Universe Models with Negative Bulk Viscosity
aa r X i v : . [ g r- q c ] J un Universe models with negative bulk viscosity
Iver Brevik • Øyvind Grøn Abstract
The concept of negative temperatures hasoccasionally been used in connection with quantum sys-tems. A recent example of this sort is reported in thepaper of S. Braun et al. [Science , 52 (2013)], wherean attractively interacting ensemble of ultracold atomsis investigated experimentally and found to correspondto a negative-temperature system since the entropy de-creases with increasing energy at the high end of theenergy spectrum. As the authors suggest, it would beof interest to investigate whether a suitable generaliza-tion of standard cosmological theory could be helpful,in order to elucidate the observed accelerated expansionof the universe usually explained in terms of a positivetensile stress (negative pressure). In the present notewe take up this basic idea and investigate a generaliza-tion of the standard viscous cosmological theory, not byadmitting negative temperatures but instead by lettingthe bulk viscosity take negative values. Evidently, suchan approach breaks standard thermodynamics, but mayactually be regarded to lead to the same kind of bizarreconsequences as the standard approach of admittingthe equation-of-state parameter w to be less than − w = − Iver BrevikDepartment of Energy and Process Engineering, Norwegian Uni-versity of Science and Technology, Trondheim, NorwayØyvind GrønOslo and Akershus University College of Applied Sciences, Fac-ulty of Technology, Art and Design, Oslo, Norway
Keywords
Viscous cosmology, negative viscosity
The absolute temperature T is in usual physics boundto be a positive quantity. Under special conditions,however, such as when high-energy states are more oc-cupied than low-energy states, the temperature calcu-lated from the thermodynamical formula1 T = (cid:18) ∂S∂U (cid:19) N , (1)can be a negative quantity. A striking example of thiskind of system has recently been found experimentally,in the form of an attractively interacting ensemble ofultracold bosons; cf. Braun (2013).This is, however, not the first example of a negative-temperature system. Thus negative temperatures areassociated with the properties of paramagnetic di-electrics; cf. for instance, Ref. Landau and Lifshitz(1980), a key factor being here that the ”magnetic spec-trum” has to lie within a finite interval of energy. Itis to be observed generally that the region of negativetemperatures lies not ”below absolute zero” but rather”above infinity”, implying that negative temperaturesare in some sense ”higher” than positive ones.An interesting idea suggested by Braun (2013) is thatthe negative-temperature model may be helpful for theconstruction of a theory of dark energy in cosmology.As is commonly accepted by now, the expansion of theuniverse is a product of a positive tensile stress, or neg-ative pressure, that the cosmic fluid displays.In which ways would it seem natural to general-ize the standard cosmological theory? One possibil-ity might be to allow for a composition of two, ormore, components in the cosmic fluid. Thus the idea of considering the fluid as a mixture of two compo-nents, one ordinary fluid component and one darkenergy component, has received substantial interest;cf., for instance, Balakin and Bochkarev (2011) andBrevik et al. (2012). The line of approach in the presentnote will however be different, namely to allow for neg-ative bulk viscosities in the cosmic fluid. We shall con-sider the fluid as a one-component one. That means,we focus attention on the dark energy component only.In the real universe there may be a composition, as al-ready mentioned, of a dark energy component and oneor more other components, corresponding to normal flu-ids and perhaps also dark matter fluids.At first sight the possibility ζ < ζ relies upon the physical requirement that the timedevelopment of entropy in a non-equilibrium system isa positive quantity. And that is just the condition thatwe wish to relax. Moreover, the characteristic propertyof the dark energy fluid implying that the parameter w occurring in the equation of state, p = wρ, (2)is less than −
