Gauge transformation in cosmological perturbation theory (${\tilde{x}}^{\mu }=x^{\mu }+{\xi }^{\mu }$)

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.