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Chemical Potential and the Nature of the Dark Energy: The case of phantom J. A. S. Lima∗ and S. H. Pereira† Departamento de Astronomia, Universidade de São Paulo Rua do Matão, 1226 - 05508-900, São Paulo, SP, Brazil The influence of a possible non zero chemical potential µ on the nature of dark energy is inves- tigated by assuming that the dark energy is a relativistic perfect simple fluid obeying theequation of state (EoS), p = ωρ (ω < 0,constant). The entropy condition, S ≥ 0, implies that the possible valuesofωareheavilydependentonthemagnitude,aswellasonthesignofthechemicalpotential. Forµ>0,theω-parametermustbegreaterthan-1(vacuumisforbidden)whilefor µ<0notonly the vacuum but even a phantomlike behavior (ω < −1) is allowed. In any case, the ratio between thechemicalpotentialandtemperatureremainsconstant,thatis,µ/T =µ0/T0. Assumingthatthe darkenergyconstituentshaveeitherabosonicorfermionicnature,thegeneralformofthespectrum 8 is also proposed. For bosons µ is always negative and the extended Wien’s law allows only a dark 0 component with ω <−1/2 which includes vacuum and the phantomlike cases. The same happens 0 in the fermionic branch for µ < 0. However, fermionic particles with µ> 0 are permmited only if 2 −1 < ω < −1/2. The thermodynamics and statistical arguments constrain the EoS parameter to n be ω < −1/2, a result surprisingly close to the maximal value required to accelerate a FRW type a universedominated bymatter and dark energy (ω.−10/21). J 2 PACSnumbers: 98.80.-k,95.36.+x ] h I. INTRODUCTION is phenomenologically described by an equation of state p p = ωρ. The case ω = −1 reduces to the cosmologi- - o cal constant. The imposition ω ≥ −1 is physically mo- The current idea of an accelerating Universe is based r tivated by the classical fluid description [6]. This hy- t on a large convergence of independent observations, and s phothesis introduces a strong bias in the ω-parameter a its explanation constitutes one of the greatestchallenges determination from observational data. In order to cir- [ for our current understanding of fundamental physics cunvent such a difficulty, superquintessence or phantom [1, 2]. In the context of a Friedmann-Robertson-Walker 1 dark energy cosmologies have been recently considered v (FRW) cosmology, dominated by pressureless matter where such a condition is relaxed [7]. In contrast to the 3 with density ρm plus an extra component of density ρ usual quintessence model, a decoupled phantom compo- 2 and pressure p, the scale factor evolution is governed by nent presents an anomalous evolutionary behavior. For 3 the equation 3a¨/a = −4πG(ρ +ρ+3p). This means 0 m instance,theexistenceoffuturecurvaturesingularities,a that a hypothetical component with large negative pres- . growthoftheenergydensitywiththe expansion,oreven 1 suremaydrivetheevolutionofanexpandingaccelerating the possibility of a rip-off of the structure of matter at 0 Universe. This exotic component is usually termed dark 8 all scales are theoretically expected. Although possess- energy or quintessence, and it represents about 70% of 0 ing such strange features, the phantom behavior is the- the total content in the Universe. The origin and the : oretically allowed by some kinetically scalar field driven v natureofdarkenergyis stillamystery,however,there is cosmology[8],aswellas,bybraneworldmodels[9],and, i no doubt that its existence is beyond the domain of the X perhaps, more important to the present work, a Phan- standard model of particle physics [3]. r tomCDMcosmologyis notruledoutby the presenttype a Among a number of possibilities to describe the dark Ia Supernovae and other observations [10, 11]. energy component, the simplest and most theoretically In a series of papers [12, 13], we have studied some appealing way is by means of a cosmologicalconstant Λ, thermodynamicsandstatisticalpropertiesofdarkenergy which acts on the FRW equations as an isotropic and with no chemical potential (µ = 0). By using standard homogeneous source with a constant equation of state thermodynamics for a relativistic simple fluid, we con- parameter p/ρ = −1. On the other hand, although cos- cluded that the case of phantom energy is ruled out be- mologicalscenarioswith a Λ term mightexplain mostof causethetotalentropyofadarkcomponentwithω <−1 the current astronomical observations,from the theoret- is negative. In addition, by combining thermodynamics ical viewpoint they are plagued with some fundamental and statistical arguments the EoS was restricted to the problems thereby stimulating the search for alternative interval −1 ≤ ω < −1/2 and a fermionic nature to the dark energy models driven by different candidates [4, 5]. dark energy particles was favored. In the XCDM scenario, the dark energy component Later on, thermodynamics arguments in favor of the phantom hypothesis were put forward by González-D´ıaz andSiguënza[14]. Theyclaimedthatthetemperatureof ∗Electronicaddress: [email protected] aphantomlikefluidisalwaysnegativeinordertokeepits †Electronicaddress: [email protected] entropy positive definite (as statistically required). This 2 new viewpoint was justified by arguing that the scalar and negative (dark energy). The cases ω = 1/3, 1, and field representationof a phantom field has a negative ki- −1 characterizes, respectively, the blackbody radiation, netic term φ˙2 which quantifies the translational kinetic a stiff-fluid and the vacuum state while ω < −1 stands energy of the associated fluid system, and, as such, its to a phantomlike behavior. temperature (a measure of the average kinetic energy) Following standard lines, the equilibrium thermody- should be negative. Although temptative to some de- namic states of a relativistic simple fluid are character- gree, both approaches have been considered in the lit- izedby anenergymomentumtensor Tαβ,a particle cur- erature (see, for instance, [15, 16] and Refs. therein). rent Nα and an entropy current Sα which satisfy the More important to the present work, they share a com- following relations monproperty,namely,thechemicalpotentialofthedark energy fluid was set to be zero from the very beginning. Tαβ =(ρ+p)uαuβ −pgαβ, Tαβ;β=0, (3) In this article we reanalyze the thermodynamic and statistical properties of the dark energy scenario by con- Nα =nuα, Nα; =0, (4) α sidering the existence of a non-zero chemical potential. In order to clarify some subtleties present in the earlier results, we rederive the physical quantities in the pres- Sα =suα, Sα; =0, (5) α ence of µ by adopting the full thermostatistic approach proposed in Refs. [12, 13]. This means that all thermo- where (;) means covariant derivative, n is the particle dynamic and statistical properties follow directly from number density, and s is the entropy density. In the the EoS plus the hypothesis that µ 6= 0. In particular, FRW background, the above conservation laws can be the temperatures of the dark energy fluids must be al- rewritten as (a dot means comoving time derivative) wayspositivedefinite. Thisisaninterestingaspectofthe presentworksincetherearemanyscalarfieldrepresenta- a˙ a˙ a˙ tionsfordarkenergyfluids,however,thethermodynamic ρ˙+3(1+ω)ρ =0, n˙ +3n =0, s˙+3s =0, (6) a a a laws are independent to what happens at a microscopic level as long as the equation of state has been defined. whose solutions can be written as: As we shall see, one of the main consequences of a negative chemical potential is that the phantom scenario is a 3(1+ω) a 3 a 3 recovered without the need to appeal to negative tem- ρ=ρ 0 , n=n 0 , s=s 0 , (7) 0 0 0 a a a peratures. In addition, a bosonic or fermionic nature of (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) the dark energy component now becomes possible. where the positive constants ρ , n , s and a are the 0 0 0 0 Thepaperisplannedasfollows. Insection2wediscuss presentday valuesofthe correspondingquantities (here- the thermodynamic constraints when a chemical poten- after present day quantities will be labeled by the index tial is introduced. In section 3 we consider a statistical “0”). Onthe otherhand,the quantitiesp,ρ, nandsare analysis by assuming that the dark energy particles are related to the temperature T by the Gibbs law massless and can have either a bosonic or a fermionic nature. Inthe conclusionsection,ajointanalysisinvolv- s ρ+p ingthethermodynamicandstatisticalconstraintsonthe nTd( )=dρ− dn, (8) EoS ω-parameter is presented n n and from Gibbs-Duhem relation [18] there are only two independent thermodynamic variables, say, n and T. II. COSMOLOGY, DARK ENERGY AND Now, by assuming that ρ = ρ(T,n) and