Cosmic expansion with matter creation and bulk viscosity
CCosmic expansion with matter creation and bulk viscosity
V´ıctor H. C´ardenas , ∗ Miguel Cruz , † and Samuel Lepe ‡ Instituto de F´ısica y Astronom´ıa, Universidad de Valpara´ıso, Gran Breta˜na 1111, Valpara´ıso, Chile Facultad de F´ısica, Universidad Veracruzana 91000, Xalapa, Veracruz, M´exico Instituto de F´ısica, Facultad de Ciencias, Pontificia UniversidadCat´olica de Valpara´ıso, Avenida Brasil 2950, Valpara´ıso, Chile (Dated: August 31, 2020)We explore the cosmological implications at effective level of matter creation effects in a dissipativefluid for a FLRW geometry; we also perform a statistical analysis for this kind of model. Byconsidering an inhomogeneous Ansatz for the particle production rate we obtain that for createdmatter of dark matter type we can have a quintessence scenario or a future singularity known aslittle rip; in dependence of the value of a constant parameter, η , which characterizes the matterproduction effects. The dimensionless age of this kind of Universe is computed, showing that thisnumber is greater than the standard cosmology value, this is typical of universes with presence ofdark energy. The inclusion of baryonic matter is studied. By implementing the construction of theparticle production rate for a dissipative fluid by considering two approaches for the expression ofthe bulk viscous pressure; we find that in Eckart model we have a big rip singularity leading to acatastrophic matter production and in the truncated version of the Israel-Stewart model such rateremains bounded leading to a quintessence scenario. For a non adiabatic dissipative fluid, we obtaina positive temperature and the cosmic expansion obeys the second law of thermodynamics. I. INTRODUCTION
The nature of the late times acceleration of the observ-able Universe is still far from our actual understanding[1]. However, this challenge has not been a reasonfor the community to remain staidly; on the contrary,this has motivated an exhaustive search for models orscenarios beyond General Relativity that attempt toroughly describe the current stage of the Universe inorder to elucidate the nature of the catalyst of suchaccelerated expansion, usually termed as dark energy.As is well known, the cosmological constant approachis a promising scenario, but it has yet to face its ownbattles. For instance, the origin of the cosmologicalconstant must be at Planck scales but its effects areobserved only at cosmological scales (the current accel-erated expansion). This difference between scales wherethe cosmological constant becomes relevant has turnedthe problem unmanageable, at Planck scales the valueof the cosmological constant is greater by approximately120 orders of magnitude than expected. A reconciliationfor the capricious behavior of the cosmological constantat different scales was proposed in Ref. [2], but thisdescription depends on a quantum formulation for thefluctuations of spacetime; nowadays there is no quantumtheory of gravity, therefore the description made wasat a semiclassical level, this scheme it lacks of a fullcharacterization for these quantum fluctuations andconsequently of their cosmic evolution. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
Another more recent but no less controversial problemof our current Universe is the so-called H tension,being H the Hubble constant, this value representsthe rate of expansion of the Universe at present time.The problem lies in the discrepancy between the valuereported for H by the Planck collaboration [3] andthe one reported in [4]. Such difference between bothvalues is not attributable to systematic errors. Thishas originated for example that schemes where darkenergy has the peculiarity of interact with dark mattermay be a response to solve this tension among otherpossible scenarios, that is, we keep looking modelsbeyond standard cosmology, see for instance [5]. Aninteresting review on the interacting scheme for the darksector can be found in [6]. Other scenarios promotethat the H tension is the result of a tension in thevalue T of the cosmic microwave background (CMB)temperature. Despite in some schemes the H tensioncan be alleviated, this implies a new paradigm regardingthe well established value for the temperature of theCMB [7]. Other proposals suggest that an enhancementin the geometric description of the Universe is a viablealternative to resolve the aforementioned tension with noneed of make use of exotic components or a reformulationof our conception of the gravitational interaction. InRef. [8] was found that the H tension can be mitigatedby considering only a non-vanishing torsional tensor todescribe the dark matter sector.In this