aa r X i v : . [ h e p - t h ] A ug Conformal anomaly around the sudden singularity
S.J.M. Houndjo Departamento de F´ısica, Universidade Federal do Esp´ırito Santo29060-900, Vit´oria, Esp´ırito Santo, Brazil
Abstract
Quantum effects due to particle creation on a classical sudden singularity have been investigatedin a previous work. The conclusion was that quantum effects do not lead to the avoidance nor themodification of the sudden future singularity. In this paper, we investigate quantum correctionscoming from conformal anomaly near the sudden future singularity. We conclude that when theequation of state is chosen to be p = − ρ − Aρ α , the conformal anomaly can transform the suddensingularity in the singularity of type III for any α > / / < α < / Pacs numbers:
There are strong evidences that the universe is in an accelereted expansion. The simplest setup todescribe this acceleration is the use of a cosmological constant, with the equation of state p Λ = − ρ Λ ,where p Λ is the pressure and ρ Λ the energy density of such a cosmological constant. However, thereare others candidates that astronomical datas allow. Namely, a dark energy component whose equationof state parameter, ω , is very close to −
1, where ω is the ratio between the pressure p and the energydensity ρ of dark energy. The relevent of this point is that dark energy can lead to quite different scenariosconcerning the futur of the universe. Precisely, if ω > −
1, dark energy corresponds to quintessence fluid;if ω = −
1, it is a cosmological constant and the universe would be asymptotically de Sitter; if ω < − e-mail: [email protected] Type I (Big Rip): for t → t s , a → ∞ , ρ → ∞ and | p | → ∞ or ρ and p are finite at t = t s . • Type II (sudden): for t → t s , a → a s , ρ → ρ s and | p | → ∞ . • Type III: for t → t s , a → a s , ρ → ∞ e | p | → ∞ . • Type IV: for t → t s , a → a s , ρ → p → H diverge.Here, a means the scale factor and t s referring the instant of the singularity. We assume the cosmicfluid to be ideal.Our attention is focalized on the sudden singularity for which a and ρ are finite but p diverges [3].This singularity occurs without violating the strong energy condition but violates the dominant one asmentioned above. Generally, when singularities appear, the relevent question to ask is: can the singularitybe modified or eliminated by taking into account some physical effects? The answer is affirmative, dueto at least two reasons. The first one is the quantum corrections due to the conformal anomaly . This isused in various cosmological applications: the dynamical Casimir efffect with conformal anomaly [7, 8],or a dark fluid with conformal anomaly [9]. The second reason, which is not the subject of this workis to take into account the bulk viscosity of the cosmic fluid. A number of studies has been carried outabout quantum effects near the sudden singularity and various answers have been found. In [4], Barrowet al have shown that quantum effects due to particle creation can not prevent nor modify the futuresudden singularity. In that case the quantum effects have been studied comparing the behaviour of theenergy density of the particle creation with the energy density of the fluid which characterizes the classicalbackground. The effects of loop quantum gravity have been investigated in cosmologies exhibiting classicalsudden singularities and they can remove the sudden singularity under certain particular conditions [5],and there is a close relationship between sudden singularities and the behaviour of Friedmann universescontaining bulk viscous stresses [6]. Brevik et al [10] have shown that the conformal anomaly correctionscauses the exponents of the future singularity to be modified. They dealt with the singularities of typeI and type III. The general structure of the singularities has been studied by Nojiri [2] and the energyconditions about each of them presented. It is important to note that many results have been foundconcerning quantum effects near the big rip. In an early work [11], it has been shown that quantumeffects due to particle creation can not prevent the big rip. It has been used the transplankian problemfor calculating the energy density of particle creation and found that this energy density tends to zerowhen the big rip is approched. The same idea, for analysing the quantum effects near the big rip hasbeen estudied in [12] where it has been used the n -wave regularization for calculating the energy densityof particle creation and found that, in this case, it tends to infinity when the big rip is approched andbecomes the dominant component of the universe. This means that the big rip can be avoided by a scalarmassless field. The opposite of the latter conclusion