GEAR-RT redshifting photons

Goal: have photons redshift

Implementation details:

• Adopted the simplified implementation of multiplying each photon group by H dt
• Does not redshift if the spectrum is constant (i.e. there is always a bin with higher energy available, so the net redshift is zero)
• Updated the CosmoUniformBox_3D example to allow for redshifting photons

Note

This implementation is overly simplified. Analytically, it aims to solve the redshifting part \int H\nu \frac{\partial E_\nu}{\partial \nu} d\nu of the equation of RT. The intuitive approach of integration by parts yields \big[E_{\nu_2} \nu_2 - E_{\nu_1}\nu_1 \big] - H\int E_\nu d\nu. The right side is simply the energy (density) already in the bin, multiplied by the Hubble parameter H, but the left side (boundary terms) requires knowledge about the exact form of the spectrum, which we do not have. For a reasonable spectrum, the sum over all photon groups will have vanishing boundary terms (i.e. the boundary terms must sum to zero).

A second approach would be to calculate the average redshifted energy (in much the same way as for the cross sections) H\langle E\rangle = H\int \nu \frac{\partial E_\nu}{d\nu} d\nu \bigg/ \int E_\nu d\nu and assume a black-body spectrum for E_\nu.

Assuming an approximate spectral shape, for both methods, only produces proper results if the spectrum in the bins actually follows the assumption. Any deviation (e.g. multiple BB spectra, absorption / emission, etc.) will produce results that are wrong.

In this implementation, I have opted for method 1 and neglect the boundary terms in each bin. That is, I have taken \int H\nu \frac{\partial E_\nu}{\partial \nu} d\nu \sim H\int E_\nu d\nu.