Gauge Transformations

  To represent the perturbations we must make a gauge choice. A gauge transformation is a change in the correspondence between the perturbation and the background represented by the coordinate shifts  

$$\begin{array} {rcl} \tilde \eta &=& \eta + TQ^{(m)}, \nonumber\\ \tilde x_i &=& x_i + L Q_i^{(m)}.\end{array}$$

T corresponds to a choice in time slicing and L a choice of spatial coordinates. Since scalar and vector quantities cannot be formed from tensor modes ($m=\pm 2$), no gauge freedom remains there. Under the condition that metric distances be invariant, they transform the metric as [17] 

$$\begin{array} {rcl}\tilde A^{(m)} &=& A^{(m)} - \dot T - {\d... ... \nonumber\\ \tilde H_T^{(m)} &=& H_T^{(m)} + kL\,.\end{array}$$

The stress-energy perturbations in different gauges are similarly related by the gauge transformations 

$$\begin{array} {rcl}\tilde \delta_f &=& \delta_f + 3(1+w_f){\... ...,, \nonumber\\ \tilde \pi_f^{(m)} &=& \pi_f^{(m)}\,.\end{array}$$

Note that the anisotropic stress is gauge-invariant. Seed perturbations are also gauge-invariant to lowest order, whereas a scalar field transforms as

$$\tilde{\delta\phi} = \delta \phi - {\dot \phi} T \,.$$ (37)

The relation between the synchronous and Newtonian gauge equations follow from these relations.