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. space-time is converted into matter will be discussed in Sec- tion 5. Section 2. Thermodynamics of Matter Creation Relation 2-7 can be written in a number of equivalent forms such as Let us consider a volume V containing N particles. For a closed system, N is constant. The corresponding thermody- p= (h/n)n [2-11] namic conservation of internal energy E is expressed by p = (np - ph)/n, [2-12] dE = dQ - p dV, [2-1] where an overdot denotes the derivative with respect to during time dr. time. It is interesting to note that the energy creation p and where dQ is the heat received by the system As exam- We rewrite relation 2-1 in the form the particle creation n determine the pressure p. may ples, let us note that p = mn implies p = 0 and, furthermore, d(p/n) = dq - pd(1/n) [2-2] p = aT4, n = bT3 implies p = p/3. with p = E/V, n = N/V, dq = dQ/N. Section 3. Alternative Cosmologies Relation 2-2 is also valid for open systems in which N is time dependent. In this case, relation 2-2 leads to a modifica- The traditional Einstein equations tion of Eq. 2-1 that explicitly takes into account the variation of the number of particles: G~,v = KTyLvl, [3-1] and homogeneous universes, involve - where h = + p [2-3] as applied to isotropic d(pV) = dQ p dV + (h/n)d(nV), p the macroscopic stress tensor T,,,, which corresponds to a is the enthalpy (per unit volume). For closed systems, adia- perfect fluid. It is characterized by a phenomenological ener- batic transformations (dQ = 0) are defined by the relation gy density p and pressure pi given by d(pV) + p dV = 0. [24] p=Toandji8(= Tj. [3-2] The extension to open systems is described by the equation In addition to the Einstein equations (3-1) we use the Bianchi identities d(pV) + p dV - (h/n)d(nV) = 0. [2-5] G'1;, = 0, In such a transformation, the "heat" received by the system is entirely due to the change in the number of particles. In which, for homogeneous and isotropic universes, lead to the our cosmological perspective (see Section 3), this change is well-known relation due to the transfer of energy from gravitation to matter. d(pV) = -pi dV, [3-3] Hence, the creation of matter acts as a source of internal energy. where V represents any comoving volume. In traditional We turn now to the second law of thermodynamics. As cosmology, one identifies this equation with an adiabatic usual, we decompose the entropy change into an entropy evolution for a closed system corresponding to Eq. 2-4. This flow deS and an entropy creation dES: involves an identification between p and the thermodynami- cal pressure p. dS = deS + dS with d1S > 0. [2-6] On the contrary, in the presence of matter creation, the To evaluate the entropy flow and the entropy production, we appropriate analysis must be performed in the context of Downloaded by guest on September 28, 2021 7430 Physics: Prigogine et aL Proc. NatL Acad Sci USA 85 (1988) open systems. This involves the inclusion of a supplemen- inequality 2-10, the simplest possible relation between irre- tary pressure Pc, as we may write Eq. 2-5 in a form similar to versible particle creation and the Hubble function H is Eq. 2-4, namely, 1 d(nR3) d(pV) = -(p + PC)dV, [3-4] R3_d_ -aH.wta0=aH2 2 0 with a 2 O. [4-2] where For a = 0 we recover the usual Robertson-Walker descrip- - h d(nV) = p + p d(nV) tion with its typical big bang singularity. However, for a #& 0, Pc- n dV n dv [5 using Eq. 3-9, we obtain Creation of matter corresponds to a supplementary pressure p = 0, p = (3/K)H2 which must be as Pc, considered part of the cosmological 1 d(nR3) aKM pressure p entering into the Einstein equations _. = ni?3 drd 3 0. [4-3] = + p P Pc, [3-6] This leads to where p is the true thermodynamic pressure. We shall now apply these general considerations concern- n(r)R3(T) = noRjeaIr/3 ing open systems to