Abstract
We derive in this chapter the perturbed Boltzmann equations for photons, massless neutrinos, CDM and baryons. We shall use these in order to track the evolution of small fluctuations in these components, and couple them to Einstein equations. For deriving the hierarchy of temperature and polarisation for photons we follow mainly Ma and Bertschinger (Astrophys J 455:7–25, 1995), Hu and White (Phys Rev D 56:596–615, 1997) and Tram and Lesgourgues (JCAP 1310:002, 2013). We shall focus most of the time on the photon perturbed Boltzmann equation which is much more laborious than the others. This is because photons are massless and interact, therefore we cannot truncate the hierarchy and a collisional term must be taken into account. The latter comes from Thomson scattering, whose cross-section depends also on the polarisation, thus further complicating the treatment of photons fluctuations, which we nonetheless will bravely face.
The Boltzmann equation plays a similar role for physicists and astronomers: no one ever talks about it, but everyone is always thinking about it
Scott Dodelson, Modern Cosmology
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Piattella, O. (2018). Perturbed Boltzmann Equations. In: Lecture Notes in Cosmology. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-95570-4_5
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DOI: https://doi.org/10.1007/978-3-319-95570-4_5
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