has been done by Pavlov [13] where he dealt with amassive scalar field, but this, in a very particular case.In this paper we study the quantum effects due to conformal anomaly on the sudden singularity.2pecifically, we seek to know if quantum effects can modify or eliminate the classical sudden singularity.We use the conformal anomaly and investigate its influence on the classical background density. Wefind that, depending on the value of the parameter α , quantum effects due to conformal anomaly cantransform the sudden singularity in the singularity of type I (big rip), or in the singularity of the typeIII or also in the big crunch.The paper is organized as follows. In the next section, we will set out the sudden singularity models.In section 3 the conformal anomaly near the singularities in general and in particular its effects near thesudden singularity will be presented. The analyse will be done in the case for which the thermodynamicalparameter ω in the equation of state p = ωρ , depends on the time . Note that it is impossible to havea constant thermodynamic parameter in the sudden singularity, since the pressure is singular while thedensity is finite. In the section 4 we present the conclusions. We considere the most general form of the sudden singularity which can occur at finite time in aFriedmann expanding universe as discussed in [3]. It has been shown that singularities of this sort canarise from a divergence of the pressure, p , at finite time despite the scale factor, a ( t ), the material density, ρ , and the Hubble parameter rate, H = ˙ a/a , remainning finite. The pressure singularity is connectedwith the divergence in the acceleration of the universe, ¨ a , at finite time. This singularity occurs withoutviolating the strong energy condition ρ + 3 p > (cid:18) ˙ aa (cid:19) = 8 πGρ, (1)¨ aa = − πG ρ + 3 p ) , (2)˙ ρ + 3 ˙ aa ( ρ + p ) = 0 . (3)In [3], it has been constructed an explicit example for seeking, over the time interval 0 < t < t s , a suitablesolution for the scale fator a ( t ), of the form a = (cid:18) tt s (cid:19) q ( a s −
1) + 1 − (cid:18) − tt s (cid:19) n . (4)Hence, the first and second derivatives of the scale factor remain˙ a = qt s (cid:18) tt s (cid:19) q − ( a s −
1) + nt s (cid:18) − tt s (cid:19) n − , (5)¨ a = q ( q − t s (cid:18) tt s (cid:19) q − ( a s − − n ( n − t s (cid:18) − tt s (cid:19) n − . (6)3n order to have the divergence in the second derivative while the scale factor and his first derivativeremain finite, one imposes 1 < n < < q ≤
1. As t → t s , we have a → a s , H → H s and ρ → ρ s where a s , H s and ρ s are all finite but p → ∞ . As t → t s , due to the fact that the density is finite andthe pressure divergent, the dominant energy condition, | p | ≤ ρ , must always be violated.The equation of state in the sudden singularity case, which we will choose to be the same even whenconformal anomaly is taken into account, is written as p = − ρ − f ( ρ ) . (7)The divergence which occurs in the pressure p must be incorporated by f ( ρ ). Since the energy density ρ is finite near the sudden singularity, we choose f ( ρ ) as f ( ρ ) = A ( ρ s − ρ ) − λ , (8) ρ = ρ s + ρ ( t s − t ) σ , (9)where λ and σ are positive constants, ρ s the energy density at the singularity time and ρ a positiveconstant. This expression of the energy density is used due to the fact that the form of the scale fator in(4) does not let tractable the expression of the energy density through the equation (1). It is clear that,as t → t s , the energy density remains finite while the pressure diverges. The study of the quantum effects near the singularities is of main interest. Here, we will studythe effect of quantum backreaction of conformal matter around the sudden singularity. The conformalanomaly, also known as trace anomaly, is the fact that the trace of the energy momentum tensor (EMT)vanishes for the classical conformally coupled field, g µν T µν = 0, while the trace of the renormalizedexpectation value , g µν h T µν i ren , is non zero.For any classical field with conformally invariant action, e.g. for conformally coupled massless scalarfields or for the electromagnetic field, the trace of the EMT identically vanishes. This can be shown bya simple calculation. If the action S [ φ, g µν ] of a general covariant theory is invariant under conformaltransformations, S [ φ, g µν ] = S [ φ, ¯ g µν ] , g µν ( x ) −→ ¯ g µν ( x ) = Ω ( x ) g µν ( x ) , (10)where Ω( x ) = 0 is an arbitrary smooth function, then the variation of the action with respect to aninfinitesimal conformal transformation must vanish. An infinitesimal conformal transformation withΩ( x ) = 1 + δ Ω( x ) yields δg µν ( x ) = 2 g µν ( x ) δ Ω( x ) , δg µν ( x ) = − g