Early Universe Thermodynamics and Evolution in Nonviscous and Viscous Strong and Electroweak epochs: Possible Analytical Solutions
AArticle
Early Universe Thermodynamics and Evolution in Nonviscousand Viscous Strong and Electroweak epochs: Possible AnalyticalSolutions
Abdel Nasser Tawfik and Carsten Greiner (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Citation:
Title. , , 0.https://doi.org/Received:Accepted:Published: Publisher’s Note:
MDPI stays neutralwith regard to jurisdictional claims inpublished maps and institutional affil-iations. Egyptian Center for Theoretical Physics, Juhayna Square off 26th-July-Corridor, 12588 Giza, Egypt Goethe University, Institute for Theoretical Physics (ITP), Max-von-Laue-Str. 1, D-60438 Frankfurt am Main,Germany
Simple Summary:
In the early Universe both QCD and EW eras play an essential role in laying seedsfor nucleosynthesis and even dictating the cosmological large-scale structure. Taking advantage ofrecent developments in ultrarelativistic nuclear experiments and nonperturbativ and perturbativelattice simulations, various thermodynamic quantities including pressure, energy density, bulk viscosity,relaxation time, and temperature have been calculated up to the TeV-scale, in which the possibleinfluence of finite bulk viscosity is characterized for the first time and the analytical dependence ofHubble parameter on the scale factor is also introduced.
Abstract:
Based on recent perturbative and non-perturbative lattice calculations with almost quarkflavors and the thermal contributions from photons, neutrinos, leptons, electroweak particles, andscalar Higgs bosons, various thermodynamic quantities, at vanishing net-baryon densities, such aspressure, energy density, bulk viscosity, relaxation time, and temperature have been calculated up tothe TeV-scale, i.e. covering hadron, QGP and electroweak (EW) phases in the early Universe. Thisremarkable progress motivated the present study to determine the possible influence of the bulk viscosityin the early Universe and to understand how this would vary from epoch to epoch. We have takeninto consideration first- (Eckart) and second-order (Israel-Stewart) theories for the relativistic cosmicfluid and integrated viscous equations of state in Friedmann equations. Nonlinear nonhomogeneousdifferential equations are obtained as analytical solutions. For Israel-Stewart, the differential equationsare very sophisticated to be solved. They are outlined here as road-maps for future studies. For Eckarttheory, the only possible solution is the functionality, H ( a ( t )) , where H ( t ) is the Hubble parameterand a ( t ) is the scale factor, but none of them so far could to be directly expressed in terms of eitherproper or cosmic time t . For Eckart-type viscous background, especially at finite cosmological constant,non-singular H ( t ) and a ( t ) are obtained, where H ( t ) diverges for QCD/EW and asymptotic EoS. Fornon-viscous background, the dependence of H ( a ( t )) is monotonic. The same conclusion can be drawnfor an ideal EoS. We also conclude that the rate of decreasing H ( a ( t )) with increasing a ( t ) varies fromepoch to epoch, at vanishing and finite cosmological constant. These results obviously help in improvingour understanding of the nucleosynthesis and the cosmological large-scale structure. Keywords:
Viscous Cosmology, Particle-theory and field-theory models of the early Universe, Mathe-matical and relativistic aspects of cosmology, Thermodynamic functions and equations of state
PACS:
1. Introduction
The current thorough knowledge on the cosmic evolution is primarily based on thestandard model of cosmology (SMC), which introduces a generic hypothesis that the cosmicbackground is isotropically and homogeneously filled up with an exclusively ideal fluid. After a r X i v : . [ g r- q c ] F e b of 27 all, we simply realize that this is an abstraction, i.e. a general description that isn’t based onreal physical situation. Apart from approaches, models, and theories, the real situation couldbe arose out from recent high-energy experiments [1–7] and cosmic observations [8–13]. Overyears, it was assumed that the impacts of the viscosity coefficients on cosmology should beweak or at least subdominant so that the inclusion of viscous concepts in the macroscopictheory of the cosmic fluid appeared as most natural improvement. It was first assumedthat the influence of viscosity in the early Universe would be the largest at the the end ofthe lepton era, i.e. during the neutrino decoupling era, at temperature (cid:39) K. Viscouscoefficients connected with particle physics have been also proposed by Misner [14,15]. Recentstudies reveal that the impact of viscosity likely sets on during the very early stages of theUniverse [16]. The present study suggests extending SMC to beyond
SMC. In bCMS, thecosmic background geometry is filled with viscous matter whatever its constituents are, sothat isotropicity and homogeneity are generalized.For the inclusion of the viscous properties, one would like to start with small perturba-tions from the thermal equilibrium. The suitable theoretical framework for this is the first-,Sec. 4.2, and the second-order cosmic relativitic fluid, Sec. 4.3. Both viscosity coefficients,the bulk viscosity ζ and the shear viscosity η can be determined. From SMC considerationsthat the Universe is spatially homogeneously expands, and cosmological observations [8–13], ζ would be taken as a dominant component, while η would be neglected. In bCMS, thisassumption could be also generalized. The motivations for viscous theories in cosmologyhave are diversified. For instance, over the last three decades, various attempts have beenreported in literature [16–22]. A direct implementation of the equations of state (EoS) deducedfrom recent lattice quantum chromodynamic calculations and/or heavy-ion collisions onphysics of the early Universe was initiated in various studies conducted by one of the au-thors [16,21,23–28]. The present paper resumes these studies, especially in light of the recentprogress enabled us to explore the very early epochs of the evolution of the Universe [16,29–32]. The procedure goes as follows. The viscous EoS introduced in Section 4.1 and takenfrom Ref. [16] shall be substituted in the Friedmann equations. This leads to sophisticateddifferential equations. Their analytical solutions turn into a very challenging mathematicaltask. By finding unambitious analytical solutions, bSMC becomes a feasible approach. In thispaper, we introduce and discuss the possible analytical solutions; the ones expressing theHubble parameter in dependence on the scale factor, i.e. functionality H ( a ( t )) , where bothquantities are also functions of the cosmic time t . We also introduce a road-map for futurestudies based on bSMC.The present script is organized as follows. The cosmic geometry and the field equationswill be reviewed in section 2. The cosmic evolution in non-viscous background geometryclassified into different epochs will be discussed in section 3. The cosmic evolution in viscousbackground geometry, section 4, is based on viscous EoS introduced in section 4.1, where thebackground fluid is described by first-order Eckart theory, section 4.2 and second-order Israel-Stewart theory, Section 4.3. The results on the possible analytical solutions, i.e. functionality H ( a ( t )) , where both H and a are also functions of the cosmic time t . will be elaborated inSection 5. Section 6 is devoted to draw the final conclusions.
2. Geometry and field equations
In curved cosmic geometry under the assumptions of SMC (homogeneity and isotropy)for cosmic space and matter, the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric reads ds = dt − a ( t ) (cid:20) dr − kr + r (cid:16) d θ + sin θ d φ (cid:17)(cid:21) , (1) of 27 where a ( t ) is the dimensionless scale factor, which describes the expansion of the Universe. k characterizes elliptical, flat (Euclidean), and hyperbolic cosmic space, where k = {−
1, 0, + } stands for negative, flat, and positive curvature, respectively. It should be noticed that if r is taken dimensionless, a ( t ) shall be given in a unit of length. In Eq. (1) and to simplify thecosmology notation, we use the natural units c = G =
1. So-far, the theory of general relativitydoesn’t inter the play. It certainly does, when the evolution of s ( t ) , the temporal evolution ofthe line element, should be tackled. Towards this end, the theory of general relativity shouldbe combined with the matter/energy content of the space-time within the cosmic geometry.The Einstein gravitational fields with finite cosmological constant are given as R µν − g µν R + Λ µν = π T µν , (2)where the indices µ , ν take discrete values 0, 1, 2, 3. The energy-momentum tensor of the bulkviscous cosmological fluid filling the very early Universe can be expressed as [18] T µν = ( ρ + p + Π ) u µ u ν − ( p + Π ) g µν , (3)where ρ is the energy density, p is the thermodynamic pressure, Π is the bulk viscous pressure,and u µ is the four velocity satisfying the normalization condition u µ u µ =
1. The bulk pressure Π can formally be included in the thermodynamic pressure p eff = p + Π . We shall discuss onhow to evaluate Π , concretely the bulk viscous pressure, in framework of Eckart (first-order),section 4.2, and Israel-Stewart (second-order) theories, section 4.3, for relativistic viscouscosmic fluid.For number density n , specific entropy s , finite temperature T , bulk viscosity coefficient ζ ,and relaxation time τ , the particle and entropy fluxes are to be related to each other as N i = ν i and S i = sN i − (cid:0) τ Π /2 ζ T (cid:1) u i , respectively. It should be emphasized that the evolution of thecosmological fluid is subject to the dynamical laws of particle number conservation N i ; i = Td ρ = d ( ρ / n ) + pd ( n ) [18]. In what follows, we assume that theenergy-momentum tensor of the cosmological fluid is locally conserved, i.e. T ki ; k =
0, where ;denotes the covariant derivative with respect to the line metric.In the proper frame, i.e. the inertial frame of reference comoving with the the fluid, thecomponents T = ρ , T = T = T = − p eff . For the isotropic and homogeneous metricgiven in Eq. (1), the Einstein field equations in natural units read H ( t ) = π ρ ( t ) − ka ( t ) + Λ H ( t ) + H ( t ) = − π (cid:104) ρ ( t ) + p eff ( t ) (cid:105) + Λ t , and H ( t ) = ˙ a ( t ) / a ( t ) is the Hubble parameter. From the expressions (4) and (5), the time evolution of Hubbleparameter can be deduced as˙ H ( t ) = − π [ ρ ( t ) + p eff ( t )] + ka ( t ) . (6)From the local conservation of the energy-momentum tensor in the Universe, followingequation has been proposed by McCrea and Milde and by Peebles with vanishing [33,34] andfinite pressure [35], respectively, as being equivalent to Newtonian mechanics,˙ ρ ( t ) + [ ρ ( t ) + p eff ( t )] H ( t ) =
0. (7) of 27
This means that the decrease in the energy content of a cube with side a ( t ) equals the energybudget due to the expansion of the Universe and the work done by the pressure on the surface.To obtain a closed system of equations, we need to propose EoS relating p to ρ . Dependingon the approach we are applying, we might also need to propose a reliable estimation for Π ,as well. In the section that follows, we introduce solution for the Friedmann equation basedon various types of EoS. We first assume vanishing bulk viscosity, Section 3. Then, we discusson the extension to finite bulk viscosity, Section 4, which then required barotropic equationsfor the pressure, Eq. (8), (9), (10), the temperature, Eq. (37), the bulk viscosity coefficient,Eqs (34), (35), (36) and the relaxation time, Eq. (39) (40), (41). These are examples on novelcontributions presented by the present script.
