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.

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.

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.

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.

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.