Viscous Cosmology for Early- and Late-Time Universe
Iver Brevik, ?yvind Grøn, Jaume de Haro, Sergei D. Odintsov, Emmanuel N. Saridakis
aa r X i v : . [ g r- q c ] J un Viscous Cosmology for Early- and Late-Time Universe
Iver Brevik a Øyvind Grøn b Jaume de Haro c Sergei D. Odintsov d,e,f,g
Emmanuel N.Saridakis h,i,j a Department of Energy and Process Engineering, Norwegian University of Science and Technology,N-7491 Trondheim, Norway b Oslo and Akershus University College of Applied Sciences, Faculty of Technology, Art and Design,St. Olavs Plass, N-0130 Oslo, Norway c Departament de Matem`atica Aplicada, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028Barcelona, Spain d ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain e Institute of Space Sciences (IEEC-CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain f Tomsk State Pedagogical University, 634061 Tomsk and Int. Lab. Theor. Cosmology, TomskState Univ. of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia g Inst. of Physics, Kazan Federal University, Kazan 420008, Russia h Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73,Athens, Greece i National Center for Theoretical Sciences, Hsinchu, Taiwan 300 j CASPER, Physics Department, Baylor University, Waco, TX 76798-7310, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , Emmanuel − [email protected]
Abstract:
From a hydrodynamicist’s point of view the inclusion of viscosity concepts inthe macroscopic theory of the cosmic fluid would appear most natural, as an ideal fluidis after all an abstraction (excluding special cases such as superconductivity). Makinguse of modern observational results for the Hubble parameter plus standard Friedmannformalism, we may extrapolate the description of the universe back in time up to theinflationary era, or we may go to the opposite extreme and analyze the probable ultimatefate of the universe. In this review we discuss a variety of topics in cosmology when it isenlarged in order to contain a bulk viscosity. Various forms of this viscosity, when expressedin terms of the fluid density or the Hubble parameter, are discussed. Furthermore, weconsider homogeneous as well as inhomogeneous equations of state. We investigate viscouscosmology in the early universe, examining the viscosity effects on the various inflationaryobservables. Additionally, we study viscous cosmology in the late universe, containingcurrent acceleration and the possible future singularities, and we investigate how one mayeven unify inflationary and late-time acceleration. Finally, we analyze the viscosity-inducedcrossing through the quintessence-phantom divide, we examine the realization of viscosity-driven cosmological bounces, and we briefly discuss how the Cardy-Verlinde formula isaffected by viscosity.
Keywords:
Viscous Cosmology, Modified Gravity, Dark Energy, Inflation ontents – 1 –
Introduction
The introduction of viscosity coefficients in cosmology has itself a long history, although thephysical importance of these phenomenological parameters has traditionally been assumedto be weak or at least subdominant. In connection with the very early universe, theinfluence from viscosity is assumed to be the largest at the time of neutrino decoupling (endof the lepton era), when the temperature was about 10 K. Misner [1] was probably thefirst to introduce the viscosity from the standpoint of particle physics; see also Zel’dovichand Novikov [2]. Nevertheless, on a phenomenological level, the viscosity concept wasactually introduced much earlier, with the first such work being that of Eckart [3].When considering deviations from thermal equilibrium to the first order in the cosmicfluid, one should recognize that there are in principle two different viscosity coefficients,namely the bulk viscosity ζ and the shear viscosity η . In view of the commonly acceptedspatial isotropy of the universe, one usually omits the shear viscosity. This is motivatedby the WMAP [4] and Planck observations [5], and is moreover supported by theoreticalcalculations which show that in a large class of homogeneous and anisotropic universesisotropization is quickly established. Eckart’s theory, as most other theories, is maintainedat first-order level. In principle, a difficulty with this kind of theory is that one becomesconfronted with a non-causal behavior. In order to prevent this one has to go to the secondorder approximation, away from thermal equilibrium.The interest in viscosity theories in cosmology has increased in recent years, for var-ious reasons, perhaps especially from a fundamental viewpoint. It is well known amonghydrodynamicists that the ideal (nonviscous) theory is after all only an approximation tothe real world. For reviews on both causal and non-causal theories, the reader may consultGrøn [6] (surveying the literature up to 1990), and later treatises by Maartens [7, 8], andBrevik and Grøn [9].The purpose of the present review is to explore how several parts of cosmological theorybecome affected when a bulk viscosity is brought into the formalism. After highlighting thebasic formalism in the remaining of the present section, in Section 2 we consider the veryearly (inflationary) universe. We briefly present the conventional inflation theory, covering“cold”, “warm” and “intermediate” inflation, and we extract various inflationary observ-ables. Thereafter we investigate the viscous counterparts in different models, dependingon the form of bulk viscosity as well as on the equation of state.In Section 3 we turn to the late universe, including the characteristic singularities inthe far future, related also to the phantom region in which the equation-of-state parameteris less than −
1. The different types of future singularities are classified, and we explorethe consequences of letting the equation of state to be inhomogeneous. A special case isthe unification of inflation with dark energy in the presence of viscosity, a topic which isdealt with most conveniently when one introduces a scalar field. Additionally, we discussholographic dark energy in the presence of a viscous fluid.In Section 4 we discuss various special topics, amongst them the possibility for theviscous fluid to slide from the quintessence region into the phantom region and then into afuture singularity, if the magnitude of the present bulk viscosity is large enough. Compari-– 2 –on with estimated values of the bulk viscosity derived from observations, indicate that thismay actually be a realistic scenario. In the same section we also discuss the viscous BigRip realization, and finally we see how the Cardy-Verlinde formula becomes generalizedwhen viscosity is accounted for, since the thermodynamic (emergent) approach to gravityhas become increasingly popular.Finally, in Section 5 we summarize the obtained results and we discuss on the advan-tages of viscous cosmology.
We begin by an outline of the general relativistic theory, setting, as usual, k B and c equalto one. The formalism below is taken from Ref. [10]. We adopt the Minkowski metric inthe form ( − + ++), and we use Latin indices to denote the spatial coordinates from 1 to 3,and Greek indices to denote spacetime ones, acquiring values from 0 to 3. U µ = ( U , U i )denotes the four-velocity of the cosmic fluid, and we have U = 1 , U i = 0 in a localcomoving frame.With g µν being a general metric tensor we introduce the projection tensor h µν = g µν + U µ U ν , (1.1)and the rotation tensor ω µν = h αµ h βν U ( α ; β ) = 12 ( U µ ; α h αν − U ν ; α h αµ ) . (1.2)The expansion tensor is θ µν = h αµ h βν U ( α ; β ) = 12 ( U µ ; α h αν + U ν ; α h αµ ) , (1.3)and has the trace θ ≡ θ µµ = U µ ; µ . The third tensor that we shall introduce is the sheartensor, namely σ µν = θ µν − h µν θ, (1.4)which satisfies σ µµ = 0. Finally, it is often useful to make use of the three tensors above inthe following decomposition of the covariant derivative of the fluid velocity: U µ ; ν = ω µν + σ µν + 13 h µν θ − A µ U ν , (1.5)where A µ stands for the four-acceleration, namely A µ = ˙ U µ = U ν U µ ; ν .The above formalism is for a general geometry. In the following we will focus onFriedmann-Robertson-Walker (FRW) geometry, which is of main interest in cosmology,whose line element is ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdϕ ) (cid:21) , (1.6)where a ( t ) is the scale factor and k = 1 , , − x µ are numerated as ( t, r, θ, ϕ ). In these coordinates the covariantderivatives of the velocity acquire the simple form U µ ; ν = Hh µν , (1.7)– 3 –ith H = ˙ a/a the Hubble parameter. The rotation tensor, the shear tensor, and thefour-acceleration all vanish, i.e ω µν = σ µν = 0 , A µ = 0 , (1.8)and the relation between scalar expansion and Hubble parameter is simply θ = 3 H. (1.9)As a next step we consider the fluid’s energy-momentum tensor T µν in the case whereviscosity as well as heat conduction are taken into account. If K is the thermal conductivity,considered in its nonrelativistic framework, then for the spacelike heat flux density four-vector we have the expression Q µ = − Kh µν ( T ,ν + T A ν ) , (1.10)with T the temperature. The last term in this expression is of relativistic origin. The coor-dinates used in (1.1) are comoving, with freely moving reference particles having vanishingfour-acceleration. Thus, one obtains the usual expression Q ˆ i = − KT , ˆ i for the heat fluxdensity through a surface orthogonal to the unit vector e ˆ i . Hence, assembling everything,in an FRW metric we can now introduce the energy-momentum tensor as T µν = ρU µ U ν + ( p − Hζ ) h µν − ησ µν + Q µ U ν + Q ν U µ , (1.11)with ρ and p the fluid’s energy density and pressure respectively, and where ζ is the bulkviscosity and η the shear viscosity.Taking all the above into consideration, we conclude that for a universe governed byGeneral Relativity in the presence of a viscous fluid, in FRW geometry the two Friedmannequations read as: H + ka = κρ H + 3 H = − κp , (1.13)with κ the gravitational constant. Note that these equations give˙ H = − ( κ/ ρ + p ) (1.14)for a flat universe. We mention that the energy density and pressure can acquire a quitegeneral form. For instance, a quite general parametrization of an inhomogeneous viscousfluid in FRW geometry is [11–13] p = w ( ρ ) ρ − B ( a ( t ) , H, ˙ H ... ) , (1.15)where w ( ρ ) can depend on the energy density, and the bulk viscosity B ( a ( t ) , H, ˙ H ... ) canbe a function of the scale factor, and of the Hubble function and its derivatives. A usualsubclass of the above general equation of state is to assume that B ( a ( t ) , H, ˙ H ... ) = 3 Hζ ( H ) , (1.16)– 4 –ith ζ ( H ) > ζ ( H ) = ζ = const. .Let us now focus on thermodynamics, and especially on the production of entropy. Thesimplest way of extracting the relativistic formulae is to generalize the known formalismfrom nonrelativistic thermodynamics. We use σ to denote the dimensionless entropy perparticle, where for definiteness as “particle” we mean a baryon. The nonrelativistic entropydensity thus becomes nk B σ , where n is the baryon number density. Making use of therelationship [14] dSdt = 2 ηT ( θ ik − δ ik ∇ · u ) + ζT ( ∇ · u ) + KT ( ∇ T ) , (1.17)where u denotes the nonrelativistic velocity and ∇ the three-dimensional Laplace operator,we can generalize to a relativistic formalism simply by imposing the effective substitutions θ ik → θ µν , δ ik → h µν , ∇ · u → H, − KT ,k → Q µ , (1.18)whereby we obtain the desired equation S µ ; µ = 2 ηT σ µν σ µν + 9 ζT H + 1 KT Q µ Q µ , (1.19)in which S µ denotes the entropy current four-vector S µ = nk B σU µ + 1 T Q µ . (1.20)More detailed derivations of these results can be found, for instance, in Refs. [15] and [16].In summary, viscous cosmology is governed by the Friedmann equations (1.12) and(1.13), along with various considerations of the fluid’s equation of state. Hence, theserelations will be the starting point of the discussion of this manuscript. In the followingsections we investigate viscous cosmology in detail. We start the investigation of viscous cosmology by focusing on early times, and in particularon the inflationary realization. Inflation is considered to be a crucial part of the universecosmological history, since it can offer a solution to the flatness, horizon and monopoleproblems [17–19]. In order to obtain the inflationary phase one needs to consider a suitablemechanism, which is either a scalar field in the framework of General Relativity [20–22],or a degree of freedom arising from gravitational modification [23, 24]. In this section wewill see how inflation can be driven by a viscous fluid.
Before proceeding to the investigation of viscous inflation, let us briefly describe the basicinflationary formulation and the relation to various observables. For convenience we reviewthe scenarios of cold and warm inflation separately.– 5 –
Cold InflationWe first start with the standard inflation realization, also called as “cold” inflation,in which a scalar field φ plays the role of the inflaton field. The Friedmann equationsare H = κ ρ = κ (cid:18)
12 ˙ φ + V (cid:19) , (2.1)¨ aa = − κ ρ + 3 p ) , (2.2)where ρ and p are respectively the energy density and pressure of the inflaton field,and V = V ( φ ) is the corresponding potential. In (2.1) we have used the fact that thescalar field can be viewed as a perfect fluid with ρ = 12 ˙ φ + V, (2.3a) p = 12 ˙ φ − V, (2.3b)and hence its equation-of-state (EoS) parameter reads p = wρ, (2.4a)with w = ˙ φ − V ˙ φ + V . (2.4b)The fluid interpolates between an invariant vacuum energy with w = − w = 1 and V = 0.The scalar-field equation of motion takes the simple form¨ φ + 3 H ˙ φ = − V ′ , (2.5)where V ′ = dV /dφ , which can be re-written as a continuity equation˙ ρ + 3 H ( ρ + p ) = 0 . (2.6)Finally, we can define the quantity N , i.e. the number of e-folds in the slow-roll era,as the logarithm of the ratio between the final value a f of the scale factor duringinflation and the initial value a ( N ) = a , namely N = ln (cid:16) a f a (cid:17) . (2.7)In inflationary theory it proves very convenient to define the so-called slow roll pa-rameters. One set of such parameters is defined via derivatives of the potential with– 6 –espect to the inflaton field. These “potential” slow roll parameters, conventionallycalled ε , η , ξ , are defined as [25] ε = 12 κ (cid:18) V ′ V (cid:19) , (2.8a) η = 1 κ V ′′ V , (2.8b) ξ = 1 κ V ′ V ′′′ V . (2.8c)Since these should be small during the slow-roll period, the potential V ( φ ) must havea flat region.One may also define the slow roll parameters in a different way, by taking the deriva-tives of the Hubble parameter with respect to the e-folding number (such an approachhas a more general application, since it can be also used in inflationary realizationsthat are driven from modified gravity, where a field and a potential are absent) [25].In particular, these horizon-flow [26–28] parameters ǫ n (with n a positive integer),are defined as ǫ n +1 ≡ d ln | ǫ n | dN , (2.9)with ǫ ≡ H ini /H and N the e-folding number, and H ini the Hubble parameter atthe beginning of inflation (inflation ends when ǫ = 1). Thus, the first three of themare calculated as ǫ ≡ − ˙ HH , (2.10) ǫ ≡ ¨ HH ˙ H − HH , (2.11) ǫ ≡ (cid:16) ¨ HH − H (cid:17) − " ... H − ¨ H ˙ H − H ˙ HH + 4 ˙ H H . (2.12)We now briefly review the formalism that is used to describe the temperature fluc-tuations in the Cosmic Microwave Background (CMB) radiation. The power spectraof scalar and tensor fluctuations are written as [29] P s = A s ( k ∗ ) (cid:18) kk ∗ (cid:19) n s − / α s ln( k/k ∗ ) , (2.13) P T = A T ( k ∗ ) (cid:18) kk ∗ (cid:19) n T +(1 / α T ln( k/k ∗ ) , (2.14)with A s = V π εM p = (cid:18) H π ˙ φ (cid:19) , (2.15) A T = 2 V π M p = ε (cid:18) H π ˙ φ (cid:19) . (2.16)– 7 –ere k is the wave number of the perturbation, and k ∗ is a reference scale usuallychosen as the wave number at horizon crossing (the pivot scale). Often one chooses k = ˙ a = aH , with a the scale factor. The quantities A s and A T are amplitudes atthe pivot scale, while n s and n T are called the spectral indices of scalar and tensorfluctuations. Moreover, − δ ns = n s − n T are called the tilts of the powerspectrum, since they describe deviations from the scale invariant spectrum where δ ns = n T = 0. The factors α s and α T are called running spectral indices and aredefined by α s = dn s d ln k , α T = dn T d ln k . (2.17)Finally, the tensor-to-scalar ratio r is defined as r = P T ( k ∗ ) P s ( k ∗ ) = A T A s . (2.18)Analysis of the observations from the Planck satellite give the result n s = 0 . ± .
006 [5, 30]. Furthermore, the observations give α s = − . ± . δ ns = 0 . n T = − . +1 . − . [31], while the BICEP/Planck data alone constrain thetensor tilt to be n T = 0 . +1 . − . .From the above equations we derive δ ns = − (cid:20) d ln P s ( k ) d ln k (cid:21) k = aH , n T = − (cid:20) d ln P T ( k ) d ln k (cid:21) k = aH , (2.19)where the quantities are evaluated at the horizon crossing ( k = k ∗ ), and as wementioned k = aH . Hence, we can finally extract the expressions that relate theinflationary observables, namely the tensor-to-scalar ratio, the scalar spectral index,the running of the scalar spectral index, and the tensor spectral index, with thepotential-related slow-roll parameters (2.10)-(2.12), which read as [25]: r ≈ ǫ, (2.20) δ ns ≈ ε − η, (2.21) α s ≈ ǫη − ǫ − ξ , (2.22) n T ≈ − ǫ. (2.23)Hence, a consistency relation between r and n T follows from Eqs. (2.13), (2.16) and(2.19), namely n T = − r . The preferred BICEP2/Planck value of r = 0 .