work we consider as alternative to describethe late times behavior of the Universe the matterproduction effects and the presence of viscosity in thecosmological fluid. We will focus on the cosmologicalimplications of the model at effective level. An interest-ing characteristic of a dissipative fluid is the generationof entropy, in the perfect fluid description the entropy a r X i v : . [ g r- q c ] A ug production and heat dissipation are lost [9]. However,a recent work showed that at cosmological scales thesecond law of thermodynamics is still valid [10]. Onthe other hand, in the literature can be found that bulkviscous effects are not ruled out at all by the obser-vational data a provide a framework in which the H tension can be weakened [11]. As additional examples,see for instance the Ref. [12], where was stated that thepresence of dissipative effects in the fluid contribute sig-nificantly to reproduce the experimental measurementsof the longitudinal polarization of hyperons produced inrelativistic heavy-ion collisions. In Ref. [13] was foundthat when the inflationary process is attenuated bydissipative causes, the inflaton interchanges its energydensity with a emerging radiation component, which canbe associated to the CMB and at late times such modelleads to a cosmological constant type evolution.Over the years the approach in which matter produc-tion effects are employed has changed slightly. At firstit was thought that these effects were important only inthe early Universe providing a natural explanation forthe reheating phase of the inflationary process [14] ordepending on the rate of matter production, the originof the Universe could be free of an initial singularity [15].However, in the meantime it has also been shown thatthe matter production effects can play an importantrole in cosmic evolution, in Ref. [16], supported bycosmological observations the authors showed that thisscheme leads to an Universe in which the dark energysector can be emulated by the particle production.Other relevant scenarios in which the production ofmatter has an important character can be found in[17], this work explores the possibility that the currentstate of the Universe is transitory and will eventuallypresent a decelerated phase, this transition is possibleby considering matter creation in their model. Anotherexample is given in [18] where the cosmological modelencloses entropic forces with matter creation.The outline of this work is as follows: in Section IIwe provide the cosmological equations for a model withmatter creation effects and viscosity. Under the adiabaticcondition for the entropy we construct the effective pa-rameter state and we explore an inhomogeneous Ansatzfor the particle production rate. We obtain the corre-sponding Hubble parameter and we calculate the age ofthis kind of Universe and we consider the inclusion ofbaryons. In Section III we perform the statistical anal-ysis of the model with baryonic matter and the use ofcurrent observational data. In Section IV we proceed inthe inverse order, with the consideration of the dissipa-tive and matter creation effects we construct the form ofthe particle production rate, Γ, we do not use the Ansatzphilosophy. We explore the well known Eckart modeland the truncated version of the causal Israel-Stewartdescription, where a cosmological constant evolution canbe emulated. In Section V we provide a description of the cosmological model when the entropy production isnot adiabatic, we find that the temperature of the fluidis positive and obeys the second law of thermodynamics.Under this approach the cosmic expansion does not havethe problem of negative entropy or temperature. Finally,in Section VI we give the final comments of our work.We will consider 8 πG = c = k B = 1 units throughoutthis work. II. MATTER CREATION AND BULKVISCOSITY
For a dissipative fluid the local equilibrium scalars suchas the particle number density and its energy densityare not altered by the dissipative effects. However, thepressure deviates from the local equilibrium pressure p eff = p + Π , (1)where Π is the bulk viscous pressure. From now on thequantity, p +Π, will denote effective pressure for the dissi-pative fluid. In this description the Friedmann equationsfor a flat FLRW spacetime can be written as follows3 H = ρ, ˙ H + H = −
16 [(1 + 3 ω ) ρ + 3Π] , (2)being ρ the energy density of the dissipative fluid, H denotes the Hubble parameter and the dot stands forderivatives with respect to time. In the previous equa-tions we have considered a barotropic equation of statebetween the density and the pressure given as, p = ωρ ,where ω is commonly known as parameter state and it isconstrained to the interval [0 , ρ + 3 H (1 + ω ) ρ = − H Π . (3)On the other hand, if matter creation exists, i.e., gravita-tional particle production, then the continuity equationfor the particle number density takes the form˙ n + 3 Hn = n Γ , (4)where the possibilities Γ >
