Abstract
This paper presents the linearised Boltzmann equation for photons for scalar, vector and tensor perturbations in flat, open and closed FLRW cosmologies. We show that E- and B-mode polarisation for all types can be computed using only a single hierarchy. This was previously shown explicitly for tensor modes in flat cosmologies but not for vectors, and not for non-flat cosmologies.
Highlights
Which drastically reduced the number of differential equations necessary for calculating the power-spectrum of anisotropies
There is another formalism that we did not mention so far, namely the covariant and gauge invariant approach of Challinor&Lasenby, see [12] and references. This approach was further expanded and refined by Challinor [13, 14] and later by Lewis [15]. This approach shares many of the advantages of the total angular momentum method, and it can be generalised to non-linear perturbations
The equations of the popular Boltzmann code CAMB is derived in this formalism [16]
Summary
We are using the (−+++) sign-convention for the metric, and greek indices are running from 0 to 3 while latin indices are running from 1 to 3. For consistency with Hu et al [8], we are omitting the Condon-Shortley phase (−1)m in the definition of the (spin-weighted) spherical harmonics and in the definition of the associated Legendre polynomials. Note that this is contrary to the conventions of both Wikipedia and Mathematica, which includes the phase for both. Where hμν is a perturbation and the spatial part of the background metric can be written as γij 1 |K |. The covariant derivative of X with respect to the spatial background metric γij will be denoted X|i.
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