05 then gives n T = − . r ≈ ǫ , (2.24) δ ns ≈ ε + ε ) , (2.25) α s ≈ − ǫ ǫ − ǫ ǫ , (2.26) n T ≈ − ǫ . (2.27)– 8 –efinitely, in cases where both the potential slow-roll parameters and the horizon flowslow-roll parameters can be used, the final expressions for the observables coincide. • Warm InflationLet us now briefly review the scenario of “warm” inflation. Usually, one is concernedwith cold inflationary models described above, for which dissipation arising from thedecay of inflaton energy to radiation is omitted. Nevertheless, this contrasts thecharacteristic feature of the so-called warm inflation, where dissipation is included asan important factor, and inflaton energy dissipates into heat [32–35]. This implies inturn that the inflationary period lasts longer than it does in the cold case. Addition-ally, no reheating at the end of the inflationary era is needed, and the transition toradiation era becomes a smooth one.The main characteristic for the warm inflationary models is that the inflaton fieldenergy ρ φ is considered to depend on the temperature T [36], in a same way as theradiation density ρ r depends on T . The first Friedmann equation writes as H = κ ρ φ + ρ r ) , (2.28)and the continuity equations for the two fluid components read˙ ρ φ + 3 H ( ρ φ + p φ ) = − Γ ˙ φ , (2.29)˙ ρ r + 4 Hρ r = Γ ˙ φ , (2.30)where Γ is a dissipation coefficient describing the transfer of dark energy into radiationand it is in general time dependent. In warm inflation the inflaton energy is thedominating component, ρ φ ≫ ρ r , and H , φ and Γ vary slowly such that ¨ φ ≪ H ˙ φ ,˙ ρ r ≪ Hρ r and ˙ ρ r ≪ Γ ˙ φ . In the slow roll epoch, the radiation is produced by darkenergy dissipation. Thus 3 H = κρ φ = κV, (2.31)(3 H + Γ) ˙ φ = − V ′ . (2.32)Defining the so-called dissipative ratio by Q = Γ3 H , (2.33)we see that in the warm inflation era Eq. (2.30) yields ρ r = 34 Q ˙ φ . (2.34)During warm inflation T > H (in geometric units), and it turns out that the tensor-to-scalar ratio is modified in comparison to the cold inflation case, namely [37] r = H/T (1 + Q ) / r, (2.35)– 9 –nd thus this ratio is suppressed by a factor ( T /H )(1 + Q ) / compared to the coldinflationary case.The slow roll parameters in the present models are calculated at the beginning t = t i of the slow roll epoch. From the definition equation (2.8) we acquire ε = − (1 + Q ) ˙ HH . (2.36)Comparing with (2.10) we see that the first slow-roll parameter of the warm inflationscenario is modified with the factor 1 + Q relative to the corresponding cold inflationparameter. Furthermore, manipulation of the above equations then yields for theparameter η η = Q Q κ Γ ′ V ′ Γ V − QH ¨ φ ˙ φ − ˙ HH . (2.37)For convenience we introduce the quantity β = Γ ′ V ′ / ( κ Γ V ), and therefore this quan-tity appears in the expression for the relative rate of change of the radiation energydensity, namely ˙ ρ r Hρ r = −
11 + Q (cid:18) η − β − ε + 2 β − ε Q (cid:19) . (2.38)Introducing also ω = TH √ πQ √ πQ , (2.39)one can find that [38] δ ns = 11 + Q (cid:26) ε − (cid:18) η − β + β − ε Q (cid:19) + ω ω (cid:20) η + β − ε π ) Q (1 + Q )(3 + 4 πQ ) ( β − ε ) (cid:21)(cid:27) . (2.40)When warm inflation is strong, Q ≫ ω ≫
1, and thus δ ns = 22 Q (cid:20)
32 ( ε + β ) − η (cid:21) , (2.41)whereas when it is weak, Q ≪
1, and therefore δ ns = 2(3 ε − η ) − ω/
41 + ω (15 ǫ − η − β ) . (2.42)Finally, the cold inflationary case corresponds to the limit Q → T ≪ H , andthen ω → δ ns → ε − η ) . (2.43)Visinelli found the following expression for tensor-to-scalar ratio in warm inflation[38]: r = 16 ε (1 + Q ) (1 + ω ) . (2.44)– 10 –ence, in the limit of strong dissipative warm inflation we have r → Q ω ε ≪ ε, (2.45)while in the limit of cold inflation we re-obtain the standard result (2.20), namely r → ε . Thus, the warm inflation models with Q ≫ ω ≫ • Intermediate inflationIntermediate inflation scenario, introduced by Barrow in 1990 [39] (see also [40, 41]),also uses a scalar field. We consider the scale factor to take the form a ( t ) = exp[ A (ˆ t α − ′ , (2.46)with 0 < α <
1, and where A is a positive dimensionless constant, while a p refersto the Planck time (ˆ t = t/ √ κ, t p = √ κ ). The reason that these models are calledintermediate, is that the expansion is faster than the corresponding one in power-law inflation, and slower than an exponential inflation (the latter corresponding to α = 1). It follows from (2.46) that H = Aαt s ˆ t α − , ˙ H = Aαt p ( α − t α − , (2.47)and since ˙ H < α <
1, the Hubble parameter decreases with time. Insertingthese equations into Eqs. (2.1) and (2.2) we obtain ρ = 3 A α t p ˆ t α − , p = Aαt p ˆ t α − [2(1 − α ) − αAt αp ] . (2.48)Since ρ + p = ˙ φ we obtain by integration, using the initial condition φ (0) = 0, that φ ( t ) = 2 t p r A − αα ˆ t α , (2.49)while since V = ( ρ − p ) we acquire V ( t ) = Aαt p ˆ t α − [3 Aαt − αp − − α )] . (2.50)Hence, eliminating t between (2.49) and (2.50) we can express the potential as afunction of the inflaton field: V ( φ ) = Aαt p (cid:20) α A (1 − α ) (cid:21) α − α (cid:18) t p φ (cid:19) α − α " α − α ) (cid:18) t p φ (cid:19) − − α ) . (2.51)For this class of models the spectral parameters are most easily calculated from theHubble slow roll parameters ε H = − ˙ HH , η H = −
12 ¨ H ˙ HH . (2.52)– 11 –he optical parameters δ ns , n r and r can be expressed in terms of the Hubble slowroll parameters to lowest order as δ ns = 2(2 ε H − η H ) , n r = − ε H , r = 16 ε H . (2.53)This gives ε H = 1 − αAα ˆ t − α , η H = 2 − α − α ) ε H . (2.54)The slow roll parameter ε H can be expressed in terms of the inflaton field as ε H = 8 (cid:18) − αα (cid:19) (cid:18) M P φ (cid:19) . (2.55)In the intermediate inflation, the e-folding number becomes N = A (ˆ t αf − ˆ t αi ) , (2.56)where ˆ t i and ˆ t f are the initial and final point of time of the inflationary era, re-spectively. In these models the beginning of the inflationary era is defined by thecondition ε H (ˆ t i ) = 1, giving t i = (cid:18) − αAα (cid:19) /α t p . (2.57)Hence, the inflationary era ends at a point of time t f = (cid:18) N α + 1 − αAα (cid:19) /α t p . (2.58)The slow roll parameters are evaluated at this point of time, giving ε H = 1 − αN α + 1 − α , η H = 2 − α N α + 1 − α ) . (2.59)Inserting the above expressions into (2.53) we can thus write δ ns ≡ − n s = 2 − αN α + 1 − α , n r = 2( α − N α + 1 − α , r = 16(1 − α ) N α + 1 − α . (2.60)Note that the curvature spectrum is scale independent, corresponding to n s = 1, for α = 2 /
3. Furthermore, n s < α < /
3. Note that the expression for n s corrects an error of Ref. [40]. For these models the r, δ ns relation becomes r = 16(1 − α )2 − α δ ns . (2.61)The constant α can be expressed in terms of N and δ ns as α = 2 − δ ns N − δ ns ≈
23 +
N δ ns . (2.62)With the Planck values δ ns = 0 .
032 and N = 60 we get α = 0 . r = 0 . r is larger than permitted by Planck observations. However, the moregeneral models with non-canonical inflaton fields studied in Refs. [40] and [41], containan adjustable parameter in the expressions for the observables, leading to agreementwith observational data. Below we shall consider warm intermediate inflation models,which lead naturally to a suppression of the curvature perturbation, resulting to asmall value of r . – 12 – .2 Viscous Inflation Having described the basics of inflation, in this subsection we will see how inflation can berealized in the framework of viscous cosmology, that is if instead of a scalar field inflationis driven by a viscous fluid [42]. We start from the two Friedmann equations (1.12) and(1.13), namely H + ka = κρ H + 3 H = − κp . (2.64)Concerning the viscosity of the fluid we consider a subclass of (1.15) and parametrize theequation of state as p = − ρ + Aρ β + ζ ( H ) , (2.65)with A , β constants, and ζ ( H ) the bulk viscosity considered with a dynamical nature ingeneral, i.e. being a function of the Hubble parameter. As a specific example we consider ζ ( H ) = ¯ ζH γ , (2.66)with ¯ ζ , γ parameters.From the Friedmann equation (2.63) and for an expanding flat universe ( H > , k = 0),we acquire H = r κρ . (2.67)Therefore, ζ ( H ) can be expressed in terms of ρ , i.e ζ ( H ) = ζ ( H ( ρ )). Thus, comparing thegeneral expression for the EoS of a fluid, namely p = − ρ + f ( ρ ) , (2.68)with (2.65) and (2.66), we deduce that f ( ρ ) = Aρ β + ζ ( H ( ρ )) = Aρ β + ¯ ζ (cid:18)r κ (cid:19) γ ρ γ/ . (2.69)We mention that f ( ρ ) is expressed as a series of powers in ρ due to the imposed assumptionthat ζ ( H ) is a power of H . Hence, this allows us to find analytical solutions and examinethe behavior of various inflationary observables.Since in fluid inflation we do not have a potential, it proves convenient to use theHubble slow-roll parameters. Inserting the Hubble function from (2.67) into (2.10)-(2.12)and then into the inflationary observables (2.24)-(2.27), after some algebra one can expressthe tilt, the tensor-to-scalar ratio and the running spectral index as [42]( δ ns , r, α s ) ≈ (6 f ( ρ ) ρ ( N ) , f ( ρ ) ρ ( N ) , − (cid:18) f ( ρ ) ρ ( N ) (cid:19) ) (2.70)= (6 ( w ( N ) + 1) ,
24 ( w ( N ) + 1) , − w ( N ) + 1) ) , (2.71)where we have also used that f ( ρ ) /ρ ( N ) = w ( N )+1. In these expressions all quantities maybe considered as functions of the e-folding number N . Hence, if we choose f ( ρ ) /ρ ( N ) =– 13 – . × − , we obtain w = − . n s , r, α s ) = (0 . , . , − . × − ).These results are consistent with the Planck data, namely n s = 0 . ± .
006 (68% CL), r < .
11 (95% CL), and α s = − . ± .
007 (68% CL), [5, 43].Let us now use the required scalar spectral index in order to reconstruct the EoS ofthe fluid through a corresponding effective potential, following [42, 44]. In order to achievethis, we first express the Friedmann equations using derivatives in terms of the e-foldingnumber N as 3 κ [ H ( N )] = ρ , (2.72) − κ H ( N ) H ′ ( N ) = ρ + p , (2.73)and similarly for the slow-roll parameters (2.8a)-(2.8c), namely δ ns = − ddN (cid:20) ln (cid:18) V ( N ) dV ( N ) dN (cid:19)(cid:21) r = 8 V ( N ) dV ( N ) dNα s = − d dN (cid:20) ln (cid:18) V ( N ) dV ( N ) dN (cid:19)(cid:21) . (2.74)Hence, one can use these quantities in order to reconstruct the equation-of-state of thecorresponding fluid. In particular, having the δ ns ( N ) as a function of N , using (2.74)we can solve for V ( N ), which will be the effective potential in an equivalent scalar-fielddescription. Then the Hubble function is related to V ( N ) through (2.72), and thus weobtain H = H ( N ). Finally, using (2.73) we can reconstruct f ( ρ ) through (2.68).Let us give a specific example of the above method, in the case where [42] δ ns = 2 N , (2.75)which is valid in Starobinsky inflation [45], and it can be satisfied in chaotic inflation [20],in new Higgs inflation [46, 47], and in models of α -attractors [48, 49], too. Combining(2.75) and (2.74) gives V ( N ) = 1( C /N ) + C , (2.76)where C ( >
0) and C are constants. Hence, using (2.76) and (2.74) we acquire r = 8 N [1 + ( C /C ) N ] = 4 δ ns C C δ ns = 4 δ ns δ ns + 2 C /C , (2.77)and thus C C = (cid:18) δ ns r − (cid:19) δ ns . (2.78)If δ ns = 0 . , r = 0 .
05 one gets C /C ≈ . α s = − N . (2.79)– 14 –hus, inserting a reasonable value N = 60 we obtain α s = − . × − , in agreementwith Planck analysis.In the case of a fluid model one uses the equation-of-state parameter from (2.68) insteadof the scalar potential. Hence, one can have (3 /κ ) ( H ( N )) = ρ ( N ) ≈ V ( N ), since thelast approximation arises from the slow-roll condition that the kinetic energy is negligiblecomparing to the potential one. Therefore, using (2.76) we obtain H ( N ) ≈ r κ C /N ) + C ] , (2.80)with ( C /N ) + C >
0. Additionally, inserting ρ ≈ V into (2.76) results to N ≈ C ρ − C ρ . (2.81)Thus, inserting (2.80) into (2.72) and (2.73) gives p = − ρ − κ H ( N ) H ′ ( N ) ≈ − ρ − C N κ H . (2.82)Finally, comparing (2.68) with (2.82) leads to f ( ρ ) ≈ − C N κ H ≈ − C (cid:0) − C ρ + C ρ (cid:1) , (2.83)where we have also used (2.72) and (2.81).We now focus on fluid inflationary models with n s and r in agreement with observa-tions. From (2.68) and (2.69) we obtain p = − ρ + f ( ρ ) = − ρ + Aρ β + ¯ ζ (cid:18)r κ (cid:19) γ ρ γ/ . (2.84)Therefore, we suitably choose the model parameters A , ¯ ζ , β , and γ , in order for relation(2.75) to be satisfied. For convenience we will focus in the regimes | C ρ | ≫ | C ρ | ≪ • Case I: | C ρ | ≫ f ( ρ ) ≈ C C ρ − C C ρ , (2.85)with C < N from (2.81). From (2.84) and (2.85) weacquire w = pρ ≈ − − (cid:18) − C C (cid:19) + 13 (cid:18) − C C (cid:19) ( − C ρ ) ≈ − N ( − − C ρ ) , (2.86)where we have also used that ( − C ) /C ≈ /N . For instance, if | C ρ | = O (10),( − C ) /C ≈ /N , and N &
60, relation (2.86) leads to w ≈ −
1, and hence the deSitter inflation can be realized, with a scale-factor of the form a ( t ) = a i exp [ H inf ( t − t i )] . (2.87)– 15 –t should be noted that for ( − C ) /C < /N , relation (2.77) for N &
73 provides atensor-to-scalar ratio r >
1, in disagreement with observations.Comparing (2.85) and (2.69) we deduce that we obtain equivalence for two combina-tions of parameters:Model (A) : A = 2 C C , ¯ ζ = − C C κ , β = 1 , γ = 4 , (2.88)and Model (B) : A = − C C , ¯ ζ = 2 C C κ , β = 2 , γ = 2 . (2.89)Hence, the corresponding fluid equation of state can be reconstructed. • Case (II): | C ρ | ≪ f ( ρ ) ≈ − C + 2 C C ρ . (2.90)Thus, (2.81) with | C ρ | ≪ C ρ ≈ N ≫ | C | /C ≪
1. Hence,(2.84) and (2.85), give w = pρ ≈ − −
13 1 C ρ + 23 (cid:18) C C (cid:19) ≈ − (cid:18) − N + 2 C C (cid:19) , (2.91)where we have used that C ρ ≈ N . Similarly to the previous subcase, (2.91) with1 /N ≪ | C | /C ≪
1, leads to w ≈ −
1, i.e to the realization of the de Sitterinflation, with a scale factor given by (2.87). Moreover, for C > C /C . /N ,and for N &
60, relation (2.77) gives r < .
