A note on accelerating cosmologies from compactifications and S-branes

We give a simple interpretation of the recent solutions for cosmologies with a transient accelerating phase obtained from compactification in hyperbolic manifolds, or from S-brane solutions of string/M-theory. In the four-dimensional picture, these solutions correspond to bouncing the radion field off its exponential potential. Acceleration occurs at the turning point, when the radion stops and the potential energy momentarily dominates. The virtues and limitations of these approaches become quite transparent in this interpretation. There is a current effort to understand the recently discovered cosmic acceleration, as well as to embed the inflation paradigm, within a fundamental framework such as string/Mtheory. Progress in this direction has been hampered by some generic arguments, which are often regarded as a no-go theorem, that forbid cosmic acceleration in the cosmological solutions obtained from compactifications of any pure supergravity model [1]. In more detail, one assumes that the internal manifold is compact, static, and non-singular. In recent work, Townsend and Wohlfarth have pointed out that it is possible to find solutions that exhibit a transient phase of acceleration if the size of the internal manifold is allowed to change in time [2]. Further extensions of this idea have appeared more recently in [3, 4, 5], where it is argued that a similar behavior is present as well in certain time-dependent solutions of supergravity with p-form fluxes, known as S(pacelike)-branes [6, 7, 8]. Earlier work on closely related cosmologies can be found in [9, 10]. Our purpose in this note is to point out that the qualitative properties of these solutions become particularly transparent by going to a reduced four-dimensional description. In this framework, the solutions are viewed as cosmologies with scalar exponential potentials, and the origin of the accelerating phase admits a simple and illuminating interpretation. The essence of the idea is as follows. The construction of these models involves compactification on a maximally symmetric space, plus possibly the fluxes of covariantly constant four-form field strengths. In the reduced description, all of these give rise to exponential potential terms for the compactification volume scalar ψ, also known as the “radion”. For sufficiently large negative ψ, the potential is a positive exponential V (ψ) ∼ +e−αψ, while for ψ → +∞, the potential goes to zero as V (ψ) ∼ ±e−βψ. Suppose now that the field starts at a large value of ψ > 0, with very large negative velocity ψ̇. This corresponds to a kinetic-dominated regime with equation of state p ' ρ. So the universe starts out in a phase of decelerated expansion. If the initial kinetic energy of ψ is large enough, then at a later time we will find the field running up the exponential potential hill +e−αψ. Eventually it will stop, due to the increase in potential energy and to cosmic friction. So around this point, the potential energy dominates: the universe enters an accelerating phase. Soon after this, the field starts rolling back down the hill, gaining kinetic energy again and entering a new decelerating phase. The endpoint of the evolution depends on the details of the potential — typically the field rolls back down to ψ →∞, but depending on whether the potential is asymptotically positive or negative, there will be a slowly decelerating expansion, or collapse. These are the attractor solutions of the system. This kind of behavior is precisely the one exhibited by the solutions of [2]. It is then clear that the acceleration is all the result of initial conditions where the field starts out with a large kinetic energy, so it can climb up the positive exponential potential and attain a brief interval of potential domination around the turning point. The expansion that one can get