LARGE volume string compactifications at finite temperature

We present a detailed study of the finite-temperature behaviour of the LARGE Volume type IIB flux compactifications. We show that certain moduli can thermalise at high temperatures. Despite that, their contribution to the finite-temperature effective potential is always negligible and the latter has a runaway behaviour. We compute the maximal temperature T-max, above which the internal space decompactifies, as well as the temperature T-*, that is reached after the decay of the heaviest moduli. The natural constraint T-* < T-max implies a lower bound on the allowed values of the internal volume V. We find that this restriction rules out a significant range of values corresponding to smaller volumes of the order V similar to 10(4)l(s)(6), which lead to standard GUT theories. Instead, the bound favours values of the order V similar to 10(15)l(s)(6), which lead to TeV scale SUSY desirable for solving the hierarchy problem. Moreover, our result favours low-energy inflationary scenarios with density perturbations generated by a field, which is not the inflaton. In such a scenario, one could achieve both inflation and TeV-scale SUSY, although gravity waves would not be observable. Finally, we pose a two-fold challenge for the solution of the cosmological moduli problem. First, we show that the heavy moduli decay before they can begin to dominate the energy density of the Universe. Hence they are not able to dilute any unwanted relics. And second, we argue that, in order to obtain thermal inflation in the closed string moduli sector, one needs to go beyond the present EFT description.