, leaving a constant growing-mode potential plus a decaying mode. During radiation domination, $c_a^2=1/3$ and the same background bracket vanishes, while $-\nabla^2/3$ supplies acoustic propagation. In the controlled superhorizon limit $k/\mathcal H\to0$, the gradient term drops out and the regular adiabatic branch has constant $\Psi$; the second solution decays as $\eta^{-5}$ in matter domination and $\eta^{-3}$ in radiation domination. These limits check both signs and normalization.

The formulas assume $k\ne0$, $\mathcal H$ regular, and divisions only where their denominators are nonzero. Homogeneous modes, bounces, interacting component equations, and nonlinear perturbations require separate treatment.

Exercises

Derive the perfect-fluid potential equation above from the trace Einstein equation and the $00$ constraint.

Solution

With $\Pi=0$, the traceless equation gives $\Phi=\Psi$. Insert $\delta p=c_a^2\delta\rho$ into the trace equation. The $00$ constraint gives

$$4\pi Ga^2\delta\rho =-3\mathcal H(\Psi'+\mathcal H\Psi)+\nabla^2\Psi.$$

Substitution collects the derivative terms into $3(1+c_a^2)\mathcal H\Psi'$, the background terms into $2\mathcal H'+(1+3c_a^2)\mathcal H^2$, and the spatial term into $-c_a^2\nabla^2\Psi$. No division by $k$ is required, so the differential identity has a homogeneous limit even though the SVT interpretation of that mode must still be handled separately.

References

• Ma, C.-P., and E. Bertschinger, “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges,” Astrophysical Journal 455, 7–25 (1995), DOI, Open PDF.
• Malik, K. A., and D. Wands, “Cosmological Perturbations,” Physics Reports 475, 1–51 (2009), DOI, Open PDF.