Equation of State for a van der Waals Universe during Reissner-Nordstrom Expansion
EEquation of State for a van der Waals
Universe during Reissner–Nordstr¨omExpansion
Emil M. Prodanov ∗ , ♠ , Rossen I. Ivanov † , ♠ , ♦ , and V.G. Gueorguiev ‡ , ♣ , ♦♠ School of Mathematical Sciences, Dublin Institute of Technology, Ireland ♣ School of Natural Sciences and Engineering, University of California – Merced,Merced CA 95343, USA ♦ On Leave of Absence from Institute for Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee, Sofia–1784, Bulgaria ∗ [email protected] † [email protected] ‡ [email protected] Abstract
In a previous work [E.M. Prodanov, R.I. Ivanov, and V.G. Gueorguiev,
Reissner–Nordstr¨om Expansion , Astroparticle Physics 27 (150–154) 2007], we proposed aclassical model for the expansion of the Universe during the radiation-dominatedepoch based on the gravitational repulsion of the Reissner–Nordstr¨om geometry —naked singularity description of particles that ”grow” with the drop of the tempera-ture. In this work we model the Universe during the Reissner–Nordstr¨om expansionas a van der Waals gas and determine the equation of state. a r X i v : . [ h e p - t h ] M a y Introduction
In 1971, Hawking suggested [1] that there may be a very large number of gravitation-ally collapsed charged objects of very low masses, formed as a result of fluctuations inthe early Universe. A mass of 10 kg of these objects could be accumulated at thecentre of a star like the Sun. Hawking treats these objects classically and his argumentsfor doing so are as follows [1]: gravitational collapse is a classical process and micro-scopic black holes can form when their Schwarzschild radius is greater than the Plancklength ( Gh/c ) − / ∼ − m (at Planck lengths quantum gravitational effects do notpermit purely classical treatment). This allows the existence of collapsed objects ofmasses from 10 − kg and above and charges up to ±
30 electron units [1]. Additionally,a sufficient concentration of electromagnetic radiation causes a gravitational collapse— even though the Schwarzschild radius of the formed black hole is smaller than thephoton’s Compton wavelength which is infinite. Therefore, when elementary particlescollapse to form a black hole, it is not the rest
Compton wavelength hc/mc that is tobe considered — one should instead consider the modified Compton wavelength hc/E ,where E ∼ kT >> mc is the typical energy of an ultra-relativistic particle that wentto form the black hole [1]. Microscopic black holes with Schwarzschild radius greaterthan the modified Compton wavelength hc/E , can form classically and independentlyon competing quantum processes.Hawking suggests that these charged collapsed objects may have velocities in the range50 – 10000 km/s and would behave in many respects like ordinary atomic nuclei [1].When these objects travel through matter, they induce ionization and excitation andwould produce bubble chamber tracks similar to those of atomic nuclei with the samecharge. The charged collapsed objects survive annihilation and, at low velocities (lessthan few thousand km/s), they may form electronic or protonic atoms [1]: the positivelycharged collapsed objects would capture electrons and thus mimic super-heavy isotopesof known chemical elements, while negatively charged collapsed objects would captureprotons and disguise themselves as the missing zeroth entry in the Mendeleev table.Such ultra-heavy charged massive particles (CHAMPS) were also studied by de Rujula,Glashow and Sarid [2] and considered as dark matter candidates.Dark Electric Matter Objects (DAEMONS) of masses just above 10 − kg and charges ofaround ±