0, Γ < T dS = d (cid:16) ρn (cid:17) + pd (cid:18) n (cid:19) , (5)we can write nT ˙ S = ˙ ρ − ρ (1 + ω ) ˙ nn , = − H Π − ρ Γ(1 + ω ) , (6)where the Eqs. (3) and (4) were considered. Note thatfor this approach the entropy is not longer a constant.However, if we assume ˙ S = 0, i.e, an adiabatic dissipa-tive fluid with particle creation in order to be in agree-ment with the standard cosmological model, we obtainthe following conditionΠ = − ρ (1 + ω ) Γ3 H , (7)we will have a negative contribution from the viscouspressure to the non equilibrium pressure if the parti-cle production rate is positive, i.e., no annihilation. Ifwe insert the previous equation in (3) we can write thecontinuity equation for the density in its standard form,˙ ρ +3 H (1+ ω eff ) ρ = 0, where the effective parameter statehas the form ω eff = ω − (1 + ω ) Γ3 H . (8)It is worthy to mention that if the dissipative fluid be-haves as standard dark matter we have, ω = 0 ( p = 0),then the effective parameter only depends of the particleproduction effects, this case was considered in Ref. [16],in such case the condition Γ > H must be fulfilled inorder to have an effective phantom behavior. A. Ansatz for the particle production rate
The most simple assumption for Γ is given by a con-stant production rate. However, in order to study the im-plications of matter production on the cosmic expansion,some specific functions of the Hubble parameter for theparticle production rate have been studied, for instanceΓ ∝ H α , being α an appropriate constant [16, 19–23]. Amore general form for Γ can be found in Ref. [24], whereΓ = Γ( ρ, p ( ρ ) , H, ˙ H, ¨ H, ... ) , (9)and the function p ( ρ ) is viable by means of the equationstate. The form given in Eq. (9) for Γ it is known asinhomogeneous particle production rate. In our case wewill consider Γ given asΓ = Γ( H, ˙ H ) = γ (cid:48) ( H ) ˙ H = dγ ( H ) dt , (10)where the prime denotes derivatives with respect to theHubble parameter by means of the chain rule. Note thatthe form of Γ given above together with Eq. (7) lead usto a direct integration integration of Eq. (3), yielding ρ = ρ a − ω ) exp [(1 + ω ) γ ( H )] , (11)then if γ ( H ) ∝ ln( H δ ) and δ being a constant, we couldwrite consistently an expression for the Hubble parame-ter by means of the first Friedmann equation. We focuson the following expression γ ( H ) = 2 (cid:18) − η (cid:19) ln (cid:18) HH (cid:19) , (12) being η a constant value and H the value of the Hubbleparameter at present time. Using the above γ functionand (10), we can writeΓ3 H = − (cid:18) − η (cid:19) (1 + q ) , (13)where q is the deceleration parameter defined as 1 + q := − ˙ H/H . As can be seen, the particle production ratecan be written as a function of the deceleration parame-ter. On the other hand, using the acceleration equation(2) together with the expression for the viscous pressureobtained from the adiabatic condition (7), one getsΓ3 H = 1 −
23 (1 + q )(1 + ω ) , (14)by equating both expressions for the quotient Γ / H , wearrive to the following result1 + q = 32 (cid:26)
11 + ω − (cid:18) − η (cid:19)(cid:27) − , (15)then, the deceleration parameter takes a constant valueif matter creation effects are introduced in a dissipativefluid. Note that if in the previous expression we con-sider the value, η = 2(1 + ω ) / (3 + 5 ω ), we have q = 0,which represents a Dirac-Milne universe. This kind ofUniverse expands at constant rate since ¨ a = 0. For stan-dard dark matter we have, q = − (1 − η/ η < / η = 2 /
3. Using the Eq. (14) we can write forthe effective parameter (8) ω eff = − q ) = − (cid:26)
11 + ω − (cid:18) − η (cid:19)(cid:27) − . (16)As expected, the effective parameter state has a contri-bution from the matter creation effects. If the follow-ing conditions are satisfied η > (1 + ω ) /ω or 0 < η < ω ) / (3+5 ω ), the effective parameter (16) will behaveas a phantom or quintessence fluid, therefore acceleratedcosmic expansion can be obtained for a dissipative fluidwith matter creation effects in presence of ordinary mat-ter. On the other hand, for ω = 0 we have ω eff = − η ,thus from the condition 0 < η < /
3, the dissipative fluidwill behave as quintessence and will exhibit phantom be-havior with η <