11 in agreement with Planck results. Onthe other hand, for C < | C | /C < /N , we need to have N &
73 in order toget r < .
11, similarly to the previous Case (I). Finally, comparing (2.90) and (2.69)we deduce that we obtain equivalence for two combinations of parameters:Model (C) : A = − C , ¯ ζ = 2 C C κ , β = 0 , γ = 2 , (2.92)and Model (D) : A = 2 C C , ¯ ζ = − C , β = 1 , γ = 0 . (2.93)Having analyzed the basic features of inflationary realization from a viscous fluid, letus examine the crucial issue of obtaining a graceful exit and the subsequent entrance tothe reheating stage [42]. In particular, we will investigate the instability of the de Sittersolution characterized by H = H inf = const. under perturbations. One starts by perturbingthe Hubble function as [50] H = H inf + H inf δ ( t ) , (2.94)– 16 –here | δ ( t ) | ≪
1. Thus, the second Friedmann equation writes as a differential equation interms of the cosmic time t , namely¨ H − κ " βA (cid:18) κ (cid:19) β H β − + (cid:16) β + γ (cid:17) A ¯ ζ (cid:18) κ (cid:19) β H β + γ − + γ ζ H γ − = 0 . (2.95)Without loss of generality we choose δ ( t ) ≡ e λt , (2.96)with λ a constant, and therefore a positive λ would correspond to an unstable de Sittersolution. This instability implies that the universe can exit from inflation. On the otherhand, a stable inflationary solution is just an eternal inflation.Inserting (2.94) and (2.96) into (2.95), and keeping terms up to first order in δ ( t ), weobtain λ − κ H Q = 0 , (2.97)with Q ≡ β (4 β − A (cid:18) κ (cid:19) β H β inf + (cid:16) β + γ (cid:17) (2 β + γ − A ¯ ζ (cid:18) κ (cid:19) β H β + γ inf + γ γ −
1) ¯ ζ H γ inf . (2.98)The solutions of (2.97) read as λ = λ ± ≡ ± √ κH inf √Q , (2.99)and therefore if Q > λ = λ + >
0, which implies the realization of asuccessful inflationary exit.Let us now check whether the four fluid models described in (2.88), (2.89), (2.92), and(2.93) above, can give rise to a graceful exit, i.e whether they can give a positive Q in(2.98). Substituting the corresponding values of A , ¯ ζ , β , and γ into (2.99), we obtain theexpressions of Q as [42]:Model (A) : Q = 2 (cid:18) C C (cid:19) (cid:18) H inf √ κ (cid:19) " − C (cid:18) H inf √ κ (cid:19) + 63 C (cid:18) H inf √ κ (cid:19) > , (2.100)Model (B) : Q = 6 (cid:18) C C (cid:19) (cid:18) H inf √ κ (cid:19) " − C (cid:18) H inf √ κ (cid:19) + 21 C (cid:18) H inf √ κ (cid:19) > . (2.101)Model (C) : Q = (cid:18) C C (cid:19) (cid:18) H inf √ κ (cid:19) " − C + 12 (cid:18) H inf √ κ (cid:19) , (2.102)Model (D) : Q = 2 (cid:18) C C (cid:19) (cid:18) H inf √ κ (cid:19) " (cid:18) H inf √ κ (cid:19) − C . (2.103)– 17 – able 1 . The equation-of-state parameter of the reconstructed viscous inflationary models of (2.88),(2.89), (2.92), and (2.93), along with the conditions for a graceful exit. The parameter C is alwayspositive, while | C ρ | ≫ C < | C ρ | ≪ p = − ρ + [2 C / (3 C )] ρ − (cid:2) C / (cid:0) C κ (cid:1)(cid:3) H No condition(i) (b) p = − ρ − (cid:2) C / (3 C ) (cid:3) ρ + [2 C / ( C κ )] H No condition(ii) (c) p = − ρ − [1 / (3 C )] + [2 C / ( C κ )] H C < C > (1 /
36) ( √ κ/H inf ) (ii) (d) p = − ρ + [2 C / (3 C )] ρ − [1 / (3 C )] C < C > (1 /
18) ( √ κ/H inf ) Hence, Models (A) and (B) have always Q >
0. On the other hand, Models (C) and (D)have Q > C <
0, while for C > Q > C > (cid:18) √ κH inf (cid:19) for Model (C) , (2.104) C > (cid:18) √ κH inf (cid:19) for Model (D) . (2.105)In summary, we can see that the models of viscous fluid inflation can have a graceful exitwithout any tuning. In Table 1 we summarize the obtained results. From the correspondingequation-of-state parameters, and comparing with (2.65), we can immediately see the terminspired by the bulk viscosity. Finally, as we described in detail above, in these modelsthe inflationary observables are in agreement with observations. In particular, the spectralindex from (2.75) is n s = 0 .
967 for N = 60. The running of the spectral index is given by α s = − /N in (2.79), leading to α s = − . × − .We close this subsection by studying the singular inflation in the above viscous fluidmodel. The finite-time singularities are classified into four types [51], and hence one can seethat Type IV singularity can be applied in singular inflation since there are no divergencesin the scale factor and in the the effective (i.e. total) energy density and pressure. Inparticular, in Type IV singularity, as t → t s , with t s the singularity time, we have a → a s , ρ → | p | →
0. Here, a s is the value of a at t = t s . Nevertheless, the higher derivativesof the Hubble function diverge.Let us consider the above viscous fluid inflationary realization, assuming that H = H inf + ¯ H ( t s − t ) q , q > , (2.106) a = ¯ a exp (cid:20) H inf t − ¯ Hq + 1 ( t s − t ) q +1 (cid:21) , (2.107)with ¯ H , q , and ¯ a the model parameters. From the two Friedmann equations (2.63),(2.64)we straightforwardly acquire ρ = 3 H κ , p = − H + 3 H κ . (2.108)– 18 –herefore, a Type IV singularity appears at t = t s , since (2.107) and (2.108) imply that as t → t s the quantities a , ρ , and p asymptotically approach finite values, while from (2.106)we deduce that higher derivatives of H diverge. From (2.106) and (2.108) we find thefollowing equation-of-state parameter of the cosmic fluid: p = − ρ + f ( ρ ) , (2.109)with f ( ρ ) = 2 q ¯ H /q κ (cid:18)r κρ − H inf (cid:19) ( q − /q . (2.110)In the case where H inf / p κρ/ H inf /H ≪ f ( ρ ) ≈ ( q − / (2 q ) ¯ H /q κ ( q +1) / q " ρ ( q − / (2 q ) − √ q − q H inf √ κ ρ − / (2 q ) . (2.111)Thus, one can clearly see from (2.111) that the function f ( ρ ) includes a linear combinationof two powers of ρ , as in (2.69) and (2.85). Hence, indeed this scenario can be realized bythe viscous fluid models reconstructed above.From (2.111), using (2.108), we find f ( ρ ) ρ ≈ = 2 q (cid:18) ¯ HH q +1 (cid:19) /q (cid:20) − ( q − q H inf H (cid:21) , (2.112)and therefore for ¯ H/H q +1 ≪ f ( ρ ) /ρ ≪
1. Thus, n s , r , and α s can be approxi-mately given by (2.70) and be in agreement with observations, which act as an additionaladvantage of singular inflation.We now examine the limit ¯ ζ = 0 in (2.65), in which the fluid equation of state in(2.65) becomes p = − ρ + Aρ β . In this limit from (2.69) we deduce that f ( ρ ) = Aρ β , i.e f ( ρ ) has only one power of ρ . However, from (2.111) and (2.112) we see that f ( ρ ) consistsof two ρ powers. Thus, f ( ρ ) can be given by (2.111) and (2.112) only if the singularinflation is realized. Hence, for a non-viscous fluid, i.e. for a fluid without the ζ ( H )-termin (2.65), singular inflation cannot be realized. From this feature we can see the importanceof the viscous term, and its significant effect on the dynamics of the early universe. Thisimportant issue will be studied in more detail in the following subsection.In summary, in the present subsection we studied the realization of inflation in a fluidframework, whose equation-of-state parameter has an additional term corresponding tobulk viscosity. Firstly, we saw that the obtained inflationary observables, namely n s , r and α s , are in agreement with Planck data. Secondly, we presented a reconstruction procedureof the fluid’s equation of state, when a specific n s is given, while the tensor-to-scalar ratio isstill in agreement with observations. Thirdly, we analyzed the stability of the inflationary,de Sitter phase, showing that a graceful exit and the pass to the subsequent thermal historyof the universe is obtained without fine tuning. Finally, we investigated the realization ofsingular inflation, corresponding to Type IV singularity, in the present viscous fluid model.Hence, viscous fluid inflation can be a candidate for the description of early universe.– 19 – .3 Viscous warm and intermediate inflation In this subsection we show how the viscous cold inflationary models considered above canbe generalized to the warm case. These kind of models are most likely more physicalthan the idealized cold ones, since they take into account the presence of massive particlesproduced from the decaying inflaton field. Moreover, an important advantage of warmscenarios is that they give rise to a much smaller tensor-to-scalar ratio than the coldmodels, and hence are easier to be in agreement with the Planck data. The presence ofmassive particles provides a natural way to explain why the cosmic fluid can be associatedwith a bulk viscosity.We abstain from using the simple equation of state p = (1 / ρ holding for radiation,and we assume instead the more general form p = wρ , where w is constant. For convenience,one can introduce the form p = ( γ − ρ with γ = 1 + w . The effective pressure becomes p eff = p + p ζ , where p ζ = − Hζ (2.113)is the viscous part of the pressure and ζ the bulk viscosity. In this case Eq. (2.30)generalizes to [52] ˙ ρ + 3 H ( ρ + p − ζH ) = Γ ˙ φ . (2.114)The usual condition about quasi-stationarity implies ˙ ρ ≪ H ( γρ − ζH ) and ˙ ρ ≪ Γ ˙ φ .We will henceforth follow the formalism of [52] for the strong dissipative case, namelyfor Q ≫ φ ) = κ / V ( φ ) , ζ = ζ ρ, (2.115)where the proportionality of ζ to ρ is a frequently used assumption (a similar analysis canbe performed for the case where Γ and ζ are assumed constants [52, 53], however we willnot go into further details and focus on the general case). From Eqs. (2.31) and (2.33) wethen have Q = √ κH . Focusing on the strong dissipative case Q ≫
1, manipulation of theequations gives the following expression for the inflaton field as a function of time: φ ( t ) = 2 κ − / p − α ) t. (2.116)This equation, predicting φ ( t ) to increase with time, is seen to be different from the corre-sponding Eq. (2.49) for cold intermediate inflation.Taking into account the expression (2.47) for H we can express the potential as afunction of time: V ( t ) = 3 A α κ − ( t/ √ κ ) α − , (2.117)which can alternatively be represented as a function of φ as V ( φ ) = 3 A α κ − " √ κφ p − α ) α − . (2.118)– 20 –ince ρ = V ˙ φ H ( γ − ζ H ) , (2.119)we see that it is necessary for the constant ζ in (2.115) to satisfy the condition ζ < γ/ H in order to make ρ positive. The density varies with time as ρ ( t ) = 2 Aα (1 − α ) κ − / ( t/ √ κ ) α − γ √ κ − ζ Aα ( t/ √ κ ) α − , (2.120)while when considered as a function of the inflaton field it reads ρ ( φ ) = 2 Aα (1 − α ) κ − / hp κ (1 − α ) φ/ i α − γ √ κ − ζ Aα hp κ (1 − α ) φ/ i α − . (2.121)Additionally, the number of e-folds becomes in this case N = √ κ √ Z φφ f V / V ′ dφ = A (1 − α ) α − A " √ κφ p − α ) α , (2.122)where φ f is the inflaton field at the end of the slow-roll epoch. Finally, the slow-rollparameters in the strong dissipative epoch ( Q ≫
1) become ε = 12 Q (cid:18) V ′ V (cid:19) , η = 1 Q " V ′′ V − (cid:18) V ′ V (cid:19) , (2.123)giving in turn for the spectral parameter δ ns δ ns = 3 α − − α ε = 3 α − αA " √ κφ p − α ) − α . (2.124)Hence, the Harrison-Zel’dovich spectrum (independent of scale) corresponds to α = 2 / In the end of subsection 2.2 we presented a brief discussion on the possibility to realizesingular inflation in the framework of viscous cosmology. Since this is an important issue,in this subsection we investigate it in detail following [54], considering more general viscousequation of states. We consider an inhomogeneous viscous equation-of-state parameter ofthe form p = − ρ − f ( ρ ) + G ( H ) , (2.125)which is a subclass of the general ansatz (1.15). Thus, when the function G ( H ) becomeszero we re-obtain the homogeneous case. An even more general equation of state would beto consider p = f ( ρ, H ) . (2.126)In the following we desire to investigate the realization of type IV singularity in inflationdriven by a fluid with the above EoS’s. – 21 –s we mentioned earlier, a type IV singularity occurs at t → t s , if the scale factor andthe effective energy density and pressure remain finite, but the higher derivatives of theHubble function diverge. A general form of the Hubble function which can describe a TypeIV singularity reads as H ( t ) = f ( t ) + f ( t ) ( t s − t ) α , (2.127)with f ( t ), f ( t ) being arbitrary differentiable functions. Hence the type IV singularityoccurs when α >
1, and without loss of generality we can consider it to take the form α = n m + 1 , (2.128)with n , m positive integers.Let us start from a simple example of type IV singularity realization, namely weconsider f ( t ) = 0 and f ( t ) = f , with f a positive parameter. In this case the twoFriedmann equations, namely ρ = κ H and p = − κ (cid:16) H + 2 ˙ H (cid:17) , become ρ = 3 f κ ( t s − t ) α (2.129) p = − κ h f ( t s − t ) α + 2 αf ( t s − t ) α − i , (2.130)and hence eliminating t s − t we get the result p = − ρ − · − α − α κ − α +12 α f /α ρ α − α . (2.131)Hence, a viscous fluid with this equation of state can generate the Hubble function (2.127)and hence the type IV singularity. Defining ˜ α ≡ α − α , a type IV singularity will occur if0 < ˜ α < (or equivalently, α > G ( H ) = 0 and f ( ρ ) = − · − α − α κ − α +12 α f /α ρ α − α , (2.132)or as an inhomogeneous one, of the form (2.125) with f ( ρ ) = 0 and G ( H ) = − ακ f /α H α − α (2.133)(since ρ = κ H ).Let us now consider a more general Hubble function inside the class (2.127), namely H ( t ) = f ( t − t ) α + c ( t − t ) β , (2.134)where c , f are constants, and α , β >