10 electron units have been studied in the Ioffe Institute and positive resultsin their detection have been reported [3] — observations of scintillations in ZnS(Ag)which are excited by electrons and nucleons ejected as the relic elementary Planckiandaemon captures a nucleus of Zn (or S).The DAMA (DArk MAtter) collaboration also report positive results [4] in the detectionof such particles using 100 kg of highly radiopure NaI(Tl) detector.Such heavy charged particles can serve as driving force for the expansion of the Uni-verse during the radiation-dominated epoch in a classical particle-scale model, whichwe recently proposed [5]. Along with this type of particles, within our model, magneticmonopoles can also play the same role for the expansion of the Universe: it has beensuggested [6] that ultra-heavy magnetic monopoles were created so copiously in theearly Universe that they outweighed everything else in the Universe by a factor of 10 .2ur particle-scale model gives the expected prediction for the behaviour of the scale fac-tor of the radiation-dominated expanding Universe, a ( τ ) ∼ √ τ , and can be consideredas a complement to the large-scale Friedmann–Lemaˆıtre–Robertson–Walker (FLRW)model (see, for example, [7, 8]) which describes the Universe as isotropic and homoge-neous, with very smoothly distributed energy-momentum sources modeled as a perfectfluid, applicable on scales much larger than galactic ones.This recently proposed [5] classical mechanism for the cosmic expansion models the Uni-verse as a two-component gas. One of the fractions is that of ultra-relativistic ”normal”particles of typical mass m and charge q with equation of state of an ideal quantumgas of massless particles. The other component is ”unusual” — these are the particlesof ultra-high masses M (of around 10 − kg and above) and charges Q (of around ± q/m ∼ (in geometrized units c = 1 = G ), while for the ”unusual” particles, M < ∼ Q . In viewof this, the general-relativistic treatment of elementary particles or charged collapsedobjects of very low masses also necessitates consideration from Reissner–Nordstr¨om (orKerr–Newman) viewpoint — for as long as their charge-to-mass ratio remains aboveunity. We also treat the ”unusual” particles classically (in line with Hawking’s argu-ments outlined earlier). That is, the ”unusual” particles are modelled as Reissner–Nordstr¨om naked singularities and the expansion mechanism is based on their gravito-electric repulsion. Instead of the Schwarzschild radius, the characteristic length that isto be considered now and compared to the modified Compton length [1], will be theradius of the van der Waals-like impenetrable sphere that surrounds a naked singularity(see [9] for a very thorough analysis of the radial motion of test particles in a Reissner–Nordstr¨om field). As shown in [5], for temperatures below 10 K, the radius of theimpenetrable sphere of an ”unusual” particle of mass 10 − kg and charge ±
10 electronunits is greater than the modified
Compton wavelength of the ”unusual” particle itself.Naked singularities have been subject of significant scrutiny for decades. In the 1950s,the Reissner–Weyl repulsive solution served as an effective model for the electron. Veryrecently, a general-relativistic model for the classical electron — a point charge withfinite electromagnetic self-energy, described as Reissner–Nordstr¨om (spin 0) or Kerr–Newman (spin 1/2) solution of the Einstein–Maxwell equations, — has been studied byBlinder [10]. Naked singularities are disliked — hence the Cosmic Censorship Conjecture[11] — but not ruled out — there is no mathematical proof whatsoever of the CosmicCensorship. At least one naked singularity is agreed to have existed — the Big Bang— the Universe itself. Of particular importance in the study of naked singularities arethe work of Choptuik [12], where numerical analysis of Einstein–Klein–Gordon solutionsshows the circumstances under which naked singularities are produced, and the workof Christodoulo [13] who proved that there exist choices of asymptotically flat initialdata which evolve to solutions with a naked singularity. The possibility of observingnaked singularities at the LHC has been studied in [14] — for example, a proton-protoncollision could result in a naked singularity and a set of particles with vanishing totalcharge or with one net positive charge — an event probably undistinguishable fromordinary particle production. In a cosmological setting, naked singularities have been3ell studied and classified — see, for example, [15].
