Non-Minimal Cosmological Model in Modified Yang--Mills Theory
aa r X i v : . [ g r- q c ] M a y Non-Minimal Cosmological Model in ModifiedYang–Mills Theory
V. K. Shchigolev ∗ , G. N. Orekhova † Ulyanovsk State University, 42 L. Tolstoy Str., Ulyanovsk 432000, Russia
Abstract – In the present paper, we consider a model of non-minimal modified Yang-Mills(Y-M) theory in the Friedmann-Robertson-Walker (FRW) cosmology, in which the Y-M fieldcouples to the scalar curvature through a function of its first invariant. We show that cosmicacceleration can be realized due to non-minimal gravitational coupling of the modified Y-Mtheory. Besides general study, we consider in detail the case of power-law coupling function.We derive the basic equations for the cosmic scale factor in our model, and provide severalexamples of their solutions.
PACS numbers : 98.80.-k, 98.80.Es, 04.30.-w, 04.62.+v
Key words : Cosmological Model, Non-Minimal Coupling, Yang-Mills Fields, AcceleratedExpansion.
Present accelerated expansion of the universe is well proved in many papers [1]-[8]. In orderto explain so unexpected behavior of our universe, one can modify the gravitational theory [9]-[14], or construct various field models of so-called dark energy which equation of state satisfies γ = p/ρ < − /
3. The most studied models consider a canonical scalar field (quintessence) [15]-[17], a phantom field, that is a scalar field with a negative sign of the kinetic term [18]-[21], or thecombination of quintessence and phantom in a unified model named quintom [22]-[25]. In suchfield dark energy scenarios, the potential choice plays a central role in the determination of thecosmological evolution.The alternative approach to the problem of accelerated expansion is the consideration of variousmodifications of the gravity theory. Among such modifications, the theories with non-minimalcoupling of a field to gravity are especially attractive [26]-[32]. For instance, scalar tensor theoriesare generalization of the minimally coupled scalar field theories in a sense that here the scalarfield is non-minimally coupled with the gravity sector of the action i.e with the Ricci scalar R .In non-minimal theories, the matter field participates in the gravitational interaction, unlike itscounterpart in the minimally coupled case where it behaves as a non gravitational source [33]-[36].In particular, the discovery of the accelerated expansion of the Universe encourages furtherdevelopment of the Y-M theory, including its non-minimal coupling to gravity. Numerous attemptshave been made to consider modified Y-M theories in cosmology as alternatives to the dark energy(see, e.g., [37] and references therein). One of the directions in such a generalization of the Y-M theory is connected with a non-minimal extension of the Y-M field theory. There are severalmotivations to study non-minimal Y-M theory (see, e.g., [38, 39] and references therein).Basically, the present study is a sequel to the paper [40], where the modified Y-M theoryis investigated under the condition of minimal coupling to gravity. Following the considerationsprovided in [40], we study a model of non-minimal modified Y-M theory, in which the Y-M fieldcouples to a function of the scalar curvature. We show that the accelerated expansion can berealized due to the non-minimal gravitational coupling of Y-M field. Besides general study, we givesome examples of exact solution for the model under consideration, which surely can not cover allpossible applications of this research. ∗ E-mail: [email protected] † E-mail: [email protected] The model equations
Let the action of our model be presented by a generalization of the modified Y-M action [40] onthe case of non-minimal coupling: S = − Z d x √− g n R κ Ψ( F aik F aik ) + Φ( F aik F aik ) o , (1)where the Yang-Mills tensor is F aik = ∂ i W ak − ∂ k W ai + f abc W bi W ck , and Ψ and Φ are the arbitrarydifferentiable functions. In the case of Ψ ≡
1, Φ = 116 π F aik F aik , this action describes the Einstein-Yang-Mills theory. We assume that the universe space-time is described by a Friedmann-Robertson-Walker (FRW) geometry: ds = N ( t ) dt − a ( t )( dr + ξ ( r ) d Ω ) , (2)where ξ ( r ) = sin r, r, sinh r in accordance with a sign of the curvature k = +1 , , −