1. Thus, two type IV singularities appear at t = t and t = t . We choose t to correspond to the inflation end and t to lie at late times.In order to simplify the expressions, we focus our analysis in the vicinity of the type IVsingularity. In this region, inserting (2.134) into the two Friedmann equations leads to ρ ≈ c ( t − t ) β κ (2.135) p ≈ − c ( t − t ) β κ − c ( t − t ) − β βκ , (2.136)– 22 –nd therefore the equation of state reads p = − ρ − c βκ (cid:18) ρκ c (cid:19) β − β . (2.137)Interestingly enough, we observe that the late-time type IV singularity is related to theearly-time type IV singularity and the corresponding equation-of-state paramater. In thesame lines, the early-time singularity is related to the effective equation of state that givesrise to the late time one.One can proceed in similar lines, and study the scenario where H ( t ) = f p t + t + f t ( − t + t ) α t + t + f ( − t + t ) β . (2.138)In this case, in the vicinity of the early-time singularity at t we obtain [54] ρ ≃ f (cid:0) t + t (cid:1) κ + 6 f f ( − t + t ) β p t + t κ + 3 f ( − t + t ) β κ ,p ≃ f t (cid:0) t + t (cid:1) / κ − f (cid:0) t + t (cid:1) κ − f f ( − t + t ) β p t + t κ − f ( − t + t ) β κ + 2 f ( − t + t ) − β βκ , (2.139)where these relations are again determined by the late-time singularity.Finally, one can study the scenario where H ( t ) = f + c ( t − t ) α ( t − t ) β , (2.140)which is reproduced by ρ = 3 f κ + 6 cf ( − t + t ) α ( − t + t ) β κ + 3 f ( − t + t ) α ( − t + t ) β κ ,p = − f κ − cf ( − t + t ) α ( − t + t ) β κ − f ( − t + t ) α ( − t + t ) β κ + 2 f ( − t + t ) − α ( − t + t ) β ακ + 2 f ( − t + t ) α ( − t + t ) − β βκ . (2.141)At both type IV singularities at t and t , the effective energy density and pressure become ρ = 3 f κ , p = − f κ , (2.142)and thus the corresponding equation-of-state parameter becomes − e -foldings N , namely [54] ρ = 3 κ ( H ( N )) (2.143) p ( N ) + ρ ( N ) = − H ( N ) H ′ ( N ) κ , (2.144)– 23 –here H ′ ( N ) = d H/ d N . Assuming that the equation of state is given by the generalansatz: p ( N ) = − ρ mat ( N ) + ˜ f ( ρ ( N )) , (2.145)then (2.144) gives ˜ f ( ρ ( N )) = − H ( N ) H ′ ( N ) κ . (2.146)Since the usual conservation equation is valid, namely ρ ′ ( N ) + 3 H ( N ) ( ρ ( N ) + p ( N )) = 0 , (2.147)with ρ ′ ( N ) = d ˜ f ( ρ ( N )) / d N , using (2.146) we find ρ ′ ( N ) + 3 ˜ f ( ρ ( N )) = 0 . (2.148)Finally, inserting (2.148) into (2.145) we acquire2 κ (cid:2) ( H ′ ( N )) + H ( N ) + H ′′ ( N ) (cid:3) = 3 ˜ f ′ ( ρ ) f ( ρ ) , (2.149)with ˜ f ′ ( ρ ( N ) ≡ d ˜ f ( ρ ) / d ρ .Now, for a given H ( t ), the slow-roll parameters ǫ , η and ξ write as [54] ǫ = − H H HH + ¨ HH ! HH ! − ,η = − HH ! − HH + ˙ H H − ¨ HH − ˙ H H + ˙ H ¨ HH − ¨ H H + 3 ¨ HH ˙ H + ... HH ˙ H ! ,ξ = 14 HH + ¨ HH ! HH ! − HH ˙ H + 3... H ˙ H + 2... HH ˙ H + 4 ¨ H H ˙ H − ¨ H ... HH ˙ H − H ˙ H + ¨ H H ˙ H + .... HH ˙ H ! . (2.150)Hence, if H ( t ) is given by (2.127), and if α >
1, i.e when a type IV singularity is obtained– 24 –t t ∼ t s , the slow-roll parameters at the vicinity of the singularity become ǫ ∼ − f ( t s ) f ( t s ) h f ( t s ) f ( t s ) f ( t s ) + ¨ f ( t s ) f ( t s ) i h ˙ f ( t s ) f ( t s ) i − , when α > − f ( t s ) f ( t s ) f ( t s ) α ( α −
1) ( t s − t ) α − h ˙ f ( t s ) f ( t s ) i − , when 2 > α > ,η ∼ − h ˙ f ( t s ) f ( t s ) i − (cid:20) f ( t s ) f ( t s ) + ˙ f ( t s ) f ( t s ) − ¨ f ( t s ) f ( t s ) − ˙ f ( t s ) f ( t s ) + ˙ f ( t s ) ¨ f ( t s ) f ( t s ) − ¨ f ( t s ) f ( t s ) + f ( t s ) f ( t s ) ˙ f ( t s ) + ... f ( t s ) f ( t s ) ˙ f ( t s ) (cid:21) , when α > − h ˙ f ( t s ) f ( t s ) i − f α ( α − α − f ( t s ) ˙ f ( t s ) ( t s − t ) α − , when 3 > α > ,ξ ∼ h f ( t s ) f ( t s ) + ¨ f ( t s ) f ( t s ) i h ˙ f ( t s ) f ( t s ) i − (cid:20) f ( t s ) f ( t s ) ˙ f ( t s ) + ... f ( t s )˙ f ( t s ) + ... f ( t s ) f ( t s ) ˙ f ( t s ) + f ( t s ) f ( t s ) ˙ f ( t s ) − ¨ f ( t s ) ... f ( t s ) f ( t s ) ˙ f ( t s ) − f ( t s ) ˙ f ( t s ) + ¨ f ( t s ) f ( t s ) ˙ f ( t s ) + .... f ( t s ) f ( t s ) ˙ f ( t s ) (cid:21) , when α > h f ( t s ) f ( t s ) + ¨ f ( t s ) f ( t s ) i h ˙ f ( t s ) f ( t s ) i − f ( t s ) α ( α − α − α − f ( t s ) ˙ f ( t s ) ( t s − t ) α − , when 4 > α > h ˙ f ( t s ) f ( t s ) i − f ( t s ) α ( α − ( α − α − f ( t s ) ˙ f ( t s ) ( t s − t ) α − , when 2 > α > . (2.151)Therefore, we deduce that if f ( t ) is a smooth function then the slow-roll parameter ǫ diverges when 2 > α >
1, whereas it remains regular for α >
2. Moreover, η diverges for3 > α >
1. Finally, ξ diverges when 2 > α > > α >
2. In summary, when α > H ( t ) = f ( t − t s ) α . Thus, the slow-roll parameters become [54] ǫ = f ( t − t s ) − α α ( − t − t s + α ) f ( t − t s ) α + α ] , (2.152) η = (cid:8) f (cid:2) f ( t − t s ) α + α (cid:3)(cid:9) − n ( t − t s ) − − α (cid:2) − t s α + 2 α − α − f ( t − t s ) α ( − α ) + f ( t − t s ) α α (1 − α + 2 α ) (cid:3) +( t − t s ) − − α (cid:0) − t + 8 tt s − t s + 4 t α − tt s α + 4 t s α − t α + 2 tt s α (cid:1) o , (2.153) ξ = ( t − t s ) − − α ( − α )( − t − t s + α )4 f [3 f ( t − t s ) α + α ] × (cid:2) t − t s ) ( α − + 3 f ( t − t s ) α ( α − ( α − α + 3 f ( t − t s ) α (1 + 2 α ) (cid:3) . (2.154)Hence, we immediately observe that the slow-roll parameters exhibit singularities at t = t s ,as mentioned above. However, such singularities in the slow-roll parameters can be viewedas rather unwanted features. – 25 –e close this subsection by making a comparison with observations. In order to achievethis it proves convenient to express the various quantities in terms of the number of e -foldings N . In particular, for a given H ( N ) the slow-roll parameters read as [54] ǫ = − H ( N )4 H ′ ( N ) H ′ ( N ) H ( N ) + H ′′ ( N ) H ( φ ) + h H ′ ( N ) H ( N ) i H ′ ( N ) H ( N ) (2.155) η = − (cid:20) H ′ ( N ) H ( N ) (cid:21) − ( H ′ ( N ) H ( N ) + 3 H ′′ ( N ) H ( N ) + 12 (cid:20) H ′ ( N ) H ( N ) (cid:21) − (cid:20) H ′′ ( N ) H ′ ( N ) (cid:21) + 3 H ′′ ( N ) H ′ ( N ) + H ′′′ ( N ) H ′ ( N ) ) (2.156) ξ = H ′ ( N ) H ( N ) + H ′′ ( N ) H ( N ) + h H ′ ( N ) H ( N ) i h H ′ ( N ) H ( N ) i (cid:26) H ( N ) H ′′′ ( N ) H ′ ( N ) + 9 H ′ ( N ) H ( N ) − H ( N ) H ′′ ( N ) H ′′′ ( N ) H ′ ( N ) + 4 H ′′ ( N ) H ( N ) + H ( N ) H ′′ ( N ) H ′ ( N ) + 5 H ′′′ ( N ) H ′ ( N ) − H ( N ) H ′′ ( N ) H ′ ( N ) − (cid:20) H ′′ ( N ) H ′ ( N ) (cid:21) + 15 H ′′ ( N ) H ′ ( N ) + H ( N ) H ′′′′ ( N ) H ′ ( N ) ) . (2.157) Therefore, we can use (2.143) and (2.144) in order to calculate the usual inflationaryobservables, namely the spectral index n s , the tensor-to-scalar ratio r and the runningspectral index a s as [54] n s − − ρ ( N ) ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) − ρ ( N ) − ˜ f ( ρ ( N )) ! + 6 ρ ( N )2 ρ ( N ) − ˜ f ( ρ ( N )) ( ˜ f ( ρ ( N )) ρ ( N )+ 12 (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + ˜ f ′ ( ρ ( N )) −
52 ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) + (cid:18) f ( ρ ) ρ ( N ) (cid:19) + 13 ρ ′ ( N )˜ f ( ρ ( N )) × (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + ˜ f ( ρ ( N )) ˜ f ′′ ( ρ ( N )) − f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) + ˜ f ( ρ ( N )) ρ ( N ) ! , (2.158) r =24 ρ ( N ) ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) − ρ ( N ) − ˜ f ( ρ ( N )) ! , (2.159) α s = ρ ( N ) ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) − ρ ( N ) − ˜ f ( ρ ( N )) ! (cid:20) ρ ( N )2 ρ ( N ) − ˜ f ( ρ ( N )) J − ρ ( N ) ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) − ρ ( N ) − ˜ f ( ρ ( N )) ! − f ′ ( ρ ( N )) − J , (2.160)– 26 –here J ≡ ˜ f ( ρ ( N )) ρ ( N ) + 12 (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + ˜ f ′ ( ρ ( N )) −
52 ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) + ˜ f ( ρ ( N )) ρ ( N ) ! + 13 ρ ′ ( N )˜ f ( ρ ( N )) × (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + ˜ f ( ρ ( N )) ˜ f ′′ ( ρ ( N )) − f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) + ˜ f ( ρ ( N )) ρ ( N ) ! , (2.161) and J ≡
452 ˜ f ( ρ ( N )) ρ ( N ) ˜ f ′ ( ρ ( N )) −
12 ˜ f ( ρ ( N )) ρ ( N ) ! + 18 ˜ f ( ρ ( N )) ρ ( N ) ! − ˜ f ′ ( ρ ( N )) −
12 ˜ f ( ρ ( N )) ρ ( N ) ! + ˜ f ′ ( ρ ( N )) −
12 ˜ f ( ρ ( N )) ρ ( N ) ! − ˜ f ′ ( ρ ( N )) −
12 ˜ f ( ρ ( N )) ρ ( N ) ! −
45 ˜ f ′ ( ρ ( N )) + 9 ˜ f ( ρ ( N )) ρ ( N )+ 3 f ′ ( ρ ( N )) − f ( ρ ( N )) ρ ( N ) + 2 ! − ˜ f ′ ( ρ ( N )) −
12 ˜ f ( ρ ( N )) ρ ( N ) ! + ˜ f ( ρ ( N )) ρ ( N ) ! − ρ ′ ( N ) ρ ( N ) × (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + ˜ f ( ρ ( N )) ˜ f ′′ ( ρ ( N )) − f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) + ˜ f ( ρ ( N )) ρ ( N ) ! + ˜ f ( ρ ( N )) ρ ( N ) ! − ( − ˜ f ( ρ ( N )) ρ ( N ) ! (cid:18) ρ ′ ( N ) ρ ( N ) (cid:19) (cid:20) (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + 2 ˜ f ( ρ ( N )) ˜ f ′′ ( ρ ( N )) −
112 ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) + 52 ˜ f ( ρ ( N )) ρ ( N ) ! + (cid:18) ρ ′′ ( N ) ρ ( N ) (cid:19) (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) + ˜ f ( ρ ( N )) ˜ f ′′ ( ρ ( N )) − f ( ρ ) ˜ f ′ ( ρ ) ρ ( N ) + ˜ f ( ρ ) ρ ! + (cid:18) ρ ′ ( N ) ρ ( N ) (cid:19) (cid:20)(cid:16) f ′ ( ρ ( N )) ˜ f ′′ ( ρ ( N )) + ˜ f ( ρ ( N )) ˜ f ′′′ ( ρ ( N )) (cid:17) ρ ( N ) − (cid:16) ˜ f ′ ( ρ ( N )) (cid:17) − f ( ρ ( N )) ˜ f ′′ ( ρ ( N )) + 6 ˜ f ( ρ ( N )) ˜ f ′ ( ρ ( N )) ρ ( N ) − ˜ f ( ρ ( N )) ρ ( N ) ! . (2.162) In order to obtain a qualitative picture of the above observables we consider an equationof state of the form p = − ρ + f ( ρ ), with f ( ρ ) = Aρ α . Since according to (2.143) and (2.144)the scale factor reads as a ( t ) = a e ρ − α − α ) A , (2.163)we can express ρ as a function of N as ρ ( N ) = [3(1 − α ) A ] − α N − α . (2.164)Additionally, since a Type IV singularity is obtained when 0 < α < , we can choose f ( ρ ) ρ ≪
1. Inserting these into (2.158)-(2.160) we finally acquire n s ≃ − N (1 − α ) , r ≃ N (1 − α ) , α s ≃ − N (1 − α ) . (2.165)– 27 –ow, the 2015 Planck results [5] provide the following values: n s = 0 . ± . , r < . , a s = − . ± . . (2.166)Hence, if in the scenario at hand we choose ( N, α ) = (60 , / n s ≃ . , r ≃ . , a s = − . . (2.167)Therefore, concerning the spectral index we acquire a good agreement, however the values ofthe tensor-to-scalar ration and of the running spectral index are not inside the observationalbounds. Nevertheless, we can obtain satisfactory agreement in more sophisticated modelsinstead of the simple example f ( ρ ) = Aρ α . According to the concordance model of cosmology the universe is currently accelerating,while it entered this era after being in a long matter-dominated epoch. This behavior,similarly to the early accelerated era of inflation, cannot be reproduced within the standardframework of general relativity and flat ΛCDM-model with dust and vacuum energy, andtherefore extra degrees of freedom should be introduced. One can attribute these extradegrees of freedom to new, exotic forms of matter, such as the inflaton field at early timesand/or the dark energy concept at late times (for reviews see [55, 56]). Alternatively, onecan consider the extra degrees of freedom to have a gravitational origin, i.e. to arise from agravitational modification that possesses general relativity as a particular limit (see [24, 57–60] and references therein). In this section we will show how the late-time acceleration canbe driven by the fluid viscosity [10, 12, 15, 42, 61–74].
We start our investigation by studying the basic scenario of late-time viscous cosmology,presenting the main properties of the viscous cosmic fluid, following [9]. As usual weassume a homogeneous and isotropic FRW universe with geodesic fluid flow, and thus thetwo Friedmann equations are given by (1.12),(1.13).Let us very briefly discuss the non-viscous case. According to the standard model thetotal energy density and pressure are ρ tot = ρ + ρ Λ , p tot = p + p Λ = − ρ Λ , (3.1)where ρ Λ = Λ / πG is the Lorentz invariant vacuum energy density and p Λ = − Λ / πG isthe vacuum pressure corresponding to a positive tensile stress. With the critical energydensity ρ c , the matter density parameter Ω M , and the Einstein gravitational constant κ defined as κρ c = 3 H , Ω M = ρρ c , κ = 8 πG, (3.2)we obtain for the scale factor [75] a ( t ) = K / s sinh / (cid:18) tt Λ (cid:19) , (3.3)– 28 –ith t Λ = H √ Ω Λ0 , K s = − Ω Λ0 Ω Λ0 , where the subscript zero refers to the present time t = t (as usual we impose a ( t ) = 1). The present age of the universe is t = t Λ arctanh p Ω Λ0 , (3.4)which leads to t Λ = 11 . × years if we insert that t = 13 . × years and Ω Λ0 = 0 . H = 23 t Λ coth (cid:18) tt Λ (cid:19) , (3.5)whereas the deceleration parameter becomes q ≡ − − ˙ HH = 12 (cid:20) − (cid:18) tt Λ (cid:19)(cid:21) . (3.6)Inserting Eq. (3.4) the present value for for the deceleration parameter of the ΛCDM-universe is ˆ q = 12 (1 − Λ0 ) . (3.7)With Ω Λ0 = 0 . q = − . t = t when deceleration turns into acceleration.The condition for this is q ( t ) = 0, and leads to t = t Λ arctanh √ , with correspondingredshift z = 1 a ( t ) − (cid:18) Λ0 − Ω Λ0 (cid:19) / − , (3.8)that is t = 7 . × years and z = 0 .
67. Finally, let t e be the time of emission of a signalthat arrives at the time t . Considering time in units of Gyr and inserting t = 13 . Λ0 = 0 .
