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RECFAST calculates the recombination of H, HeI, and HeII in the early Universe; this involves a line-by-line treatment of each atomic level. It differs in comparison to previous calculations in two major ways: firstly, the ionization fraction x_e is approximately 10% smaller for redshifts <~800, due to non-equilibrium processes in the excited states of H, and secondly, HeI recombination is much slower than previously thought, and is delayed until just before H recombines. RECFAST enables fast computation of the ionization history (and quantities such as the power spectrum of CMB anisotropies which depend on it) for arbitrary cosmologies.
MAH calculates the posterior distribution of the "minimum atmospheric height" (MAH) of an exoplanet by inputting the joint posterior distribution of the mass and radius. The code collapses the two dimensions of mass and radius into a one dimensional term that most directly speaks to whether the planet has an atmosphere or not. The joint mass-radius posteriors derived from a fit of some exoplanet data (likely using MCMC) can be used by MAH to evaluate the posterior distribution of R_MAH, from which the significance of a non-zero R_MAH (i.e. an atmosphere is present) is calculated.