3. Cosmic evolution in non-viscous approach
By combining recent non-perturbative and perturbative calculations with other degreesof freedom (dof), such as photons, neutrinos, leptons, electroweak particles, and Higgsbosons, various thermodynamic quantities for almost net-baryon-free cosmic matter havebeen calculated up to the TeV-scale, i.e. covering quantum chromodynamic (QCD) andelectroweak (EW) eras of the early Universe [16]. It was found that while the EoS relatingthe pressure p to the energy density ρ for the hadronic matter is simple, the one for QCDand that EW matter are rather complicated. It is worth highlighting that in the cosmologicalcontext, the various thermodynamic quantities should be translated into time-dependingquantities. When the cosmic time elapses, the spacial dimensions of the Universe expand, andaccordingly thermodynamic quantities characterizing the background geometry vary. TheEoS proposed in Fig. 1 [16] are preliminary depending on the energy density, whose decreasemight be - for simplicity - taken as a scale for increasing cosmic time and vice versa. Hadron : p ( t ) = α + β ρ ( t ) , (8) QCD / EW : p ( t ) = α + β ρ ( t ) + γ ρ ( t ) δ , (9)where α = ± β = ± α = ± β = ± γ = − ± δ = − ± ρ ( t ) , the asymptotic behaviorbecomes very close to that of an ideal gas limit, Asymp . : p ( t ) = γ ρ ( t ) , (10)where γ = ± α and β could easily be - due to its large uncertainty - reexpressedwith vanishing α . Nevertheless, in the present calculations, we keep α finite. In section 3,we present solutions for the Friedmann equations, Eqs. (4), (5), (6), with the various Eqs. (9)-(10), which as mentioned characterize various types of cosmic backgrounds correspondingvarious epochs of the early Universe. The results obtained for the dependence of the Hubbleparameter on the scale factor are presented in Fig. 4.At vanishing bulk viscosity, Π ( t ) =
0, the effective pressure p eff ( t ) = p ( t ) + Π ( t ) canbe simplified as the thermodynamic pressure p ( t ) . Then, the Friedmann equation (6) can berewritten as ¨ a ( t ) a ( t ) − ˙ a ( t ) + π [ ρ ( t ) + p ( t )] a ( t ) − k =
0, (11)which can be solved if combined with set of closed equations, such as Eq. (4) and suitable EoS.Accordingly, we have various solutions characterizing the various eras in the early Universe. of 27 (2+1+1+1)q, g, γ, ν , e, µ, τ , W +- , Z , H p [ G e V / f m - ] ρ [GeV/fm ]Nature539(2016)69: SU(5)LQCDNature539(2016)69: SU(6)g JCAP07(2015)035
Fig. 1.
The pressure is depicted as a function of the energy density. Both quantities are given in GeV/fm units. The dashed lines present the various parameterizations (see text). By substituting Eq. (4) and Eq. (8) into Eq. (11), we get¨ a ( t ) a ( t ) + C ˙ a ( t ) + C a ( t ) + C k =
0, (12)where the variables C and C are given in Tab. 1. C and C are functions of the coefficientsobtained in the parameterized EoS, which in tern vary from epoch to epoch. For the sake ofsimplicity, the coefficients are conjectured remaining constant within each epoch. We noticethat the value of C - in the hadron era - is finite but not necessarily unity. This assures that k ,the curvature contact, remains finite.• At vanishing k , which is the case at β = − a ( t ) = c cosh (cid:20)(cid:113) C ( − C )( t + c ) (cid:21) ( − C ) , (13)where c and c are integration constants which can be fixed at boundary and initialconditions. For instance, at t = H ( t ) =
0, Eq. (14), then, c = − t . In general c has thedimension of the cosmic time t and therefore varies with the evolution of the Universe.Hence, c is finite and the cosmological parameters of the hadron epoch ( ∼ − − − sor ∼ × MeV − ), for instance Eq. (14) can be estimated, numerically. Onthe other hand, for the scale factor, a ( t ) , we still need to estimate the other integrationconstant c . Having an analytical expression for the scale factor, Eq. (13), then the Hubbleparameter can then be obtained, H ( t ) = ˙ a ( t ) a ( t ) = (cid:115) C + C tanh (cid:20)(cid:113) C ( + C )( t + c ) (cid:21) . (14) assuming that 6.58 × − s = eV − of 27 • At non-vanishing k , there is no direct analytical solution for a ( t ) . But when assumingthat u = ˙ a ( t ) and substituting this into Eq. (12), du ( a ( t )) da ( t ) + C u ( a ( t )) a ( t ) + C a ( t ) + C ka ( t ) =
0, (15)a solution for ˙ a ( t ) can be proposed u ( a ( t )) = ˙ a ( t ) = c a ( t ) − C − C + C a ( t ) − kC + a ( t ) . (16)The physical solution is the one assuring that, a ( t ) < (cid:20) c C ( + C ) (cid:21) ( − C ) . (17)Hence, the Hubble parameter can be deduced as H ( t ) = (cid:26) c a ( t ) − C − C + kC + a ( t ) (cid:27) a ( t ) . (18)By solving the second-order differential equation (16), an analytical expression for thescale factor a ( t ) can also be deduced, a ( t ) = (cid:34)(cid:115) C + kc ( C + ) (cid:35) − ( + C ) , (19)whose time dependence is given by the time dependence of the corresponding EoS,namely C and C , which are listed in Tab. 1. As discussed they have an indirect timedependence through the coefficients of the corresponding EoS. But within one era, theyare conjectured to remain constant. The latter might be the only way possible to gain ananalytical solution. Then, H ( t ) , Eq. (18), can be rewritten as H ( t ) = c (cid:34)(cid:115) C + kc ( C + ) (cid:35) − + C − ( + C ) + C + kC + (cid:34)(cid:115) C + kc ( C + ) (cid:35) − ( + C ) . (20)Apparently, all coefficients involved in can be determined. Based on the EoS outlined in Eq. (9), and the dependence of energy density on the Hubbleparameter, Eq. (4), then Eq. (11) can be reexpressed as¨ a ( t ) a ( t ) + C ˙ a ( t ) + C a ( t ) + C k + πγ a ( t ) ρ δ =
0, (21)where C and C are variables depending on the corresponding EoS, Tab. 1. The last term inlhs of this expression gives another difference with Eq. (12). The other terms remaining can be of 27 estimated as outlined in the previous section. Now we focus on the the contributions addedby this term,4 πγ a ( t ) ρ δ = πγ a ( t ) ( − δ ) (cid:18) π k (cid:19) δ (cid:20) − (cid:18) Λ a ( t ) k − ˙ a ( t ) k (cid:19)(cid:21) δ . (22)For δ = − (cid:39) − (cid:20) − (cid:18) Λ a ( t ) k − ˙ a ( t ) k (cid:19)(cid:21) − = − Λ a ( t ) k − ˙ a ( t ) k + · · · . (23)Thus, we might approximate the entire bracket to first terms outlined. This result can also beobtained when assuming that the exponent δ approaches unity. Then, Eq. (21) becomes¨ a ( t ) a ( t ) + C ˙ a ( t ) + C a ( t ) + C k + C a ( t ) (cid:18) − C a ( t ) k − ˙ a ( t ) k (cid:19) =
0. (24)As done while solving Eq. (15), we assume that u ( a ( t )) = ˙ a ( t ) . Then, the approximatedEq. (24) becomes du ( a ( t )) da ( t ) + C u ( a ( t )) a ( t ) + C a ( t ) + C k a ( t ) − + C a ( t ) (cid:18) − C a ( t ) k − u ( a ( t )) k (cid:19) =
0, (25)which can be solved as u ( t ) = ˙ a ( t ) = C (cid:26) k [ C + C − ( + C ) C ] − C C a ( t ) + c a ( t ) − C e C k a ( t ) − C k [ − C − C + ( + C ) C ] e C k a ( t ) Ein − C (cid:18) C k a ( t ) (cid:19)(cid:27) , (26)where c is another integration constant to be fixed for boundary conditions and the exponen-tial integral represents a special case of the incomplete gamma function Ein n ( x ) = ( x ) n − Γ [ − n , y ] . (27)For equation (26), there is no analytical solution. Nevertheless, the Hubble parameter, H ( t ) = ˙ a ( t ) / a ( t ) , can be constructed as H ( t ) = C a ( t ) (cid:26) k [ C + C − ( + C ) C ] − C C a ( t ) + c a ( t ) − C e C k a ( t ) − C k [ − C − C + ( + C ) C ] e C k a ( t ) Ein − C (cid:18) C k a ( t ) (cid:19)(cid:27) . (28) C and C are given in Tab. 1. The results obtained for the Hubble parameter as a function ofthe scale factor shall be presented in Fig. 4, in which the exponential function, Eq. (27), is - forthe sake of simplicity - assigned to the unity. of 27 Again when substantiating Eq. (4) and Eq. (10) into Eq. (11), we get¨ a ( t ) a ( t ) + C ˙ a ( t ) + C a ( t ) + C k =
0, (29)which apparently looks almost identical to Eq. (12) in section 3.1. The possible analyticalsolution reads u ( t ) = ˙ a ( t ) = c a ( t ) − C − (cid:20) C C + + k (cid:21) a ( t ) , (30)for which the Hubble parameter can be given as H ( t ) = (cid:26) c a ( t ) − C − (cid:20) C C + + k (cid:21) a ( t ) (cid:27) a ( t ) . (31)The results of this expression are given in Fig. 4. By integrating (30), an expression for thescale factor can be obtained a ( t ) = (cid:34)(cid:115) C + k ( + C ) c ( + C ) (cid:35) − ( + C ) , (32)which helps in constructing the corresponding Hubble parameter H ( t ) = (cid:20) C + k ( + C ) c ( + C ) (cid:21) + C − C + k ( + C ) + C (cid:34)(cid:115) C + k ( + C ) c ( + C ) (cid:35) − + C + (cid:34)(cid:115) C + k ( + C ) c ( + C ) (cid:35) − + C − C . (33)The various coefficients characterizing the various EoS and also combining cosmologicalconstant, C , C , C , C , are listed in Tab. 1. Accordingly, analytical solutions similar to Eq.(13) are obtained. The cosmological constant Λ is conjectured to count for the dark energycomponent [36]. For Eq. (29), expressions for Hubble parameter similar to (14) can then bederived. Also, with the variable change u = ˙ a ( t ) , (18) can be obtained, as well. Section C C C C Hadron ( + β ) − πα − ( − β ) Λ QCD / EW ( + β ) − πα − ( − β ) Λ π (cid:0) π k (cid:1) δ γ Λ Asymp . 3.3 ( + γ ) − − ( + γ ) Λ Tab.