Consider a ”normal” particle of specific charge q/m , and an ”unusual” particle of charge Q such that sign( Q ) q/m ≥ −
1. If the ”normal” particle approaches the ”unusual”particle from infinity, the field of the naked singularity is characterized by three regions[5, 9]:
U(r) r q''' > q''q''/m > M/Q - repulsion onlyM/Q > q'/m > q/mM/Q > q/m ! - 1 Figure 1: The potential of the interaction between an ”unusual” particle of mass M and charge Q > q/m ≥ −
1. If the specificcharge of the probe is smaller than −
1, then, as shown in [9], the probe will reachthe singularity. Note that electrically neutral probes, in addition to the attraction,also suffer repulsion, while probes of specific charge q/m > M/Q are alwaysrepelled (the gravitational attraction cannot overcome the electric repulsion). Theform of this potential is derived later (4) in this section. The vertical asymptoteto each graph is at r = r ( T ) — the radius of the van der Waals-like impenetrablesphere surrounding the naked singularity. When minima are present, they arelocated at the critical radius r = r c — where attraction and repulsion interchange— see equation (6). (a) Impenetrable region — between r = 0 and r = r ( T ).For an incoming test particle, the condition for reality of the kinetic energy leadsto the existence of two turning radii [5, 9] with a forbidden region in-between. Theupper (outside) radius, which we denote r ( T ), can be thought of as a radius ofan ”impenetrable” sphere surrounding the naked singularity. It depends on theenergy of the incoming particle (or the temperature T of the ”normal” fractionof the Universe): the higher the energy (or the temperature), the deeper the4ncoming particle will penetrate into the gravitationally repulsive field of the nakedsingularity.(b) Repulsive region — between the turning radius r ( T ) and the critical radius r c ≥ r ( T ).The critical radius r c is where the repulsion and attraction interchange (we deter-mine r c later in this section). As the temperature drops, the ”unusual” particles”grow” (incoming particles have lower and lower energies and turn back fartherand farther from the naked singularity). When the temperature gets sufficientlylow, the radius of the ”unusual” particles r ( T ) grows to r c (but not beyond r c , asthe region r > r c is characterized by attraction and an incoming particle cannotturn back while attracted). This means that incoming particles have such lowenergies that they turn back immediately after they encounter the gravitationalrepulsion. Incoming particles of charge q such that qQ > M m do not even experi-ence attraction — we shall see that the repulsive region for such particles extendsto infinity (the gravitational attraction will not be sufficiently strong to overcomethe electrical repulsion).(c) Attractive region — from the critical radius r c to infinity. Again, there is nogravitationally attractive region for an incoming particle such that qQ > M m .As shown in [9], when an incoming particle has sufficiently large charge which is alsoopposite in sign to that of the naked singularity: sign( Q ) q/m < − , the particle willcollide with the naked singularity. When the naked singularity ”captures” such parti-cle, its charge Q decreases and its mass M increases. If sufficient number of incomingparticles are captured, Q will eventually become equal to M — the naked singularitywill pick a horizon and turn into a black hole. This black hole will evaporate quickly af-terwards. We will assume that our ”unusual” particles have survived such annihilation.We will also assume that these super-heavy charged particles have survived annihilationthrough all other different competing mechanisms — for example, they could recombineinto neutral particles or decay before or after that (see Ellis et al. [16] on the astrophys-ical constraints on massive unstable neutral relic particles and Gondolo et al. [17] onthe constraints of the relic abundance of a dark matter candidate — a generic particleof mass in the range of 1 − TeV, lifetime greater than 10 − years, decayinginto neutrinos).An interesting general-relativistic effect (with no classical analogue) is related to theability of naked singularities to capture probes of charge having the same sign. This isassociated with the inner turning radius which we denote by ρ ( T ). On Figure 2, thecurves representing the two turning radii, r ( T ) and ρ ( T ), are given as functions ofthe specific charge q/m of the probe for different temperatures. The forbidden regionis between the two curves. As can be seen, the lower curve ρ ( T ) corresponds to aturning radius (capturing) of a radially outgoing probe with charge having the samesign as the centre. This has no classical analogue and we argue that it could serve as apossible mechanism for the formation of the ”unusual” particles in the extremely densevery early Universe. Moreover, this can also allow the extension of the range of validity5f our model to account