7, we acquire the useful expression t e = 11 . . − . ] . (3.9)After this brief introduction we now proceed to the investigation of the viscous case,that is we switch on the viscosity in the equation-of-state parameter of the cosmic fluid.For convenience we assume a flat geometry. Without loss of generality we consider thesimplest ansatz (1.16), and thus the two Friedmann equations (1.12),(1.13) become3 H = κρ + Λ , (3.10)˙ H + H = κ ζH − ρ − p ) + 13 Λ , (3.11)while the conservation equation reads˙ ρ + 3 H ( ρ + p ) = 9 ζH , (3.12)with p = wρ, (3.13)– 29 –here in its simplest version w is a constant. Finally, similarly to Ω M = ρ/ρ c , it provesconvenient to introduce the density parametersΩ ζ = κζH , Ω Λ = Λ κρ c , (3.14)where the critical density follows from 3 H = κρ c . Thus, we can express the currentdeceleration parameter as q = 12 (1 + 3 w ) −
32 [Ω ζ + (1 + w )Ω Λ0 ] . (3.15)If the cosmic fluid is cold, i.e. with w = 0, as is often assumed, we obtainΩ ζ = 13 (1 − q ) − Ω Λ0 . (3.16)In principle, this equation enables one to estimate the viscosity parameter Ω ζ if one hasat hand accurate measured values of q and Ω Λ0 . It follows from Eqs. (3.7) and (3.16) thatΩ ζ = 23 (ˆ q − q ) . (3.17)Hence Ω ζ is proportional to the deviation of the measured deceleration parameter fromthe standard ΛCDM-value as given in Eq. (3.7). This means that one needs to measure thedeceleration parameter very accurately in order to obtain information about the viscositycoefficient from its relation to the deceleration parameter. One has so far not been able todetermine Ω ζ in this way.However, we can indicate its present status. Ten years ago D. Rapetti et al. [76]gave kinematical constraints on the deceleration parameter using type Ia supernovae- andX-ray cluster gas mass fraction measurements, obtaining q = − . ± .
14 at the 1 σ confidence level. Inserting q > − .
95 in Eq. (3.17) we obtain Ω ζ < .
27. However,some years later Giostri et al. [77] used SN Ia and BAO/CMB measurements and found − . < q < − .
20 with one light curve fitted, and − . < q < − .
36 with another.Note that if measurements give q < − .
55 then Ω ζ <
0, which is unphysical.We mention though an interesting study of Mathews et al. [78], in which the productionof viscosity was associated with the decay of dark matter particles into relativistic particlesin a recent epoch with redshift z < ζ = ζ . Equation (3.11)gives ˙ H = − H + 32 Ω ζ H H, (3.18)which upon integration with H ( t ) = H leads to H = Ω ζ H − (1 − Ω ζ ) e Ω ζ H ( t − t ) . (3.19)– 30 –nother integration with a ( t ) = 1 gives a = " e Ω ζ H ( t − t ) − (1 − Ω ζ )Ω ζ . (3.20)This implies that the age of the universe when expressed in terms of the present Hubbleparameter H becomes t = 43Ω ζ H arctanh (cid:18) Ω ζ − Ω ζ (cid:19) . (3.21)Hence, it is seen that for early times, in which Ω ζ H t ≪
1, the viscosity can be neglected,and we obtain a ≈ (cid:20) H ( t − t ) (cid:21) , (3.22)corresponding to the evolution of a dust universe. At late times Ω ζ H t ≫
1, the expansionbecomes exponential with H = κζ , a ∝ exp( κζ ) , ρ = 3 κζ , and thus the universe entersinto a late inflationary era with accelerated expansion. A drawback of this model is howeverthat the time when the bulk viscosity becomes dominant is predicted to be unrealisticallylarge.Let us now consider briefly the following model, which has attracted attention, namelythe one where viscosity is considered to be [66, 80, 81] ζ = ζ + ζ ˙ aa + ζ ¨ aa . (3.23)It is based on the physical idea that the dynamic state of the fluid influences its viscosity.We then obtain a ˙ H = − bH + cH + d, (3.24)where a = 1 − κζ , b = 32 [1 + w − κ ( ζ + ζ )] , c = 3 κζ , d = 12 (1 + w )Λ . (3.25)Integrating this equation with a (0) = 0 , a ( t ) = 1 and assuming κ ( ζ + ζ ) < w ≥ b > bd + c >
0, we obtain H ( t ) = c b + ab ˆ H coth( ˆ Ht ) , (3.26)with ˆ H = bda + c a . The age of the universe in this model becomes t = 1ˆ H arctanh a ˆ H bH − c ! , (3.27)and thus viscosity increases the age of the universe. Hence, assuming that κζ ≪ H theincrease of the age due to viscosity is roughly Ω ζ t .– 31 –et us return to the solution (3.26) and apply it to the case where the universe doesnot contain any matter but only dark energy with w = −
1. Moreover, we assume a linearviscosity ( ζ = ζ = 0) and therefore b = 0. Cataldo et al. [82] found that in this case˙ H = 3 κζ H, (3.28)and thus integration with a ( t ) = 1 gives H ( t ) = H exp (cid:20) ζ H t − t ) (cid:21) , (3.29) a ( t ) = exp (cid:26) ζ (cid:20) e ζ H ( t − t ) − (cid:21)(cid:27) . (3.30)Hence, a universe dominated by viscous dark energy with constant viscosity coefficientexpands exponentially faster comparing to the corresponding universe without viscosity.One may now ask the question how does the introduction of a bulk viscosity confrontwith the observed acceleration of the universe. There have been several works dealing withthis issue, for instance see Refs. [83–85]. In the model of Avelino and Nucamendi [85] itwas considered that ζ = ζ = 0, w = 0, Ω M = 1, Ω Λ = 0, and therefore the scale factorcan be written as a ( t ) = (cid:18) − Ω ζ Ω ζ (cid:19) / (cid:16) e Ω ζ H t − (cid:17) / , (3.31)which satisfies the boundary conditions a (0) = 0 , a ( t ) = 1. The age of the universe in thismodel becomes t = 43Ω ζ H arctanh (cid:18) Ω ζ − Ω ζ (cid:19) = − ζ H ln(1 − Ω ζ ) . (3.32)Such a universe model was actually considered earlier, by Brevik and Gorbunova [86, 87]and by Grøn [88], and is also similar to the model of Padmanabhan and Chitre consideredabove [79]. The Hubble parameter reads as H ( t ) = Ω ζ H − e − (3 / ζ H t , (3.33)and it approaches a de Sitter phase for t ≫ / Ω ζ H , with a constant Hubble parameterequal to Ω ζ H . The deceleration parameter is q = 32 exp[(3 / ζ H t ] − , (3.34)and its value at present is q ( t ) = (1 − ζ ) / . (3.35)Hence Ω ζ = 13 (1 − q ) . (3.36)– 32 –ssuming that accurate measurements will verify the ΛCDM model, so that q = − . ζ = 0 .
7. This means that for the universe model to be realistic,there must exist a physical mechanism able to produce a viscosity of this magnitude.The expansion thus starts from a Big Bang with an infinitely large velocity, but decel-erates to a finite value. As usual, when t = t determined by q ( t ) = 0 there is a transitionto an accelerated eternal expansion, namely at t = 2 ln(3 / ζ H , (3.37)at which time the scale factor is a ( t ) = (cid:18) − Ω ζ ζ (cid:19) / , (3.38)and the corresponding redshift is z = (cid:18) ζ − Ω ζ (cid:19) / − . (3.39)Under the assumption that this model contains a mechanism producing viscosity so thatΩ ζ = 0 .
7, this equation gives z = 0 .
8. This is larger than the corresponding value inthe ΛCDM model. Hence the transition to accelerated expansion happens earlier if theacceleration of the expansion is driven by viscosity than by dark energy.We deduce that the bulk viscosity must have been sufficiently large, namely Ω ζ > / a ( t ) <
1. Finally, notethat for this universe model, with spatial curvature k = 0, the matter density is equal tothe critical density, namely ρ = 3 H κ = 3Ω ζ H κ (cid:2) − e − (3 / ζ H t (cid:3) , (3.40)and thus the matter density approaches a constant value, ρ → (3 /κ )Ω ζ H .In the aforementioned study of Avelino and Nucamendi [85] supernova data were usedin order to estimate the value of Ω ζ , giving the best fit for a universe containing dustwith constant viscosity coefficient. The result was that Ω ζ = 0 .
64 had to be several or-ders of magnitude greater than estimates based upon kinetic gas theory [10]. However,as an unorthodox idea we may mention here the probability for producing larger viscosityvia dark matter particles decaying into relativistic products [89]. Additionally, the com-parison between the magnitude of bulk viscosity and astronomical observations were alsoperformed in a recent paper by Normann and Brevik [71], using the analyses of variousexperimentally-based sources [90, 91]. Various ansatzes for the bulk viscosity were ana-lyzed: (i) ζ =constant, (ii) ζ ∝ √ ρ , and (iii) ζ ∝ ρ . The differences between the predictionsof the options were found to be small. As a simple estimate based upon this analysis, wesuggest that ζ ∼ Pa s (3.41)– 33 –an serve as a reasonable mean estimate for the present viscosity. With H = 67 . − Mpc − this corresponds to Ω ζ = 0 . p = wρ . In this case Eq. (3.24) reduces to˙ H = − bH , (3.42)where b = (1 + w ). For such a universe there is a Big Rip at t R = t + 23(1 + w ) H . (3.43)On the other hand, in Ref. [82] a fluid was considered to have w < − ξ , implying b < d = 0. In this case (3.24) reduces to˙ H = −
32 (1 + w ) H + 32 Ω ζ H H. (3.44)The Hubble parameter, scale factor and density for this universe are respectively extractedto be H = H w Ω ζ + (cid:16) − w Ω ζ (cid:17) e − Ω ζ ( t − t ) , (3.45) a = (cid:20) − w Ω ζ + 1 + w Ω ζ e Ω ζ H ( t − t ) (cid:21) w ) , (3.46)and ρ = ρ h w Ω ζ + (cid:16) − w Ω ζ (cid:17) e − Ω ζ ( t − t ) i . (3.47)Thus, in this case there is a Big Rip singularity at t R = t + 23Ω ζ H ln (cid:18) − Ω ζ w (cid:19) . (3.48)Similar models, with variable gravitational and cosmological “constants” have been inves-tigated by Singh et al. [93, 94]. Furthermore, one can go beyond isotropic geometry andstudy viscous fluids in spatially anisotropic spaces, belonging to the Bianchi type-I class.The interested reader might consult, for instance, the discussion in Ref. [9]. In this subsection we examine the appearance of singularities in viscous cosmology. Itis well-known that in FRW geometry, when the equation of state modeling the mattercontent is a linear equation with an equation of state parameter greater than −
1, theBig Bang singularity appears at early times, where the energy density of the universediverges. Moreover, dealing with nonlinear equations of state one can see that other kindof singularities such as Sudden singularity [95–97] or Big Freeze [51, 98–100] appear.In fact, the future singularities are classified as follows [51] (see also [101] for a moredetailed classification): – 34 –
Type I (Big Rip): t → t s , a → ∞ , ρ → ∞ and | p | → ∞ . • Type II (Sudden): t → t s , a → a s , ρ → ρ s and | p | → ∞ . • Type III (Big Freeze): t → t s , a → a s , ρ → ∞ and | p | → ∞ . • Type IV (Generalized Sudden): t → t s , a → a s , ρ → | p | → H diverge.Similarly to the future ones, one can define the past singularities: • Type I (Big Bang): t → t s , a → ρ → ∞ and | p | → ∞ . • Type II (Past Sudden): t → t s , a → a s , ρ → ρ s and | p | → ∞ . • Type III (Big Hottest): t → t s , a → a s , ρ → ∞ and | p | → ∞ . • Type IV (Generalized past Sudden): t → t s , a → a s , ρ → | p | → H diverge.For the simple case of a linear equation of state p = wρ it is well-known that for anon-phantom fluid ( w > −
1) one obtains a Big Bang singularity, while for a phantom fluid( w < −
1) [102–106] the singularity is a future Type I (Big Rip). Hence, in order to obtainthe other type of singularities one has to consider phantom fluids modeled by non-linearequations of state of the form p = − ρ − f ( ρ ) , (3.49)where f is a positive function. The simplest model is obtained taking f ( ρ ) = Aρ α with A >
0. In this case from the conservation equation ˙ ρ = − H ( ρ + p ) and the Friedmannequation H = κρ one obtains the dynamical equation˙ ρ = √ κAρ α + , (3.50)whose solution is ρ = (cid:20) √ κA ( t − t )(1 − α ) + ρ − α (cid:21) − α when α = ρ e √ κA ( t − t ) when α = . (3.51)Furthermore, in order to obtain the evolution of the scale factor we will integrate theconservation equation, resulting in a = a exp (cid:18) Z ρρ ¯ ρd ¯ ρf (¯ ρ ) (cid:19) , (3.52)which using (3.51) leads to a = a exp h A (1 − α ) ( ρ − α − ρ − α ) i when α = 1 a (cid:16) ρρ (cid:17) A when α = 1 . (3.53)Once we have calculated these quantities, we have the following different situations(see also [51]): – 35 –. When α < t s = t − √ κA ρ − α − α < t , implying that the pressure diverges at t = t s .2. When α = 0 there are no singularities. The dynamics is defined from t s = t − √ κA √ ρ (where the energy density is zero) up to t → ∞ .3. When 0 < α < there are two different cases:(a) − α is not a natural number. One has a past Type IV singularity at t s = t − √ κA ρ − α − α , since higher derivatives of H diverge at t = t s .(b) − α is a natural number. In that case there are not any singularites and thedynamics is defined from t s = t − √ κA ρ − α − α to t → ∞ .4. When α = there are no singularities in cosmic time.5. When < α <
1, one has future Type I singularities, since in this case ρ , p and a diverge at t s = t − √ κA ρ − α − α > t .6. When α = 1 the equation of state is linear, and thus we obtain a Big Rip singularity.7. When α >
1, the energy density and the pressure diverge but the scale factor remainsfinite at t = t s , implying that we have a future Type III singularity.The remarkable case appears when 0 < α < and with − α being a natural number.In this case, from the Friedmann equation H = κρ and the solution (3.51) one obtains H = r κ " √ κA t − t )(1 − α ) + ρ − α n , (3.54)with n = − α . As we have already seen, this solution describes a universe in the expandingphase driven by a phantom fluid, which is defined from t s = t − √ κA ρ − α − α (where H = 0)up to t → ∞ . However, solution (3.54) could be extended analytically back in time. Thereare two different cases: When n is odd, this extended solution describes a universe drivenby a phantom field that goes from the contracting to expanding phase, bouncing at time t s . On the contrary, when n is even the universe moves always in the expanding phase,and before t s it is driven by a non-phantom field, while after t s the universe enters in aphantom era. We will explain this phenomenon in more detail in the next subsection.Motivated by the introduction of bulk viscous terms in an ideal fluid one can considera subclass of the general equation of state of (1.15) of the form p = − ρ − f ( ρ ) + G ( H ) . (3.55)Then, the conservation equation becomes ˙ ρ = 3 H [ f ( ρ ) − G ( H )], and using the Friedmannequation (1.12) in the expanding phase leads to˙ ρ = 3 H (cid:20) f ( ρ ) − G (cid:18)r κρ (cid:19)(cid:21) ≡ HF ( ρ ) , (3.56)– 36 –hich reveals that this formalism is equivalent with considering a fluid with an effectiveequation of state given by p = − ρ − F ( ρ ) = − ρ − f ( ρ ) + G (cid:18)r κρ (cid:19) . (3.57)It is clear that, in general, the equation of state (3.55) does not lead to a universecrossing the phantom barrier. A simple way to obtain transitions from the non-phantomto the phantom regime is to explicitly consider an inhomogeneous equation of state of theform F ( ρ, p, H ) = 0, for example [12, 74]( ρ + p ) − C ρ (cid:18) − H H (cid:19) = 0 , (3.58)with C and H some positive constants. Inserting this into the square of the equation˙ H = − κ ( ρ + p ), one obtains the bi-valued dynamical equation˙ H = 94 C H (cid:18) − H H (cid:19) . (3.59)From this equation, since there are two square roots and the effective equation of stateparameter is given by w eff ≡ − − H H , one can see that there are two different dynamics:one which corresponds to the branch with ˙ H <
H > H ( t ) = 169 C H ( t − t − )( t + − t ) , (3.60)where we have introduced the notation t ± = ± C H . It is easy to check that H ( t ) isonly defined for times between t − and t + , since at t ± the Hubble function H diverges (weobtain a Big Bang at t − and a Big Rip at t + ). Moreover, it is a decreasing function for t ∈ ( t − ,
0) and an increasing one for t ∈ (0 , t + ), implying that at t = 0 the universe crossesthe phantom divide (it passes from the non-phantom to the phantom era).Another interesting example arises from the equation of state( ρ + p ) + 16 H κ t ( H − H ) ln (cid:18) H − HH (cid:19) = 0 , (3.61)where t , H , H are parameters satisfying H > H >