1: The parameters defining different solutions for Hadron, QCD/EW and Asympt.phases corresponding to various EoS, (12), (21), and (29), respectively. of 27
4. Cosmic evolution in viscous approaches
The recent results for the bulk viscosity are based on non-perturbative and perturbativecalculations with as much quark flavors as possible. By combining these calculations withadditional dof, such as photons, neutrinos, leptons, electroweak particles, and Higgs bosons,various thermodynamic quantities including bulk viscosity, for almost net-baryon-free cosmicmatter, have been calculated up to the TeV-scale [16]. The dependence of the bulk viscosity onthe energy density [28] is depicted in the top panel of Fig. 2, in which both quantities are givenin the physical units. As discussed, such a barotropic dependence straightforwardly allows fordirect cosmological implications [19,21,23,26], where ρ ( t ) can be directly substituted by H ( t ) ,Eq. (4). These wide values of ρ ( t ) which are accompanied by a wide range of temperaturescover quantum chromodynamic (QCD) (Hadron and QGP) and electroweak (EW) phases inthe early Universe. Accordingly, the dependence of the bulk viscosity on the energy densitycan be parameterized. Hadron − QGP : ζ ( t ) = d + d ρ ( t ) + d ρ ( t ) d , (34) QCD : ζ ( t ) = e + e ρ ( t ) e , (35) EW : ζ ( t ) = f + f ρ ( t ) f . (36)The various fit parameters are given as follows. For Hadron-QCD: d = − ± d = ± d = ± d = ± e = ± e = ± e = ± f = ± f = ± f = ± × − .While ζ ( t ) vs. ρ ( t ) is much structured in the hadron era, there are three domains to beemphasized (from low to large energy density).• The first one is the hadron-QGP domain (Hadron-QGP), which spans over ρ ( t ) (cid:47)
100 GeV/fm . At the beginning, there is a rapid increase in ζ ( t ) , i.e. ζ (cid:104) ,at ρ ( t ) (cid:39) , which is then followed by a slight increase in ζ ( t ) . For example,at ρ ( t ) (cid:39)
100 GeV/fm , zeta ( t ) reaches ∼
130 GeV . It is apparent that the hadron-parton phase transition seems to take place at ρ ( t ) (cid:47) [37,38]. At this value, ζ ( t ) (cid:47) .• The second domain, the QGP epoch, seems to be formed, at 0.5 (cid:47) ρ ( t ) (cid:47)
100 GeV/fm ,i.e. a much wider ρ ( t ) than that of the hadron domain. Thus, we could conclude thatover this wide range of ρ ( t ) , the bulk viscosity is obviously not only finite but ratherlargely supporting the RHIC discovery of strongly correlated viscous QGP [3,4,6]. Athigher ρ ( t ) , we observe a tendency of a linear increase in ζ ( t ) with further increasing ρ ( t ) . Thus, the second domain is the one where 100 (cid:39) ρ ( t ) (cid:47) × GeV/fm and80 (cid:39) ζ ( t ) (cid:47) GeV . In light on this observation, we conclude that the phase transitionfrom QCD to EW domain is very smooth.• The third domain is also characterized by an almost linear increase in ζ ( t ) with increasing ρ ( t ) . For 10 (cid:47) ρ ( t ) (cid:47) GeV/fm , there is a nearly steady increase in ζ ( t ) from 10 to10 GeV .For temperatures ranging from a few MeV to TeV and energy densities up to 10 GeV/fm ,we have taken into consideration almost all possible contributions to the bulk viscosity. Withthese we mean the thermodynamic quantities calculated in non-perturbation and perturba-tion QCD with up, down, strange, charm, and bottom quark flavors. The second type ofcontributions is the guage bosons, the entire gluonic sector. We have also included photons, W ± , and Z , charged leptons (neutrino, electron, muon, and tau), and scalar Higgs particle.The third type of contributions is the vacuum and thermal condensations. We have included condensations for up, down, strange and charm quarks. We merely still miss the vacuum andthe thermal bottom quark condensates, besides the entire gravitational, the neutral leptons,and the top quark sector to compile the entire standard model for elementary particles. (a) (2+1+1+1)q, g, γ, ν , e, µ, τ , W +- , Z , H ζ [ G e V ] ρ [GeV/fm ]Nature539(2016)69: SU(5)LQCDNature539(2016)69: SU(6)g JCAP07(2015)035 (b) (2+1+1+1)q, g, γ, ν , e, µ, τ , W +- , Z , H T [ G e V ] ρ [GeV/fm ]Nature539(2016)69: SU(5)LQCDNature539(2016)69: SU(6)g JCAP07(2015)035
Fig. 2.
Top panel depicts the energy-density dependence of the bulk viscosity. Bottom panel illustratesthe temperature as a function of energy density. The parameterizations are depicted as curves.
The dependence of temperature T ( t ) on the energy density ρ ( t ) , the barotropic equationof state, is depicted in bottom panel of Fig. 2. We notice that T ( t ) almost linearly depends on ρ ( t ) . A best parametrization reads T ( t ) = α + β ρ ( t ) γ , (37)where α = ± β = ± × − , and γ = ± × − . We notice thatat low ρ ( t ) the temperature looks a little bit structured. While with increasing ρ ( t ) , thetemperature goes almost linearly with increasing ρ ( t ) , especially at very high temperatures,where ρ ( t ) becomes related in T , i.e. ideal gas.The third quantity, for which we need to propose a barotropic EoS, is the relaxationtime, τ ( t ) . We assume to apply the phenomenological model presented in refs. [18,19,26,39],which is based on dissipative relativistic fluid. This model was assumed to characterize theevolution of the Universe with a flat homogeneous isotropic Friedmann-Robertson-Walkergeometry filled with viscous cosmic fluid, but still valid for other types of curvature andcosmic backgrounds. Accordingly, we have τ ( t ) = ζ ( t ) ρ ( t ) . (38) × -5 × -4 × -3 (2+1+1+1)q, g, γ, ν , e, µ, τ , W +- , Z , H τ [ G e V - ] ρ [GeV/fm ]Nature539(2016)69: SU(5)LQCDNature539(2016)69: SU(6)g JCAP07(2015)035
Fig. 3.
The energy-density dependence of the relaxation time. The parameterizations, Eqs. (39), (40),(41), are depicted as curves.