µν ( x ) δ Ω( x ) . (11)4sing the definition T µν ≡ √− g δS m δg µν , (12)where S m is the action of the matter field, we find0 = δS m = Z d x δSδg µν ( x ) δg µν ( x ) = Z d x √− gT µν g µν δ Ω . (13)This relation should hold for arbitrary functions δ Ω( x ), therefore the integrand must vanish for all x , T µµ ( x ) ≡ T µν g µν ( x ) ≡ . (14)This conclusion holds for any classical generally covariant and conformally invariant field theory, but failsfor quantum fields. This failing is due to the fact that when one compute EMT in some quantum vac-uum state, divergencies appear and it necessary to renormalize the quantum EMT. In a renormalizationprocess, the divergence parts wich appear in the expression of the quantum energy momentum tensoris discarded. Consequently, the trace of the energy momentum tensor turns out to be non null. Vari-ous tecnics of regularization can be used for renormalizing the quantum energy momentum tensor andspecifically, for a massless conformal scalar field in a conformally invariant space-time, the renormalizedenergy momentum tensor is [14, 15, 16, 17, 18] h T µν i ren = − π " (cid:18) − R ; µν + R αµ R αν − RR µν (cid:19) + g µν (cid:18) (cid:3) R − R αβ R αβ + 13 R (cid:19) (cid:21) , (15)whose the trace is g µν h T µν i ren = − π (cid:18) (cid:3) R − R αβ R αβ + 13 R (cid:19) = − π (cid:0) R αβγδ R αβγδ − R αβ R αβ + (cid:3) R (cid:1) . (16)Since the quantum corrections usually contain the powers of the curvature or higher derivatives terms,such correction terms plays important role near the singularity. We will use here the general form of theconformal anomaly contribution as backreaction as in [10].The conformal anomaly T A in its general form is written as follow: T A = b (cid:18) F + 23 (cid:3) R (cid:19) + b ′ F + b ′′ (cid:3) R , (17)where F is the square of the Weyl tensor and F the Gauss-Bonnet invariant, which are defined as F = C αβγδ C αβγδ = 13 R − R αβ R αβ + R αβγδ R αβγδ , (18)5 = R − R αβ R αβ + R αβγδ R αβγδ . (19)In general, when we have N scalars, N / spinors, N vector fields, N gravitons and N HD higherderivative conformal scalars, the coefficients b and b ′ are given by the expressions b = N + 6 N / + 12 N + 611 N − N HD π ) (20) b ′ = N + 11 N / + 62 N + 1411 N − N HD π ) (21)whereas b ′′ is an artibrary constant whose value depends on the regularization method used. For theusual matter we have b > b ′ <
0, except for higher derivative conformal scalars.Quantum effects due to the conformal anomaly act as a fluid with energy density ρ A and pressure p A . The total energy density is ρ tot = ρ + ρ A . The conformal anomaly also known as trace anomaly T A is given by T A = − ρ A + 3 p A . (22)The density and the pressure connected with the conformal anomaly obey to the energy conservation lawin the Friedmann universe as in (3): ˙ ρ A + 3 ˙ aa ( ρ A + p A ) = 0 (23)Putting (22) into (23), one finds T A = − ρ A − ˙ ρ A H = − ρ A − adρ A da , (24)which, multiplied by a da , leads to d ( a ρ A ) dt = − a dadt T A = − a HT A . (25)Integrating, one obtains ρ A = − a Z a HT A dt. (26)In terms of the Hubble parameter, the conformal anomaly (17) is given by T A = − b ˙ H + 24 b ′ (cid:16) − ˙ H + H ˙ H + H (cid:17) − (4 b + 6 b ′′ ) (cid:16) H (3) + 7 H ¨ H + 4 ˙ H + 12 H ˙ H (cid:17) . (27)Taking into account the conformal anomaly, the equation (1) turns out to be3 κ H = ρ + ρ A , (28)6here κ = 8 πG . Due to the fact that we are dealing with the sudden singularity, the Hubble parameter H is finite, but the first and higher derivatives diverge initially before the introduction of the conformalanomaly. Also, each higher derivative of H is more singular than the early one. This means that thedominant term in (27) is − (4 b + 6 b ′′ ) H (3) . In the same way, 3 / (8 πG ) H << | ρ A | , and the equation (28)leads to ρ ≈ − ρ A . Hence, we have ˙ ρ ≈ − (4 b + 6 b ′′ ) HH (3) . (29)When one takes quantum effects into account, the energy density of the fluid stop to be finite by a backreaction, since ρ A is a divergente quantity and ρ ≈ − ρ A . In this case, we assume a new form for the stateequation as p = − ρ − f ( ρ )= − ρ − Aρ α (30)where we impose the condition α > /
2. This choice will be clear latter. The constant A is chosen to bepositive for guaranting to the pressure to be negative. Since we know now that the energy density in thepresence of the conformal anomaly is divergent, one chooses the energy density in a new form as ρ = ρ ( t s − t ) − σ , (31)where σ continues being positive constant.Using the conservation law (3) and the equation (30), one obtains H = ˙ ρ f ( ρ )= ˙ ρAρ α . (32)Hence, H ∝ ( t s − t ) − σ ( α − . (33)Using (33), the product H (3) H behaves as H (3) H ∝ ( t s − t ) − σ ( α − , (34)which also is the behaviour of ˙ ρ through the equation (29). Integrating (34), one obtains ρ ∝ ( t s − t ) − σ ( α − . (35)Identifying (35) with (31), we obtain the parameter σ as σ = 42 α − . (36)7ince we need to have the parameter σ as a positive quantity, it is necessary to have the condition α > / p ∝ ( t s − t ) − α α − . (37)With the condition α > /
