Proc. Nati. Acad. Sci. USA Vol. 85, pp. 7428-7432, October 1988 Physics Thermodynamics of cosmological matter creation I. PRIGOGINE*t, J. GEHENIAUt, E. GUNZIGt, AND P. NARDONEt *Center for Statistical Mechanics, University of Texas, Austin, TX 78712; and tFree University of Brussels, Brussels, Belgium Contributed by I. Prigogine, June 3, 1988 ABSTRACT A type of cosmological history that includes ergy of these produced particles is then extracted from that large-scale entropy production is proposed. These cosmologies of the (classical) gravitational field (1-4). But these semiclas- are based on reinterpretation of the matter-energy stress ten- sical Einstein equations are adiabatic and reversible as well, sor in Einstein's equations. This modifies the usual adiabatic and consequently they are unable to provide the entropy energy conservation laws, thereby including irreversible mat- burst accompanying the production of matter. Moreover, the ter creation. This creation corresponds to an irreversible ener- quantum nature of these equations renders the various re- gy flow from the gravitational field to the created matter con- sults highly sensitive to quantum subtleties in curved space- stituents. This point of view results from consideration of the times such as the inevitable subtraction procedures. thermodynamics ofopen systems in the framework ofcosmolo- The aim of the present work is to overcome these prob- gy. It is shown that the second law of thermodynamics requires lems and present a phenomenological model of the origin of that space-time transforms into matter, while the inverse the instability leading from the Minkowskian vacuum to the transformation is forbidden. It appears that the usual initial present universe. We propose a phenomenological macro- singularity associated with the big bang is structurally unsta- scopic approach that allows for both particles and entropy ble with respect to irreversible matter creation. The corre- production in the early universe. We shall indeed show that sponding cosmological history therefore starts from an insta- the thermodynamics of open systems (5, 6*), as applied to bility of the vacuum rather than from a singularity. This is cosmology, leads naturally to a reinterpretation, in Ein- exemplified in the framework of a simple phenomenological stein's equations, of the matter stress-energy tensor (7, 8), model that leads to a three-stage cosmology: the first drives the which then takes into account matter creation on a macro- cosmological system from the initial instability to a de Sitter scopic level. To do this, we extend the concept of adiabatic regime, and the last connects with the usual matter-radiation transformation from closed to open systems. This will then Robertson-Walker universe. Matter as well as entropy cre- apply to systems in which matter-creation occurs. ation occurs during the first two stages, while the third in- These considerations lead to an extension of thermody- volves the traditional cosmological evolution. A remarkable namics as associated with cosmology. Traditionally, in addi- fact is that the de Sitter stage appears to be an attractor inde- tion to the geometrical state ofthe universe, the two physical pendent of the initial fluctuation. This is also the case for all variables describing the cosmological fluid are the energy- the physical predictions involving the present Robertson- density p and the pressure p. Einstein's equations are then Walker universe. Most results obtained previously, in the solved assuming an equation of state p = p(p). In our case, framework of quantum field theory, can now be obtained on a however, a supplementary variable, the particle density n, macroscopic basis. It is shown that this description leads quite enters naturally into the description. This leads to an en- naturally to the introduction of primeval black holes as the largement of traditional cosmology, which we shall develop intermediate stage between the Minkowski vacuum and the in Section 3. An important conclusion is that, in these cir- present matter-radiation universe. The instability at the ori- cumstances, creation of matter can occur only as an irrevers- gin of the universe is the result of fluctuations of the vacuum in ible process, corresponding to an irreversible transfer of en- which black holes act as membranes that stabilize these fluctu- ergy from the gravitational field to the created matter. More ations. In short, black holes will be produced by an "inverse" precisely, the transition from traditional cosmology (involv- Hawking radiation process and, once formed, will decompose ing only adiabatic transformations for closed systems) to adi- into "real" matter through the usual Hawking radiation. In abatic transformations for open systems, leads to modifica- this way, the irreversible transformation of space-time into tion of the expression for energy conservation (Section 2). matter can be described as a phase separation between matter As a result, a supplementary effective pressure Pc, related to and gravitation in which black holes play the role of "critical particle creation, appears. This pressure is always negative nuclei." or zero, according to the second law.