Cosmological Constraints on Unstable Particles: Numerical Bounds and
  Analytic Approximations
  Many extensions of the Standard Model predict large numbers of additional
unstable particles whose decays in the early universe are tightly constrained
by observational data. For example, the decays of such particles can alter the
ratios of light-element abundances, give rise to distortions in the cosmic
microwave background, alter the ionization history of the universe, and
contribute to the diffuse photon flux. Constraints on new physics from such
considerations are typically derived for a single unstable particle species
with a single well-defined mass and characteristic lifetime. In this paper, by
contrast, we investigate the cosmological constraints on theories involving
entire ensembles of decaying particles --- ensembles which span potentially
broad ranges of masses and lifetimes. In addition to providing a detailed
numerical analysis of these constraints, we also formulate a set of simple
analytic approximations for these constraints which may be applied to generic
ensembles of unstable particles which decay into
electromagnetically-interacting final states. We then illustrate how these
analytic approximations can be used to constrain a variety of toy scenarios for
physics beyond the Standard Model. For ease of reference, we also compile our
results in the form of a table which can be consulted independently of the rest
of the paper. It is thus our hope that this work might serve as a useful
reference for future model-builders concerned with cosmological constraints on
decaying particles, regardless of the particular model under study.