1. To study themodel based on action (1) in metrics (2), we substitute this metrics into action (1) and take intoaccount that R = − a ¨ aN − a ˙ a ˙ N + ˙ a Na N for this metrics. As a result, we can obtain the following effective Lagrangian per unit solid angle: L eff = 38 πG (cid:16) a ¨ aN + a ˙ a N − a ˙ a ˙ NN + kaN (cid:17) Ψ( I ) ξ − Φ( I ) a N ξ , (3)where I = F aik F aik . At the same time, the generalized Wu-Yang ansatz for SO Yang-Mills fieldcan be written down as [41]: W a = x a W ( r, t ) er , W aµ = ε µab x b K ( r, t ) − er + (cid:16) δ aµ − x a x µ r (cid:17) S ( r, t ) er . Substituting K ( r, t ) = P ( r ) cos α ( t ) , S ( r, t ) = P ( r ) sin α ( t ) , W ( r, t ) = ˙ α ( t )into this ansatz, we get the following components of Y-M tensor [41] F = F = F = 0 , F = e − P ′ ( r ) (cid:16) m cos α + l sin α (cid:17) , F = e − P ′ ( r ) sin θ (cid:16) m sin α − l cos α (cid:17) , F = e − sin θ (cid:16) P ( r ) − (cid:17) n , (4)presented in the orthonormalized isoframe n = (sin θ cos φ, sin θ sin φ, cos θ ) , l =(cos θ cos φ, cos θ sin φ, − sin θ ) and m = ( − sin φ, cos φ, r . As it has been noted in [41], Y-M field (4) possesses only magnetic compo-nents. From formulas (2) and (4), it is easy to find that the Y-M invariant I = F aik F aik has thefollowing expression: I = 2 e a ξ h P ′ + ( P − ξ i . (5)Varying the Lagrangian density (3) over P ( r ) and taking into account (5), we obtain the followingEuler–Lagrange equation for the Y-M field: n P ′′ − ( P − Pξ oh Q ( t )Ψ ′ + a Φ ′ i + P ′ ∂I∂r h Q ( t )Ψ ′′ + a Φ ′′ i = 0 , (6)where Φ ′ ≡ d Φ( I ) /dI, Ψ ′ ≡ d Ψ( I ) /dI and the following notation is temporarily introduced: Q ( t ) = 38 πG (cid:16) a ¨ aN + a ˙ a N − a ˙ a ˙ NN + kaN (cid:17) . The particular solution of the Y-M equation in the FRW metrics has been obtained in [41]. Ithas the form P ( r ) = ξ ′ ( r ) = cos r, cosh r for k = +1 , − ∂I∂r = 0. It is easy to prove the latter, as for this solution the Y-M invariant depends only on time: I = I ( t ) = 6 e a ( t ) . (7)Varying Lagrangian over a ( t ) and N ( t ) with the subsequent choice of gauge N = 1, one can obtainthe following equations for our model: h aa + (cid:16) ˙ aa (cid:17) + ka i Ψ( I ) − I Ψ ′ ( I ) h aa − (cid:16) ˙ aa (cid:17) + ka i + 16 (cid:16) ˙ aa (cid:17) I Ψ ′′ ( I ) == 8 πG h Φ( I ) − I Φ ′ ( I ) i , (8) h(cid:16) ˙ aa (cid:17) + ka i Ψ( I ) − (cid:16) ˙ aa (cid:17) I Ψ ′ ( I ) = 8 πG I ) , (9) Several examples of exact solution (I)
The arbitrariness of differentiable functions Ψ( I ) and Φ( I ) essentially complicates the generalanalysis of the model equations (8),(9). Therefore, we will consider some special cases. Let us beginwith the power-law dependencies: Φ( I ) = A I n , Ψ( I ) = B I m , (10)where A and B are some dimensional constants, free parameters of the model. The substitutionof expressions (10) into equations (8), (9) leads to the following set of equations:(1 − m ) h aa + (cid:16) ˙ aa (cid:17) (1 − m ) + ka i = 8 πG AB (3 − n ) I n − m , (11) (cid:16) ˙ aa (cid:17) (1 − m ) + ka = 8 πG AB I n − m . (12)As one can see from these equations, the case m = 1 / m = 1 / n = 3 /
4, i.e. the power of I in equation (12) is equalto n − m = 1 / A and B of the form: k = 8 √ πG e AB . Thus, the scale factor remains uncertain, or arbitrary. It is easy to prove thatthe same unclearly interpreted result will turn out, if functions (10) are directly substituted intoLagrangian (3) together with n = 3 / , m = 1 / A (6 /e ) / a − ( t ) , Ψ = A (6 /e ) / a − ( t )).Varying then it over a ( t ) and N ( t ), we again arrive at the same relationship between A and B . Asfor the equation for the second derivative of the scale factor, it will be satisfied identically for any a ( t ). Putting this case aside, we consider our model with m = 1 / aa = 8 πG AB [2( n − m ) − m − I n − m (13)for the second derivative of the scale factor.It follows from (7) that I >
0. Therefore, the necessary condition of accelerated expansion ofthe universe (¨ a > AB [2( n − m ) − m − > . The latter inequality must be solved for m > /
4, or m < /
4. The result of solving is presentedin Table 1. 3a
A/B > m > / n − m > / A/B > m < / n − m < / A/B < m > / n − m < / A/B < m < / n − m > / Table 1.
