Abstract
We explore simple but novel bouncing solutions of general relativity that avoid singularities. These solutions require curvature k = +1, and are supported by a negative cosmological term and matter with −1 < w < −1/3. In the case of moderate bounces (where the ratio of the maximal scale factor a + to the minimal scale factor a − is $ \mathcal{O}(1) $ ), the solutions are shown to be classically stable and cycle through an infinite set of bounces. For more extreme cases with large a + /a −, the solutions can still oscillate many times before classical instabilities take them out of the regime of validity of our approximations. In this regime, quantum particle production also leads eventually to a departure from the realm of validity of semiclassical general relativity, likely yielding a singular crunch. We briefly discuss possible applications of these models to realistic cosmology.
Highlights
Tμν vμvνFor a suitable class of vectors vμ, where Tμν is the stress-energy tensor of the sources supporting the Universe, one can prove that the Universe must be geodesically incomplete (“singular”)
Two questions have recurred often in theoretical cosmology [1,2,3,4,5,6,7,8,9]: 1) is the Universe eternal or did it have a beginning at some definite time in the past?, and 2) is it possible to make Universes with one or more “bounces” where the scale factor crunches and bangs
Even in scenarios where the current ΛCDM cosmology was preceeded by a phase of slow-roll inflation [11,12,13], with eternal inflation occurring on even larger scales, it is striking [14] that the initial singularity remains, independent of the energy condition assumed
Summary
For a suitable class of vectors vμ, where Tμν is the stress-energy tensor of the sources supporting the Universe, one can prove that the Universe must be geodesically incomplete (“singular”). Even in scenarios where the current ΛCDM cosmology was preceeded by a phase of slow-roll inflation [11,12,13], with eternal inflation occurring on even larger scales, it is striking [14] that the initial singularity remains, independent of the energy condition assumed. For k = +1, the strong energy condition (SEC) (where vμ in (0.1) is futurepointing timelike) must be assumed.. For k = +1, the strong energy condition (SEC) (where vμ in (0.1) is futurepointing timelike) must be assumed.3 This condition is violated by macroscopic sources in our world, as well as in many completely consistent theoretical toy models. The minimal model which oscillates has three components: positive curvature, a negative cosmological constant (energy density = Λ < 0), and a “matter” source with equation of state in the range p = wρ, − 1 < w < −1/3.
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