0. The corresponding bi-valueddynamical equation is ˙ H = − H t ( H − H ) ln (cid:18) H − HH (cid:19) , (3.62)which has two fixed points, namely H and H − H . As we have already explained, when˙ H <
H >
H < H to H − H , it reaches H = H and– 37 –hen it enters in the other branch ( ˙ H >
0) going from H − H to H . In fact, in [12, 74]the authors found the following solution: H ( t ) = H − H exp (cid:18) − t t (cid:19) , (3.63)which satisfy all the properties described above.A final remark is in order: One can indeed consider the more general equation ofstate given in (1.15), namely of the form F ( ρ, p, H, ˙ H , ¨ H, · · · ) = 0, containing higher orderderivatives of the Hubble parameter. In this case, using the Friedmann equations theequation of state becomes the dynamical equation F H κ , − Hκ − H κ , ˙ H , ¨ H, · · · ! = 0 . (3.64)A non-trivial example is the following equation of state [12, 74]: p = wρ − G − κ ˙ H + G ˙ H , (3.65)where G and G are constant. Then, the dynamical equation becomes − H (1 + w ) κ = − G + G ˙ H . (3.66)We look for periodic solutions of the form H ( t ) = H cos(Ω t ) depicting an oscillatoryuniverse. Inserting this expression into (3.66) we obtain the algebraic system: G = G Ω H , G = 3 H (1 + w ) κ , (3.67)whose solution is given by H = s κG w ) , Ω = s w ) κG , (3.68)provided that G (1 + w ) > G (1 + w ) >
0. On the other hand, when G (1 + w ) < H ( t ) = H cosh(Ω t ), obtaining H = s κG w ) , Ω = s − w ) κG . (3.69) In this subsection we analyze how one can describe in a unified way the early (inflationary)and late time acceleration, in the framework of viscous cosmology. The simplest way tounify early inflationary epoch with the current cosmic acceleration is by using scalar fields[107]. Starting with the action S = Z d x √− g (cid:26) κ R − ω ( φ ) ∂ µ φ∂ µ φ − V ( φ ) (cid:27) , (3.70)– 38 –here ω and V are functions of the scalar field φ , and focusing on flat FRW geometry, oneobtains the following dynamical equation ω ( φ ) ¨ φ + 12 ω ′ ( φ ) ˙ φ + 3 Hω ( φ ) ˙ φ + V ′ ( φ ) = 0 . (3.71)The relevant fact, is that given a function f ( φ ) the equation (3.71) has always the solution φ = t and H = f ( t ), provided that (for details see [12, 74]) ω ( φ ) = − κ f ′ ( φ ) , (3.72) V ( φ ) = 1 κ (cid:2) f ( φ ) + f ′ ( φ ) (cid:3) . (3.73)An interesting example is obtained when one considers the function f ( φ ) = H (cid:18) φ s φ + φ s φ s − φ (cid:19) , (3.74)where H and φ s are the two positive parameters of the model. In this case one has ω ( φ ) = 2 H φ s ( φ s − φ ) κφ ( φ s − φ ) , (3.75) V ( φ ) = H φ s κφ ( φ s − φ ) (3 H φ s − φ s + 2 φ ) , (3.76)whose dynamics is given by H = H t s t ( t s − t ) , a = a (cid:18) tt s − t (cid:19) H t s , (3.77)where we have introduced the notation t s = φ s . Since H diverges at t = 0 and t = t s , thedynamics is defined in (0 , t s ). In fact at t = 0 one has a = 0, which means that we obtaina Big Bang singularity, while at t = t s the scale factor diverges, implying that we have aBig Rip singularity. On the other hand, the derivative of the Hubble parameter reads as˙ H = H t s t ( t s − t ) (2 t − t s ) , (3.78)that is the universe lies in the non-phantom regime when 0 < t < t s /
2, while it lies in thephantom phase for t s / < t < t s . Hence, we conclude that this model could describe thecurrent cosmic acceleration.In order to examine the behavior at early times, we note that near t = 0 one canmake the approximation a = a (cid:16) tt s (cid:17) H t s , and thus its second derivative at early times isapproximately ¨ a = a H t s ( H t s − t . (3.79)From this we deduce that if one chooses H t s > f ( φ ) = H sin( νφ ) , (3.80)with H and ν positive parameters. A straightforward calculation leads to ω ( φ ) = − H νκ cos( νφ ) (3.81) V ( φ ) = 2 κ (cid:2) H ν cos( νφ ) + H sin ( νφ ) (cid:3) . (3.82)In this case one obtains a non-singular oscillating universe, whose dynamics is given by H = H sin( νt ) , (3.83) a = a exp (cid:20) − H ν cos( νt ) (cid:21) . (3.84)This solution depicts a universe that bounces at time t = nπν where n is an integer, and since˙ H = H ν cos( νt ) one can easily check that the universe lies in the phantom regime when πν (cid:0) − + 2 n (cid:1) < t < πν (cid:0) + 2 n (cid:1) , while it is in the non-phantom phase when πν (cid:0) + 2 n (cid:1)
1, which implies that the Hubble parameter varies from infinity to ξ . Moreover,since w eff = − ξ H − ξ H , (3.87) w eff ∼ = − H ≫ ξ ) and late ( H ∼ = ξ ) times, from which we deduce that thisviscous fluid model unifies inflation with the current cosmic acceleration. Additionally, w eff is positive when ξ √ − √ < H < √ √ , having the maximum value w eff = 1 at H = ξ .Thus, in summary, in the scenario at hand the universe starts from an inflationary epoch,it evolves through a Zel’dovich fluid ( w eff = 1), radiation- ( w eff = 1 /
3) and matter-( w eff = 0) dominated epochs, and finally it enters into late-time acceleration tendingtowards a de Sitter phase. – 40 –he solution of equation (3.86) is H = ξ (cid:16) e − ξ t + 1 (cid:17) , (3.88)and the scalar field that induces this dynamics, if one chooses ω ( φ ) ≡
1, has the followingHiggs-style potential (for details see [109]): V ( φ ) = 27 ξ κ (cid:18) φ − κ (cid:19) . (3.89)We stress here that this unified model for inflation and late-time acceleration leads toinflationary observables, namely the spectral index, its running and the ratio of tensorto scalar perturbations, that match at 2 σ Confidence Level with the observational dataprovided by Planck 2015 announcements [5] (for a detailed discussion see [109, 110]).We close this section by considering a very simple quintessential-inflation potentialwhich unifies inflation with late time acceleration, namely [111] V ( φ ) = ( (cid:0) H E − Λ3 (cid:1) (cid:0) φ − κ (cid:1) for φ ≤ φ E Λ κ for φ ≥ φ E , (3.90)where φ E ≡ − q κ H E q H E − Λ3 , and with H E > H = ( − H E + Λ for H ≥ H E − H + Λ for H ≤ H E , (3.91)whose solution has the following expression H ( t ) = ( (cid:0) − H E + Λ (cid:1) t + 1 t ≤ q Λ3 3 H E + √
3Λ tanh( √ t )3 H E tanh( √ t )+ √ t ≥ , (3.92)with the corresponding scale factor a ( t ) = a E e h ( − H E +Λ ) t + t i t ≤ a E h H E √ sinh( √ t ) + cosh( √ t ) i t ≥ . (3.93)We mention that this dynamics arises also from a universe filled with a fluid with the simplelinear equation of state of the form p = ( − ρ + 2 ρ E − κ ρ ≥ ρ E ρ − κ ρ ≤ ρ E , (3.94)where ρ E = H E κ . Equivalently it can arise from a viscous fluid, since effectively, choosing w = 1 and the following viscosity coefficient ζ = ( κ (cid:16) H − H E H (cid:17) H ≥ H E H ≤ H E , (3.95)– 41 –nd inserting it into (3.11) one obtains the dynamics (3.91). Finally, for this model theeffective equation-of-state parameter is given by w eff = ( − H (cid:0) H E − Λ (cid:1) H ≥ H E − H H ≤ H E , (3.96)which shows that for H ≫ H E one has w eff ( H ) ∼ = − H ∼ = H E , the equation-of-state parameter satisfies w eff ( H ) ∼ = 1 (deflationary perioddominated by a Zel’dovich fluid), and lastly for H ∼ = q Λ3 one also acquires w eff ( H ) ∼ = − In this subsection we investigate the cosmological scenario of holographic dark energywith the presence of a viscous fluid. Holographic dark energy [112, 113] is a scenario in thedirection of incorporating the nature of dark energy using some basic quantum gravitationalprinciples. It is based on black hole thermodynamics [114] and the connection of theultraviolet cut-of of a quantum field theory, which induces the vacuum energy, with thelargest distance of this theory [115]. Determining suitably an IR cut-off L , and imposingthat the total vacuum energy in the maximum volume cannot be greater than the mass ofa black hole of the same size, one obtains the holographic dark energy, namely ρ DE = 3 c κL , (3.97)with κ the gravitational constant, set to κ = 1 in the following for simplicity, and c aparameter. The holographic dark energy scenario has interesting cosmological applications[74, 116–119]. Concerning the ultraviolet cut-off one uses the future event horizon L f L f = a Z ∞ t dta . (3.98)However, one can generalize the model using the quadratic Nojiri-Odintsov cut-off L definedas [74, 120, 121] cL = 1 L f (cid:0) α + α L f + α L f (cid:1) (3.99)with c , α , α and α constants, or the generalized Nojiri-Odintsov cut-off defined inRefs. [74] and [122].In this subsection we will consider the scenario in which generalized holographic darkenergy interacts with a viscous fluid, following [123]. In particular, we consider a darkmatter sector with a viscous equation of state of the form p DM = − ρ DM + ρ αDM + χH β , (3.100)where ρ DM and p DM are respectively the dark matter energy density and pressure, andwith α , χ and β the model parameters. Furthermore, we allow for an interaction betweenviscous dark matter and holographic dark energy:˙ ρ DE + 3 Hρ DE (1 + w DE ) = − Q, (3.101)– 42 – ρ DM + 3 Hρ DM (1 + w DM ) = Q, (3.102)with w DE and w DM respectively the equation-of-state parameters of the dark energy anddark matter sectors, and where Q is a function that determines the interaction. One canimpose the following form for Q [123] Q = 3 Hb ( ρ DE + ρ DM ) , (3.103)where b is a constant, although more complicated forms could also be used [124]. Finally,the first Friedmann equation reads as H = 13 ρ eff , (3.104)where the effective (total) energy density is given by ρ eff = ρ DE + ρ DM .We start our analysis by investigating the non-interacting scenario, that is setting Q (i.e. b ) to zero. In this case, using (3.97),(3.98),(3.100) and (3.104), the decelerationparameter q ≡ − − ˙ H/H is found to be q = − √ Ω de ˙ L f ( α + 2 α L f ) − ˆ q H L f , (3.105)where ˆ q = H Ω de ( ˙ L f + 1) + L f (cid:0) H + p DM (cid:1) . Moreover, the evolution of the dark matterdensity parameter Ω DM is determined by the differential equationΩ ′ DM = 2Ω / DE ˆ L f − √ Ω DE ˆ L f − ˆ A H L f , (3.106)with ˆ L f = ˙ L f ( α +2 α L f ) and ˆ A = Ω DE [ H ( ˙ L f +1)+ L f p DM ]+ H Ω DE ( ˙ L f +1), and whereprimes denote differentiation with respect to N = ln a . Finally, for the non-interacting case(3.97) and (3.101) lead to w DE = − L HL = − L f HL f − L f ( α + 2 α L f ) (cid:16) α + α L f + α L f (cid:17) . (3.107)As one can see, the deceleration parameter q starts from positive values, it decreases, and itbecomes negative marking the passage to late-time accelerated phase [123]. The role of theviscosity parameter χ is significant, since larger positive χ leads the transition redshift z tr (from deceleration to acceleration) to smaller values and the present deceleration parameterto negative values closer to zero. Hence, the larger the fluid viscosity is the more difficultit is for the universe to exhibit accelerated expansion. Lastly, an important feature is thatthe dark energy equation of state parameter can exhibit the phantom divide-crossing, ascan be seen from (3.107) [74, 123, 125]. In Tables 2 and 3 we present the various calculatedvalues for different choices of the model parameters, where the features described aboveare obvious. – 43 – able 2 . The present-day values of the deceleration parameter q , of the dark energy equation-of-state parameter w DE and its derivative w ′ DE , the statefinder parameters ( r, s ) and the value of thetransition redshift z tr , for the non-interacting model, for several values of the viscosity parameter χ in (3.100), and with α = 1 . α = 0 . α = 0 . α = 0 .
25. We have set H = 0 . DM = 0 .
27. From [123]. χ q ( w ′ DE , w DE ) ( r, s ) z tr − . − .
766 (0 . , − . . , − . . − . − .
666 (0 . , − . . , − . . . − .
599 (0 . , − . . − . . . − .
533 (0 . , − . . , − . . . − .
433 (0 . , − . . , − . . Table 3 . The present-day values of the deceleration parameter q , of the dark energy equation-of-state parameter w DE and its derivative w ′ DE , the statefinder parameters ( r, s ) and the valueof the transition redshift z tr , for the non-interacting model, for several values of the parameter α in (3.100), and with χ = − . α = 0 . α = 0 . α = 0 .
25. We have set H = 0 . DM = 0 .
27. From [123]. α q ( w ′ DE , w DE ) ( r, s ) z tr . − .
508 (0 . , − . . , − . − . − .
554 (0 . , − . . , − . − . − .
595 (0 . , − . . − . . . − .
666 (0 . , − . . , − . . . − .
682 (0 . , − . . , − . . Q in (3.101),(3.102).In this case, using (3.97),(3.98),(3.100) and (3.104), for the deceleration and matter densityparameters we find q = L (cid:2) (1 − b ) H + p DM (cid:3) − √ Ω DE ˆ L f − H Ω DE ( ˙ L f + 1)2 H L , (3.108)and Ω ′ DM = A + 2Ω / DE ˆ L f − √ Ω DE ˆ L f + H Ω DE ( ˙ L f + 1) H L f , (3.109)with A = Ω DE { H [(3 b − HL f − − L f p DM } , while for the dark-energy equation-of-state parameter we obtain w DE = − bH L + 2 √ Ω DE ˆ L f + H Ω DE ( ˙ L f + 1)3 H L f Ω DE . (3.110)As we observe, the deceleration parameter q exhibits the transition from deceleration toacceleration, and the role of the positive interaction parameter b in (3.103) is to make z tr larger and the present value of q more negative [123]. This is expected since larger positive b implies larger positive Q in (3.101),(3.102) and thus larger energy transfer to the dark– 44 –nergy sector. Moreover, the role of the viscosity parameter χ is as in the non-interactingcase, i.e the larger the χ is the more difficult it is for the universe to exhibit acceleratedexpansion. Lastly, the dark energy equation-of-state parameter w DE can exhibit the phan-tom divide-crossing, too. In Table 4 we present the various calculated values for differentchoices of the model parameters, where the features described above are obvious. Table 4 . The present-day values of the deceleration parameter q , of the dark energy equation-of-state parameter w DE and its derivative w ′ DE , the statefinder parameters ( r, s ) and the value of thetransition redshift z tr , for the interacting model, for several values of the interaction parameter β of (3.103), and with χ = 0 . α = 1 . α = 0 . α = 0 . α = 0 .
25. We have set H = 0 . DM = 0 .
27. From [123]. b q ( w ′ DE , w DE ) ( r, s ) z tr . − .
533 (0 . , − . . , − . . . − .
548 (0 . , − . . , − . . . − .
578 (0 . , − . . , − . . . − .
608 (0 . , − . . , − . . . − .
634 (0 . , − . . , − . . In this section we discuss various topics of viscous cosmological theory, focusing on inves-tigations in which the present authors have taken part. As a brief remark to the materialcovered below we think it is appropriate to underscore the great power of the hydrody-namical formalism when applied to quite different problems in cosmology. The formalismrobustness is in general striking. Definitely, in view of the considerably large activity in thefield of viscous cosmology, there are many aspects that cannot be discussed here. For in-stance, instead of assuming a one-component fluid model, one might consider an extensionof the model in order to encompass two different fluid components. We may here mentionthe recent study of Ref. [126], where the cosmic fluid was considered to be constituted ofa dark matter component endowed with a constant bulk viscosity, and a non-viscous darkenergy component. In other related works [73, 127], viscous coupled-fluid models were in-vestigated when the equation of state was assumed to be inhomogeneous. Furthermore, in[128] the authors studied the important self-reproduction problem of the universe, namelythe graceful exit from inflation, where it was shown how inflation without self-reproductioncan actually be obtained by imposing restrictions on the value of the thermodynamic pa-rameter in the equation of state. Finally, we mention the very different approach which– 45 –onsists in applying particle physics theory and the relativistic Boltzmann equation in orderto derive expressions for the bulk and the shear viscosities, and the corresponding entropyproduction, in the specific lepton-photon era, where the temperature dropped from 10 Kto 10 K. Calculations of this kind were recently given in Ref. [129], [130] and [131].