Figure 3 shows the energy-density dependence of the relaxation time τ ( t ) . Bearing inmind the linear dependence of the energy density ρ ( t ) on the temperature T , bottom panelof Fig. 2 and Eq. (37), the temperature dependence τ ( t ) can be almost straightforwardlyestimated. As done in the present script, we would like to distinguish between hadron-QGP(squares), QCD (circles) and electroweak eras (diamonds). In the hadron-QGP era, thereis a very rapid decrease in τ ( t ) with increasing rho ( t ) . The QCD epoch is characterized bya slower decline in τ ( t ) with increasing rho ( t ) . The relaxation time within the electroweakepoch starts and ends with a slow decrease, while in the middle EW era, τ ( t ) rapidly decreaseswith increasing ρ ( t ) . This region likely characterizes the electroweak phase transition . Thedependence of τ ( t ) on rho ( t ) is proposed as follows. Hadron − QGP : τ ( t ) = g + g exp ( − g ρ ( t ) g ) , (39) QCD : τ ( t ) = h + h h + log ( h ρ ( t )) , (40) EW : τ ( t ) = k ρ ( t ) k log ( k ρ ( t )) , (41)where g = ± × − , g = ± g = ± g = ± h = − × − ± × − , h = ± h = ± × , h = ± × , k = × − ± × − , k = ± k = × − ± × − .In the sections that follow, we apply well-know theories for relativistic dissipative fluidto the cosmic background. We start with the Eckart relativistic theory of a simple dissipativefluid, which is used to simplify the nuclear motion arising in the second Born–Oppenheimerapproximation. The cosmic relevance of this theory is remarkable because it introduces the so-called Eckart frame, which is a frame of orthonormal vectors following a vibro-rotating object.The orientation of this frame is governed by the so-called Eckart conditions assuring minimalCoriolis interaction. Second we apply the Israel-Stewart theory as this theory is conjectured tosolve Eckart theory’s lack of causality and its obvious instabilities by introducing a second-order term to the entropy. In this case, τ ( t ) is conjectured to play the role of an order parameter. For the electroweak phase transition, other thermodynamic order parametersshould be proposed and then analyzed.2 of 27 For cosmological context, the first theory of relativistic dissipative fluid has been pre-sented by Eckart [40] and Landau and Lifshitz [41]. It was pointed out that regardless thechoice of EoS, the equilibrium states of this theory are found unstable [42]. In this theory, onlythe first-order deviation from the equilibrium is taken into consideration. But this leads to thesuperluminal velocities of the dissipative signals, i.e. signals propagate through the relativisticdissipative fluid with velocities exceeding the speed of light c and hence the theory violatesthe causality principle [43]. Moreover, it was shown that the resulting equilibrium states areunstable [44]. All these severe problems are originated from the fact that the Eckart theorymerely considers first-order deviations from the equilibrium leading to parabolic differentialequations, Eq. (45). The applicability of this theory can only be thought for quasi-stationaryphenomena, i.e. temporally and spatially slowly varying, which are characterized by meanfree-path and mean collision-time.The Eckart theory introduces a linear relationship between the bulk viscous pressureand the rate of expansion of the Universe [45]. Obviously, this feature - despite the severecontains - makes it possible to work out an analytical method for the cosmic parameters in theexpanding Universe. For bulk viscous cosmic fluid, whose energy-momentum tensors aregiven as T µν = ( ρ + p + Π ) u µ u ν − ( p + Π ) δ νµ , (42)the line element in flat homogeneous isotropic Friedmann-Robertson-Walter metric reads ds = dt + a ( t ) (cid:104) dx + dy + dz (cid:105) . (43)When applying Eckart theory on modeling such a cosmic fluid, we assume averaged 4-velocity fields u α with u α u α = n α = n u α . For unbalancedcreation/annihilation processes in gravitational fields, n α ; α = n + H n =
0, (44)with Hubble parameter H = u α ; α . In this theory, the entropy current is given as S α = s n u α . (45)As discussed, this is a non-conserved quantity. The covariant form of second law of thermo-dynamics reads S α ; α ≥
0. The divergence of this quantity is given as TS α ; α = − H Π .With respect to the proposed cosmic fluid, the temporal evolution can be related to thedynamical laws of the particle number conservation N i ; i =
0. Gibbs equation implies that Td ρ = d ( ρ / n ) + pd ( n ) . Then, the covariant entropy current assures a linear first-order relationship between the thermodynamical flux Π ( t ) and the corresponding H ( t ) . It is worthhighlighting that H ( t ) in this context plays the role of a force [46] Π ( t ) = − ζ ( t ) H ( t ) . (46)Having an estimation of the bulk viscous pressure Π , we can now substitute Eq. (46) in Eq. (6), ˙ H ( t ) = − π [ ρ ( t ) + p ( t ) − ζ ( t ) H ( t )] + ka ( t ) , (47) which can be rewritten as¨ a ( t ) a ( t ) − [ + πζ ( t )] ˙ a ( t ) + π [ ρ ( t ) + p ( t )] a ( t ) − k =
0. (48)For the present calculations, we start with Eq. (48), in which we substitute with thebarotropic EoS of the pressure and the bulk viscosity. They can then be related to the Hubbleparameter, Eq. (4). Due to the various barotropic EoS introduced, Eq. (8), Eq. (9), Eq. (10)and section 4.1, the evolution of the various cosmological parameters obviously differ fromepoch to epoch. Equation (48) combines extended assumptions and ingredients of SMC. Thisis viscous cosmic background.The sections that follow elaborate details on the dependence of H ( t ) on a ( t ) , where bothquantities are functions of the cosmic time t . Such a limitation is merely based on the currentlyavailable analytical solutions. When lifting such mathematical limitations, bSMC emerges asa proper cosmological approach.4.2.1. Hadron-QGP eraSubstituting with the pressure, Eq. (8), and the bulk viscosity, Eq. (34) into Eq. (48) leadsto a second-order differential equation¨ a ( t ) a ( t ) − D ˙ a ( t ) + D a ( t ) + D k − D (cid:20) − D a ( t ) − ˙ a ( t ) k (cid:21) − d a ( t ) − ˙ a ( t ) − d a ( t ) − ˙ a ( t ) =
0, (49)where the various coefficients of the different EoS are included in the following parameters D = + π d − ( + β ) − d Λ , D = πα − Λ ( + β ) , D = ( + β ) − D = − d + d π − d d k − d , D = Λ k .For Eq. (49), there is no analytical solution. But if assuming that u ( a ( t )) = ˙ a ( t ) , this can bereduced to u (cid:48) ( a ( t )) − D u ( a ( t )) a ( t ) + D a ( t ) + D ka ( t ) − − D (cid:20) + (cid:18) u ( a ( t )) k − Λ k (cid:19)(cid:21) d a ( t ) − − d u ( a ( t )) a ( t ) − − d ku ( a ( t )) a ( t ) − =
0, (50) where u (cid:48) ( a ( t )) = du ( a ( t )) / da ( t ) . With binomial expansion, the squared bracket can beapproximated to unity. Such an assumption leads to an analytical solution, u ( a ( t )) = ˙ a ( t ) = − A k ( d k ) A k K ( t ) d k M (cid:104) D − k ( D + D − )+ A k , 2 + A k , − kd a ( t ) (cid:105) [ D − k ( D + D − ) + A ] − ( k + A ) a ( t ) − A k + A k ( d k ) A k K ( t ) (cid:26) ( D + k − kD + A ) M (cid:20) D − k ( D + D − ) + A k , 1 + A k , − kd a ( t ) (cid:21)(cid:27) a ( t ) − A k − c A k K ( t ) d k M (cid:104) − − D + k ( D + D − )+ A k , 2 − A k , − kd a ( t ) (cid:105) [ − D + k ( D + D − ) + A ] − ( A − k ) − c A k K ( t ) [ k ( D − ) − D + A ] M (cid:20) − − D + k ( D + D − ) + A k , 1 − A k , − kd a ( t ) (cid:21) a ( t ) , (51) where K ( t ) = d k (cid:26) A k ( d k ) A k M (cid:20) k ( D − k ( D + D − ) + A ) , 1 + A k , − kd a ( t ) (cid:21) a ( t ) − A k + c A k M (cid:20) − k ( − D + k ( D + D − ) + A ) , 1 − A k , − kd a ( t ) (cid:21)(cid:27) , A = (cid:104) d ( D + D D ) k + [ D − ( D − ) k ] (cid:105) (52) Even for the resultant differential equation (51) there is no analytical solution in termsof the cosmic time t . The only possible solution is ˙ a ( t ) as a function of a ( t ) . Such a solution[impeded in Eq. (53)] leads to an analytical expression for H ( t ) as a function of a ( t ) , i.e.functionality, which in turn depends on regularized confluent hypergeometric function whoseasymptotic limit reads M ( { a , b , z } ) ∼ Γ ( b )( e z z a − b + ( − ) − a / Γ ( b − a )) [47]. Two of the threeregular singularities of M ( { a , b , z } ) are conjectured to merge into an irregular singularity andtherefrom the conjugate "confluent" emerges. The Hubble parameter reads H ( t ) = − A k ( d k ) A k K ( t ) d k M (cid:104) D − k ( D + D − )+ A k , 2 + A k , − kd a ( t ) (cid:105) [ D − k ( D + D − ) + A ] − ( k + A ) a ( t ) − − A k + A k ( d k ) A k K ( t ) (cid:20) ( D + k − kD + A ) M (cid:20) D − k ( D + D − ) + A k , 1 + A k , − kd a ( t ) (cid:21)(cid:21) a ( t ) − A k − c A k K ( t ) d k M (cid:104) − − D + k ( D + D − )+ A k , 2 − A k , − kd a ( t ) (cid:105) [ − D + k ( D + D − ) + A ] − ( A − k ) a ( t ) − − c A k K ( t ) [ k ( D − ) − D + A ] M (cid:20) − − D + k ( D + D − ) + A k , 1 − A k , − kd a ( t ) (cid:21)(cid:41) . (53) It is worth highlighting that Kummer confluent hypergeometric functions, for instance,which are common standard forms of the confluent hypergeometric functions M , have aregular singular point, at z ≡ − kd /2 a ( t ) = z ≡− kd /2 a ( t ) = ∞ . Thus, the curvature parameter k and the scale factor a ( t ) define whetherregular or irregular singular point appears. At vanishing and finite cosmological constant, theresults of H ( t ) vs. a ( t ) are shown in Fig. 5. This reduces the certainty of the proposed solution but this seems to be the only approximation possible! The inclusion of higher terms simply preventsany analytical solution.5 of 27 a ( t ) a ( t ) − E ˙ a ( t ) + E a ( t ) + E k − E (cid:20) + (cid:18) ˙ a ( t ) k − Λ k (cid:19)(cid:21) e ˙ a ( t ) a ( t ) e + E (cid:20) + (cid:18) ˙ a ( t ) k − Λ k (cid:19)(cid:21) δ a ( t ) − e =