for the inflation of the Universe: if charge non-conservation ofthe naked singularities occurs (naked singularities picking up charge), then acceleratedexpansion can be achieved: a ( τ ) ∼ e Hτ or a ( τ ) ∼ τ n , with n > q/m-1 +1r ! (T), ρ ! (T) r ! (T ') r ! (T ''), T '' > T ' ρ ! (T ') ρ ! (T ''), T '' > T ' Figure 2: Plotted as functions of the specific charge q/m of a radially moving testparticle in the field of a positively charged naked singularity, the two temperature-dependent curves r ( T ) and ρ ( T ) represent the outer and inner turning radii,respectively. The region between the curves r ( T ) and ρ ( T ), for any given tem-perature T , is not allowed as it is characterized by negative kinetic energy. Theexplicit form of these curves is given in (5) later in this section (see also [5, 9]). Our expansion model assumes that initially, at extremely high energies and pressures ofthe very early Universe, the ”normal” particles are within the gravitationally repulsiveregions of the ”unusual” particles with radial coordinates just above the upper turningradius r ( T ). The particles from the ”normal” fraction ”roll down” the gravitation-ally repulsive potentials of the ”unusual” particles and in result the Universe expands.The addition of a new class of particles (the ”unusual”) in the picture of the Universedoes not challenge our current understanding of the physical laws governing the Uni-verse. The ”unusual” particles interact purely classically with the ”normal” componentof the Universe and this classical interaction results in the appearance of a repulsiveforce. Our aim is to offer a possible explanation for the expansion of the Universewhile conforming with the well established theoretical models. As shown in [5], duringthe Reissner–Nordstr¨om expansion, the standard relation between the scale factor ofthe Universe a and the temperature T holds: aT = const. Also, during the Reissner–Nordstr¨om expansion, the time-dependance of the scale factor is: a ( τ ) ∼ √ τ (see [5] fordetails). Such is the behaviour of the scale factor during the expansion of the Universethroughout the radiation-dominated era, obtained by the standard cosmological treat-ment.On a large scale, the Universe is isotropic and homogeneous and for a FLRW Universe([7, 8]), the energy-momentum sources are modeled as a perfect fluid, specified by anenergy density and isotropic pressure in its rest frame. This applies for matter knownobservationally to be very smoothly distributed. On smaller scales, such as stars oreven galaxies, this is a poor description. In our picture, the Universe has global FLRWgeometry, but locally it has Reissner–Nordstr¨om geometry. The compatibility of localReissner–Nordstr¨om geometry with global FLRW geometry has been well established:in 1933, McVittie [18] proposed a metric embedding a Schwarzschild solution [19] in aFLRW universe. In 1993, Kastor and Traschen (KT) [20] found a solution desribing a6ystem of an arbitrary number of charged black holes in the background of a de Sitteruniverse [21]. The case of vanishing cosmological constant in the KT solution corre-sponds to the static Majumdar–Papapetrou (MP) solution [22], while the solution withpositive cosmological constant is highly dynamical and describes black holes exchangingradiation with the background until becoming extreme ( | Q | = M ). A spinning versionof the MP solution with naked singularities was found in [23] and [24]. In 1999, theKT solution was extended [25] to multi-Kerr-Newman-de Sitter black holes. Metric forReissner–Nordstr¨om black holes in an expanding/contracting FLRW universe was ob-tained in [26]. The interplay between cosmological expansion and local attraction in agravitationally bound system is studied in [27] where new exact solutions are presentedwhich describe black holes perfectly comoving with a generic FLRW universe.Returning to the local Reissner–Nordstr¨om geometry, on the level of the interaction be-tween the ”unusual” particles and the ”normal” particles of the Universe, the density andpressure variables should be different from those used in the large-scale geometry. Weare going to complement the entire radiation-dominated era with Reissner–Nordstr¨omexpansion and model the interaction between the ”unusual” particles and the ”normal”particles as interaction between the components of a van der Waals gas. Modeling theUniverse as a van der Waals phase is possible in the light of the deep analogies betweenthe physical picture behind the Reissner–Nordstr¨om expansion and the classical vander Waals molecular model: atoms are surrounded by imaginary hard spheres and themolecular interaction is strongly repulsive in close proximity, mildly attractive at inter-mediate range, and negligible at longer distances. The laws of ideal gas must then becorrected to accommodate for such interaction: the pressure should increase due to theadditional repulsion and the available volume should decrease as atoms