Newtonian gauge

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.

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.

For pressureless matter in conformal time and Newtonian gauge,Its linear curl implies that vorticity decays as $a^{-1}$.

For pressureless matter in Newtonian gauge in cosmology with a common metric potential $\Phi$, let $\theta =\nabla \cdot \mathbf{v}$. The linear equations ${\delta }_m^{\prime }=-\theta +3{\Phi }^{\prime }$ and ${\theta }^{\prime }=-\mathcal{H}\theta -{\nabla }^2\Phi$, where primes denote conformal time, yieldThis follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial Laplacian.

On subhorizon scales, the Newtonian potential sourced by the matter density contrast obeys