1, is a counterintuitive property of thesame kind. So, we think that the possibility of allow-ing for negative values of ζ is not so unreasonable af-ter all, in view of the general bizarre properties of thedark energy fluid. As we will show in the next sec-tion, the reversal of the sign of ζ implies in a naturalway that the entropy of the fluid decreases with time.Thus we are discussing a viscosity-induced, rather thana temperature- induced, violation of the conventionalsecond law in thermodynamics.Further examples of universe models with negativebulk viscosity are investigated in sections 4 and 5.While positive viscosity accelerates the expansion of theuniverse, negative viscosity will in general decrease it.In the summary section, section 6, we trace out someconnections with other lines of approach investigated incontemporary cosmology. Whereas we keep T positiveand find the entropy change with respect to time to benegative, there are other approaches in which T is nega-tive and the corresponding entropy change positive. Insome sense there is an equivalence. It is instructive to start from nonrelativistic theory. Let u i denote the components of the fluid velocity. Theentropy density is S = nσ , where n is the particle(baryon) density and σ the entropy per particle (we use geometric units). Then, if η denotes the shear viscosityand ζ as before the bulk viscosity, we have dSdt = 2 ηT ( θ ik − δ ik ∇ · u ) + ζT ( ∇ · u ) + κT ( ∇ T ) , (3)where θ ik = u ( i ; k ) and κ is the thermal conductivity; ζ (as well as η and κ ) are positive quantities in theconventional theory.This expression can readily be generalized into a rel-ativistic language. We here need some definitions. Let U µ = ( U , U i ) be the four-velocity of the fluid; in co-moving coordinates U = 1 , U i = 0. If g µν is the gen-eral metric, the projection tensor is h µν = g µν + U µ U ν , (4)the rotation tensor is ω µν = h αµ h αν U [ α ; β ] , (5)the expansion tensor is θ µν = h αµ h βν U ( α ; β ) , (6)and finally the shear tensor is σ µν = θ µν − h µν θ, (7)where θ = θ µµ = U µ ; µ is the scalar expansion.With the spacelike heat flux density vector definedas Q µ = − κh µν ( T ,ν + T A ν ) , (8)where A ν = U α U ν ; α is the four-acceleration, we cannow make the effective substitutions θ ik → θ µν , δ ik → h µν , ∇ · u → θ, − κT ,k → Q µ , (9)whereby S µ ; µ = 2 ηT σ µν σ µν + ζT θ + 1 κT Q µ Q µ . (10)Here S µ is the entropy current four-vector S µ = nσU µ + 1 T Q µ . (11)(The sketchy derivation above follows Brevik and Heen(1994); more complete treatments can be found inWeinberg (1971) and Taub (1978).)Assume now spatial isotropy, implying η = 0, andassume that there is no heat flux, Q µ = 0. Then S µ ; µ = ζT θ . (12) We consider henceforth the spatially flat FRW space-time, with metric ds = − dt + a ( t )( dr + r d Ω ) , (13)implying that θ = 3 H with H the Hubble parameter.As S µ ; µ = n ˙ σ in the local rest frame, we have˙ σ = ζnT θ = 9 ζnT H . (14)Thus, if T still means the conventional positive temper-ature in the cosmic fluid, one gets ˙ σ < ζ <
0. Thespecific entropy σ has to decrease with increasing timein this model. A more detailed insight into the physics of this modelcan be achieved by considering the behavior near thephantom divide more closely. This divide is definedas the case w = −
1. As is known from observationsNakamura et al. (2010); Amanullah (2010), w lies closeto − w = − . +0 . − . . (15)A characteristic property of most of the phantom darkenergy models is the occurrence of the Big Rip futuresingularity: once the phantom divide is crossed so asto give w < −
