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.

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.

In the Newtonian pressureless approximation, let $\mathbf{x}$ denote comoving positions, $\eta$ conformal time, $a$ the scale factor, and $\mathbf{v}=d\mathbf{x}/d\eta$ the physical peculiar velocity. With $\mathcal{H}=a^{\prime }/a$ and gravitational potential $\varphi$, the Euler equation isAt linear order, define the comoving-coordinate curl ${\mathbf{\omega }}_c={\nabla }_x\times \mathbf{v}$. The gradient force has zero curl, soThe physical spatial derivative is ${\nabla }_{\mathrm{phys}}=a^{-1}{\nabla }_x$. Therefore the physical vorticity is ${\mathbf{\omega }}_{\mathrm{phys}}=a^{-1}{\mathbf{\omega }}_c\propto a^{-2}$. Stating which derivative defines the vorticity prevents confusing these two dilution laws.

This smooth single-stream model omits pressure forces and velocity dispersion. Shell crossing or additional nongravitational forces can invalidate its simple description; the linear decay law is not a universal assertion about every rotating cosmic structure.^[ref]

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