The Universe Originating from an Empty Planck-Size Torus

Abstract

We consider a Universe with a three-torus topology which before inflation is devoid of any matter or radiation. Its pre-inflationary evolution is driven solely by Casimir energies of the existing fields, with a radiation-like equation of state. We show that, under natural assumptions and with the appropriate number of fermions and bosons in the theory, such a Universe can start its evolution from a Planck size at the Planck time. Moreover, assuming typical parameters for inflation and reheating, the three-torus Universe can be several Hubble radii across at present time, which is precisely the size hinted by the anomaly in the cosmic microwave background observed at low multipole moments. We derive a relation between the size of the Universe, the number of e-folds of inflation, and the energy density decrease during reheating, which we then use to determine the parameter values consistent with the anomaly and the lower bound on the size of the Universe from the Planck satellite.