0, (54)with the coefficients E = + π e − ( + β ) , E = πα − Λ ( + β ) , E = ( + β ) − E = − e + e π − e e k − e , E = − δ δ π − δ γ k − δ .Assuming that u ( a ( t )) = ˙ a ( t ) and applying the same approximation given in Eq. (23), theprevious differential equation can be reduced to u (cid:48) ( a ( t )) − E u ( a ( t )) a ( t ) + E a ( t ) − k E a ( t ) + E (cid:20) − C a ( t ) k − u ( a ( t )) k (cid:21) u ( a ( t )) a ( t ) + e + E (cid:20) − C a ( t ) k − u ( a ( t )) k (cid:21) a ( t ) − δ =
0. (55)An analytical solution is only possible when both squared brackets are replaced by unity u ( a ( t )) = ˙ a ( t ) = (cid:26)(cid:20) c e a ( t ) E − E k L − E e (cid:18) − E e a ( t ) − e (cid:19) − E L − E + e e (cid:18) − E e a ( t ) (cid:19) a ( t ) (cid:21) a ( t ) δ − E L − E − + δ e (cid:18) − E e a ( t ) − e (cid:19) a ( t ) (cid:41) e − E e a ( t ) − e e a ( t ) − δ , (56)where L ν ( z ) = (cid:90) ∞ e − zt t ν dt . (57) Thus, the corresponding Hubble parameter reads H ( t ) = a ( t ) (cid:26)(cid:20)(cid:20) c e a ( t ) E − E k L − E e (cid:18) − E e a ( t ) − e (cid:19) − E L − E + e e (cid:18) − E e a ( t ) (cid:19) a ( t ) (cid:21) a ( t ) δ − E L − E − + δ e (cid:18) − E e a ( t ) − e (cid:19) a ( t ) (cid:21) e − E e a ( t ) − e e a ( t ) − δ . (58)The results of H ( t ) vs. a ( t ) are depicted in Fig. 5.4.2.3. EW (asymptotic) eraFor pressure, Eq. (10), and bulk viscosity, Eq. (36), Eq. (48) leads to¨ a ( t ) a ( t ) − F ˙ a ( t ) − F a ( t ) + F k − F (cid:20) (cid:18) ˙ a ( t ) k − Λ k a ( t ) (cid:19)(cid:21) f a − f ( t ) ˙ a ( t ) =
0, (59)where F = + π f − ( + γ ) , F = Λ ( + γ ) , F = ( + γ ) − F = − f + f π − f f k − f .Applying the same substitution, u ( a ( t )) = ˙ a ( t ) , and taking into account the first term of thebinomial expansion as unity, we get an analytical solution, functionality, u ( a ( t )) = ˙ a ( t ) = e − F f a ( t ) − f f (cid:26) c f a ( t ) F − F k L − F f (cid:18) − F f a ( t ) − f (cid:19) + F L − F + f f (cid:18) − F f a ( t ) − f (cid:19) a ( t ) (cid:27) . (60)Accordingly, the Hubble parameters is given as H ( t ) = a ( t ) e − F f a ( t ) − f f (cid:26) c f a ( t ) F − F k L − F f (cid:18) − F f a ( t ) − f (cid:19) + F L − F + f f (cid:18) − F f a ( t ) − f (cid:19) a ( t ) (cid:27)(cid:21) . (61)The dependence of H ( t ) on a ( t ) is presented in Fig. 5. In order to fix the acausality and instability problem of Eckart theory, Israel and Stewarthave introduced a relativistic second-order theory for relativistic fluid [48,49]. With extended irreversible thermodynamics, this theory was then developed by Hiscock and Lindblom[50]. This theory is also characterized by a deviation from equilibrium as defined by Eckarttheory. Quantities such as bulk stress, heat flow, and shear stress are treated as independentdynamical variables. Accordingly, 14 dynamical fluid variables have to be estimated. Therole that this type of causal thermodynamics would play in the general theory of relativitywas reported in ref. [18]. A general algebraic form for S α including a second-order term in thedissipative thermodynamical flux Π [48,49,51] reads S α = s n u α + τζ Π u α T , (62)where τ is the relaxation time. Similar to Eckart theory, the corresponding number flux couldbe given as N α = N u α . (63)For the evolution of the bulk viscous pressure, we adopt the causal evolution equationin the simplest way, i.e. linear in Π satisfying the H -theorem [18]. Accordingly, the entropyproduction remains nonnegative, S i ; i = Π / ζ T ≥ τ ˙ Π + Π = − ζ H − (cid:101) τ Π (cid:18) H + ˙ ττ − ˙ ζζ − ˙ TT (cid:19) , (64)where (cid:101) is a parameter controlling the type of considered theory. (cid:101) = (cid:101) = τ =
0. In order to have a closed system from Eqs. (4), (7) and (64), we have tointroduce EoS for the pressure p ( t ) , the temperature T ( t ) , bulk viscosity coefficient ζ (t), andthe relaxation time τ ( t ) , respectively, section 4.1.In the sections that follow, we elaborate the consequences of the various barotropic EoSfor p ( t ) , T ( t ) , ζ ( t ) , and τ ( t ) in strong of electroweak epochs of the early Universe. We getsophisticated differential equations. We hope that this concrete mathematical problem findsresonances among mathematicians. Despite their apparent challenging analytical solutions,we separately derive them in the appendices. A future work shall be devoted in order topropose numerical solutions for all these differential equations.
5. Results
The present section summarizes the results of the possible analytical solutions outlined insections 3 and 4.2. They are only limited to the Hubble parameter in dependence on the scalefactor for non-viscous, section 3, and Eckart-type viscous cosmic backgrounds, section 4.2. Asintroduced, from the corresponding EoS, we could differentiate between the various epochsof the early Universe. Nevertheless, we did not emphasize when each epoch starts or whenends, i.e. in terms of the cosmic time. Such a concrete limitation becomes only possible whenthe initial and the final conditions are precisely determined. This is not precisely available.As alternatives, we would be able to propose for each epoch an interval of cosmic energydensities, which in turn could be related to an interval of the Hubble parameter. The lattercan hen be given as functions of the scale factor; the proposed analytical solutions. On theother hand, such a concrete limitation would be only urgently needed, when a completeor an inter-epochal picture is to be drawn. The results discussed in the sections that followare not limiting the evolution of the Hubble parameter within the successive epochs. Theycover a wider range than than of the corresponding epoch. Accordingly, we conclude that the evolution during the successive epochs characterized by electroweak and strong interactionswould not be monotonic. a ( t ) H ( t ) IdealHadronQCDEWAsympt a ( t ) H ( t ) IdealHadronQCDEWAsympt
Fig. 4.
The dependence of the Hubble parameter on the scale factor in non-viscous cosmic background isdepicted for finite (top) and vanishing cosmological constant (bottom panel). The equations of state forhadron, QCD-EW and asymptotic limit are presented as dashed, dotted, long dashed curves, respectively.
Figure 4 depicts the dependence of the Hubble parameter on the scale factor at finite(top panel) and vanishing cosmological constant (top panel). The results for the equations ofstate characterizing hadron, QCD-EW, and asymptotic limit are presented as dashed, dotted,long dashed curves, respectively. We also draw the ideal gas results as tiny dashed curves.There is a rapid decrease in H ( t ) with increasing a ( t ) . The various epochs (the different EoS)show miscellaneous rates. Relative to the ideal gas EoS, hadron and asymptotic EoS lookvery similar, especially, at finite cosmological constant (top panel). At large a ( t ) , both hadronand asymptotic EoS become almost identical. The QCD/EW EoS shows a slightly differentbehavior, especially at large a ( t ) , where H ( t ) diminishes.At vanishing cosmological constant (bottom panel), the rate of decreasing H ( t ) with theincrease in a ( t ) is larger than the one observed in the top panel. Here, only QCD/EW EoSlooks similar to the ideal gas EoS, while both hadron and asymptotic epochs look almost identical. At small a ( t ) , both have a similar decrease as the one of ideal and QCD/EW EoS,while, at large a ( t ) , their corresponding H ( t ) vanishes.As discussed, each EoS should be restrictively utilized within a specific interval ofthe cosmic time characterizing the corresponding cosmic epoch. Due to the mathematicaldifficulties associated with the resulting differential equations so that the proposed analyticalsolutions are restricted to the functionality H ( a ( t )) but not in terms of the cosmic time t ,directly, we are left with a unique alternative. This is relating the various epochs to an intervalof energy densities, as introduced in ref. [16,28] and section 4.1. It is obvious that even thisoption is an approximation. Thus, we leave the results drawn in Fig. 4 unchanged. Theconclusion which could be drawn here is that the cosmic evolution [ H ( t ) vs. a ( t ) ] seems notmonotonic, especially over the successive asymptotic, EW-QCD and hadron epochs. a ( t ) H ( t ) IdealHadronQCDEWAsympt a ( t ) H ( t ) IdealHadronQCDEWAsympt
Fig. 5.
The same as in Fig. 4 but here for viscousv cosmic geometry (Eckart theory).
Figure 5 presents the same as in in Fig. 4 but here for viscous cosmic geometry (Eckarttheory). Comparing with the results depicted in Fig. 4, the dependence of H ( t ) vs. a ( t ) inviscous background geometry looks very different. Hadronic EoS is associated with non-singularity. Finite cosmological constant likely assures non-singularity in both quantities,while vanishing cosmological constant is associated with non-singular Hubble parameter.The QCD-EW EoS results in diverging Hubble parameter within a tiny range of the scaleparameter. At lower a ( t ) , we find that H ( t ) remains almost vanishing. For the asymptotic EoS, at finite cosmological constant and low a ( t ) , H ( t ) vanishes. Then, H ( t ) gets positivesmall values. At higher a ( t ) , H ( t ) becomes non-physical. Again, within the short range of a ( t ) , H ( t ) diverges. For the asymptotic EoS, at vanishing cosmological constant and low a ( t ) , H ( t ) vanishes. Then increasing a ( t ) , the resulting H ( t ) very slightly linearly decreases. Then, H ( t ) diverges within the short range of a ( t ) .