1, the universe is inevitably driven intoa singularity (the scale factor becoming infinity) at afinite time in the future. This was first observed byCaldwell (2002), and has later been re-examined by anumber of researchers (for a recent review including alsomodified gravity, see, for instance, Nojiri and Odintsov(2011)).These early theories assumed the cosmic fluid to benonviscous. Once bulk viscosity is included, the the-ory becomes richer and more flexible. One importantproperty, on which we shall focus attention in the fol-lowing, is that on the basis of a conventional positivevalue of ζ it becomes possible for the fluid to slide fromthe quintessence region ( − < w < − /
3) through thephantom divide into the phantom region and thus after-wards into the future singularity. This was first pointedout in Brevik and Gorbunova (2005). Whether a tran-sition through the ”point” w = − ζ .Consider now the Friedmann equations for the flatspace, θ = 3 κρ, (16) 2 ˙ θ + θ = − κ ( p − ζρ ) . (17)Together with the conservation equation for energy,˙ ρ + ( ρ + p ) θ = ζθ , (18)they provide a set of equations enabling us to derive thegoverning equation for the scalar expansion, or equiva-lently, for the energy density. Imagine first the generalcase for which w = w ( ρ ). Then, if the function f ( ρ ) isdefined via1 + w ( ρ ) = − f ( ρ ) /ρ, (19)we can write the governing equation for ρ in the form˙ ρ − p κρ f ( ρ ) − κρζ ( ρ ) = 0 , (20)which has the solution t = 1 √ κ Z ρρ dρ √ ρf ( ρ )[1 + √ κ ζ ( ρ ) √ ρ/f ( ρ )] . (21)Here we have taken t = t = 0 as the initial point, ρ meaning ρ ( t ).We limit ourselves in the following to the case when f ( ρ ) = αρ , with α a constant. Thus p = wρ = − (1 + α ) ρ. (22)We next need to model the form of the bulk viscos-ity. Probably the most interesting form from a physicalpoint of view is to put the viscosity proportional to thescalar expansion. Therewith we allow for an increase ofthe viscosity in the case of increasingly vigorous move-ments in the cosmic fluid. Let us assume that ζ = τ θ, (23)with τ a constant. This choice has been analyzed re-peatedly also before; cf., for instance, Brevik and Gorbunova(2005); Grøn (1990); Brevik and Grøn (2013). Equa-tion (21) then yields t = 1 √ κ α + 3 κτ (cid:18) √ ρ − √ ρ (cid:19) . (24)This shows that the fate of the universe is criticallydependent of the sign of the prefactor. The conditionfor a Big Rip ( ρ = ∞ ) to occur, is that α + 3 κτ > . (25)In conventional viscous cosmology, as explored inBrevik and Gorbunova (2005), even if the universestarts from a state lying in the quintessence region ( α < τ > τ <
0, the situa-tion becomes reversed. Even if the fluid starts withinthe phantom region ( α >
0) it is possible, if the neg-ative viscosity becomes large enough in magnitude, toabandon the singularity by making the expression onthe left hand side of (25) negative. The fluid goes backto the quintessence region w > −
1, and becomes thusinfinitely thinned in the far future, ρ →
0. The role ofthe phantom divide as a kind of a one-way ”membrane”is in this way no longer upheld.
We here use the results of Mostafapoor and Grøn(2011), but allow for the possibility that the bulk vis-cosity is negative. Let us first consider a flat universemodel with dust, and LIVE in which the interaction ofthe dust and the vacuum energy with stress and thuswith negative absolute temperature, is modeled by neg-ative viscosity. The Raychaudhury equation may thenbe written˙ H + 32 H − κζH − κ ρ Λ = 0 , (26)where κ = 8 πG is Einstein’s gravitational constant and ρ Λ is the density of the Lorentz Invariant Vacuum En-ergy, LIVE, which is constant and may be representedby the cosmological constant.We shall first consider the case where the bulk vis-cosity is constant and negative, ζ = ζ <
