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, quite strikingly, with the appropriate number of fermions and bosons in the theory, such a Universe evolving from Planck size at Planck time has the correct critical energy density today. 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 Planck.

Figures (1)

• Figure 1: The allowed parameter space (shaded in blue) for the number of $e$-folds of inflation $N$ vs. energy density drop during reheating $D$, for a three-torus Universe starting its evolution from Planck size at Planck time and with its shape described by metric (\ref{['special_metric']}). The red shaded region is excluded based on searches for repeated patterns in the CMB, whereas the orange shaded region is not favored by the low multipole anomaly present in the data. The star indicates the benchmark point discussed in the text with $N=71.5$ and $D = 10^5$, for which the current size of the Universe is $1.3\times 10^{27}\ \rm m$.