p = p(T,n) and THERMODYNAMICS combining the thermodynamic identity [17] Let us now consider that the Universe is described bythehomogeneousandisotropicFriedmann-Robertson- ∂p ∂ρ T =ρ+p−n , (9) Walker (FRW) geometry (c=1) (cid:18)∂T(cid:19) (cid:18)∂n(cid:19) n T ds2 =dt2−a2(t) dr2 +r2dθ2+r2sin2θdφ2 , (1) withtheconservationlawsasgivenby(6),onemayshow (cid:18)1−κr2 (cid:19) that the temperature satisfies where κ = 0,±1 is the curvature parameter and a(t) T˙ ∂p n˙ a˙ is the scale factor. In what follows we consider that the = =−3ω . (10) T (cid:18)∂ρ(cid:19) n a mattercontent(oratleastoneofitsnoninteractingcom- n ponents is a fluid described by the EoS Therefore, assuming that ω 6=0 we obtain p=ωρ, (2) where p is the pressure, ρ is the energy density and ω a T ω1 a −3ω n=n ⇔ T =T . (11) constantparameterwhichmaybepositive(whiteenergy) 0(cid:18)T (cid:19) 0(cid:18)a (cid:19) 0 0 3 Thetemperaturesappearingintheaboveexpressionsare positiveregardlessofthe valueofω. Inparticular,inthe radiation case (ω =1/3), one finds aT =a T as should 0 0 beexpected. Ascomparedtothiscase,theuniquediffer- ence is that the dark energy fluid (even in the phantom regime)becomes hotter inthe courseofthe cosmological adiabaticexpansionsinceitsequation-of-stateparameter isanegativequantity. Aphysicalexplanationforthisbe- havior is that thermodynamic work is being done on the system [12, 13]. It should be stressed that the derivation of the temperature evolution law presented here is fully independent of the entropy function, as well as, of the chemical potential µ. The above expressions also imply that for 1 any comoving volume of the fluid, the product TωV re- mains constantin the course of expansionand must also FIG. 1: The allowed intervals of ω values (heavy lines) and characterizetheequilibriumstates(adiabaticexpansion) forbidden(dashedlines)fornull,positiveandnegativechem- regardless of the value of µ. Further, by inserting the ical potentials. Note that a large portion of the dark branch temperature law into the energy conservation law (7), ω <0 is always thermodynamically permitted. However, for one obtains the energy density as function of the tem- µ≥0, thephantomlikebehavior(ω<−1)is thermodynami- perature cally forbidden. 1+ω T ω At this point, the fundamental question is: How the ρ=ρ . (12) 0 (cid:18)T (cid:19) chemical potential modifies the entropy constraints [12, 0 13] derived in the previous papers? Now,inorderto determine the chemicalpotentialand In order to show that we compute explicitly the en- its influence on the thermodynamics of dark energy, we tropy of dark energy for a comoving volume V. As consider the Euler relation [18] remarked before, the entropy function should scale as S ∝Tω1V. Actually, Ts=p+ρ−µn, (13) whereµin generalcanalsobe afunction ofT andn [19, (1+ω)ρ −µ n T 1/ω 20]. By combining the above expression with equations S(T,V)≡sV = 0 0 0 V =s V , 0 0 (2), (7) and (11) we obtain: (cid:20) T0 (cid:21)(cid:18)T0(cid:19) (16) whichremainsconstantasexpected(seediscussionbelow a −3ω T Eq.(11)). However, in order to keep the entropy S ≥ 0 µ=µ =µ , (14) 0 0 (as statistically required), the following constraint must (cid:18)a (cid:19) (cid:18)T (cid:19) 0 0 be satisfied: where µ n ω ≥ω =−1+ 0 0 , (17) min 1 ρ µ = [(1+ω)ρ −T s ]. (15) 0 0 0 0 0 n 0 which introduces a minimal value to the ω-parameter, This straightforward thermodynamic result has some below which the entropy becomes negative. This is a re- interesting consequences. In principle, the chemical po- markable expression and its consequences are apparent. tential may be either positive or negative, and it also For instance, consider that µ = 0 (no chemical poten- 0 dependsonthevaluesoftheEoSω-parameter. Inpartic- tial). Inthiscase,thesmallestvalueoftheω−parameter ular,µisalwaysnegative(µ <0)inthecaseofphantom is ω =−1 andthe previous analysisby Lima and Al- 0 min energy,and becomes evenmore negative in the courseof caniz[13]isfullyrecovered,thatis,thephantomdomain time (T grows with the scale factor during the cosmic (ω <−1) is thermodynamically forbidden. However, for evolution). It is also known that µ is zero in the case a negative chemical potential, the phantomlike regime of photons (ω = 1/3) because they are their own an- becomes thermodynamically allowed thereby recovering tiparticles [21]. In this case, (15) yields correctly that the hypothesis of phantom energy without appealing to 3s T =4ρ as should be expected. In general, if µ=0, negative temperature as proposed in the literature [14]. 