0. For a Dirac-Milne Universe we willhave ω eff = − / H = ρ = ρ a − ω ) exp( γ ( H )), rearranging properly the termsappearing in this expression after inserting the Eq. (12);we obtain for normalized Hubble parameter E ( z ) = Ω / − ∆) ρ (1 + z ) ω ) / − ∆) , (17)where we have defined ∆ := (1 + ω )(1 − /η ), Ω ρ corre-sponds to the standard definition of the density parame-ter, Ω ρ := ρ / H and we also used the relation betweenthe scale factor and the redshift, 1 + z = a − , besides E ( z ) := H ( z ) /H . Note that for η = 1 we have a nullcontribution from matter creation effects and we recoverthe standard cosmology. In the standard dark mattercase we have E ( z ) = Ω / − ∆) ρ (1 + z ) / − ∆) , (18)for the quintessence scenario we have 0 < η < /
3, there-fore −∞ < ∆ < − /
2, then as we approach to the farfuture, we have for the normalized Hubble parameter, E ( z → − →
0. For the phantom regime we have η <
0, thus ∆ >
1, which leads to a singular scenarioas we approach to the far future, E ( z → − → ∞ . Asimilar behavior is obtained for the energy density as weapproach to the far future, for quintessence ρ → ρ → ∞ for phantom. It is worthy to mention that in thephantom scenario the singular nature takes place only atthe far future and not for a specific value of the redshift,therefore this kind of singularity corresponds to a littlerip. From the current observational data it is not possibleto determine if the final fate of the Universe is a futuresingularity or not but if consistency with the supernovais demanded, then this kind of singular model could rep-resent a viable alternative to the ΛCDM model [25–27].If we consider the Eq. (17) and H ( a ) = d ln a/dt , we cancompute the dimensionless age of the Universe H t ( a ) = (cid:90) a da (cid:48) a (cid:48) H ( a (cid:48) ) , (19)as stated in Ref. [28], the value for the dimensionlessage at present time, H t , is around 1 for the ΛCDMmodel and this is independent of the transition from thedecelerated to accelerated stage, the authors call this factas synchronicity problem given that it appears we areliving in a special time. In Fig. (1) we show the behaviorof the dimensionless age of the Universe if we consider theEq. (17) with a quintessence behavior. We also considerthe value ω = 1 / . ± . ρ , according to thelatest Planck collaboration results, this case correspondsto the dashed line in the figure. On the other hand, ifwe consider that created matter is the only componentof the Universe we must have, Ω ρ = 1, this correspondsto the solid line of the plot, this latter case is closer tothe value 1 at present time ( a = 1). Note that at presenttime the value of the dimensionless age of the Universe isincreased in both cases, however, the obtained values areclose to 1; the augmentation in the age of the Universeat present time by dark energy was discussed in Ref. [29]for phantom dark energy. B. Including baryonic matter
It is worthy to mention that if we consider only thematter creation effects we are left with a constant de-celeration parameter in our description, this can be seen H t a H t ( a ) FIG. 1: Age of the Universe with quintessence behavior. Thedashed line corresponds to Ω ρ ≈ . ρ = 1. The value ω = 1 / in Eq. (15). Despite that dark energy behavior can beobtained for certain values of the parameters η and ω ,the model itself is not consistent with the ΛCDM modeland the type Ia supernovae data; this issue can be alle-viated with the inclusion of baryons as done in Ref. [30].In this new scenario the form of the normalized Hubbleparameter can be expressed as follows (cid:18) HH (cid:19) = Ω B (1 + z ) + (1 − Ω B )(1 + z ) ω ) (cid:18) HH (cid:19) , (20)where we take into account the normalization condition,Ω B + Ω CM = 1, i.e., for the late times description ofthe Universe we only consider the contribution from thegravitationally created matter and baryons. From theprevious expression it is not possible to find the normal-ized Hubble parameter analytically. However, if we con-sider the acceleration equation (2) with the assumption, p B = 0, and the equations (7), (12) together with thestandard relationship between the redshift and the scalefactor given above; we can write the following differentialequation for the normalized Hubble parameter(1 + z ) E dEdz − E −
13 (1 + ω )(1 − Ω B )(1 + z ) (cid:18) − η (cid:19) d ln Edz = 12 (cid:2) (1 + 3 ω )(1 − Ω B ) + Ω B (1 + z ) (cid:3) , (21)from which E ( z ) can be obtained by numerical integra-tion. We will use this equation to constraint the param-eters of the model with the use of current cosmologicalobservations later. In this we will have at effective levelfrom the contribution of baryons and matter creation ef-fects ω eff = η −
11 + Ω B H (1 − Ω B ) η (1 + z ) − η ) , (22)where we have considered the equation (7) for the pres-sure together with the assumption that created matterbehaves as standard dark matter, ω = 0, and p B = 0. III. OBSERVATIONAL CONSTRAINTS
