We describe a novel technique that renders theories of N axions tractable, and more generally can be used to efficiently analyze a large class of periodic potentials of arbitrary dimension. Such potentials are complex energy landscapes with a number of local minima that scales as √ N !, and so for large N appear to be analytically and numerically intractable. Our method is based on uncovering a set of approximate symmetries that exist in addition to the N periods. These approximate symmetries, which are exponentially close to exact, allow us to locate the minima very efficiently and accurately and to analyze other characteristics of the potential. We apply our framework to evaluate the diameters of flat regions suitable for slow-roll inflation, which unifies, corrects and extends several forms of "axion alignment" previously observed in the literature. We find that in a broad class of random theories, the potential is smooth over diameters enhanced by N 3/2 compared to the typical scale of the potential. A Mathematica implementation of our framework is available online.

We describe a systematic framework for periodic potentials of arbitrary dimension, such as those governing multiple axions. A novel type of alignment renders even complex theories analytically tractable. Theories with ∼100 axions and random parameters have an exponential number of metastable vacua. Decay from a metastable minimum can occur via a thin-wall instanton and allows for a sufficient period of slow-roll inflation that ends in a vacuum containing axion dark matter and a cosmological constant, both consistent with current observations. Hence, this model can reproduce many macroscopic features of our Universe without tuned parameters.

We study the cosmology of complex multi-axion theories. With O(100) fields and GUT scale energies these theories contain a vast number of vacua, inflationary trajectories and a natural dark matter candidate. We demonstrate that the vacua are stable on cosmological timescales. In a single theory, both large-and small-field inflation are possible and yield a broad range of cosmological observables, and vacuum decay can be followed by a relatively large number (> 60) of efolds of inflation. Light axions stabilized by gravitational instantons may constitute a natural dark matter candidate that does not spoil an axion solution to the strong CP problem.

We study bubble universe collisions in the ultrarelativistic limit with the new feature of allowing for nontrivial curvature in field space. We establish a simple geometrical interpretation of such collisions in terms of a double family of field profiles whose tangent vector fields stand in mutual parallel transport. This provides a generalization of the wellknown flat field space limit of the free passage approximation. We investigate the limits of this approximation and illustrate our analytical results with numerical simulations.