Reflections on reflexions -- I. Light echoes in Type Ia supernovae

In the last 10 yr, observational evidence has been steadily growing about a possible connection between Type Ia supernovae (SNe Ia) properties and the environment where they explode. In this paper, I discuss, from a theoretical point of view but with an observer's perspective, the usage of light echoes (LEs) to probe the circumstellar matter around SNe Ia because, in principle, they give us a unique opportunity for obtaining a three-dimensional description of the SN environment. In turn, this can be used to check the often suggested association of some SNe Ia with dusty/star-forming regions, which would point to a young population for the progenitors. After giving a brief introduction to the LE phenomenon in the single scattering approximation, I derive analytical and numerical solutions for the optical light and colour curves for a few simple dust geometries. A fully three-dimensional multiple scattering treatment has also been implemented in a Monte Carlo (MC) code, which I have used to investigate the effects of multiple scattering. In particular, I have explored in detail the LE colour dependency from time and dust distribution, because this is a promising tool to determine the dust density and derive the effective presence of multiple scattering from the observed properties. Finally, again by means of Monte Carlo simulations, I have studied the effects of multiple scattering on the LE linear polarization, analysing the dependences from the dust parameters and geometry. Both the analytical formalism and MC codes described in this paper can be used for any LE for which the light curve of the central source is known.