6. Conclusions
Based on recent progress achieved, especially the numerical and experimental studies onhadron, parton, and EW matter, the main conclusion of the present study is that the analyticalsolutions for EoS, in which as much as possible contributions from both standard modelfor elementary particles and standard model for cosmology are taken into consideration,are sophisticated. The only possible analytical solutions are the ones relating the Hubbleparameter to the scale factor, functionality, in non-viscous and Eckart-type-viscous cosmicbackgrounds. For Israel-Stewart-viscosity, the resulting differential equations are challengingtasks for mathematicians. We have outlined these differential equations as road-maps forfuture studies.Recent non-perturbative and perturbative calculations with as much as possible quarkflavors at almost physical masses have been combined with the thermal contributions fromphotons, charged neutrinos, leptons, electroweak particles ( W ± and Z bosons), and the scalarHiggs bosons. Various thermodynamic quantities, including pressure, energy density, bulkviscosity, relaxation time, and temperature for almost net-baryon-free cosmic matter could becalculated up to the TeV-scale, i.e. covering hadron, QGP and electroweak (EW) phases.In equivalence with Newtonian mechanics and based on Friedman solutions and theconservation of the energy-momentum tensor, McCrea and Milde and by Peebles derived thetemporal evolution of the energy density, i.e. an equation of motion, with vanishing and finitepressure, that dictates that the decrease in the energy content of the Universe is given by theenergy budget due to the expansion and the work done by the pressure. We have followedthe same procedure and in order to have a closed set of equations, we have integrated withvarious equations of state, such as pressure vs. energy density. For the present study, wehave introduced a reliable estimation for the bulk pressure, for which we have taken intoconsideration Eckart (first order) and Israel-Stewart (second order) theories for relativisticfluid. For the latter, we found that the resulting differential equations are higher-orderednonlinear nonhomogeneous so that no analytical solution could be proposed, so far. For theearlier, the only possible solutions relates the Hubble parameter with the scale factor, but noneof them could be directly given in terms of the cosmic time.The present study has a potential to be extended to cover new standard inflationarycosmology with baryosynthesis and dark matter, for which reliable barotropic EoS are unfor-tunately missing. Taking into consideration the possible influence processes of the beyondstandard model is also conditioned to reliable barotropic EoS. Despite the observational con-straints on the cosmological evolution at earlier stages are still challenging, another extensionto cover light element abundance with BBN predictions could subject to a future study. Wewould like to suggest concrete predictions and/or observable features of the effects of bulkviscosity in the early cosmological evolution. A framework of new standard cosmology wouldbe rather the ultimate goal. The present script is designed to pave a path towards these goals. Acknowledgments:
The work of AT was supported by the ExtreMe Matter Institute (EMMI) at the GSIHelmholtz Centre for Heavy Ion Research, Visiting Professor 2019.
Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A Relativistic viscous fluid in the expanding early Universe
Appendix A.1 Israel-Stewart second-order theory
Appendix A.1.1 Hadron epochWhen starting with the continuity equation, Eq. (7), which can be rewritten as Π ( t ) = − ˙ ρ ( t ) H ( t ) − [ ρ ( t ) + p ( t )] , (A1)and substituting with the corresponding EoS, Eq. (8), where ρ ( t ) can be replaced as in Eq. (4),we obtain an expression for the viscous stress tensor, Π ( t ) = − π ˙ H ( t ) − ( + β ) π H ( t )+ (cid:20) − ( + β ) (cid:21) k π a − ( t ) + ( + β ) Λ π − α , (A2)which is valid for Eckart as well as for Israel-Stewart theories. For the latter, we take intoconsideration the second-order entropy fulfilling causality and stability conditions. Then, thetime derivative of the viscous stress tensor reads˙ Π ( t ) = − (cid:20) − ( + β ) (cid:21) k π a − ( t ) H ( t ) − ( + β ) π H ( t ) ˙ H ( t ) − π ¨ H ( t ) . (A3)Having both expressions, Eqs. (A2) and (A3), we still need additional EoS for τ ( t ) , ζ ( t ) and T ( t ) and their temporal evolutions to solve the differential equation resulting from Eq. (64).As τ ( t ) , ζ ( t ) and T ( t ) are expressed in dependence on ρ ( t ) , given Eq. (7), their temporalevolutions shall be depending on ˙ ρ ( t ) , which in turn can be expressed in dependence on the scale factor a ( t ) . Then Eq. (64) leads to a sophisticated third-order inhomogeneous differentialequation (cid:34) d − d Λ π + d π k + ˙ a ( t ) a ( t ) + − d d (cid:18) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:19) d (cid:35) ˙ a ( t ) a ( t ) + (cid:101) π (cid:20) ( − πα + Λ ( + β )) a ( t ) − k + ˙ a ( t ) ( + β ) − − a ( t ) ¨ a ( t ) (cid:21) g + g e − − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g ˙ a ( t ) a ( t ) − k + ˙ a ( t ) − a ( t ) ¨ a ( t ) Λ a ( t ) − ( k + ˙ a ( t ) ) − g g g g (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:17)(cid:105) g g + g e − − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g − β γ (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:17)(cid:105) γ γ α + β (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:17)(cid:105) γ − (cid:20) × d d ( k + ˙ a ( t ) ) + a ( t ) (cid:20) − d d Λ + π d d (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:17)(cid:105) d (cid:21)(cid:21) × d d ( k + ˙ a ( t ) ) + a ( t ) (cid:20) + d π d − d d Λ + π d (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:17)(cid:105) d (cid:21) + e − − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g π a ( t ) [ − πα + Λ ( + β )] a ( t ) e − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g + (cid:104) k + ˙ a ( t ) (cid:105) ( + β ) g + g e − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g ˙ a ( t ) − e − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g k + ˙ a ( t ) ( + β ) − + β g + g e − g g (cid:20) π (cid:18) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g ˙ a ( t ) ¨ a ( t ) a ( t ) − e − − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g ¨ a ( t ) + g + g e − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) g ... a ( t ) a ( t ) =
0. (A4)
Despite the obvious assessment that there is no analytical solution to be proposed, we statethis expression here and hope that interested mathematicians become interested in such highlycomplicated physical problems.Appendix A.1.2 QGP epochAs discussed, for cosmic relativistic fluid, no matter whether Eckart and Israel-Stewarttheories are applied, the viscous stress tensor Π ( t ) can be deduced for a given EoS, Eq. (9),which in turn can be expressed in terms of the Hubble parameter, Eq. (4) Π ( t ) = − π ˙ H ( t ) − ( + β ) π H ( t ) + (cid:18) − ( + β ) (cid:19) k π a − ( t )+ ( + β ) Λ π − α − γ (cid:20) π (cid:18) H ( t ) + k a ( t ) − Λ (cid:19)(cid:21) δ , (A5)Then, the time derivative of Π ( t ) is given as˙ Π ( t ) = − π ¨ H ( t ) − ( + β ) π H ( t ) ˙ H ( t ) − (cid:18) − ( + β ) (cid:19) k π a − ( t ) H ( t ) − γ δ (cid:20) π (cid:18) H ( t ) + k a ( t ) − Λ (cid:19)(cid:21) δ − (cid:16) H ( t ) ˙ H ( t ) − ka − ( t ) H ( t ) (cid:17) . (A6) This results in a highly sophisticated third-order inhomogeneous differential equation − α − − δ γ (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) a ( t ) (cid:19)(cid:21) δ + Λ a ( t ) − [ k + ˙ a ( t ) ] π a ( t ) ( + β ) + k + ˙ a ( t ) ¨ a ( t ) π a ( t ) + (cid:34) d − d Λ π + d π k + ˙ a ( t ) a ( T ) + − d d (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( T ) (cid:19)(cid:21) d (cid:35) ˙ a ( t ) a ( t ) + (cid:101) × − − δ π a ( t ) g + g e − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( T ) (cid:19)(cid:21) g (cid:20) − δ k + ˙ a ( t ) ( + β ) − + a ( t ) (cid:20) + β − δ Λ − π (cid:32) δ α + γ (cid:20) π (cid:18) − Λ + ( k + ˙ a ( t ) ) a ( T ) (cid:19)(cid:21) δ (cid:33)(cid:35) − + δ a ( t ) ¨ a ( t ) (cid:35)(cid:26) − k + ˙ a ( t ) − a ( t ) ¨ a ( t ) Λ a ( t ) − [ k + ˙ a ( t ) ] − g g g g (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( T ) (cid:17)(cid:105) g g + g e − g g (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) ) a ( T ) (cid:19)(cid:21) g − β γ (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( T ) (cid:17)(cid:105) γ γ α + β (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( T ) (cid:17)(cid:105) γ − (cid:20) × d d [ k + ˙ a ( t ) ] − + a ( t ) (cid:18) − d d Λ + d d π (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) ) a ( T ) (cid:17)(cid:105) d (cid:19)(cid:21)(cid:20) × d d [ k + ˙ a ( t ) ] − + a ( t ) (cid:18) + d d π − d d Λ + d π (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:17)(cid:105) d (cid:19)(cid:21) ˙ a ( t ) + g + g e − g g − Λ − ( k + ˙ a ( t ) ) a ( T ) π g π a ( t ) ˙ a ( t ) (cid:40) k + ˙ a ( t ) − a ( t ) ¨ a ( t )[ a ( t ) ] − + k + ˙ a ( t ) − a ( t ) ¨ a ( t )[ β ˙ a ( t ) ] − + k + ˙ a ( t ) − a ( t ) ¨ a ( t )[ δ − δ γ δ ˙ a ( t ) ] − − Λ + k + ˙ a ( t ) a ( T ) π δ − − k + ˙ a ( t ) − a ( t ) ¨ a ( t )[ a ( t ) ˙ a ( t ) ¨ a ( t )] − + (cid:34) a ( t ) + ˙ a ( t ) [ k − a ( t ) ¨ a ( t )] + a ( t ) ¨ a ( t ) − k [ a ( t ) ¨ a ( t )] − + a ( t ) ˙ a ( t ) ... a ( t ) (cid:35) ˙ a ( t ) (cid:41) =
0, (A7) for which the analytical solution a very challenging task.
