Arrow of time in dissipationless cosmology

It is generally believed that a cosmological arrow of time must be associated with entropy production. Indeed, in his seminal work on cyclic cosmology, Tolman introduced a viscous fluid in order to make successive expansion/contraction cycles larger than previous ones, thereby generating an arrow of time. However, as we demonstrate in this letter, the production of entropy is not the only means by which a cosmological arrow of time may emerge. Remarkably, systems which are dissipationless may nevertheless demonstrate a preferred direction of time provided they possess attractors. An example of a system with well defined attractors is scalar-field driven cosmology. In this case, for a wide class of potentials (especially those responsible for inflation), the attractor equation of state during expansion can have the form $p \simeq -\rho$, and during contraction $p \simeq \rho$. If the resulting cosmology is cyclic, then the presence of cosmological hysteresis, $\oint p~dV \neq 0$ during successive cycles, causes an arrow of time to emerge in a system which is formally dissipationless. An important analogy is drawn between the arrow of time in cyclic cosmology and an arrow of time in an $N$-body system of gravitationally interacting particles. We find that, like the $N$-body system, a cyclic universe can evolve from a single past into two futures with oppositely directed arrows of time.