Conditions for the free parameters
A/B, m and n − m , corresponding tocorresponding to accelerated expansion according to equation (13).It is necessary to emphasize that equation (13) is employed for the analysis of the acceleratedmode of evolution, but is a differential consequence of equation (12). Therefore, the model dynamicsis defined by the only independent equation (12). As one aims to solve this equation at the conditionof accelerated expansion, it is necessary to take into account Table 1.From Table 1, it follows that in the case of standard Y-M Lagrangian ( n = 1) the acceleratedexpansion is possible, if (Ia) A/B > m ∈ (1 / , / A/B < m < / m = 0) corresponds to (Id). However,equation (12) at A/B < , m = 0 , n = 1 and ( k = +1) has no the real solution, and the case (Ia)does not correspond to m = 0. The latter simply means the absence of the accelerated mode inEinstein-Yang-Mills minimal theory [37]. The trivial solution with accelerated expansion can beobtained in the case A/B < , m = 0 , n = 1, and for the negative sign of curvature. Taking intoaccount (7), and supposing A = 1 / π, B = −
1, we can express equation (12) and its solution asfollows: ˙ a = 1 − Ge a , a ( t ) = r Ge + t , (14)where the constant of integration is equal to zero for the sake of simplicity. For this solution, theacceleration equals ¨ a = ( G/e ) / [( G/e ) + t ] / >
0, and it decreases to zero with time. It isinteresting that this model of non-massive Y-M field with linear dependance on I = F aik F aik leadsto the accelerated expansion. However, due to equation (12), this model violates the weak energycondition. Therefore this example of solution (14) is exclusively illustrative.Let us consider now two examples of exact solution for this model which are not so trivial butrather simple. a ) Let n = 1 , m = 3 /
4, that is we will consider the case (Ic) from Table 1. Then equation (12)can be written down as follows:2 ˙ a − k = C a, where C = 8 πG (cid:18) e (cid:19) / (cid:12)(cid:12)(cid:12) AB (cid:12)(cid:12)(cid:12) . The obvious solution for this equation is: a ( t ) = C t − C ) − kC , where C ≥ k/C . This model experiences constant acceleration: ¨ a = C/
4, and the Hubbleparameter is equal H = 2( t − C )( t − C ) − k/C . b ) We now consider an example of exact solution with n = m that corresponds to (Ib) and (Ic)in Table 1. In these cases, equation (12) can be written down as follows: M ˙ a − δ k = C a , (15)where C = 8 πG (cid:12)(cid:12)(cid:12) AB (cid:12)(cid:12)(cid:12) , M = | m − | , δ = (cid:26) +1 for m > / , − m < / . For the positive sing of curvature ( k = +1) and m > /
4, as well as for the negative sign ofcurvature ( k = −
1) and m < /
4, the exact solution for equation (15) is equal a ( t ) = 1 √ C sinh (cid:16)r CM t + C (cid:17) , C is an integration constant. For the case of negative curvature ( k = −
1) and m > /
4, orpositive curvature ( k = +1) and m < /
4, we can obtain the following solution for equation (15): a ( t ) = 1 √ C cosh (cid:16)r CM t + C (cid:17) . where C is an arbitrary constant. It is interesting that in these cases, as it can be observed fromthe special feature of solutions and equation (15), this model behaves similarly to the FRW modelin which the only source of gravity is the effective cosmological constant Λ = 3 C/M . (II) To involve in our consideration the widely discussed modifications of Y-M theory (see, forexample, [36]-[39], [42]), we assume that function Ψ( I ) = B I m , and Φ( I ) arbitrarily depends on I . Then the main equations of our model can be written down as follows:(1 − m ) h aa + (cid:16) ˙ aa (cid:17) (1 − m ) + ka i = 8 πG B [3Φ( I ) − I Φ ′ ( I )] I m , (16) (cid:16) ˙ aa (cid:17) (1 − m ) + ka = 8 πG B Φ( I ) I m . (17)Combining these equations, it is possible to obtain, instead of (16), the following equation forthe second derivative of the scale factor:¨ aa = 8 πG B I − m [(1 + 2 m )Φ( I ) − I Φ ′ ( I )]1 − m . (18)It is easy to verify that (18) is a differential consequence of equation (17) which remains the onlyindependent equation of our model. Nevertheless, from equation (18) it is possible to obtain anecessary condition for the accelerated regime (see Table 2).IIa B > m > / I ) > B > m < / I ) < B < m > / I ) < B < m < / I ) > Table 2.
