Axions arise in many theoretical extensions of the Standard Model of particle physics, in particular the string axiverse. If the axion masses, $m_a$, and (effective) decay constants, $f_a$, lie in specific ranges, then axions contribute to the cosmological dark matter and dark energy densities. We compute the background cosmological (quasi-)observables for models with a large number of axion fields, $n_{rm ax}sim mathcal{O}(10-100)$, with the masses and decay constants drawn from statistical distributions. This reduces the number of parameters from $2n_{rm ax}$ to a small number of hyperparameters. We consider a number of distributions, from those motivated purely by statistical considerations, to those where the structure is specified according to a class of M-theory models. Using Bayesian methods we are able to constrain the hyperparameters of the distributions. In some cases the hyperparameters can be related to string theory, e.g. constraining the number ratio of axions to moduli, or the typical decay constant scale needed to provide the correct relic densities. Our methodology incorporates the use of both random matrix theory and Bayesian networks.
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