Appendix A.1.3 QCD-EW epochIn this era, we assume that the EoS, Eq. (9), so that an equation very similar to (A5)and (A6) shall be obtained. For the time derivative of the bulk stress, we apply with relatedbarotropic EoS, (37), (36), and (41) for T ( t ) , ζ ( t ) and τ ( t ) , respectively, − α + Λ a ( t ) − ( k + ˙ a ( t ) ) π a ( t ) ( + β ) − δ γ (cid:20) − π (cid:18) Λ − ( k + ˙ a ( t ) a ( t ) (cid:19)(cid:21) δ + k + ˙ a ( t ) − a ( t ) ¨ a ( t ) π a ( t ) + (cid:34) e + − e e (cid:18) − π (cid:20) Λ − ( k + ˙ a ( t ) ) a ( t ) (cid:21)(cid:19) e (cid:35) ˙ a ( t ) a ( t ) + (cid:101) × − − δ π a ( t ) h + h h + log (cid:18) h [ − Λ a ( t ) + ( k + ˙ a ( t ) ] π a ( t ) (cid:19) (cid:34) − δ k + ˙ a ( t ) ( + β ) − + a ( t ) (cid:34) δ ( + β ) Λ − π (cid:32) δ α + γ (cid:20) π (cid:18) − Λ + ( k + ˙ a ( t ) ) a ( t ) (cid:19)(cid:21) δ (cid:33)(cid:35) − + δ a ( t ) ¨ a ( t ) (cid:35) − h (cid:2) k + ˙ a ( t ) − a ( t ) ¨ a ( t ) (cid:3)(cid:20) h + log (cid:18) h [ − Λ a ( t ) + ( k + ˙ a ( t ) ) ] π a ( t ) (cid:19)(cid:21)(cid:20) h + h h + h log (cid:18) h [ − Λ a ( t ) + ( k + ˙ a ( t ) ) ] π a ( t ) (cid:19)(cid:21) [ Λ a ( t ) − [ k + ˙ a ( t ) ]] − e e (cid:2) k + ˙ a ( t ) − a ( t ) ¨ a ( t ) (cid:3)(cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:17)(cid:105) e [ Λ a ( t ) − ( k + ˙ a ( t ) ] (cid:16) e e + e (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:17)(cid:105) e (cid:17) − β γ (cid:2) k + ˙ a ( t ) − a ( t ) ¨ a ( t ) (cid:3)(cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:17)(cid:105) γ [ Λ a ( t ) − a ( k + ˙ a ( t ) ] (cid:16) γ α + β (cid:104) π (cid:16) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:17)(cid:105) γ (cid:17) ˙ a ( t ) − (cid:20) h + log (cid:18) h [ − Λ a ( t ) + ( k + ˙ a ( t ) ] π a ( t ) (cid:19)(cid:21) − π a ( t ) [ Λ a ( t ) − ( k + ˙ a ( t ) ] (cid:40)(cid:34) h + h h + h log (cid:32) h (cid:2) − Λ a ( t ) + ( k + ˙ a ( t ) (cid:3) π a ( t ) (cid:33)(cid:35)(cid:20) a ( t ) ( + β ) − + δ ˙ a ( t ) (cid:32) × + δ ( + β ) − − a ( t ) (cid:34) δ Λ ( + β ) − − πγ δ (cid:20) π (cid:18) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:19)(cid:21) δ (cid:35) − δ β a ( t ) ¨ a ( t ) (cid:33) + − δ ˙ a ( t ) (cid:32) × δ k ( + β ) − − ka ( t ) (cid:34) δ Λ ( + β ) − − πγ δ (cid:20) π (cid:18) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:19)(cid:21) δ (cid:35) − δ k β a ( t ) ¨ a ( t )+ a ( t ) (cid:34) δ β Λ − πγ δ (cid:20) π (cid:18) − Λ + ( k + ˙ a ( t ) a ( t ) (cid:19)(cid:21) δ (cid:35) ¨ a ( t ) (cid:33) + a ( t ) ... a ( t )( − k + Λ a ( t ) ) − − a ( t ) ˙ a ( t ) ... a ( t ) (cid:35)(cid:41) =
0. (A8)
Appendix A.1.4 EW (asymptotic) epochSubstituting with ρ ( t ) , Eq. (4), and the corresponding EoSe, Eq. (10), in Eq. (7), theviscous stress tensor and its temporal evolution, respectively, become Π ( t ) = − π ˙ H ( t ) − ( + γ ) π H ( t )+ (cid:20) − (+ γ ) (cid:21) k π a − ( t ) + ( + γ ) Λ π , (A9)˙ Π ( t ) = − π ¨ H ( t ) − ( + γ ) π H ( t ) ˙ H ( t ) − (cid:20) − ( + γ ) (cid:21) π a − ( t ) H ( t ) . (A10) Then, the cosmic evolution, Eq. (64), can be expressed as (cid:34) f + − f f (cid:18) π (cid:20) Λ − k + ˙ a ( t ) a ( t ) (cid:21)(cid:19) f (cid:35) ˙ a ( t ) a ( t ) + (cid:101) × a − − k k π a ( t ) (cid:18) π (cid:20) − Λ + k + ˙ a ( t ) a ( t ) (cid:21)(cid:19) k log (cid:20) k π a ( t ) (cid:104) − Λ a ( t ) + (cid:104) k + ˙ a ( t ) (cid:105)(cid:105)(cid:21)(cid:20) Λ a ( t ) ( + γ ) − − k + ˙ a ( t ) ( + γ ) − − a ( t ) ¨ a ( t ) (cid:21) ˙ a ( t ) a ( t ) + k [ k + ˙ a ( t ) − a ( t ) ¨ a ( t )] Λ a ( t ) − a ( t )[ k + ˙ a ( t ) ] + [ k + ˙ a ( t ) − a ( t ) ¨ a ( t )] Λ a ( t ) − [ a ( t ) + ˙ a ( t ) ∗ ] a ( t ) log (cid:20) k [ − Λ a ( t ) + [ k + ˙ a ( t ) ]] π a ( t ) (cid:21) − k + ˙ a ( t ) − a ( t ) ¨ a ( t ) Λ a ( t ) − [ k + ˙ a ( t ) ] a ( t ) f f (cid:16) π (cid:104) − Λ + k + ˙ a ( t ) a ( t ) (cid:105)(cid:17) f f f + f (cid:16) π (cid:104) − Λ + k + ˙ a ( t ) a ( t ) (cid:105)(cid:17) f − β γ (cid:16) π (cid:104) − Λ + k + ˙ a ( t ) a ( t ) (cid:105)(cid:17) γ γ α + β (cid:16) π (cid:104) − Λ + k + ˙ a ( t ) a ( t ) (cid:105)(cid:17) γ + − − k π a ( t ) k Λ a ( t ) ( + γ ) − + k ˙ a ( t )( + γ ) − log (cid:20) k π a ( t ) (cid:104) − Λ a ( t ) + (cid:104) k + ˙ a ( t ) (cid:105)(cid:105)(cid:21) (cid:16) π (cid:104) − Λ + k + ˙ a ( t ) a ( t ) (cid:105)(cid:17) k [ k + ˙ a ( t ) ] − − (cid:34) k k ( + γ ) − + k ˙ a ( t ) ( + γ ) − + k γ log (cid:34) k (cid:2) − Λ a ( t ) + (cid:2) k + ˙ a ( t ) (cid:3)(cid:3) π a ( t ) (cid:35)(cid:18) π (cid:20) − Λ + k + ˙ a ( t ) a ( t ) (cid:21)(cid:19) k ˙ a ( t ) ¨ a ( t ) (cid:35) a ( t ) − (cid:34) k ¨ a ( t ) + k log (cid:34) k (cid:2) − Λ a ( t ) + (cid:2) k + ˙ a ( t ) (cid:3)(cid:3) π a ( t ) (cid:35)(cid:18) π (cid:20) − Λ + k + ˙ a ( t ) a ( t ) (cid:21)(cid:19) k ... a ( t ) (cid:35) a ( t ) (cid:41) =
0. (A11)
References
1. Heinz, U.W. The Little bang: Searching for quark gluon matter in relativistic heavy ion colli-sions.
Nucl. Phys. , A685 , 414–431, [arXiv:hep-ph/hep-ph/0009170]. doi:10.1016/S0375-9474(01)00558-9.2. Tawfik, A.M.; Ganssauge, E. Levy stable law description of the intermittent behavior in Pb + Pbcollisions at 158/A-GeV.
Acta Phys. Hung. , A12 , 53, [arXiv:hep-ph/hep-ph/0012008].3. Gyulassy, M.; McLerran, L. New forms of QCD matter discovered at RHIC.
Nucl. Phys. , A750 , 30–63, [arXiv:nucl-th/nucl-th/0405013]. doi:10.1016/j.nuclphysa.2004.10.034.4. Heinz, U.; Shen, C.; Song, H. The viscosity of quark-gluon plasma at RHIC and the LHC.
AIP Conf.Proc. , , 766–770, [arXiv:nucl-th/1108.5323]. doi:10.1063/1.3700674.5. Adamczyk, L.; others. Energy Dependence of Moments of Net-proton Multiplicity Distributionsat RHIC. Phys. Rev. Lett. , , 032302, [arXiv:nucl-ex/1309.5681]. doi:10.1103/Phys-RevLett.112.032302.6. Ryu, S.; Paquet, J.F.; Shen, C.; Denicol, G.; Schenke, B.; Jeon, S.; Gale, C. Effects of bulk viscosityand hadronic rescattering in heavy ion collisions at energies available at the BNL RelativisticHeavy Ion Collider and at the CERN Large Hadron Collider. Phys. Rev. , C97 , 034910,[arXiv:nucl-th/1704.04216]. doi:10.1103/PhysRevC.97.034910.7. Bzdak, A.; Esumi, S.; Koch, V.; Liao, J.; Stephanov, M.; Xu, N. Mapping the Phases of QuantumChromodynamics with Beam Energy Scan.