2, the quantity − α α − is negative. This means that the divergence in thepressure remains. Note that, when the quantum correction becomes important, its works to furnish anegative energy density ρ A which pratically cancels with the energy density ρ of the dark energy. Theconformal anomaly contribution leads to a divergent energy density while the pressure continues beinginfinite. Although, the important question to be asked is to know what happens about the scale factorafter the introduction of the conformal anomaly. There is only two possibilities: the scale factor canremain finite as in the sudden singularity case or become infinite. Thus, the quantum effects due to theconformal anomaly can transform the sudden singularity in the singularity of type III, in the singularityof type I (the big rip) or in the big crunch depending on the value of α . Note that the singularity oftype III is a singularity for which, at a finite time, the scale fator is finite while the pressure and theenergy density are divergent. Thus, it is clear that quantum effects coming from the conformal anomalywork to modify the singular nature of the universe. But when one analyses the behaviour of the Hubbleparameter after the introduction of the conformal anomaly, one see that it behaves as H ∝ ( t s − t ) α − α − , (38)and its rate behaves as ˙ H ∝ ( t s − t ) − α − . (39)This allow us to find the conditions on α for obtainning the results coming from the conformal anomalyeffects.Hence, we see that for 1 / < α < /
2, the Hubble parameter turns out to be divergent and for α > / α > /
2, therate of the Hubble parameter ˙ H is divergent. When the Hubble parameter diverges, this means that wehave three possibilities: • ˙ a diverges and a is constant, or • ˙ a = 0 and a →
0, or • ˙ a and a diverge but ˙ a diverges more than a .In the first case, it is clear that the scale factor is finite and besides to this, the energy density and thepressure of the fluid are divergent: this is the singularity of the type III. In the second case, since thescale factor goes toward zero, the singularity is the big crunch. In the third case, the scale factor isdivergent, knowing that the energy density and the pressure of the fluid are also divergent: this is the8ig rip. Hence, for 1 / < α < /
2, we conclude that the conformal anomaly can transform the suddensingularity in the singularity of type III or in the singularity of type I (big rip) or also in the big crunch.When the Hubble parameter is constant, we assume that the unique possibility is: • ˙ a and a are finite but a = 0.Since the scale factor is finite is this case, we conclude that for α > /
2, the conformal anomaly cantransform the sudden singularity only into the singularity of type III.
We have evaluated the quantum effects due to the conformal anomaly around the sudden singularity.The sudden singularity is a singularity which occurs at finite time, by the divergence appearing in thepressure while the scale factor and the energy density remain finite. This means also that the firstderivative of the Hubble parameter is singular while the proper Hubble parameter remains finite. Sincethe conformal anomaly brings some higher derivative contribution of the Hubble parameter, this leads toa divergent energy density without eliminate the singular behaviour of the pressure and the state equationis assumed to be p = − ρ − Aρ α . Thus, quantum effects coming from the conformal anomaly can modifythe nature of the sudden singularity in an expanding universe. Depending on the latter form of the scalefactor (remaining finite or becoming singular) after the conformal anomaly effects, the sudden singularitycan be transformed in the singularity of type III, or in the singularity of type I (the big rip) or also in thebig crunch. We see that for any α > / / < α < /
2. We conclude also that the conformal anomaly makes the Hubble parameter tobe divergent for 1 / < α < /
2, which is not the case near the sudden singularity without the conformalanomaly contribution. Moreover, we mention that the singular behaviour of the energy density comesfrom the negative energy contribution of the conformal anomaly wich cancels the dark energy one.
Acknowledgement:
The author thanks CNPq (Brazil) for partial financial support and also Prof.J. C. Fabris for his comments and discussions.
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