In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.

This final chapter discusses spectral distortions of the CMB. We first introduce the relevant collision processes in a universe with photons and non-relativistic electrons: Compton scattering, Bremsstrahlung and double Compton scattering. We derive the corresponding collision terms and Boltzmann equations. For Compton scattering this leads us to the Kompaneets equation for which we present a detailed derivation. We introduce timescales corresponding to these three collision processes and determine at which redshift a given process freezes, i.e., becomes slower than cosmic expansion. We also discuss the generation of a chemical potential in the CMB spectrum by a hypothetical particle decay and by Silk damping of small scale fluctuations. Finally, we study the Sunyaev{Zel’dovich effect of CMB photons which pass through hot cluster gas.

In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.

In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.

This chapter is devoted to parameter estimation. We first discuss the physical dependence of CMB anisotropies on cosmological parameters. After a section on CMB data we then treat in some detail statistical methods for CMB data analysis. We discuss especially the Fisher matrix and explain Markov chain Monte Carlo methods. We also address degeneracies, combinations of cosmological parameters on which CMB anisotropies and polarization depend only weakly. Because of these degeneracies, cosmological parameter estimation also makes use of other, non CMB related, observations especially observations related to the large scale matter distribution. We summarize them and other cosmological observations in two separate sections.

In this chapter we present an introduction to the vast subject of non-Gaussian perturbations. We mainly concentrate on the bispectrum and the trispectrum. We define some standard shapes of the bispectrum in Fourier space and translate them to angular space. For a description of arbitrary N-point function in the sky we introduce a basis of rotation-invariant functions on the sphere in Appendix 4. This chapter has been added in the second edition.

In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.

Here, we develop cosmological perturbation theory. This is the basics of CMB physics. The main reason why the CMB allows such an accurate determination of cosmological parameters lies in the fact that its anisotropies are small and can be determined mainly within first-order perturbation theory. We derive the perturbations of Einstein’s equations and the energy {momentum conservation equations and solve them for some simple but relevant cases. We also discuss the perturbation equation for light-like geodesics. This is sufficient to calculate the CMB anisotropies in the so-called instant recombination approximation. The main physical efffects that are missed in such a treatment are Silk damping on small scales and polarization. We then introduce the matter and CMB power spectrum and draw our first conclusions for its dependence on cosmological and primordial parameters. For example, we derive an approximate formula for the position of the acoustic peaks. In the last section we discuss fluctuations not laid down at some initial time but continuously sourced by some inhomogeneous component, a source, such as, for topological defects example, / that / may form during a phase transition in the early universe.

We derive the perturbed Boltzmann equation for CMB photons. After a brief introduction to relativistic kinetic theory, we first derive the Liouville equation, i.e. the Boltzmann equation without collision term. We also discuss the connection between the distribution function and the energy{momentum tensor. We then derive the collision term, i.e. the right-hand side of the Boltzmann equation, due to Thomson scattering of photons and electrons. In this first attempt we neglect the polarization dependence of Thomson scattering. This treatment however includes the finite thickness of the last scattering surface and Silk damping. The chapter ends with a list of the full system of perturbation equations for a ΛCDM universe, including massless neutrinos.

In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.

In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.

We introduce weak lensing due to foreground structures with the aim of treating lensing of CMB anisotropies and polarization. This second-order effect is especially important on small scales but has to be taken into account for ℓ ≳ 400 if we want to achieve an accuracy of better than 1%. We first derive the deflection angle and the lensing power spectrum. Then we discuss lensing of CMB fluctuations and polarization in the at sky approximation, which is sufficiently accurate for angular harmonics with ℓ ≳50 where lensing is relevant.

This chapter is devoted to the initial conditions. Here we explain how the unavoidable quantum fluctuations are amplified during an inflationary phase and lead to a nearly scale-invariant spectrum of scalar and tensor perturbations. We also calculate the small non-Gaussianities generated during single field in ation and discuss the initial conditions for mixed adiabatic and iso-curvature perturbations.

In this chapter we present the analysis of the large scale matter distribution within linear perturbation theory in a fully relativistic way. We take into account that only directions and redshifts are observable while lengths scales are always inferred from a cosmological model. We first introduce the traditional density and redshift space distortion contribution to the observed uctuations and then proceed to discuss the smaller lensing and large scale relativistic terms. We express the clustering properties of matter in terms of directly observable quantities and study their scale and redshift dependence. We also discuss ‘intensity mapping’ a new observational technique which will hopefully bear fruit in the near future. Also this chapter has been newly added in the second edition.

In a series of appendices we give a list of physical constants and conversion factors; we present some mathematical details and the special functions needed in the main text; we derive the Boltzmann equation for a curved universe and we compute the fluctuations of the luminosity distance. We also solve some of the exercises given in the main text.