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The exact numerical treatment of inflationary models

The precision reached by the recent CMB measurements gives new insights into the shape of the primordial power spectra of the cosmological perturbations. In the context of inflationary cosmology, this implies that the CMB data are now sensitive to the form of the inflaton potential. Most of the current approaches devoted to the derivation of the inflationary primordial power spectra, or to the inflaton potential reconstruction problem, rely on approximate analytical treatments that may break down for exotic models. In this article, we numerically solve the inflationary evolution of both the background and all the perturbed quantities to extract the primordial power spectra exactly. Such a method solely relies on General Relativity and linear perturbation theory. More than providing a tool to test analytical approximations, one may consider, without complications, the treatment of non-standard inflationary models as those involving several fields, eventually non-minimally coupled to gravity. The usefulness of the exact numerical approach to deal with CMB data is illustrated by analysing the WMAP third year data in the context of single field models. For this purpose, we introduce a new inflationary related parameter encoding the basic properties of the reheating era. This reheating parameter has significant observable effects and provides a self-consistency test of inflationary models. As a working example, the marginalised probability distributions of the reheating and potential parameters associated with the small field models are presented.