cosmology. In the case of an isotropic and and homogeneous universe, we choose for V the value R(T) = (1 + C(eaIcMTr/6 - 1))2/3, V=R=(T) where then 9 (KMn°1/2 \ 3 Pc =3nH (h + 3Hn), [3-7] C=KMa / where R(T) is the Robertson-Walker function and H = RI? The universe starts with Ro = 1, at X = 0, with a particle is the Hubble function. density no describing the initial fluctuation. It therefore fol- Because of the thermodynamic inequality 2-10, this exten- lows that the presence of dissipative particle creation (a :& 0) sion of the Einstein equations to open systems now includes leads to the disappearance ofthe big bang singularity. In oth- the second law of thermodynamics. This implies that, in the er words, the singularity of the Einstein equations is struc- presence of matter creation, the usual cosmological Einstein turally unstable with respect to irreversible particle creation. equations become Hence, a cosmology that includes particle creation starts from an instability (no #& 0) and not from a singularity. Kp= 3H2 + k/IR2 [3-8] After a characteristic time ) = (h/n)(p + p). [3-9] Tc = 6/aKM [4-4] The corresponding cosmologies are more general, because the universe reaches a de Sitter regime. they involve three functions-p, p, n-rather than p and p The de Sitter era corresponds to only. We have, for instance, a class ofde Sitter spaces with p = = 0, arbitrary pressure p, and Pc = -h. Rd(r) = C2/ea#]MT/9 = CV3e2r/3rc Therefore our approach "rehabilitates" the de Sitter uni- Hd = aKM/9 = verse, which is now compatible with the existence of matter 213rc endowed with a usual equation of state. We may even con- nd = (KM/27)a. [4-5] sider classes of different de Sitter universes, such as "inco- herent" de Sitter universes (p = mn, n = cst, p = 0) and It is remarkable that all the de Sitter physical quantities such "radiative" de Sitter spaces (T = cst, n = bT3, p = aT4). as Hd, nd, pd are independent of C. This cosmological state It has been suggested that the expansion of the universe therefore appears to be an attractor independent ofthe initial provides the arrow of time. A transition from an expanding fluctuation. No essential change would be introduced if in- universe into a contracting one would then invert the arrow stead of relation 4-2 we started with a more general kinetic of time. We do not confirm this idea, as inequality 2-10 im- equation involving an expansion in powers of H, starting plies only that with H2. We expect the zero spatial curvature to be the natural h + 3Hn > 0, [3-10] starting point, which is then taken over to the de Sitter uni- which is compatible with H . 0, H = 0, and H s 0. Howev- verse as well as into the matter-radiation universe that fol- er, in the case of a de Sitter universe, in which h = 0 by lows. virtue of Eq. 3-9, relation 3-10 reduces to H - 0. Then only The de Sitter stage survives during the decay time Td of its an expanding de Sitter universe is thermodynamically possi- constituents and then connects continuously to a usual (adia- ble. batic) matter-radiation Robertson-Walker universe. Let us investigate the consequences of the transition be- Section 4. Thermodynamic Properties of the Present tween the two regimes. Although our considerations up to Universe this point are quite general, the matching of the de Sitter universe to a Friedman-Lemaitre universe involves supple- mentary assumptions. We indeed expect the post-de Sitter Let us consider a homogeneous and isotropic universe with universe to be of a great complexity, involving a variety of zero spatial curvature. The first Einstein equation (3-9) re- particles and evolving gradually to the present matter-radia- duces to tion universe. To avoid the introduction of unknown param- Kp= 3H2. [4-1] eters, we directly match the de Sitter universe to a matter- radiation universe. This matching is most easily performed