0. Thenthe general solution of this equation with a (0) = 0 and a ( t ) = 1 may be written H ( t ) = κ ζ + α coth (cid:18) αt (cid:19) , (27a) α = (cid:18) κ ζ + 13 κρ Λ (cid:19) / (27b) a ( t ) = β exp (cid:20) κζ t − t ) (cid:21) sinh (cid:18) αt (cid:19) , (28a) β = (cid:20) ρ M − H ζ ρ Λ + (3 / κζ (cid:21) / , (28b)where ρ M is the present density of cold dark energywhich has been assumed to be pressure less dust, and H = H ( t ) is the present value of the Hubble param-eter.Note that the value of α is independent of the sign of ζ . The Hubble parameter has an infinitely large initialvalue and decreases towards H ∞ = κζ / α , which ispositive for all values of ζ . It is seen that when ζ isconstant the sign of ζ does change the behavior of theuniverse qualitatively. The age of the universe is t = (2 / α )arcsinh(1 /β ) / , (29)showing that negative viscosity makes the age smaller.We shall then consider the case where the bulk vis-cosity is proportional to the Hubble parameter with anegative constant of proportionality, ζ = ζ H, ζ < H + 32 (1 − κζ ) H − κ ρ Λ = 0 . (30)Solving this equation with the boundary condition that H ( t ) = H gives H ( t ) = H coth (cid:18) H t (cid:19) , (31a) H = H s Ω Λ0 − κζ , (31b) H = H p (1 − κζ )Ω Λ0 . (31c)The scale factor is a ( t ) = " sinh (cid:0) H t (cid:1) sinh (cid:0) H t (cid:1) − κζ . (32)The age of this universe model is t = 23 H p Ω Λ0 (1 − κζ ) arccoth r − κξ Ω Λ0 . (33)In the present case the age-redshift relationship, i.e. therelationship between the time t of emission and the time t of observation of radiation with redshift z is t = t arcsinh (cid:20) √ Ω Λ0 (1+ z )
32 (1 − κζ √ − κζ − Ω Λ0 (cid:21) arctanh q Ω Λ0 − κζ . (34)The corresponding expression for the standard universemodel without viscosity Grøn (2002) is obtained byputting ζ = 0 . For this universe model the equation of continuitytakes the form˙ ρ M + 3(1 − κζ ) Hρ M − κζ ρ Λ H = 0 . (35) Inserting the expression (31) for H we find that Eq. (35)has the general solution ρ M ( t ) = c sinh (cid:0) H t (cid:1) + ρ M ∞ , ρ M ∞ = κζ ρ Λ − κζ ,c = ( ρ M − ρ M ∞ )sinh (cid:18) H t (cid:19) , (36)where ρ M = ρ (0) and ρ M ∞ = lim | t →∞ ρ ( t ). Note that ρ M ∞ < ζ <
0. Hence there exists an instant t where the density of the dust vanishes, after whichthis model is no more physically realistic. The equation ρ ( t ) = 0 leads tosinh (cid:18) H t (cid:19) = r − ρ M ρ M ∞ sinh (cid:18) H t (cid:19) . (37)The scale factor and the Hubble parameter at this in-stant are a ( t ) = (cid:18) − ρ M ρ M ∞ (cid:19) − κζ , (38) H ( t ) = (cid:18) H ρ M − H ρ M ∞ ρ M − ρ M ∞ (cid:19) / . (39)Here there is no singularity at the instant t , only atransition to an unphysical state with negative massdensity. We shall here consider some viscous universe modelsthat have been studied earlier with positive bulk vis-cosity, and investigate how their physical properties be-come changed when the viscosity changes sign.If the vacuum energy is removed in the universemodel described by Eqs. (27) and (28), and the fluid isassumed to obey an equation of state p = wρ with con-stant value of w , the scale factor with a (0) = 0 , a ( t ) =1 is given by Treciokas and Ellis (1971) a (1+ w ) = 23 ζ (cid:18) exp ( 32 ζ t ) − (cid:19) , (40a) t = 23 ζ ln (cid:18) ζ (cid:19) . (40b)Hence with ζ > a | t →∞ = ∞ , but when ζ < a | t →∞ = − / ζ . Some years ago, Murphy (1973) considered a class ofviscous universe models dominated by a viscous fluidwith p = wρ and bulk viscosity ζ = γρ where γ is con-stant. Then the rate of change of the Hubble parameteris˙ H = 32 H (3 γH − − w ) . (41)The solution of this equation with a ( t ) = 1 , H ( t ) = H is given by the equation2 γ w ln (cid:18) γ − wH (cid:19) + 23 (cid:18) H − H (cid:19) = (1 + w )( t − t ) , (42a) H = 1 + w γ − . (42b)On the other hand inserting H = ˙ a/a in the last factorof Eq. (41) and then integrating with the same bound-ary conditions we obtain3 γ ln a + 23 (cid:18) H − H (cid:19) = (1 + w )( t − t ) . (43)Comparison between Eqs. (42) and (43) gives a (1+ w ) = 3 γ − wH . (44)For positive viscosity ( γ >
0) these equations describean expanding universe model, but for negative viscosity( γ <