In this section we study to what extend the evolutionimplied by Eq. (21) describes appropriately the obser-vations. In particular, we use the latest type Ia super-novae sample called Pantheon [31] consisting in 1048 datapoints. The data gives us the apparent magnitude m atmaximum from which we can compute the distance mod-ulus µ = m − M with M the absolute magnitude for typeIa supernovae. Here we compute the residuals µ − µ th andminimize the quantity χ = ( µ − µ th ) T C − ( µ − µ th ) , (23)where µ th = 5 log ( d L ( z ) / pc ) gives the theoreticaldistance modulus, d L ( z ) is the luminosity distance givenby d L ( z ) = (1 + z ) CH (cid:90) dzE ( z ) , (24) C is the covariance matrix released in [31], and the ob-servational distance modulus takes the form µ = m − M + α X − α Y, (25)where m is the maximum apparent magnitude in bandB, X is related to the widening of the light curves, and Y corrects the color. usually, the cosmology – speci-fied here by µ th – is constrained along with the parame-ters M , α and α . These nuisance parameters are thenmarginalized to obtain the posterior probabilities for ourparameters of interest: w , Ω B and η . In the statisticalstudy we use a prior for Ω B and we assume w = 0 as-suming the contribution be as dark matter. Then, thefree parameters to fit are Ω B and η . Using the valueΩ B = 0 . ± . B = 0 . ± . η = 0 . ± . σ and 2 σ are shown in Fig. (2). If we insert the con-strained values for Ω B and η in Eq. (22) together withthe Hubble constant reported in Ref. [3] one gets thatat present time the effective parameter lies in the in-terval [ − . , − . ω eff ( z → − → η − − . ± . Ω B η FIG. 2: Confidence contours at 1 σ and 2 σ for the η and Ω B parameter in the case of Eq. (21) keeping w = 0. IV. CONSTRUCTING Γ FROM VISCOUSMODELS
In this section we construct the particle productionrate from bulk viscous considerations. We will not con-sider a specific Ansatz for this term. In general, the bulkviscous pressure must obey the following transport equa-tion [33, 34]Π = − ξ ( ρ ) H − τ ˙Π − ζ τ Π (cid:34) H + ˙ ττ − ˙ ξξ − ˙ TT (cid:35) , (26)where ξ ( ρ ) is the bulk viscosity coefficient and giventhat it is a function of the energy density we must have ξ ( ρ ) ≥
0; the usual assumption for this term is ξ = ξ ρ s ,with ξ and s constants, T is the barotropic temperaturewhich is also a function of the energy density by meansof the integrability Gibbs condition. On the other hand, τ represents the relaxation time for bulk viscous effectsand is given by τ = ξ/c b ( ρ + p ) = ξ ρ s − /c b (1 + ω ) for abarotropic equation of state and we must have 0 < c b ≤ (cid:15) (1 − ω ),where (cid:15) is a constant parameter. From the Friedmannequations (2), we can write for the bulk viscous pressureΠ = − H − ω ) H . (27)In order to illustrate some results, in the following sec-tions we will study two cases separately. A. Eckart model
If we consider τ = 0 in Eq. (26) we obtain the Eckartmodel. This approach have been studied exhaustivelydespite its superluminal propagation problem. However,represents a manageable framework for viscous effects,see for instance the references contained in [35], wherethe Eckart framework is studied in the late and earlyUniverse and tested with recent observational data. Fromthe Eq. (26) with τ = 0, the Friedmann equation (2) andthe standard definition for the bulk viscous coefficient wecan write Π = − ξ ( ρ ) H = − s +1 ξ H s +1 , (28)if we consider the previous expression in equation (7) wearrive to an explicit form for the particle production rategiven as Γ( H )3 H = 3 s ξ (1 + ω ) H s − / , (29)note that in order to maintain gravitational particle pro-duction, Γ >
0, we must have ξ > s = 1 / √ ξ / (1 + ω ) or Γ = √ ξ for the standard darkmatter case. Now, using the Eq. (7) and (27) we canwrite Γ3 H = 1 + 23(1 + ω ) (cid:32) ˙ HH (cid:33) , (30)and by means of (29) the previous expression takes thefollowing form˙ HH = − (cid:110) ω − s ξ H s − / (cid:111) . (31)If we consider the standard dark matter case, ω = 0,together with the case s = 1 /
2, then the Eq. (31) can beintegrated straightforwardly, yielding H ( t ) = H (cid:34) √ (cid:18) √ − ξ (cid:19) H ( t − t ) (cid:35) − , (32)where H denotes evaluation of H ( t ) at t = t . The ex-pansion of the Universe is guaranteed for 0 < ξ < / √ ξ > / √ H ( t ) = 23 √ (cid:18) ξ − / √ (cid:19) ( t s − t ) − , (33)where we have defined t s = t + 1 H (cid:20) √ ξ − / √ (cid:21) . (34) Notice that t s is a constant value and will represent avalue for the time at the future at which the Hubble pa-rameter becomes singular. According to the classificationfor future singularities given in [36], we have a big rip sin-gularity at t = t s . Thus inserting the Hubble parameter(33) in Eq. (29) with s = 1 / ω = 0, one getsΓ( t ) = 3 √ ξ H = (cid:18) ξ ξ − / √ (cid:19) ( t s − t ) − . (35)As can be seen, the particle production rate becomes sin-gular as we approach to t s . This result is contradictorysince in the phantom scenario as the Universe reaches thefuture singularity, matter or any structure must be disag-gregated. However, we must keep in mind that have beenshown that phantom scenarios are unstable, i.e., at quan-tum level they have an unbounded negative energy thatleads to the absence of a stable vacuum state. On theother hand, in order to avoid this kind of problem in Ref.