Moreover, we assume the simple relation p = Mn between p in conformal coordinates, namely, and n. To complete the problem we need one more relation between n and R(T). In accordance with the thermodynamic ds2 = R2(t)(dt2 - d12) = dT2 - R2(r)dl2, [4-6] Downloaded by guest on September 28, 2021 7431 Physics: Prigogine et al. Proc. NatL Acad ScL USA 85 (1988) evaporation time (15). For N = 2 helicity states (photons) where t is the conformal time coordinate. one has: In these coordinates the de Sitter Robertson-Walker func- and one type of massive scalar particle, tion becomes = = (640/81lr)_(M3/M4) Td (640/8127r)KM3 [416 Rd(T) = Cl/3eHd' Rd(t) = C2'3/(1 - C2Hdt)- [4-7] 2.5 (M/MP)3'PS [4-16] The present adiabatic matter-radiation regime is character- energy where Tp = 2.7 x 10-43 sec is the Planck time. The explicit ized by the matter energy density Pb and the radiation from relations 4-5 and 4- related to the Robertson-Walker function by values for a, cc, and Syb follow then density py, 12. These are KPb = 3a/R3, Kpy = 3b/R4, and py = (ir2/15)p7, [4-8] C = 2/311d = (20/1T2)12(M2/M ) 1.42 (M/Mp)2rp where a and b are constants related to the total number Nb of a = 9Hd/KM = baryons and photons Ny in a volume R3 and T is the black- 9(ir2/45)112(Mp/M3) body radiation temperature. and finally _3a ~~~~2t(3)145\~4b1 Syb = 7.31 X 10-20(M/Mp)el 1778M/Mp. Nb = nbR3 = = 3= 7r2 (72 (K [4-91 KMb My n-yR IT\7T/2 \ More generally, when one takes into account the N helicity where mb is chosen to be the proton mass (4.75 x 1015 m-1) states associated with the massless particles present in the (the system of units used here is such that h = c = kB = 1 cosmological medium as well as one type of massive scalar corresponding to a Planck mass Mp = 1.234 x 1034 m-l). particle, the value of Syb is expressed as The solution of the Einstein equation 4-1 (for a #& 0) leads directly to Sb = 7.31 X 10-20(2/N)(M/Mp) R(t) = (a/4)t2 - (b/a). x exp[1.1778(M/Mp)3V'N1/Vi(N + 1)]. The continuous connection (up to the first derivative, in- very sensitive to the value of time Td The value of Syb is of course volving a discontinuity in the pressure p) at the decay the mass M. (or in conformal time td) then gives N = 2 N = 1OO C3(1 - C2&3Hdtd) 1 = (a/4)t2 - (b/a) = M/Mp = 40* Syb = 8.46 X 102 M/Mp = 200 Syb 8.97 X 102 - = (a/2)td HdC413(1 C213Hdtd)2 M/Mp = 50*Syb = 1.38 X 108 M/Mp = 250 Syb = 2.64 x 108 1- = e -~HdTd 1o13 1 - C213Hdtd= e- [4-10] M/Mp = 60+ Syb = 2.16 X 1013 M/Mp = 300 Syb = 7.45 x This enables us to determine a and b (when eHdd >> 1) With N = 2, the correct observed value for the specific en- tropy Syb is obtained with values of the mass very close to a - 2HdC2e2Hd'r b HdC8/3e4Hdd.H [4-11] the quantum mechanically predicted mass (53.3 Mp) (16). In = for are obtained with This implies that the (constant) specific entropy Syb per contrast, for N 100 similar values Syb = 5. is a mass increased roughly by a factor baryon The present value of the black body radiation temperature fly t(3) f45\ 3/4 can be obtained in our model if we specify the present value Syb = = 2(-22) Km4mbHd 2eHdd. [4-12] of the Hubble function HP. Indeed, one can deduce HP from the expression for the present adiabatic Robertson-Walker Hence the specific entropy per baryon is entirely fixed by function R(t): the knowledge of the two characteristic times m and Td (see -2 a a 2 relation 4-5). Rather than Sybs which involves the baryon - b Hp----,: tp t p [4-17] mass mb explicitly, it is interesting to deduce from relations 2 4 a 4-8 and 4-11 the value of the adiabatic invariant py/Tpb: In the present matter-dominated cosmological stage, where py _ b3/4 ( 2rK\1/4 [4-13] Pb = 3HP/K, we have TPb a 45 (aT4pt2>>bpa Similarly, as shown below, the black body radiation tem- This implies that perature of the present universe can be obtained in terms of times. - == these characteristic