are no longerentities with zero own volumes (see, for example, [28]).As an interesting development in a similar vein, one should point out the work [29] (seealso the references therein) which studies van der Waals quintessence by consideringa cosmological model comprising of two fluids: baryons, modelled as dust (large-scalestructure fluid) and dark matter with a van der Waals equation of state (backgroundfluid). Van der Waals equation of state for ultra-relativistic matter has been studied by[30].During the Reissner–Nordstr¨om expansion, once the temperature drops sufficiently lowso that r ( T ) becomes equal to r c , the ”normal” particle with charge q , such thatsign( Q ) q/m ≥ − qQ < mM , will be expelled beyond r = r c (as r ( T ) < r always) — into the region of gravitational attraction. Due to its ultra-high energy, the”normal” particle will overcome the gravitational attraction and will escape unopposedto infinity. Thus the gravitationally attractive region is of no importance for such par-ticles and for them we can assume that the potential of the naked singularity is infinityfrom r = 0 to r = r ( T ) and zero from r = r c to infinity.For ”normal” particles such that qQ > mM , the potential gradually drops to zero to-wards infinity (there is no attraction for these probes). For ultra-high temperatures, theenergy E of a ”normal” particle is of the order of kT . At temperatures below 10 K, thedominant term in the energy E becomes the particle’s rest energy mc (throughout thepaper we use geometrized units) and, as we shall see, the turning radius r ( T ) becomes7nfinitely large below such temperature. As we model the entire radiation-dominatedepoch with Reissner–Nordstr¨om repulsion, at Recombination (the end of this epoch: t recomb ∼
300 000 years), the free ions and electrons combine to form neutral atoms( q = 0) and this naturally ends the Reissner–Nordstr¨om expansion — a neutral ”nor-mal” particle will now be too far from an ”unusual” particle to feel the gravitationalrepulsion (the density of the Universe will be sufficiently low). During the expansion,the volume V of the Universe is proportional to the number N of ”unusual” particlestimes their volume (one can view the impenetrable spheres of the naked singularitiesas densely packed spheres filling the entire Universe). At Recombination, V ∼ t recomb .Therefore, at Recombination, the radius r ( T ) of an ”unusual” particle will be of theorder of R c = N − / t recomb . During the expansion, a ”normal” particle is never fartherthan r ( T ) from an ”unusual” particle. We will request that once r ( T ) becomes equalto R c = N − / t recomb , then the potential of the interaction between a naked singularityand a particle of charge q , such that qQ > mM , becomes zero.In this paper we use a standard treatment [28] to model the van der Waals phase of theUniverse as a real gas and, using the virial expansion, we obtain the gas parameters.Combining the van der Waals equation with aT = const, we find the equation of statedescribing the classical interaction between the ordinary particles in the Universe andthe “unusual” particles.Consider the Reissner–Nordstr¨om geometry [31, 8] in Boyer–Lindquist coordinates [32]: ds = − ∆ r dt + r ∆ dr + r dθ + r sin θ dφ . (1)where: ∆ = r − M r + Q , M is the mass of the centre, and Q — the charge of thecentre. We will be interested in the case of a naked singularity only, namely: Q > M .The radial motion of a test particle of mass m and charge q in Reissner–Nordstr¨omgeometry can be modeled by an effective one-dimensional motion of a particle in non-relativistic mechanics with the following equation of motion [5, 9] (see also [33] forSchwarzschild geometry) :˙ r (cid:104) − (cid:16) − qm QM (cid:15) (cid:17) Mr + 12 (cid:16) − q m (cid:17) Q r (cid:105) = (cid:15) − , (2)where (cid:15) = E/m is the specific energy (energy per unit mass) of the three-dimensionalmotion. The expression in the square brackets is the effective non-relativistic one-dimensional potential and the specific energy of the effective one-dimensional motion is(1 / (cid:15) − U ( r ) of the three-dimensional motion. In the rest frame of the probe( ˙ r = 0), equation (2) becomes a quadratic equation for the energy (cid:15) . The bigger rootof this equation is exactly the gravitational potential energy U ( r ) plus the rest energy m (see also [34]). Namely: U ( r ) = qQ + m √ ∆ r − m = qQr + m (cid:115) − Mr + Q r − m . (3)8ince M ∼ Q ∼ − cm, expression (3) for the potential energy U ( r ), for distancesabove 10 − cm, can be approximated by: U ( r ) = − mMr + qQr + m − M + Q ) 1 r . (4)From now on, we will use this pseudo-Newtonian potential to mimic general-relativisticeffects with a classical theory.Motion is allowed only when the kinetic energy is real. Equation (2) determines theregion ( r − , r + ) within which motion is impossible. The turning radii are given by [5, 9]: r ± = M(cid:15) − (cid:34) (cid:15) qm QM − ± (cid:115)(cid:16) (cid:15) qm QM − (cid:17) − (1 − (cid:15) ) (cid:16) − q m (cid:17) Q M (cid:35) . (5)We identify the impenetrable radius r ( T ) of an “unusual” particle as the bigger root r + and the inner turning radius ρ ( T ) as the smaller root r − . The expansion mechanism isbased on the fact that r ( T ) is inversely proportional to the temperature, namely, thenaked singularity drives apart all neutral particles and particles of specific charge q/m such that sign( Q ) q/m ≥ − (cid:15) → kT drops below m , or below 10 K), then the turning radius r ( T )tends to infinity.At the point where gravitational attraction and repulsion interchange, there will be noforce acting on the incoming particle. That is, this is the point where the derivative ofthe potential (4) vanishes: r c = M (cid:16) Q M − (cid:17)(cid:16) − qm QM (cid:17) − . (6)Obviously, the critical radius r c for an incoming particle charged oppositely to the”unusual” particle ( qQ <
0) will be smaller than the critical radius for a neutral ( q =0) incoming particle (neutral particles suffer repulsion) as the region of gravitationalrepulsion will be reduced by the additional electrical attraction. When the incomingprobe has charge with the same sign as that of the ”unusual” particle and qQ > mM ,then r c does not exist. This means that there will be a region of repulsion only —the gravitational attraction will not be sufficiently strong to overcome the electricalrepulsion.Finally, the potential energy of a charged probe in the field of an “unusual” particle canbe written as follows: U ( r ) = ∞ , r < r ( T ) , − mMr + qQr + m ( − M + Q ) r , r ( T ) ≤ r ≤ R ,0 , r > R , (7)9here: R = (cid:40) r c , sign( Q ) q/m ≥ − qQ ≤ mM ,R c , qQ > mM . (8)Obviously, the expansion beyond r c will be due to those particles that satisfy qQ > mM . Next, we consider the thermodynamics of a real gas. The virial expansion relates thepressure p to the particle number N , the temperature T and the volume V [28]: p = N kTV (cid:104)
NV F ( T ) + (cid:16) NV (cid:17) G ( T ) + · · · (cid:105) , (9)where the correction term F ( T ) is due to two-particle interactions, the correction term G ( T ) is due to three-particle interactions and so forth. We will ignore all interactionsinvolving more than two particles. The correction term F ( T ) is [28]: F ( T ) = 2 π ∞ (cid:90) λ ( r ) r dr = β − αkT , (10)where λ ( r ) is given by: λ ( r ) = 1 − e − U ( r ) kT . (11)Then “van der Waals” equations is [28]: p + (cid:16) NV (cid:17) α = N kTV (cid:16)
NV β (cid:17) . (12)In the limit N β/V →
0, this equation reduces to the usual van der Waals equation [28]: (cid:104) p + (cid:16) NV (cid:17) α (cid:105)(cid:16) − NV β (cid:17) = N kTV . (13)We now assume that the “unusual” particles leave “voids” in the Universe where “nor-mal” particles cannot enter. Thus, the effective space left for the motion of the “normal”component of the gas is reduced by
N β , where β is the “volume” of an “unusual” par-ticle and N is the number of “unusual” particles. We will also pretend that “unusual”particles are not present and that the potential in which the “normal” particles move isnot due to the “unusual” particles, but rather to the two-particle interactions betweenthe “normal” component of the gas. In essence, we “remove” N “unusual” particles outof all particles and we are dealing with a gas of n “normal” particles. The ”van derWaals” equation (12) then becomes: p + (cid:16) NV (cid:17) α = nkTV (1 + NV β ) , (14)10or the potential determined in (7), we have: λ ( r ) = 1 − e − U ( r ) kT = , r < r ( T ) , U ( r ) kT , r ( T ) ≤ r ≤ R , , r > R . (15)We then get: β = 2 π r ( T ) (cid:90) r dr = 2 π r ( T ) = 12 v ( T ) , (16) α = 2 π R (cid:90) r ( T ) U ( r ) r dr = πmM (cid:16) − Q M (cid:17) [ R − r ( T )]+ πmM (cid:16) − qm QM (cid:17) [ R − r ( T )] , (17)where v ( T ) is the “volume” of an ”unusual” particle. Note that both α and β dependon the temperature via the particle’s radius r ( T ).We have shown [5] that for our expansion model, the standard relation between thescale factor of the Universe a and the temperature T holds: aT = const. Let ρ denotethe density of the Universe. Then, as the volume V of the Universe is proportional tothe third power of a and as V ∼ /ρ , we have T ∼ ρ / . Therefore, T /V ∼ ρ / .The volume V of the Universe during the van der Waals phase is proportional to thevolume v ( T ) of the ”unusual” particles times their number N . Using equation (16),namely: β = v ( T ), it immediately follows that N β/V is, essentially, constant.Equation (14) is the equation of state for the van der Waals phase of the expandingUniverse and can be written: as: p = ηρ / − αβ . (18)Here η is some constant. The second term depends on the temperature via α and β and becomes irrelevant towards the end, as α → r ( T ) → R . Note also that thecorrection term − α/β is positive as α is negative. Acknowledgements
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