Appendix C: Gauge-transformation table

The table uses one passive convention throughout. When comparing formulas across references, the sign of the gauge generator and the sign convention for the scalar spatial metric perturbation should be translated before individual terms are compared.

A tilde is reserved for observed or background-inferred quantities. For the passive relabelling

\[ x^\mu\mapsto x^\mu+(T,\partial^iL), \]

the conventions used in these notes give

Table C.1: Gauge transformations in the passive convention of Eq. 2.1.

Quantity	Transformation
\(A\)	\(A\mapsto A-\mathcal HT-T'\)
\(B\)	\(B\mapsto B+L'-T\)
\(D\)	\(D\mapsto D-\mathcal HT\)
\(E\)	\(E\mapsto E-L\)
\(v\)	\(v\mapsto v+L'\)
\(\delta_m\)	\(\delta_m\mapsto\delta_m+3\mathcal HT\)
\(\delta_g\)	\(\delta_g\mapsto\delta_g-(\bar n_g'/\bar n_g)T\)
\(\Delta x^\mu\)	\(\Delta x^\mu\mapsto\Delta x^\mu+\xi^\mu\)
\(\Delta\ln a\)	\(\Delta\ln a\mapsto\Delta\ln a+\mathcal HT\)
\(I_A\)	\(I_A\mapsto I_A-aT\)
\(\mathcal T=HI_A+\Delta\ln a\)	invariant
\(\delta_m^{\mathrm{pt}}=\delta_m+3HI_A\)	invariant
\(\delta_m^{\mathrm{or}}=\delta_m-3\Delta\ln a\)	invariant
\(\delta\boldsymbol{\mathcal J}+\mathcal L_{\Delta x}\overline{\boldsymbol{\mathcal J}}\)	invariant
\(\mathcal C,\mathcal B_i,\mathcal A_{ij}\)	invariant as complete ruler distortions

For any scalar background \(\bar S(\eta)\),

\[ \delta S\mapsto\delta S-\bar S'T. \tag{C.1}\]

The signs of the density transformations are fixed by the background evolution. The table uses the boundary convention \((aT)_{\mathrm{ini}}=0\). Without it, the \(I_A\) row reads \(I_A\mapsto I_A-aT+(aT)_{\mathrm{ini}}\); see Section 2.5.

C.1 Gauge-invariant combinations

The gauge transformations listed in Table C.1 immediately identify the scalar shear variable

\[ \sigma \equiv B+E'. \]

Using

\[ B\mapsto B+L'-T, \qquad E\mapsto E-L, \]

one finds

\[ \sigma\mapsto (B+L'-T)+(E'-L')=\sigma-T. \]

The cancellation of the spatial gauge generator \(L\) is automatic, so only the time shift \(T\) remains.

The two Bardeen potentials adapted to the metric convention adopted in these notes are therefore

\[ \boxed{ \Psi_B = A-\mathcal H\sigma-\sigma' } \]

and

\[ \boxed{ \Phi_B = D-\mathcal H\sigma. } \]

Substituting the transformation rules from Table C.1 shows that both combinations are invariant under an arbitrary scalar gauge transformation.

In the Newtonian (longitudinal) gauge,

\[ B=E=0, \]

so that

\[ \Psi_B=A, \qquad \Phi_B=D. \]

C.1.1 Remark on conventions

Many cosmology references instead write the spatial metric as

\[ g_{ij}=a^2(1-2\Phi)\delta_{ij}, \]

whereas these notes use

\[ g_{ij}=a^2(1+2D)\delta_{ij}. \]

Accordingly,

\[ D=-\Phi_{\mathrm{common}}, \]

where \(\Phi_{\mathrm{common}}\) denotes the potential in the more common convention.

More generally, the definition of the scalar shear variable depends on the complete metric convention (including the sign of \(g_{0i}\), the definition of \(E\), and the passive/active gauge convention). Gauge-transformation formulae should therefore always be translated into a common convention before comparing expressions from different references.