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A<sc>bstract</sc> We study the landscape of axion theories in compactifications of type IIB string theory on orientifolds of Calabi-Yau threefolds. In a sample of approximately 400,000 geometries we find that in the regime of perturbative control there are only a handful of distinct axion minima per geometry, despite there being infinitely many instanton contributions to the potential with unbounded charges. The ensemble we consider has numbers of axion fields ranging from 1 to 491, but the median number of distinct minima is 1, the mean number is 1.9 and the largest is 54. These small numbers of minima occur because the leading axion charge matrix is quite sparse, while the subleading corrections are increasingly exponentially suppressed as the charges increase. On their own, such potentials are nowhere near rich enough to be of interest anthropically. This is in stark contrast to potentials for which the charge matrix is less sparse or the hierarchies between the instanton contributions are less steep, where one can find$$ \mathcal{O}\left({10}^{500}\right) $$ $O\left({10}^{500}\right)$minima for$$ \mathcal{O}(500) $$ $O\left(500\right)$axions. To generate a sufficiently large landscape from string compactifications our results indicate that one would need to rely on varying flux or topology, or to develop tools that allow one to go beyond the regime we can control with current techniques. 

During slow-roll inflation, nonperturbative transitions can produce bubbles of metastable vacuum. These bubbles expand exponentially during inflation to superhorizon size, and later collapse into black holes when the expansion of the Universe is decelerating. Estimating the rate for these transitions during a time-dependent slow-roll phase requires the development of new techniques. Our results show that in a broad class of models, the inflationary fine-tuning that gives rise to small density fluctuations causes these bubbles to appear only during a time interval that is short compared to the inflationary Hubble time. As a result, despite the fact that the final mass of the black hole is exponentially sensitive to the moment bubbles form during inflation, the resulting primordial black hole mass spectrum can be nearly monochromatic. If the transition occurs near the middle of inflation, the mass can fall in the “asteroid” range ${10}^{17}–{10}^{22}\mathrm{g}$in which all known observations are compatible with black holes comprising 100% of dark matter.