0) the Hubble parameter must be negative for thescale factor to be positive. Hence the universe modelcontracts.As an example of anisotropic universe models weshall finally consider the effect of negative viscosity ofsome simple universe models of Bianchi type I. The lineelement has the form ds = − dt + X i =1 a i ( t )( dx i ) . (45)The directional Hubble parameters are H i = ˙ a i /a i , andthe anisotropy parameter is A = 13 X i =1 (cid:18) ∆ H i H (cid:19) , (46a)∆ H i = H i − H, H = 13 ( H + H + H ) . (46b)In a universe filled with LIVE with constant density ρ Λ we have; cf. Mostafapoor and Grøn (2013) κ ( ρ + ρ Λ ) = 32 (2 − A ) H . (47) If ρ Λ = 0 and in the case of a relativistically rigid fluidwith w = 1 (a so-called Zel’dovich fluid), with constantbulk viscosity ζ , the anisotropy parameter as a functionof time is A = A e − ζ ( t − t ) . (48)Hence, a positive viscosity leads to decay of theanisotropy, while a negative viscosity increases theanisotropy. This is a characteristic behavior for moregeneral viscous anisotropic universe models. IncludingLIVE the Hubble parameter is H ( t ) = κζ H coth(3 ˆ Ht ) , ˆ H = (cid:18) κζ (cid:19) + κρ Λ . (49)It is seen that the sign of the viscosity does not influ-ence ˆ H , but the average Hubble factor H is smallerwith negative viscosity than with a positive one. Neg-ative viscosity acts like a decelerating force upon theexpansion of the universe. Our purpose in this paper has been to investigate sit-uations in cosmology where the entropy decreases withincreasing time. Specifically, we have achieved this bytaking the bulk viscosity ζ to be less than zero (in ac-cordance with spatial isotropy, the shear viscosity hasbeen put equal to zero). The ansatz ζ < ζ in conventional cosmology is based uponthe requirement that the change of entropy in a non-equilibrium system is positive, and that is just the prop-erty that we wish to relax. One may also observe theanalogy with the phantom era in the expansion of theuniverse, meaning that the parameter w in the equationof state p = wρ is less that −
1. In both cases, bizarrethermodynamic behaviors are encountered.We have shown that in a generalization of the ΛCDMuniverse model with negative bulk viscosity, the viscos-ity contributes with an attractive gravity, and hencetends to decrease the expansion. It turns out that ina model where the negative coefficient of bulk viscosityis proportional to the density of the fluid, expansion isnot allowed. Therefore, after all, even if negative vis-cosity is a theoretical possibility, it does not seem to bea favored property of the cosmic fluid.Finally, it is of interest to put our developments intoa wider perspective by comparing them with some otherworks in modern cosmology. • Our formalism allows for the presence of a negativebulk viscosity ζ but keeps the temperature T positive. We emphasize that the entropy four-vector (11) doesnot depend upon the sign of ζ at all. What changessign with ζ , is the change of entropy with time; cf.Eq. (10). • In many cases, the inclusion of bulk viscosity in cos-mological theory does not lead to significant changes.For instance, the Cardy-Verlinde formula for entropy,cf. Verlinde (2000), has been found to apply undervarious conditions in the presence of viscosity, even inthe case of a multicomponent fluid obeying an inho-mogeneous equation of state; cf. Brevik and Odintsov(2002); Brevik (2002a); Brevik et al. (2010). A some-what stronger influence from cosmology is experi-enced, as mentioned above, in cases where the vis-cosity is large enough to make the fluid pass throughthe phantom barrier w = − • In other cases, when dealing with dark energy,one expects that the the entropy itself can be nega-tive. Thus, Nojiri and Odintsov (2005) considered theeffect of a dark energy ideal fluid by inserting an in-homogeneous Hubble-parameter dependent term in thelate-time universe. Remarkably enough, a thermody-namical dark energy model was found in which, despitepreliminary expectations (Brevik et al. (2004)), the en-tropy of the phantom epoch could be positive. This wascaused by crossing of the phantom barrier. Theories ofthis kind are wider in scope, and generally differentfrom, the one presented by us above.
References