The Generalized Second Law of Thermodynamics in Cosmology

A classical and quantum mechanical generalized second law of thermodynamics in cosmology implies constraints on the effective equation of state of the universe in the form of energy conditions, obeyed by many known cosmological solutions, and is compatible with entropy bounds which forbid certain cosmological singularities. In string cosmology the second law provides new information about the existence of non-singular solutions, and the nature of the graceful exit transition from dilaton-driven inflation. Typeset using REVTEX 1 The existence of cosmological singularities and their nature has been intensely investigated, relying on the celebrated singularity theorems of Hawking and Penrose [1], who concluded that if sources in Einstein’s equations obey certain energy conditions, cosmological singularities are inevitable. Entropy considerations were brought in only much later, when Bekenstein [2] argued that if the entropy of a visible part of the universe obeys the usual entropy bound from nearly flat space situations [3], certain cosmological singularities are thermodynamically unacceptable. Recently, Veneziano [4] suggested that a black hole larger than a cosmological horizon cannot form, and therefore the entropy of the universe is always bounded. This suggestion is related, although not always equivalent, to the application of the holographic principle [5] in cosmology [6–9]. I propose a concrete classical and quantum mechanical form of a generalized second law (GSL) of thermodynamics in cosmology, valid also in situations far from thermal equilibrium, discuss various entropy sources, such as thermal, geometric and quantum entropy, apply GSL to study cosmological solutions, and show that it is compatible with entropy bounds. GSL allows a more detailed description of how, and if, cosmological singularities are evaded. The proposed GSL is different from GSL for black holes [10], but the idea that in addition to normal entropy other sources of entropy have to be included has some similarities. The starting point of our classical discussion is the definition of the total entropy of a domain containing more than one cosmological horizon [4]. For a given scale factor a(t), and a Hubble parameter H(t) = ȧ/a, the number of cosmological horizons within a given comoving volume V = a(t) is simply the total volume divided by the volume of a single horizon, nH = a(t) /|H(t)| (we will ignore numerical factors, use units in which c = 1, GN = 1/16π, h̄ = 1 and discuss only flat, homogeneous, and isotropic cosmologies). If the entropy within a given horizon is S , then the total entropy is given by S = nHS H . Classical GSL requires that the cosmological evolution, even when far from thermal equilibrium, must obey dS ≥ 0, in addition to Einstein’s equations. In particular, nH∂tS H + ∂tnHS H ≥ 0. (1)