A significant amount of research has been spent in order to study the behavior of the cosmicfluid in the far future. In such an examination, and as we discussed in detail in subsection3.2 above, there may appear various kinds of singularities: the Big Rip [61, 62], the LittleRip [67, 132, 133], the Pseudo-Rip [134], the Quasi-Rip [135], as well as other kinds of softsingularities (for instance the so-called type IV finite time singularities [136]).In the framework of viscous cosmology, the value of the (effective) bulk viscosity atpresent time is naturally an important ingredient of such investigation. Recent observationsfrom the Planck satellite have given us a better ground for estimating the bulk viscosityvalue ζ = ζ at present time t = t . As discussed already in the previous Sections, referringto [71], as well as to several other theoretical and experimental manuscripts, the estimate ζ ∼ Pa s (4.1)was suggested as a reasonable (logarithmic) mean value. However, the corresponding un-certainty is quite large; there have appeared proposals ranging from about 10 Pa s toabout 10 Pa s, depending on analyses of different sources.We will follow the discussion of [126], in which two different cosmological models wereanalyzed: (1) a one-component dark energy model where the bulk viscosity ζ was associatedwith the cosmic fluid as a whole, and (2) a two-component model where ζ was associatedwith a dark matter component ρ m only, the latter component assumed to be non-viscous.For convenience, we focus on the one-component scenario.We assume the simple equation of state p = wρ , with w = const. , and hence the twoviscous Friedmann equations acquire the usual form, namely3 H = κρ, H + 3 H = − κ [ p − Hζ ( ρ )] , (4.2)and the energy conservation equation reads as˙ ρ + 3 H ( ρ + p ) = 9 H ζ ( ρ ) . (4.3)Solving this equation in the regime around w = −
1, i.e expanding as w = − α andassuming that α is small, we obtain t = 1 √ κ Z ρρ dρρ / [ − α + √ κ ζ ( ρ ) / √ ρ ] , (4.4)where t = 0, and the integration extends into the future. For the bulk viscosity we willconsider the form adopted in the literature, namely ζ = ζ (cid:18) HH (cid:19) λ = ζ (cid:18) ρρ (cid:19) λ , (4.5)– 46 –ith λ a constant. In the following we examine two options for the value of λ , which areboth physically reasonable. • Case (i): λ = 1 / ζ ∝ √ ρ ) . In this case, from Eq. (4.4) we obtain t = 23 H X (cid:18) − √ Ω (cid:19) , (4.6)where for convenience we have introduced the dimensionless quantities X = Ω ζ − α, Ω = ρρ . (4.7)The point that worths attention here is that even if the fluid is initially in thequintessence region α > t = 0 it will, if X >
0, inevitably be driven into aBig Rip singularity ( ρ → ∞ ) after a finite time [69, 71, 86, 87] t s = 23 H X , ( ζ ∝ √ ρ ) . (4.8)If on the other hand the combination of equation-of-state parameter α and viscosity ζ is such that X <
0, then the cosmic fluid becomes gradually diluted as ρ ∝ /t in the far future. • Case (ii): λ = 0 ( ζ = ζ = const. ) . In this case we obtain the solution t = 23Ω ζ H ln " X − α + Ω ζ / √ Ω , ( ζ = ζ ) , (4.9)which implies an energy density of the formΩ = ρρ = (cid:26) Ω ζ α + (Ω ζ − α ) exp [ − (3 / ζ H t ] (cid:27) . (4.10)Hence in the far future ρ → const. , which implies H → const. , which is just the deSitter solution. Let us denote the limiting value of the density by ρ dS . Then ρ dS = ρ (cid:18) Ω ζ α (cid:19) = 3 κζ α . (4.11)From this expression we deduce that both α and X are important for the future fateof the cosmic fluid.Thus, this case may be defined as a pseudo-Rip in accordance with the definitiongiven by Frampton et al. [134], since the limiting value of the density reached afteran infinite span of time is finite. – 47 –e close this subsection by providing some values for the inflationary observables, inorder to compare with the 2015 Planck observations. In particular, from Table 5 of [5] wehave w = − . +0 . − . . Thus, α = 1 + w will be lying within two limits, i.e. between α min = − . , α max = +0 . . (4.12)As mentioned above, we took ζ = 10 Pa s, i.e. Ω ζ = 0 .
01, to be a reasonable mean valueof the present viscosity. Then, according to (4.7) we have X ( α max ) = − . , X ( α min ) = +0 . . (4.13)Hence, we recover the cases 2 and 3 above: the future de Sitter energy density will becomelower than ρ . This subsection is a continuation of the previous one, and is motivated by the followingquestion: is the value of ζ , as inferred from the analysis of recent observations, actuallylarge enough to permit the crossing of the phantom divide, i.e. the transition from thequintessence region to the phantom region? To analyze this question we have to considermore carefully the uncertainties in the data found from different sources. We will presentsome material discussing this point, following the recent work [69].Assume that the bulk viscosity varies with energy density as ζ ∝ √ ρ . The conditionfor phantom divide crossing, as noted above, is that the quantity X defined in Eq. (4.7)has to be positive. In the analysis of Wang and Meng [90] various assumptions for the bulkviscosity in the early universe were considered, and the corresponding theoretical curvesfor H = H ( z ) were compared with a number of observations. The detailed comparison isquite complicated, but for our purpose it is sufficient to note that the preferred value ofthe magnitude Ω ζ is (compare also with the discussion in [71]):Ω ζ = 0 . , (4.14)corresponding to ζ ∼ × Pa s , (4.15)which is a rather high value. In this context, we may compare with the formula for thebulk viscosity in a photon fluid [15], namely ζ = 4 a rad T τ f (cid:20) − (cid:18) ∂p∂ρ (cid:19) n (cid:21) , (4.16)where a rad = π k B / ~ c is the radiation constant and τ f the mean free time. If we esti-mate τ f = 1 /H (the inverse Hubble radius), we obtain ζ ∼ Pa s, which is considerablylower. In summary, it seems that one has to allow for a quite wide span in the value of thepresent bulk viscosity. All suggestions in the literature can be encompassed if we write10 Pa s < ζ < Pa s , i.e. − < Ω ζ < . . (4.17)– 48 –e can now rewrite the condition for phantom divide crossing as ζ > H κ α = (1 . × ) α, (4.18)where we have inserted H = 67 .
80 km s − Mpc − = 2 . × − s − . As noted above,from the observed data we derive the maximum value of α to be α max = 0 . ζ > H κ α max = 6 . × Pa s , or Ω ζ > . . (4.19)Thus, comparison between (4.17) and (4.19) implies that, on the basis of available data, aphantom divide crossing is actually possible even if α = α max . In this subsection we investigate the realization of bouncing solutions in the framework ofviscous cosmology following [137] (see also [138]). Bouncing cosmological evolutions offera solution to the initial singularity problem [139]. Such models have been constructed inmodified gravity constructions, such as in the Pre-Big-Bang [140] and in the Ekpyrotic[141] scenarios, in f ( R ) gravity [142–144], in f ( T ) gravity [145], in braneworld scenarios[146, 147], in loop quantum cosmology [148, 149] etc. Additionally, non-singular bouncescan be obtained using matter forms that violate the null energy condition [150, 151].In order to be more general, in the following we will allow also for a spatial curvature,and hence the two Friedmann equations write as H + ka = κρ − (2 ˙ H + 3 H ) κ = p , (4.21)with k = − , , p = w ( ρ ) ρ − B ( a ( t ) , H, ˙ H ... ) , (4.22)where w ( ρ ) can depend on the energy density, but the bulk viscosity B ( a ( t ) , H, ˙ H ... ) isallowed to be a function of the scale factor, and of the Hubble function and its derivatives.Thus, the fluid stress-energy tensor writes as T µν = ρu µ u ν + h w ( ρ ) ρ + B ( ρ, a ( t ) , H, ˙ H ... ) i ( g µν + u µ u ν ) , (4.23)with u µ = (1 , , ,
0) the four velocity. Hence, the standard conservation law ˙ ρ +3 H ( ρ + p ) =0 leads to ˙ ρ + 3 Hρ (1 + w ( ρ )) = 3 HB ( ρ, a ( t ) , H, ˙ H ... ) (4.24)We now proceed to the investigation of simple bounce solutions in the above framework,and we discuss the properties of the viscosity of the fluids that drive such solutions. A firstexample is the bounce with an exponential scale factor of the form a ( t ) = a e α ( t − t ) n (4.25) H ( t ) = 2 nα ( t − t ) n − , (4.26)– 49 –ith n a positive integer and a , α positive parameters. We consider t > t < t we have a contracting universe and when t > t expansiontakes place. We mention that if n is non-integer then singularities may arise, while thesimplest case n = 1 / H ( t ) = const. (in general for n = m/
2, with m an odd integer, the bounce is absent). Finally, note that for the ansatz(4.26) we have ¨ aa = H + ˙ H = 2 nα ( t − t ) n − (cid:2) nα ( t − t ) n + (2 n − (cid:3) , (4.27)and hence we obtain (early-time) acceleration after the bounce, which is a significantphenomenological advantage.Inserting the bouncing scale factor (4.26) into (4.20) we acquire ρ = 3 κ (cid:20) n α ( t − t ) n − + ka e α ( t − t ) n (cid:21) . (4.28)Since for k = − k = 0 and k = +1 cases where it is always positive definite. As we observe, in the flat case ρ decreasesin the contracting phase, it becomes zero at t = t , and it increases in the expanding regime.On the other hand, for k = +1, and when n >
1, there is a region around the bouncingpoint where ρ increases in the contracting phase, it reaches the value ρ = 3 / ( a κ ) at t = t ,and then it decreases (these can be seen by examining the derivatives of (4.28)). However,for t ≫ t , the energy density starts to increase. This behavior may have an importanteffect on the cosmological parameter Ω = 1 + ka H , which for the bouncing scale factor(4.26) becomes Ω = 1 + ka α ( t − t ) n − e α ( t − t ) n , (4.29)and thus it exhibits a decreasing behavior. Such a post-bounce acceleration, with thesimultaneous decrease of ρ and of Ω may be compatible with the inflationary phenomenol-ogy, in which at the end of inflation Ω is very close to 1. Definitely, in order to stop theaforementioned early-time acceleration we need to add additional fluids that could becomedominant and trigger the transition to the matter era.Let us now analyze what kind of fluids with equation of state given by (4.22) canproduce the bouncing solution (4.26). We first consider an inhomogeneous but non-viscousfluid, namely we assume B ( a ( t ) , H, ˙ H ... ) = 0. In this case, for the flat geometry, equations(4.20) and (4.22) lead to p = − ρ − ρ ( n − n − (cid:20) κ (2 nα ) (cid:21) n n − (cid:18) n − nα (cid:19) , (4.30)and thus to w ( ρ ) = − − ρ − n (2 n − (cid:20) κ (2 nα ) (cid:21) n n − (cid:18) n − nα (cid:19) . (4.31)As we mentioned earlier, if the exponent of ρ in (4.31) is negative then we obtain thebounce realization, however if it is positive then we have the appearance of a singularity.– 50 –s a second example we switch on viscosity, considering B ( a ( t ) , H, ˙ H ... ) = 3 Hζ ( H ) , (4.32)with ζ ( H ) > w = −
1. In this case, for the flat geometry, equations (4.20) and (4.22) lead to p = − ρ − Hζ ( H ) ,ζ ( H ) = (cid:18) κ (cid:19) n − n − (2 nα ) n − (cid:18) n − (cid:19) H − n − . (4.33)Nevertheless, for k = +1 the equation of state for the fluid becomes complicated andtherefore it is necessary to go beyond (4.32) and consider a viscosity that depends on thescale factor too. Such a case could be p = − ρ − Hζ ( H, a ( t )) , (4.34)which then leads to ζ ( H, a ( t )) = (cid:18) κ (cid:19) n − n − (2 nα ) n − (cid:18) n − (cid:19) H − n − − k κHa ( t ) . (4.35)As we can see, and as expected, for large scale factors the above relation coincides with(4.33), and therefore we can treat the closed geometry as the flat one.Since we have analyzed the exponential bounce, we now proceed to the investigationof other bouncing solutions. In particular, we will focus on the power-law bouncing scalefactor of the form a ( t ) = a + α ( t − t ) n , (4.36) H ( t ) = 2 nα ( t − t ) n − a + α ( t − t ) n , (4.37)with n a positive integer, a , α positive parameters, and t > aa = 2 n (2 n − α ( t − t ) n − a + α ( t − t ) n , (4.38)which implies that the post-bounce expansion is accelerated. Inserting (4.37) into the firstFriedmann equation (4.20) we acquire ρ = 3 κ [ a + α ( t − t ) n ] (cid:20) n α ( t − t ) n − + ka + α ( t − t ) n (cid:21) . (4.39)Since for the open universe case ρ can become negative (in particular, ρ = − / ( a κ ) at t = t ), we focus on the k = 0 and k = +1 cases, where ρ is positive definite. Taking thederivative of (4.39) we find˙ ρ = − n ( t − t ) n − α [2 n ( t − t ) n α ( a (1 − n ) + ( t − t ) n α ) + k ( t − t ) ]3( a + α ( t − t ) n ) κ , (4.40)– 51 –hus near the bounce point we have˙ ρ ( t → t ) ≃ n ( t − t ) n − α (2 n − a κ , (4.41)from which we deduce that the energy density decreases in the contracting phase beforethe bounce and increases immediately after it. Nevertheless, for | t | ≫ t we have˙ ρ ( | t | ≫ t ) = − n ( t − t ) − n − [2 n ( t − t ) n α + k ( t − t ) ]3 α κ , (4.42)which implies that after a suitable amount of time in the expanding phase ρ starts decreas-ing again. Finally, the cosmological parameter Ω = 1 + ka H behaves asΩ = 1 + k n α ( t − t ) n − , (4.43)and therefore it exhibits a decreasing behavior. Hence, similarly to the case of the expo-nential bounce analyzed earlier, such behaviors could be interesting for the description ofthe post-bouncing universe and the correct subsequent thermal history, since it will leavea universe with Ω very close to 1 and a decreasing ρ .Lastly, let us investigate what kind of fluids with equation of state given by (4.22) canproduce the power-law bouncing solution (4.37). Considering an inhomogeneous viscousfluid with equation of state p = − ρ − Hζ ( a ( t ) , H ) , (4.44)and inserting (4.37), we find the bulk viscosity as ζ ( a ( t ) , H ) = (2 n − a ( t )3 n ( a ( t ) − a ) κ . (4.45)Note that away from the bouncing point, namely when a ( t ) ≫ a , the bulk viscositybecomes ζ ( H, a ( t ) ≫ a ) ≃ (2 n − nκ = const. . (4.46)Hence, if 0 < ζ < /
3, which corresponds to n > /
2, then the bounce can be realized. Onthe other hand if 2 / < ζ then, as we mentioned earlier, singularities might appear.In summary, in this subsection we saw that viscous fluids can offer the mechanismto violate the null energy condition, which is the necessary requirement for the bouncerealization. Hence, various bouncing solutions can be realized, driven by fluids with suitablyreconstructed viscosity. As specific examples we studied the exponential and the power-lawbounces, which are also capable of describing the accelerated post-bouncing phase, withthe additional establishment of the spatial flatness. These features reveal the capabilitiesof viscosity. – 52 – .4 Inclusion of isotropic turbulence In this subsection we discuss turbulence issues in the framework of viscous cosmology. Fromhydrodynamics point of view the inclusion of turbulence in the theory of the cosmic fluidseems most natural, at least in the final stage of the universe’s evolution when the fluidmotion may well turn out to be quite vigorous. The local Reynolds number must thenbe expected to be very high. On a local scale this brings the shear viscosity concept intoconsideration, as it has to furnish the transport of eddies over the wave number spectrumuntil the local Reynolds number becomes of order unity, marking the transfer of kineticenergy into heat. Due to the assumed isotropy in the fluid, we must expect that the typeof turbulence is isotropic when looked upon on a large scale. According to standard theoryof isotropic turbulence in hydrodynamics we then expect to find a Loitziankii distributionfor low wave numbers (energy density varying as k ), whereas for higher k we expect aninertial subrange in which the energy distribution is E ( k ) = αǫ / k − / , (4.47)where α denotes the Kolmogorov constant and ǫ is the mean energy dissipation per unitmass and unit time. When k reaches the inverse Kolmogorov length η K , i.e. k → k L = 1 η L = (cid:16) ǫν (cid:17) / , (4.48)with ν the kinematic viscosity, then the dissipative region is reached.In the following we will consider a dark fluid developing into the future from thepresent time t = 0, when turbulence is accounted for. We will perform the analysis intwo different ways: either assuming a two-fluid model with one turbulent constituent, orassuming simply a one-component fluid, following [9, 68, 152, 153].We start by considering a two-component model, where the effective energy is writtenas a sum of two parts, namely ρ eff = ρ + ρ turb , (4.49)with ρ denoting the conventional energy density. Taking ρ turb to be proportional to thescalar expansion θ = 3 H , and calling the proportionality factor τ , we acquire ρ eff = ρ (1 + 3 τ H ) . (4.50)Additionally, the effective pressure p eff is split in a similar way as p eff = p + p turb . (4.51)For both components we assume homogeneous equations of state, namely p = wρ, p turb = w turb ρ turb , (4.52)The Friedmann equations can thus be written (recall that κ = 8 πG ) as H = 13 κρ (1 + 3 τ H ) , (4.53)– 53 –¨ aa + H = − κρ ( w + 3 τ Hw turb ) , (4.54)leading to the following governing equation for H :(1 + 3 τ H ) ˙ H + 32 γH + 92 τ γ turb H = 0 , (4.55)where we used the standard notation γ = 1 + w, γ turb = 1 + w turb . (4.56)Finally, when the energy dissipation is assumed to be ǫ = ǫ (1 + 3 τ H ) , (4.57)the energy balance may be written as˙ ρ + 3 H ( ρ + p ) = − ρǫ (1 + 3 τ H ) . (4.58)In summary, the input parameters in this model are { w, w turb , τ } , all of them assumedto be constants. In the following we analyze the cases of two specific choices for w and w turb . • The case w turb = w < − H = H Z , Z = 1 + 32 γH t. (4.59)Hence, we have a Big Rip singularity after a finite time t s = 23 | γ | H , (4.60)and we obtain correspondingly a = a Z / γ , ρ = 3 H κ Z Z + 3 τ H . (4.61)In the vicinity of t s , using that Z = 1 − t/t s , we find H ∼ t s − t , a ∼ t s − t ) / | γ | , (4.62) ρ ∼ t s − t , ρ turb ρ ∼ t s − t , (4.63)which reveal the same kind of behavior for H and a as in conventional cosmology,nevertheless the singularity in ρ has become more weak. The physical reason for thisis obviously the presence of the factor τ .– 54 –t is interesting to see how these solutions compare with our assumed form (4.58)for the energy equation. The left hand side of Eq. (4.58) can be calculated, andwe obtain in the limit t → t s (details omitted here) the following expression for thepresent energy dissipation: ǫ = 12 | γ | τ . (4.64)This result could hardly have been seen without calculation; it implies that thespecific dissipation ǫ is closely related to the EoS parameter γ and the parameter τ . • The case w < − , w turb > − w < − , w turb > − − < w turb <
