  We revisit the idea that the inflaton may have dissipated part of its energy
into a thermal bath during inflation, considering monomial inflationary
potentials and three different forms of dissipation rate. Using a numerical
Fokker-Planck approach to describe the stochastic dynamics of inflationary
fluctuations, we confront this scenario with current bounds on the spectrum of
curvature fluctuations and primordial gravitational waves. We also obtain
purely analytical approximations that improve over previously used ones in the
small dissipation regime for the amplitude of the spectrum and its tilt. We
show that only our numerical Fokker-Planck method is accurate, fast and precise
enough to test these models against current data. We advocate its use in future
studies of warm inflation. We also apply the stochastic inflation formalism to
this scenario, finding that the resulting spectrum is the same as the one
obtained with standard perturbation theory. We discuss the origin and
convenience of using a commonly implemented large thermal correction to the
primordial spectrum and the implications of such a term for a specific
scenario. Improved bounds on the scalar spectral index will further constrain
warm inflation in the near future.