Phys. Rept. , , 1–87, [arXiv:nucl-th/1906.00936].doi:10.1016/j.physrep.2020.01.005.8. Komatsu, E.; others. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:Cosmological Interpretation. Astrophys. J. Suppl. , , 18, [arXiv:astro-ph.CO/1001.4538]. doi:10.1088/0067-0049/192/2/18.9. Ade, P.A.R.; others. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. , , A13, [arXiv:astro-ph.CO/1502.01589]. doi:10.1051/0004-6361/201525830.10. Aghanim, N.; others. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys. , , A6, [arXiv:astro-ph.CO/1807.06209]. doi:10.1051/0004-6361/201833910.11. Akrami, Y.; others. Planck 2018 results. X. Constraints on inflation. Astron. Astrophys. , , A10, [arXiv:astro-ph.CO/1807.06211]. doi:10.1051/0004-6361/201833887.12. Aghanim, N.; others. Planck 2018 results. V. CMB power spectra and likelihoods. Astron. Astrophys. , , A5, [arXiv:astro-ph.CO/1907.12875]. doi:10.1051/0004-6361/201936386.
13. Akrami, Y.; others. Planck 2018 results. VII. Isotropy and Statistics of the CMB.
Astron. Astrophys. , , A7, [arXiv:astro-ph.CO/1906.02552]. doi:10.1051/0004-6361/201935201.14. Misner, C.W. The isotropy of the universe. The Astrophysical Journal , , 431.15. Zeldovich, Ya.B.; Novikov, I.D. RELATIVISTIC ASTROPHYSICS. VOL. 2. THE STRUCTURE ANDEVOLUTION OF THE UNIVERSE ; 1983.16. Tawfik, A.N.; Mishustin, I. Equation of State for Cosmological Matter at and beyond QCD andElectroweak Eras.
J. Phys. , G46 , 125201, [arXiv:hep-ph/1903.00063]. doi:10.1088/1361-6471/ab46d4.17. Gron, O. Viscous inflationary universe models.
Astrophys. Space Sci. , , 191–225. doi:10.1007/BF00643930.18. Maartens, R. Dissipative cosmology. Class. Quant. Grav. , , 1455–1465. doi:10.1088/0264-9381/12/6/011.19. Tawfik, A.; Harko, T.; Mansour, H.; Wahba, M. Dissipative Processes in the Early Universe: BulkViscosity. Uzbek J. Phys. , , 316–321, [arXiv:gr-qc/0911.4105].20. Tawfik, A.; Wahba, M. Bulk and Shear Viscosity in Hagedorn Fluid. Annalen Phys. , , 849–856, [arXiv:hep-ph/1005.3946]. doi:10.1002/andp.201000056.21. Tawfik, A.; Wahba, M.; Mansour, H.; Harko, T. Hubble Parameter in QCD Universe for finite BulkViscosity. Annalen Phys. , , 912–923, [arXiv:gr-qc/1008.0971]. doi:10.1002/andp.201000103.22. Adamczyk, L.; others. Constraining the initial conditions and temperature dependent viscositywith three-particle correlations in Au+Au collisions. Phys. Lett. B , , 81–88, [arXiv:nucl-ex/1701.06497]. doi:10.1016/j.physletb.2018.10.075.23. Tawfik, A.; Harko, T. Quark-Hadron Phase Transitions in Viscous Early Universe. Phys. Rev. , D85 , 084032, [arXiv:astro-ph.CO/1108.5697]. doi:10.1103/PhysRevD.85.084032.24. Tawfik, A. The Hubble parameter in the early universe with viscous QCD matter and finitecosmological constant.
Annalen Phys. , , 423–434, [arXiv:gr-qc/1102.2626]. doi:10.1002/andp.201100038.25. Tawfik, A.; Magdy, H. Thermodynamics of viscous Matter and Radiation in the Early Universe. Can. J. Phys. , , 433–440, [arXiv:gr-qc/1109.6469]. doi:10.1139/p2012-037.26. Tawfik, A.; Wahba, M.; Mansour, H.; Harko, T. Viscous Quark-Gluon Plasma in the Early Universe. Annalen Phys. , , 194–207, [arXiv:gr-qc/1001.2814]. doi:10.1002/andp.201000052.27. Tawfik, A. Thermodynamics in the Viscous Early Universe. Can. J. Phys. , , 825–831,[arXiv:gr-qc/1002.0296]. doi:10.1139/P10-058.28. Tawfik, A.N.; Greiner, C. Bulk viscosity at high temperatures and energy densities . [arXiv:hep-ph/1911.02797].29. Laine, M.; Schroder, Y. Quark mass thresholds in QCD thermodynamics. Phys. Rev. , D73 , 085009, [arXiv:hep-ph/hep-ph/0603048]. doi:10.1103/PhysRevD.73.085009.30. Laine, M.; Meyer, M. Standard Model thermodynamics across the electroweak crossover.
JCAP , , 035, [arXiv:hep-ph/1503.04935]. doi:10.1088/1475-7516/2015/07/035.31. D’Onofrio, M.; Rummukainen, K. Standard model cross-over on the lattice. Phys. Rev. , D93 , 025003, [arXiv:hep-ph/1508.07161]. doi:10.1103/PhysRevD.93.025003.32. Borsanyi, S.; others. Calculation of the axion mass based on high-temperature lattice quantumchromodynamics.
Nature , , 69–71, [arXiv:hep-lat/1606.07494]. doi:10.1038/nature20115.33. McCrea, W.H.; Milne, E.A. Newtonian universes and the curvature of space. The quarterly journalof mathematics , pp. 73–80.34. McCrea, W.H.; Milne, E.A. Newtonian Universes and the Curvature of Space.
General Relativityand Gravitation , , 1949–1958. doi:10.1023/A:1001949117817.35. Peebles, P.J.E. Principles of Physical Cosmology ; 1993.36. Shalyt-Margolin, A.E. Entropy In The Present And Early Universe, New Small Parameters AndDark Energy Problem.
Entropy , , 932–952, [arXiv:gr-qc/0911.5597]. doi:10.3390/e12040932.37. Tawfik, A. QCD phase diagram: A Comparison of lattice and hadron resonance gas modelcalculations. Phys. Rev. , D71 , 054502, [arXiv:hep-ph/hep-ph/0412336]. doi:10.1103/Phys-RevD.71.054502.38. Tawfik, A. The Influence of strange quarks on QCD phase diagram and chemical freeze-out:Results from the hadron resonance gas model.
J. Phys. , G31 , S1105–S1110, [arXiv:hep-ph/hep-ph/0410329]. doi:10.1088/0954-3899/31/6/068.
39. Pun, C.S.J.; Gergely, L.A.; Mak, M.K.; Kovacs, Z.; Szabo, G.M.; Harko, T. Viscous dissipativeChaplygin gas dominated homogenous and isotropic cosmological models.
Phys. Rev. , D77 , 063528, [arXiv:gr-qc/0801.2008]. doi:10.1103/PhysRevD.77.063528.40. Eckart, C. The Thermodynamics of Irreversible Processes. 1. The Simple Fluid.
Phys. Rev. , , 267–269. doi:10.1103/PhysRev.58.267.41. Landau, L.; Lifshitz, E. Fluid Mechanics ; Number v. 6, Elsevier Science, 1987.42. Osada, T. Modification of Eckart theory of relativistic dissipative fluid by introducing extendedmatching conditions.
Phys. Rev. , C85 , 014906, [arXiv:nucl-th/1111.1276]. doi:10.1103/Phys-RevC.85.014906.43. Piattella, O.F.; Fabris, J.C.; Zimdahl, W. Bulk viscous cosmology with causal transport theory.
JCAP , , 029, [arXiv:astro-ph.CO/1103.1328]. doi:10.1088/1475-7516/2011/05/029.44. Israel, W. Thermo field dynamics of black holes. Phys. Lett. , A57 , 107–110. doi:10.1016/0375-9601(76)90178-X.45. Coley, A.A.; van den Hoogen, R.J. Qualitative analysis of viscous fluid cosmological modelssatisfying the Israel-Stewart theory of irreversible thermodynamics.
Class. Quant. Grav. , , 1977–1994, [arXiv:gr-qc/gr-qc/9605061]. doi:10.1088/0264-9381/12/8/015.46. Tawfik, A.; Magdy, H.; Ali, A.F. Effects of quantum gravity on the inflationary parameters andthermodynamics of the early universe. Gen. Rel. Grav. , , 1227–1246, [arXiv:gr-qc/1208.5655].doi:10.1007/s10714-013-1522-0.47. Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables , ninth dover printing, tenth gpo printing ed.; Dover: New York, 1964.48. Israel, W. Nonstationary irreversible thermodynamics: A Causal relativistic theory.
Annals Phys. , , 310–331. doi:10.1016/0003-4916(76)90064-6.49. Israel, W.; Stewart, J.M. Thermodynamics of nonstationary and transient effects in a relativistic gas. Physics Letters A , , 213–215. doi:10.1016/0375-9601(76)90075-X.50. Hiscock, W.A.; Lindblom, L. Stability and causality in dissipative relativistic fluids. Annals ofPhysics , , 466–496. doi:10.1016/0003-4916(83)90288-9.51. Chattopadhyay, S. Israel-Stewart approach to viscous dissipative extended holographic Ricci darkenergy dominated universe. Adv. High Energy Phys. ,2016