0 0 0 necessarilytherelations T =(1+ω)ρ mustbeobeyed, Note also that for a positive chemical potential not even 0 0 0 whichisjustthepresentdayexpressionofsT =(1+ω)ρ a cosmological constant (ω = −1) is possible. In figure as required by (13). 1, we summarize the basic thermodynamic results. 4 1 1 ω=−0.4 (a) (b) ω=−0.3 0 0 ω=−0.1 cosmological y=1.0 ω=0.1 constant K1 ω=0.2 K1 y=0.37 ω=1/3 y=1.0 K2 ω=1/2 K2 ω=−3 y=0.135 ω=1 y=0.37 ω=−3/2 K3 K3 y=0.050 ω=−1 y=0.135 K4 K4 y=0.018 .blackbody ω=−0.75 radiation y=0.050 K5 K5 ω=−0.6 y=0.007 y=0.018 y=0.002 y=0.007 K6 K6 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 x x FIG. 2: Solutions for the bosonic case (µ ≤ 0). a) For given values of the pair (x,y), the straight lines and curves represent the l.h.s. side and the r.h.s. of (21), respectively. The intersection points between the curves and the straight lines represent the desired physical solutions for ω > 0. The standard radiation solution (µ = 0 and ω = 1/3) is indicated by a black point. b) Curves representing the right hand side of (21) for different ω values in the region ω <−1/2 (lower half plane) and −1/2 < ω ≤ 0 (upper half plane). The cosmological constant case (ω = −1) is indicated. For both diagrams the fugacity y=exp(µ/kBT)≤1. III. DARK ENERGY WITH A CHEMICAL POTENTIAL: STATISTICAL BEHAVIOR ǫω ye−x =−ǫ+ x, (20) (cid:20)1+2ω(cid:21) Another interesting feature of a dark energy component with a non zero chemical potential is related to its where x = hc/k λT and y = exp(µ/k T) ≡ B B spectral distribution. The generalized Wien-type spec- exp(µ /k T ) is a constant fugacity. When µ = 0 the 0 B 0 0 trum for dark energy with µ = 0 has already been dis- above expressions reduces to the one obtained in [13]. cussedinthe literature[12,13]. Adifferent approachfor The solution of the above algebraic equation can be de- the phantomlike behavior involving the modulus of the rived both numerically and graphically. We are only in- temperature has also been proposed [14]. In the present terested in solutions with positive x, because the tem- case, since the temperature is positive, we simply add perature is always positive. Due to the presence of the the chemical potential µ to the spectrum previously de- chemical potential, the analysis of the above condition rived[12, 13]. More precisely, we postulate the following will be done separately for bosons and fermions. spectral distribution: αν1/ω A. Bosons ρ(T,ν)= , (18) e(hν−µ)/kBT +ǫ Let us now analyze the bosonic case (ǫ = −1). It where ǫ = +1 stands for the Fermi-Dirac distribution provesconvenienttorewritecondition(20)inthefollow- and ǫ=−1 to the Bose-Einsteinone, and α is a positive ing form: and ω-dependent constant. Here it is important to note that for bosonsthe chemicalpotential is alwaysnegative or null, while for fermions it may be positive or negative ωx lny−x=ln 1− . (21) [21]. (cid:18) 1+2ω(cid:19) Adirectconsequenceof(18) is relatedto the displace- ment Wien’s law. The wavelength for which the distri- For each value of µ, the left hand side (l.h.s.) of the bution attains its maximum value is determined by the aboveexpressionisacollectionofstraightlineswithslope condition equalto−1. Sinceµisalwaysnegativeornullforbosons, it follows that 0 < y ≤ 1 so that −∞ < lny ≤ 0. This means that the l.h.s. of (21) is a collection of parallel hc 1.438 straight lines on the lower half plane. The first straight λ T = = , (19) m k x′(ω) x′(ω) line is the trivial solution with zero chemical potential. B Note also that the right hand side (r.h.s.) of (21) in- where x′(ω) is the root of the transcendent equation volves the singularity for ω = −1/2, and, as such, must 5 y=148 y=403 2 y=54.6 y=20.1 y=7.39 ω=−0.7 cosmological constant ω=−1 1 y=2.72 ω=−3/2 ω=−4 0 y=1.0 ω=1 K1 y=0.37 ω=0.7 ω=0.5 ω=0.4 K2 y=0.14 FIG. 4: Thermodynamic and statistical constraints. The allowed (heavy lines) and forbidden (dashed lines) values of ω 0 1 2 3 4 5 for the bosonic and fermionic cases with µ < 0 and µ > 0. x The phantom branch ω < −1 is excluded for the fermionic case with ω >0. Note that the dark branch −1<ω <−1/2 for bosons is