[37] was proposed the interaction of phantom particlesand standard matter, at least gravitationally. In conse-quence, the gravitational interaction allows processes asthe spontaneous production from the vacuum of a pairof phantom particles and a pair of photons, to mentionsome. In this case the phase space integral is divergent,this indicates a catastrophic instability. This instabil-ity can be avoided by imposing a non Lorentz invariantmomentum space cutoff, but despite this correction thedensity number of photons and phantom particles can bewritten as, n ∝ (Γ × age of the Universe), and accord-ing to the diffuse gamma ray background observationsthe production rate of photons, Γ, leads to higher valuesfrom the typical ones for the energy of the produced pho-tons. Then, any cosmic ray experiment at earth shouldbe detecting more events than normal and more energeticthan those detected until now. These bounds for Γ and n together with other considerations of Ref. [37] suggestthat the origin of phantom must come from new physicsbeyond the standard model of particles. See also Refs.[38, 39] for similar discussions on the topic. B. Israel-Stewart approach
Setting ζ = 0 in the transport equation (26) leadsto the truncated version of the Israel - Stewart theory[33, 34]. This effective model has been widely studied atcosmological level in several contexts, see the referencesin [40]. Therefore, if we consider the truncated transportequation (26) together with the Friedmann equations (2)and the continuity equation for the energy density (3), weobtain a second order differential equation for the Hubbleparameter¨ H + 3 H ˙ H (1 + ω ) + 92 (cid:15) (cid:0) − ω (cid:1) (cid:26) (1 + ω )3 s ξ H − s − (cid:27) H + (cid:15) (1 − ω )3 s − ξ ˙ HH − s ) = 0 . (36)Taking the value s = 1 / x = ln( a/a ), we can solve for the Hubbleparameter in the standard dark matter case H ( z ) = H (cid:115) cos ( β [ c + ln (1 + z )])cos( βc ) (1 + z ) α , (37)where c is an integration constant given as c = 1 β arctan (cid:18) { α − (1 + q ) } β (cid:19) . (38)The form written in Eq. (37) for the Hubble parameteris obtained from the condition H ( z = 0) and the firstderivative H (cid:48) ( z = 0) given that we are solving a secondorder differential equation, for simplicity in the notationwe have defined α = 3 / √ (cid:15)/ ξ together with β = 1 / (cid:113) (cid:15)/ξ ( √ − ξ − ξ / (cid:15) − (cid:15)/ ξ ), which arealso constants. On the other hand, q is the decelerationparameter evaluated at z = 0. Note that at present time z = 0, the Hubble parameter (37) takes the value H .In Fig. (3) we depict the normalized Hubble pa-rameter, H ( z ) /H . In Ref. [41] the velocity of bulkviscous perturbations was constrained to the interval,10 − (cid:28) c b ( (cid:15) ) (cid:46) − , in order to obtain a similar behav-ior as the ΛCDM model at perturbative level, thereforewe will consider these values for the parameter (cid:15) andthe interval [10 − , − ] for the constant ξ . As com-mented before, the solution (37) depends on the valueof the deceleration parameter at present time, we willconsider its definition coming from the ΛCDM model q ( z ) = − Λ , / Ω m, )(1 + z ) − ]) − togetherwith the normalization condition Ω Λ , + Ω m, = 1 andthe recent value reported by the Planck collaboration forΩ m, [3]. With these considerations we obtain − . ≤ q ≤ − . ξ = 10 − , (cid:15) = 10 − , − and the aforemen-tioned values for q , as can be seen the model coincideswith the ΛCDM model from the present time ( z = 0) tothe far future ( z = −
1) and also in the past ( z > ξ to 10 − , in this case we can see that the normalizedHubble parameter tends to zero (dotted lines). As thevalue of the parameter (cid:15) grows the model deviates signif-icantly from the ΛCDM model, but, this could representan unstable scenario for the viscous model.Then, if we consider the Eq. (30) with ω = 0, we canobtain for the particle production rateΓ( z ) = 3 H ( z ) − z ) dH ( z ) dz , = H (1 + z ) α (cid:115) cos ( β [ c + ln (1 + z )])cos ( βc ) [3 − α + β tan ( β [ c + ln (1 + z )])] . (39) - - - - - z H ( z ) H - - - - - z H ( z ) H FIG. 3: Hubble parameter in the truncated Israel-Stewartmodel.
