Unification of Cosmology and Second Law of Thermodynamics: Solving Cosmological Constant Problem, and Inflation

Abstract

We seek here to unify the second law of thermodynamics with the other laws, or at least to put up a law behind the second law of thermodynamics. Assuming no fine tuning, concretely by a random Hamiltonian, we argue just from equations of motion -- but {\em without} second law -- that entropy cannot go first up and then down again except with the rather strict restriction S_{large} \le S_{small 1} + S_{small 2}. Here S_{large} is the "large" entropy in the middle era while S_{small 1} and S_{small 2} are the entropies at certain times before and after the S_{large} - era respectively. From this theorem of "no strong maximum for the entropy" a cyclic time S^1 model world could have entropy at the most varying by a factor two and would not be phenomenologically realistic. With an open ended time axis (-\infty, \infty) ={\bf R} some law behind the second law of thermodynamics is needed if we do not obtain as the most likely happening that the entropy is maximal (i.e. the heat death having already occurred from the start). We express such a law behind the second law -- or unification of second law with the other ones -- by assigning a probability weight $P$ for finding the world/the system in various places in phase space. In such a model $P$ is almost unified with the rest as P = exp (-2 ~S_{Im}) with S_{Im} going in as the imaginary part of the action. We derive quite naturally the second law for practical purposes, a Big Bang with two sided time directions and a need for a bottom in the Hamiltonian density. Assuming the cosmological constant is a dynamical variable in the sense that it is counted as "initial condition" we even solve in our model the cosmological constant problem \underline{without} any allusion to anthropic principle.