Pre-bangian origin of our entropy and time arrow

I argue that, in the chaotic version of string cosmology proposed recently, classical and quantum effects generate, at the time of exit to radiation, the correct amount of entropy to saturate a Hubble (or holography) entropy bound (HEB) and to identify, within our own Universe, the arrow of time. Demanding that the HEB be fulfilled at all times forces a crucial “branch change” to occur, and the so-called string phase to end at a critical value of the effective Planck mass, in agreement with previous conjectures. CERN-TH/99-32 February 1999 The origin of the present entropy of our Universe, S0, is one of the deepest cosmological mysteries. The 2.7 K cosmic microwave background (CMB), if it indeed fills our observable Universe uniformly, contributes a gigantic 10 to S0. However, as repeatedly emphasized by many people, most notably by Roger Penrose [1], such an amount falls very short of what entropy could have been expected to be, even if we go back to the Planckian era, i.e. to t = tP ∼ 10 s after the big bang. Since entropy can only grow, the entropy of our Universe at t = tP , SP , must be smaller than S0; yet, on the basis of the energy content and of the size of the Universe RP at t ∼ tP ≡ lP/c, we might have expected 1 SP ∼ EPRP/ch̄ ∼ ρPR4 P ∼ (RP/lP ) 4 ∼ 10 . (1) The fact that the entropy of our Universe must have been at least 30 orders of magnitude smaller than the value in (1) would nicely “explain” our arrow of time, by identifying the beginning of the Universe with this state of incredibly small entropy near the Planck time [1]. In order to solve the problem, Penrose [1] invokes, without much justification, a new “Weylcurvature hypothesis”. The expected value given in Eq. (1) coincides with the so-called Bekenstein entropy bound (BEB) [2], which states that, for any physical system of energy E and physical size R, entropy cannot exceed SBB = ER/ch̄. This bound is saturated by a black hole of mass E and size equal to its Schwarzschild radius R = GE. If the newly born Universe were a single black hole its Schwarzschild radius would have been much larger than RP , and an even higher entropy, O(10 ), would have resulted. What could have made the initial entropy much smaller than SBB is instead the possibility that the Universe, right after the big bang, was already in a very ordered, homogeneous state. But this is just restating the puzzle in terms of the usual homogeneity problem of standard (non-inflationary) cosmology [3]. The way the two problems are related can be made explicit by introducing a stronger bound on entropy, which, unlike Bekenstein’s general bound, should apply to the special case of (fairly) homogeneous cosmological situations. We shall call it the “Hubble entropy bound” and formulate it as follows: Consider a sufficiently homogeneous Universe in which a (local) Hubble expansion (or contraction) rate can be defined, in the sinchronous gauge, as: H ∼ 1/6 ∂t(log g) , g ≡ det (gij) , (2) with H varying little (percentage-wise) over distances O(H). In this case H, the socalled Hubble radius, is known to correspond to the scale of causal connection, i.e. to the scale within which microphysics can act. In such a context it is hard to imagine that a black hole larger thanH can form, since, otherwise, different parts of its horizon would be unable to hold together. Thus, the largest entropy we may conceive is the one corresponding to having just one black hole per Hubble volume H. Using the Bekenstein–Hawking formula Throughout this paper we will stress functional dependences while ignoring numerical factors.