In this case we have that if at some stage of the cos-mic evolution the argument β ( c + ln(1 + z )) of thetangent function takes the value π (1 + 2 n ) / n =0 , ± , ± , ... , the particle production rate diverges. Fromthe previous equation we can write the quotientΓ3 H = 1 − α + β β [ c + ln (1 + z )]) . (40)We show the behavior of the above expression in Fig.(4) with ξ = 10 − , (cid:15) = 10 − , − and q = − . , − . ω = 0the quotient remains positive and tends to 1 at the farfuture ( z = − > ω eff given in (8), we can seethat the cosmological viscous fluid with matter produc-tion effects from the present time ( z = 0) to the futurewill behave as quintessence dark energy and at the farfuture will emulate a cosmological constant behavior, atthis stage the matter production stops. If we considerthe value ξ = 10 − the resulting behavior for Γ / H issimilar as the one shown in Fig. (4). Therefore, the trun-cated Israel-Stewart model with matter production doesnot allow a phantom scenario. - - z Γ H FIG. 4: Quotient Γ / H . To end this section, we discuss the following Ansatzfor the Hubble parameter H ( t > t s ) = | A | t s (cid:18) tt s − (cid:19) − , (41)this Ansatz was proposed in Ref. [42] for the full Israel-Stewart model in order to study some of its thermody-namics properties. One interesting feature of this Ansatzis that the cosmic evolution starts from an initial singu-larity given at, t = t s , that posses the characteristics ofa big bang and represents an expanding universe since | A | >
0. Inserting the Ansatz (41) in the truncated dif-ferential equation of the Israel-Stewart model (36), weobtain a quadratic equation for | A | , the solutions will begiven as | A | ± = (cid:0) √ (cid:15)/ξ (cid:1) ± (cid:113) (cid:15) ξ − √ (cid:15)ξ + 36 (cid:15) √ (cid:15)/ξ − (cid:15) , (42)then | A | is a constant given in terms of the parameters (cid:15) and ξ . Thus, using the equation (30) with the Ansatz(41), we obtain the following expression for the particleproduction rateΓ( t ) = 1 t s (cid:18) tt s − (cid:19) − (cid:8) | A | ± − (cid:9) , (43)always that the condition 3 | A | ± > > H = 1 − | A | ± . (44)Adopting the values for the parameters ξ and (cid:15) as in theprevious case, we obtain some different cases for (44).With ξ = [10 − , − ] and (cid:15) = [10 − , − ] we havethat Γ / H ≈ | A | + , then bymeans of Eq. (8) with ω = 0, we find that the dissipa-tive fluid with matter production behaves as a cosmolog-ical constant along the cosmic evolution. On the otherhand, for | A | − with the same set of values for ξ and (cid:15) we have that depending on the election of these values,the quotient can be positive but (cid:28)
1, negative or zero.Using the Eq. (8) with ω = 0 we can see that for a posi-tive quotient the effective parameter is excluded from thequintessence region and for negative quotient ω eff turnspositive leading to a decelerated expansion. For the caseof null particle production we have q = 1 / | A | + , but as in the case discussed previously,no phantom regime is allowed. V. NON ADIABATIC EXPANSION
The results obtained in the previous sections emergefrom the adiabatic condition for the entropy, i.e. ˙ S = 0,leading to a constant entropy in time. However, if weconsider non adiabaticity the thermodynamics descrip-tion of the cosmological model becomes more consistent[43]. From Eqs. (4), (10) and (12) we can compute theparticle number density, yielding n ( H ) = n V (cid:18) HH (cid:19) ( − η ) , (45)where V is the Hubble volume given as ( a/a ) . Notethat for H = H , we simply have the constant densitynumber, n /V , and for an expanding Universe this den-sity is always positive. With the inclusion of dissipativeeffects we must consider the following evolution equationfor the temperature [9]˙ TT = − H (cid:18) ∂p∂ρ (cid:19) + n ˙ S (cid:18) ∂T∂ρ (cid:19) , (46)the integrability condition ∂ S∂T ∂n = ∂ S∂n∂T , (47)still holds, therefore we can write n ∂T∂n + ( ρ + p ) ∂T∂ρ = T ∂p∂ρ . (48)If we consider these results together with Eqs. (4), (6)and a barotropic equation of state in the expression (46),we obtain˙ TT = − Hω ρ (1+ ω ) + Γ3 H Π ρ (1+ ω ) ( − Γ3 H ) + Γ3 H ( − Γ3 H ) , (cid:124) (cid:123)(cid:122) (cid:125) Ξ( t ) (49)by considering the relationship between the redshift andthe scale factor the equation for the temperature can bewritten as 1 T dTdz = 3 ω (1 + z ) − Ξ( z ) , (50)if we integrate T ( z ) = T exp (cid:20) ω (cid:90) z Ξ( z (cid:48) )1 + z (cid:48) dz (cid:48) (cid:21) . (51)It is worthy to mention that the obtained expression forthe temperature is definite positive, in the case wherewe have null contribution from dissipative and mattercreation effects, Π = Γ = 0, the temperature takes thestandard definition obtained in a single fluid description, T ( z ) = T (1 + z ) ω and for the standard dark mattercase the temperature takes a constant value, as in theΛCDM model.On the other hand, if we consider the expression (6),we can write˙ S = 9 H nT (cid:20) −
23 (1 + q ) + (1 + ω ) (cid:18) − Γ3 H (cid:19)(cid:21) , (52)and the expression (27) was considered for the bulk vis-cous pressure. Now, inserting Eq. (13) in the previousresult one gets˙ S = 9 H nT (cid:26)
23 (1 + q ) (cid:20) (1 + ω ) (cid:18) − η (cid:19) − (cid:21) + (1 + ω ) (cid:27) . (53)As can be seen the entropy production has contribu-tions from matter creation and dissipative effects. Forthe standard dark matter case we have˙ S = H nT (cid:110) −