now possible. FIG. 3: Curves representing the right hand side of (22), for different ω values in theregion ω<−1/2 and ω>0. fermionic case reads be separately analyzed. Apart from this point, we have ωx 3 different intervals, namely: ω > 0, ω < −1/2 and lny−x=ln −1+ . (22) (cid:18) 1+2ω(cid:19) −1/2<ω ≤0. In Figures 2a and 2b we display the main results. The analysis on the l.h.s. of (22) is similar to the When ω > 0 we have a collection of curves represented bosoniccase,theunique differenceisthatµ canbe pos- 0 in Fig. 2a. All of them cross the straight lines in some itive. In this case, we have y >1 and lny >0. As in the point x > 0 thereby indicating the solutions of the al- previouscase,thediscussiononther.h.s. of(22)depends gebraic equation (21). The standard radiation solution on the ω values. For the cases ω < −1/2 and ω > 0 we (µ=0andω =1/3)isindicated,however,itshouldalso have the curves represented in Figure 3. We see that all be remarkedthe theoretical possibility of a radiationso- curves crossing the straight lines for some positive value lution with µ6= 0. For ω < −1/2 we have the collection ofx yield a validsolutionfor (22). Note alsothat onthe ofcurvesrepresentedonthelowerhalfplaneofFigure2b, interval −1/2 < ω < 0, all the curves are in the nega- superposedto the straightlines. This analysisshowthat tive x-axis (negative temperatures),and, therefore,none all these curves cross the straight lines in some positive of them cross the straight lines (two reasons for the the x value, indicating a solution to the algebraic expression interval be a forbidden region). (21). Finally, on the interval −1/2<ω ≤0 we have the set of curves represented on the upper half plane of Fig- ure2btherebyshowingtheabsenceofphysicalsolutions. IV. CONCLUDING REMARKS In summ, a simple graphic analysis shows that there are two intervals of ω for which the condition (21) has a Inthis paperwehaveinvestigatedthe thermodynamic solution, namely, ω > 0 and ω < −1/2, while the inter- and statistical properties of a dark energy fluid with val −1/2 < ω ≤ 0 is statistically forbidden. Therefore, equation of state, p = ωρ, by assuming that its chem- unlikethe previousanalyzeswithµ=0[12, 13],the EoS ical potential is different from zero. ω < −1/2 for bosons now becomes possible when the In Figure 4, we summarize the main results of the chemical potential is negative. This includes the phan- present analysis by combining both approaches. As dis- tom dark energy as a physical possibility. cussed in section 2, the regions with S < 0 are thermo- dynamicallyforbidden. Notealsothatmanydarkenergy fluids satisfy the combined constraints regardless of the B. Fermions µ sign, that is, a large interval of negativeω values is allowed by thermodynamic and statistical considerations. The analysis of the fermionic case (ǫ = +1) is similar However,aphantomlikebehavior(ω <−1)ispermitted to the bosonic one, but now the chemical potential can only for µ<0, and the corresponding massless particles be eitherpositiveornegative. Thecondition(21)forthe can have either a bosonic or fermionic nature. 6 Itwasalsoproved(seealsoFig. 4)thattheEoSparam- atures. Perhaps more interesting, unlike the results for eter of a dark energy fluid obeying a generalized Wien’s µ=0 which favored only a fermionic nature to the dark law always satisfies the constraint ω < −0.5 (a thermo- energy fluid (phantom and nonphantom), it was demon- statistics limit). This upper limit is surprisingly close to strated that a bosonic kind of dark energy becomes pos- the maximal value of the EoS ω-parameter necessary to sible from a thermostatistics viewpoint. accelerate the present universe. Actually, in order to accelerate a FRW universe dominated by matter and dark energy, the EoS parameter must satisfies the inequality, ω < −(1 + Ω /Ω )/3. Therefore, for Ω ∼= 0.3 and Ω ∼=0.7, as inmdicaxted by the presentobsermvations [1, 2], Acknowledgments x one finds the dynamic constraint ω .−10/21. Finally, it should be stressed that for µ = 0 one finds The authors would like to thank V. C. Busti, J. V. ω = −1 (see Eq. (17)) in accordance to the results Cunha, J. F. Jesus, A. C. Guimaraes, R. Holanda and min previouslyderivedbyLimaandAlcaniz[13]. Thepresent R. C. Santos for helpful discussions. 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