§ Moreover, it is shown that the big bang singularity, pres- ent in traditional cosmology, is structurally unstable with re- Section I. Introduction spect to irreversible matter creation. Such a cosmology starts from an instability (10-12) of the Minkowski vacuum Very few physical theories are in such a paradoxical situa- and not from a singularity. We specify these properties in tion as cosmology. On the one hand, our universe is charac- Section 3, in the framework of a simple phenomenological terized by a considerable entropy content, mainly in the model of irreversible particle production. This model pro- form of black body radiation. On the other hand, Einstein's vides a cosmological history that evolves in three stages equations are adiabatic and reversible, and consequently (Section 4): (i) A creation period that drives the cosmologi- they cannot provide, by themselves, an explanation of the cal system from an initial fluctuation of the vacuum to a de origin of cosmological entropy. Sitter space; (ii) the de Sitter space exists during the decay As is well known, matter may be produced quantum me- chanically in the framework of Einstein's equations. The en- tRecently, two of us (J.G. and I.P.) have considered the problem of a redefinition of matter-density and pressure in the stress tensor. To some extent, the work reported here continues this attempt. The publication costs of this article were defrayed in part by page charge §It should be emphasized that our approach differs from that used payment. This article must therefore be hereby marked "advertisement" by Hoyle to take into account matter creation with a "C-field" in accordance with 18 U.S.C. §1734 solely to indicate this fact. (see, for example, ref. 9). 7428 Downloaded by guest on September 28, 2021 Physics: Prigogine et aL Proc. NatL Acadl ScL USA 85 (1988) 7429 time of its constituents; and (iii) a phase transition turns the start from the total differential of the entropy de Sitter space into the usual Robertson-Walker universe, Td(sV) = d(pV) + p dV - d(nV), [2-7] which extends to the present. 1L A fundamental fact is that the de Sitter regime appears as where At is the chemical potential an attractor whose properties are independent of the initial fluctuation. This implies, in turn, that all the physical param- s = S/V pn = h-Ts with A .O,2 s 2 O. [2-8] eters characterizing the present Robertson-Walker stage are For closed systems and adiabatic transformation, relation 2- independent of this initial fluctuation. In particular, the spe- 7 leads to cific entropy per baryon S = ny/nb depends only on two characteristic times of the theory: the creation period time rc dS = 0 and diS = 0. [2-9] and the de Sitter decay time Td, which are obtained in Sec- Let us consider the effect of matter creation. We consider tion 4. expect that we still into homogeneous systems and we therefore Our model takes the second law of thermodynamics have dS = 0. In contrast, matter creation contributes to the account from the beginning. Indeed, the energy transfer entropy production. We have therefore from space-time curvature to matter is treated as an irre- versible process, leading to a burst of entropy associated T djS = T dS = (h/n)d(nV) - ,u d(nV) of matter. Therefore, the distinction be- with the creation = . 0. [2-10] tween space-time and matter is provided by entropy cre- T(s/n)d(nV) ation. The latter occurs only during the two first cosmologi- This inequality, in the cosmological context developed in cal stages while, as is well known, the Robertson-Walker Section 3, implies that space-time can produce matter, universe evolves adiabatically on the cosmological scale. while the reverse process is thermodynamically forbidden. It is interesting that "mini black holes" seem to play an The relation between space-time and matter ceases to be essential role in these fluctuations that lead from the Min- symmetrical, since particle production, occurring at the ex- kowski vacuum to the present universe. It is tempting to re- pense of gravitational energy, appears to be an irreversible late this to the fact that black holes play the role of mem- process. A situation somewhat similar corresponds to sys- branes stabilizing the vacuum fluctuations. In this way, the tems in which macroscopic kinetic energy can be trans- primeval stage corresponding to the formation of the uni- formed into internal energy. This kinetic energy then ap- verse can be viewed as a phase separation between gravita- pears as a source term in the entropy balance equation (5, 6). tion and matter. A few remarks concerning this mechanism The microscopic interpretation of this process in which are presented in Section 5.