23 (1+ q ) η (cid:111) , quintessence , H nT (cid:110) | q | η (cid:111) , phantom , (54)where we have considered that at effective level bothcases can appear. In order to be in agreement with sec-ond law of thermodynamics, ˙ S >
0, the conditions T > n >
0, must be satisfied together with the follow-ing cases: (i) in the quintessence scenario the conditions1 > q ) / η and η > η < η > η < > | q | / η . Thus, under the non adiabatic condi-tion for the entropy the cosmic expansion is free of thenegative entropy or temperature problem [44]. VI. FINAL REMARKS
In this work we studied the cosmic evolution thatemerges from the consideration of matter creation effectsin a viscous fluid under the adiabatic condition forthe entropy, i.e., constant entropy. As first approachwe adopted an inhomogeneous Ansatz for the particleproduction rate, Γ. In this first scheme we obtainedthat the model at effective level could describe phantomor quintessence regimes depending on the values of theparameters η and ω , being the case of interest the darkmatter type behavior for created matter denoted by ω = 0. For this special case phantom and quintessencebehaviors are still present. On the other hand, we com-puted the dimensionless age of this kind of Universe fortwo cases: created matter represents the whole contentof Universe and secondly created matter represents theactual matter of the Universe; for both cases we gotthat the age for these kind of universes deviates from 1but remains close to this value, this is characteristic ofuniverses with presence of dark energy. However, whenthe deceleration parameter is computed in this firstdescription we obtain a constant parameter as result,therefore a model of this kind it is not consistent with thecosmological observations or the ΛCDM model. In orderto fix this issue we considered the inclusion of baryonicmatter and performed the statistical analysis for themodel in order to constraint the parameters Ω B and η with ω = 0. Therefore, with the constrained parameterswe proved that this model behaves as quintessence darkenergy along the cosmic evolution.As second approach we dropped the Ansatz philosophyfor the particle production rate and we constructed suchterm with the consideration of two descriptions for thebulk viscous pressure: the Eckart model and the trun-cated version of the Israel-Stewart formalism. The par-ticle production rate has a relevant role in the cosmicevolution since ω eff = − Γ3 H , for ω = 0, as discussed previously. If Γ > H , thephantom scenario is allowed and the case Γ < H couldrepresent quintessence or decelerated expansion. Inthe Eckart description a big rip singularity appears,then in this case the Γ term also exhibits a singularbehavior leading to a catastrophic matter production.This scenario contradicts the expected behavior of thephantom regime where matter or any structure mustbe diluted. We infer that this conduct is due to thefact that the Eckart formalism is non-causal, thereforeit is not appropriate to describe late times in anytype of Universe. On the other hand, if the truncatedversion of the Israel-Stewart model is considered wehave several possibilities to study, however, in this workwe focused only on two cases: an analytical solutionfor the Hubble parameter emerging from this approach0and an Ansatz for the Hubble parameter that leads toa cosmic evolution with an initial singularity with theproperties of a Big Bang. It is worthwhile to mentionthat in this scenario the catastrophic matter productionis not present. For the analytical solution of the Hubbleparameter we obtained that the corresponding Γ termleads to a quintessence dark energy evolution and atthe far future the model could mimic the ΛCDM model.This scenarios are possible if the parameter (cid:15) (a constantparameter responsible of characterizing the velocity ofbulk viscous perturbations) lies in a specific intervalsuch that at perturbative level the viscous model isclose to ΛCDM, if we consider other values for (cid:15) wecould have other possibilities for the cosmic evolution ofthe model but our description could reveal an unstablebehavior. When the Ansatz for the Hubble parameter isconsidered, we have two branches characterized by | A | ± ;for the branch | A | + the cosmological constant expansionis emulated by our model and if we consider the branch | A | − decelerated expansion is obtained. Then, this lattercase is not favored and must be discarded.We must emphasize that it seems that the scenarioof dark matter type production plus other cosmological effects does not allow the crossing to the phantomregime. The case of dark matter type production wasdiscussed by the authors in Ref. [16].Finally, we discuss the cosmic expansion of our modelif we implement the non adiabaticity condition. In thisdescription we obtain that the temperature of the viscousfluid is positive definite and in the case of dark mattertype behavior, ω = 0, we got that such temperaturetakes a constant value as in the ΛCDM model even ifΠ and Γ are different from zero. For Π = Γ = 0, werecover the single fluid description at thermodynamicslevel. Then, the model obeys the second law if we takeinto account some conditions for the parameters of themodel. These phantom or quintessence scenarios havepositive entropy and temperature. Acknowledgments
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