Metric Tensor

  The ultimate source of CMB anisotropies is the gravitational redshift induced by the metric fluctuation $h_{\mu\nu}$ 

$$g_{\mu\nu} = a^2(\eta_{\mu\nu} + h_{\mu\nu}) \, ,$$ (27)

where the zeroth component represents conformal time $d\eta = dt/a$ and, in the flat universe considered here, $\eta_{\mu\nu}$ is the Minkowski metric. The metric perturbation can be further broken up into the normal modes of scalar, vector and tensor types as in §IIC. Scalar and vector modes exhibit gauge freedom which is fixed by an explicit choice of the coordinates that relate the perturbation to the background. For the scalars, we choose the Newtonian gauge (see e.g. [15,17])  

$$h_{00} = 2\Psi Q^{(0)}, \qquad h_{ij} = 2\Phi Q^{(0)}\delta_{ij}\, ,$$ (28)

where the metric is shear free. For the vectors, we choose

 

h_0i = -V Q_i⁽¹⁾, (29)

and the tensors

 

h_ij = 2 H Q_ij⁽²⁾. (30)

Note that tensor fluctuations do not exhibit gauge freedom of this type.