Conditions for m and function Θ( I ) = 2 I Φ ′ ( I ) − (1+2 m )Φ( I ), correspondingto accelerated expansion according to equation (18).Let us then consider the case of non-Abelian Lagrangian of the Born-Infeld type (BI): L NBI = β π (cid:16)s F aik F aik β − ( ˜ F aik F aik ) β − (cid:17) , where β is the critical intensity of BI-field, ˜ F aik is a dual Y-M tensor. From formulas (4) and metrics(2), it follows that the second invariant of Y-M field for our solution ˜ F ik F ik = 0. So, we can writedown Φ( I ) as Φ( I ) = 116 πα (cid:16)p αI − (cid:17) , (19)where α = 1 / β . From the latter, it is easy to find that2 I Φ ′ ( I ) − (1 + 2 m )Φ( I ) = 116 πα [(2 m + 1)( p αI − − mαI ] p αI . (20)In view of inequalities 2 αI > √ αI >
1, one can find that expression (20) will be positive,only if m ∈ (0 , / a ( t ) > a cr = h α m e (1 − m ) i / . B < m remains arbitrary. However, the acceleration conditionsare different for m < / m > /
4. The first case corresponds to (IIb) with m ∈ ( − / , / B >
0. The cosmic acceleration is possible while 0 < a ( t ) < a cr . In the case (IIc) (i.e. for m > / , B < m requires a thoroughresearch. Therefore, we are going to give more details and consequences of the model consideredhere in our further investigation. Conclusion
In summary, the modified non-minimal Y-M theory in FRW non-flat cosmology are studied inthis paper. First of all, we have derived the set of main equations which determines the modeldynamics: (6), (8), (9). Throughout the last section of the paper, non-minimal coupling to gravityis described by the factor Φ( I ) in the Einstein-Hilbert sector of action (1). The non-trivial solutionof the modified Y-M equation (8) proposed by one of the authors (V.K.S) earlier allows us to buildseveral modifications of accelerated cosmic expansion in the frame of non-minimal coupling. Besidesgeneral study, we have considered in detail the power-law dependence of Φ( I ) on its argument. Wehave derived the basic equations for the cosmic scale factor in our model, and have provided severalexamples of their solutions. This work implies that the cosmological applications of modified Y-Mtheory with non-minimal coupling to gravity may have more fruitful phenomena, which is worthstudying further. References [1] S. Perlmutter et al., Astrophys. J. 517, 565 (1999).[2] C. B. Netterfield et al., Astrophys. J. 571, 604 (2002).[3] N. W.Halverson et al., Astrophys. J. 568, 38 (2002).[4] S. Bridle, O. Lahab, J. P. Ostriker and P. J. Steinhardt, Science 299, 1532 (2003).[5] D. N. Spergel et al., Astrophys. J. Suppl. Ser. 148, 175 (2003).[6] C.L. Bennett, et al., Astrophys. J. Suppl. , 1 (2003).[7] M. Tegmark, et al.[SDSS Collaboration], Phys. Rev. D , 103501 (2004).[8] S.W. Allen, et al.,Mon. Not. Roy. Astron. Soc.
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