Hp 8(atp3)-' Rp (aHp-2)1/3. [4-18] At this point, it is tempting to consider the "proto-parti- cles" of mass M as black holes as suggested earlier (13) on It results from relation 4-8 for the radiation energy density py the basis of quantum mechanical considerations. In the that framework of the present model, the unusual feature of the black hole model is that both Hd and Td can be expressed in ~- (45 bl/4)23)4 [4-19] terms of a single parameter, namely, the mass M of the pro- TP= T a1/3HP duced particles. The value of Hd is (14) Hd= (1r2/45)(K2M4)-' [4-141 Tp(K) 2.82 x1-9 which corresponds to (\75kmsMp 2/3/ M \ 0.3926M/Mp. [4-20] Hd = (Or2/45)"2(M3/M2). [4-15] . are as mini black holes (13, The present value ofHp is bounded by 50 km sl-Mpc Hp As these particles interpreted 1 = 3.09 x 14), their decay time is consequently identified with their < 100 km sl-Mpc (Mpc, million parsec; parsec Downloaded by guest on September 28, 2021 7432 Physics: Prigogine et aL Proc. NatL Acad ScL USA 85 (1988) 1016 m). The observed black body temperature is 2.7 K. Then, these black holes decay through the usual Hawking For any N we have the following relation between the mechanism: absorption of a negative energy flow and emis- black body temperature and the specific entropy: sion of a positive one, as shown below. Tp= 6.74 x 0-35S (.5 kHXMp ) K. [4-21] This delivers for Hp = 75 and Syb = 108, for example, the value Tp - 3.13 K. Section 5. The Cosmological Instability: A Phase Separation Between Matter and Gravitation In essence, we have here a model of initial instability as a phase separation between matter and gravitation. This is in To understand the observed large-scale structure of the agreement with the quantum mechanical description of the world, we need not only some unification program for the vacuum in terms of two interacting lagrangians as presented forces involved but also a mechanism for the'existence ofthe previously (1-4). These considerations have been limited to irreversible processes prevalent at all levels of physical de- an homogeneous and isotropic universe; they have not yet scription. We believe that this irreversibility comes from the been extended to more general models. dissymetrical relation between space-time and matter. In We thank Larry Shepley for discussions. We also thank the Rob- this view, creation of matter introduces a broken time sym- ert A. Welch Foundation and the Instituts Internationaux de Phy- metry into the Einstein equations. It is interesting that dissi- sique et Chimie fonde par Ernest Solvay for their help during the pation eliminates the need for an initial singularity, which is preparation of this paper. here replaced by the instability of the Minkowski back- ground. This instability results (10-12) from' fluctuations of 1. Brout, R., Englert, F. & Gunzig, E. (1978) Ann. Phys. 115, 78- the geometrical Minkowskian background associated to 106. quantum mechanical fluctuations of matter fields. This leads 2. Brout, R., Englert, F. & Gunzig, E. (1979) Gen. Relativ. Grav- to the appearance of unstable massive particles of mass m it. 1, 1-5. << M, creating a background of negative energy. If a suffi- 3. Brout, R., Englert, F., Frere, J.-M., Gunzig, E., Nardone, P., cient number of such masses appear in the same adequate Spindel, P. & Truffin, C. (1980) Nucl. Phys. B 170, 228-264. 4. Brout, R., Englert, F. & Spindel P. (1979) Phys. Rev. Lett. 43, volume element, this would give rise to the formation of a 417-420. black hole. We expect that these black holes act as mem- 5. Prigogine, I. (1947) Open Systems Etude Thermodynamique branes that stabilize such fluctuations. In other words, the des Phe'nomenes Irreversibles (Dunod, Paris). formation of black holes would correspond to the absorption 6. Glansdorff, P. & Prigogine, I. 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