0, in which the turbulent pressure will benegative as before. However, it also covers the region w turb >
0, where the turbulentpressure becomes positive as in ordinary hydrodynamics.The governing equation (4.55) can be solved with respect to t as t = 23 | γ | (cid:18) H − H (cid:19) − τ | γ | (cid:18) γ turb | γ | (cid:19) ln (cid:20) | γ | − τ γ turb H | γ | − τ γ turb H H H (cid:21) , (4.65)showing that the kind of singularity encountered in this case is of the Little Rip type.As t → ∞ , the Hubble function H approaches the finite value H crit = 13 τ | γ | γ turb . (4.66)Physically, γ turb plays the role of softening the evolution towards the future singular-ity.We close this subsection by investigating the case of a one-component scenario. Inparticular, instead of assuming the fluid to consist of two components as above, we canintroduce a one-component model in which the fluid starts from t = 0 as an ordinary viscousnon-turbulent fluid, and then after some time, marked as t = t ∗ , it enters a turbulent stateof motion. This picture is definitely closer to ordinary hydrodynamics.Let us follow the development of such a fluid, assuming as previously that w < −
1, inorder for the fluid to develop towards a future singularity. After the sudden transition toturbulent motion at t ∗ , we have that w → w turb and correspondingly p turb = w turb ρ turb .Similarly to the two-component scenario, we assume w turb > −
1, and for simplicity weassume that ζ is a constant.We can now easily solve the Friedmann equations, requiring the density of the fluid tobe continuous at t = t ∗ . It is convenient to introduce the “viscosity time”, namely t c = (cid:18) κζ (cid:19) − . (4.67)Hence, for 0 < t < t ∗ we obtain [86, 87]: H = H e t/t c − | γ | H t c ( e t/t c − , (4.68)– 55 – = a (cid:2) − | γ | H t c ( e t/t c − (cid:3) / | γ | , (4.69) ρ = ρ e t/t c (cid:2) − | γ | H t c ( e t/t c − (cid:3) , (4.70)whereas for t > t ∗ we acquire: H = H ∗ γ turb H ∗ ( t − t ∗ ) , (4.71) a = a ∗ (cid:2) γ turb H ∗ ( t − t ∗ ) (cid:3) / γ turb , (4.72) ρ = ρ ∗ (cid:2) γ turb H ∗ ( t − t ∗ ) (cid:3) . (4.73)Thus, the density ρ at first increases with time, and then decreases again until it goes tozero as t − when t → ∞ . Note that in the turbulent region, p ∗ = w turb ρ ∗ will even begreater than zero in the case where w turb > As discussed in detail in subsection 3.2 above, it is well known that there exist several theo-ries for singularities in the future universe [51, 101]. Amongst them, the Little Rip scenarioproposed by Frampton et al. [132, 133] (for nonviscous fluids) is an elegant solution, whichwe will consider in more detail in this subsection, generalized to the case of viscous fluids.The essence of the original model, as well as of its viscous counterpart, is that the darkenergy is predicted to increase with time in an asymptotic way, and therefore an infinitespan of time is required to reach the singularity. This implies that the equation-of-stateparameter is always w <
1, but w → − p = − ρ − A √ ρ − ξ ( H ) , (4.74)where A is a constant and ξ ( H ) a viscosity function (not the viscosity itself). This is aninhomogeneous equation of state. Assuming a spatially flat FRW universe the first andsecond Friedmann equations write as H = κ ρ, ¨ aa + 12 H = κ ρ + A √ ρ + ξ ( H )] , (4.75)while the conservation equation for energy, namely T ν ; ν = 0, becomes˙ ρ − A √ ρH = 3 ξ ( H ) H. (4.76)Let us study separately the non-viscous and viscous cases.– 56 – (I) Non-viscous case.For comparison, we start from the non-viscous case ξ ( H ) = 0 [132]. Setting thepresent scale factor a equal to one, we obtain t = 1 √ κ A ln ρρ . (4.77)This relation reveals the Little Rip property: the singularity ρ → ∞ is not reached ina finite time. Additionally, the density ρ can be expressed as a function of the scalefactor as ρ ( a ) = ρ (cid:18) A √ ρ ln a (cid:19) . (4.78)Using the first Friedmann equation we can also express a as a function of t , namely a ( t ) = exp ( √ ρ A " exp √ κ At ! − . (4.79) • (II) Viscous case.Let us now switch on the viscous term in (4.74). In this case the second Friedmannequation, as well as the energy conservation equation, will change. We shall considerhere only the simplifying ansatz where the viscosity function is constant, namely ξ ( H ) ≡ ξ = const . (4.80)This choice is motivated mainly from mathematical reasons. Then, from the govern-ing equations above, it follows that t = 2 √ κ A ln ξ + A √ ρξ + A √ ρ . (4.81)Inverting this equation we acquire ρ ( t ) = "(cid:18) ξ A + √ ρ (cid:19) exp √ κ At ! − ξ A . (4.82)Hence, the state ρ → ∞ can indeed be reached, however it requires an infinite timeinterval. This is precisely the Little Rip characteristic, now met under viscous condi-tions. The term ξ /A multiplying the exponential tends to promote the singularity,as mentioned. The influence from the last term ξ /A becomes negligible at largetimes. In this subsection we will discuss the connection of viscous cosmology with thermody-namics. The apparent deep connection between general relativity, conformal field theory(CFT), and thermodynamics, has aroused considerable interest for several years. In thefollowing we will consider one specific aspect of this subject, namely to what extent the– 57 –ardy-Verlinde entropy formula remains valid if we allow for bulk viscosity in the cosmicfluid. For simplicity we will assume a one-component fluid model, and we assume the bulkviscosity ζ to be constant. For more details, the reader may consult Refs. [86, 87, 154–156],and additionally the related Ref. [157].We start with the Cardy entropy formula for an (1+1) dimensional CFT: S = 2 π r c (cid:16) L − c (cid:17) , (4.83)where c is the central charge and L the lowest Virasoro generator [158, 159]. Comparingwith the first Friedmann equation for a closed universe ( k = +1) when Λ = 0, namely H = 8 πG ρ − a , (4.84)we deduce (as pointed out by Verlinde [160]) that formal agreement is achieved if we choose L → Ea, c → π VGa , S → HV G , (4.85)where E = ρV is the energy in the volume V . One noteworthy fact is evident already atthis stage: the correspondence is valid also if the fluid possesses viscosity, since there is noexplicit appearance of viscosity in the first Friedmann equation. Moreover, the equation ofstate for the fluid is so far not involved.In order to highlight the physical importance of the formal substitutions (4.85), let usconsider the thermodynamic entropy of the fluid. As is known, there exist several defini-tions, the Bekenstein entropy, the Bekenstein-Hawking entropy, and the Hubble entropy.We will consider only the last quantity here, called S H . Its order of magnitude can beeasily estimated by observing that the holographic entropy A/ G ( A is the area) of a blackhole with the same size as the universe may be written in the form S H ∼ H − G ∼ HV G , (4.86)since A ∼ H − and hence V ∼ H − . Various arguments have been provided to assume theuniverse’s maximum entropy to be identified with the entropy of a black hole having thesame size as the Hubble radius [161–164]. Nevertheless, more precise arguments of Verlinde[160] lead to the replacement of the factor 4 in the denominator with a factor 2, that is S H ∼ HV G . (4.87)Therefore, one can see that this relation coincides with the last relation of (4.85), indicatingthat the formal substitutions above have a physical basis.Consider now the Casimir energy E C , defined in this context to be E C = 3( E + pV − T S ) . (4.88)We may make use of scaling arguments for the extensive part E E and the Casimir part E C that make up the total energy E . These arguments finally give (details omitted here)– 58 – ( S, V ) = E C ( S, V ) + E C ( S, V ). An essential point is the property of conformal invari-ance, that the products E E a and E C a are volume independent and depend only on S .Hence, we acquire E E = α πa S / , E C = β πa S / , (4.89)where α, β are constants. Their product arises from CFT arguments as √ αβ = 3 for n = 3spatial dimensions. From the formulae above we obtain S = 2 πa p E C (2 E − E C ) , (4.90)which is the Cardy-Verlinde formula. With the substitutions Ea → L and E C → c/
12 it isseen that expressions (4.89) and (4.83) are in agreement, apart from a numerical prefactor.This is caused by our assumption about n = 3 spatial dimensions instead of the n = 1assumption in the Cardy formula.The above arguments were made for a radiation dominated, conformally invariant,universe. Hence, the question that arises naturally is whether the same arguments applyto a viscous universe too. The subtle point here is the earlier pure entropy dependence ofthe product Ea , which is now lost. To analyze this question we may consider the followingequation, holding for a k = 1 , Λ = 0 universe with EoS p = ρ/
3, namely ddt ( ρa ) = 9 ζH a . (4.91)This is essentially an equation for the rate of change of the quantity Ea . Let us comparethis relation with the entropy production formula n ˙ σ = 9 H T ζ, (4.92)where n is the particle number density and σ the entropy per particle. As we observe, bothtime derivatives in (4.92) and (4.91) are proportional to ζ . If ζ is small we can insert theusual solution for the scale factor of the nonviscous case, namely a ( t ) = q (8 πG/ ρ in a sin η ,with η the conformal time (“in” denotes the initial time). As the densities ζ − ρa and ζ − nσ can then be regarded as functions of t (recall that ζ =constant), we conclude that ρa can be regarded as a function of nσ . This implies in turn that Ea can be regarded asa function of S . This property, originally based upon CFT, can thus be carried over to theviscous case too, assuming that the viscosity is small.At this stage we should pay attention to the following conceptual point. The specificentropy σ in (4.92) is a conventional thermodynamic quantity, whereas the identification S → HV / (2 G ) in (4.85) is based on the holographic principle. The latter entropy isidentified with the Hubble entropy S H , and thus we can set nσ H = H/ (2 G ), with σ H thespecific Hubble entropy. The quantity σ H is holography-based, whereas the quantity σ isnot. Finally, note that the same kind of arguments can be also applied in the more generalsituation where the EoS has the form p = ( γ − ρ, (4.93)– 59 –ith γ a constant. For the non-viscous case this analysis was performed by Youm [165],with the result S = " πa γ − √ αβ p E C (2 E − E C ) γ − . (4.94)Lastly, in this case the application to weak viscosity can also be performed as in [86, 87, 154],and when γ = 4 / From a hydrodynamicist’s point of view the inclusion of viscosity concepts in the macro-scopic theory of the cosmic fluid seems most natural, as an ideal fluid is after all anabstraction (unless the fluid is superconducting). Modern astronomical and cosmologicalobservations permit us to look back in history, evaluating the Hubble parameter up to aredshift z of about 2. Armed with such observational data, and having at one’s disposal theformalism of FRW cosmology with bulk viscosity included, one would like to extrapolatethe description of the universe back in time up to the inflationary era, or go to the oppositeextreme and analyze the probable ultimate fate of the universe, which might well be inthe form of a Big Rip singularity. In the present review we have undertaken this quiteextensive program.After fixing the notation in subection 1.1, we began in Section 2 with a presentation ofthe theory of the inflationary epoch, covering cold as well as warm inflation in the presenceof bulk viscosity. We investigated in detail the viscosity effects on the various inflationaryobservables, showing that they can be significant. A point to be noted in this context isthat viscous effects may be represented by a generalized and inhomogeneous equation ofstate.In Section 3 we turned to viscous theory in the late universe. We considered thephantom era with its characteristic singularities. Additionally, we discussed how one candescribe in a unified way the inflationary and late-time acceleration in the framework ofviscous cosmology. The simplest way to achieve this task is to introduce scalar fields.Moreover, we investigated the cosmological scenario of holographic dark energy in thepresence of a viscous fluid, a subject which is related to black hole thermodynamics.In the final Section 4 of our review we dealt with specific topics. We classified variousoptions for the ultimate fate of the universe. We gave an analysis of whether the magnitudeof bulk viscosity derived from observations is sufficient to drive the cosmic fluid from thequintessence into the phantom region. Numerical estimates indicated that such a transitionmight well be possible. Furthermore, we investigated viscous bounce cosmology, and wemade use of isotropic turbulence theory from hydrodynamics to describe the late cosmicfluid. Moreover, we discussed the Little Rip occurrence in the presence of viscosity. Finally,we examined how viscosity influences the Cardy-Verlinde formula, which is a topic thatrelates cosmology with thermodynamics, and falls within the emergent gravity program.We close this work mentioning that it would be both interesting and necessary to applythe cosmography formalism [55, 166–168] in order to impose constraints on the viscosity– 60 –arameters. Contrary to standard observational constraints, the advantage of cosmog-raphy is that it is model-independent, since one expands the scale factor independentlyof the solution of the cosmological equations. In particular, one introduces H = a dadt , q = − a d adt H − , j = a d adt H − , s = a d adt H − , l = a d adt H − , known respectively as Hub-ble, deceleration, jerk, snap and lerk parameters [55]. One can show that these parametersare related through [166–168],˙ H = − H (1 + q )¨ H = H ( j + 3 q + 2) , ... H = H [ s − j − q ( q + 4) − ,d H/dt = H [ l − s + 10( q + 2) j + 30( q + 2) q + 24] , which can easily lead to the distance - redshift relation [55]. Hence, using the Friedmannequation in the case of viscous cosmology, we can relate the above cosmographic quantitieswith the present value of viscosity parameter. This could be a significant advantage, sincethe obtained constraint would be model-independent, and thus more robust that the onesdiscussed in subsection 4.1. Nevertheless, the detailed investigation of this subject liesbeyond the scope of this review, and it is left for a future project.Mostly, this review is based on a theoretical approach. We have however providedinformation concerning quantities related to observations, giving estimations on the infla-tionary observables, as well as on the magnitude of the current bulk viscosity itself.In summary, from the above analysis one can see the important implications andthe capabilities of the incorporation of viscosity, which make viscous cosmology a goodcandidate for the description of Nature. Acknowledgments
This work is supported by MINECO (Spain), Project FIS2013-44881, FIS2016-76363-Pand by CSIC I- LINK1019 Project (S. D. O. and I. B.) and by Ministry of Education andScience of Russia, Project N. 3.1386.2017.
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