Deflation at Turnaround for Oscillatory Cosmology

It is suggested that dark energy in a brane world can help reconcile an infinitely cyclic cosmology with the second law of thermodynamics. A cyclic cosmology is described, in which dark energy with constant equation of state leads to a turnaround at finite future time, when entropy is decreased by a huge factor equal to the inverse of its enhancement during the initial inflation. Thermodynamic consistency of cyclicity requires the arrow of time to reverse during contraction. Entropy reduction in the contracting phase is infinitesimally smaller than entropy increase during expansion. Introduction One of the oldest questions in theoretical cosmology is whether an infinitely oscillatory universe which avoids an initial singularity can be consistently constructed. As realized by Friedmann [1] and especially by Tolman [2, 3] one principal obstacle is the second law of thrmodynamics which dictates that the entropy increases from cycle to cycle. If the cycles thereby become longer, extrapolation into the past will lead back to an initial singularity again, thus removing the motivation to consider an oscillatory universe in the first place. This led to the abandonment of the oscillatory universe by the majority of workers. Nevertheless, an oscillatory universe is an attractive alternative to the Big Bang. One new ingredient in the cosmic make-up is the dark energy discovered only in 1998 and so it natural to ask whether this new component in the Friedmann equation can avoid the difficulties with entropy which have dogged previous attempts. Some work has been started to exploit the dark energy in allowing cyclicity possibly without apparently the need for inflation in [4–7]. Another new ingredient is the use of branes and a fourth spatial dimension as in [8–11] which have examined the consequences for cosmology. The Big Rip and replacement of dark energy by modified gravity have been explored in [12, 13]. If the dark energy has a super-negative equation of state, ωΛ = pΛ/ρΛ < −1, it leads to a Big Rip at a finite time where there exist extraordinary conditions with regard to density and causality as one approaches the Big Rip. In the present article we explore whether these exceptional physical conditions can assist in providing a truly infinitelycyclic entropy density in an oscillatory universe of time periodicity t = t ( mod τ). We shall consider the situation where if, as we approach the Big Rip, the expansion stops just short of the rip and there is a turnaround at t = tT (mod τ ) when the scale factor is deflated to a very tiny fraction (f in our notation) of itself. For the deflation there is a consistency condition, written in Eq.(10). Entropy is extensive so a fraction (1 − f ) is jettisoned at turnaround. One key ingredient is that the turnaround takes place a sufficiently short time before the Big Rip would have occurred, at a time when the universe is fractionated into many causal patches [13]. We then proceed to investigate the contraction phase which occurs with a very much smaller universe than in the expansion phase. A bounce at t = τ ( mod τ) takes place a short time before what would have been the Big Bang. Then, immediately after the bounce, entropy is injected as usual by inflation [14] where the scale factor is enhanced by factor E and hence entropy by E. Inflation is thus an essential part of the present scenario which is one distinction from the work of [4–7]. For cyclicity of the entropy, S(t) = S(t + τ) to be consistent with thermodynamics it is insufficient that the huge inflationary enhancement E be completely compensated by deflation at turnaround. Additionally, it is necessary for the thermodynamic arrow of time to reverse during contraction. This is one shortcoming of the proposal. The decrease in entropy during contraction is infinitesimal, being at most 10 (sic) of the entropy increase during expansion. The parameters f and E are related by consistency of the