Duality linking standard and tachyon scalar field cosmologies
P.P. Avelino, D. Bazeia, L. Losano, J.C.R.E. Oliveira, A.B. Pavan
aa r X i v : . [ a s t r o - ph . C O ] J un Duality linking standard and tachyon scalar field cosmologies
P.P. Avelino
Centro de F´ısica do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal andDepartamento de F´ısica da Faculdade de Ciˆencias da Universidade do Porto,Rua do Campo Alegre 687, 4169-007 Porto, Portugal
D. Bazeia and L. Losano
Departamento de F´ısica, Universidade Federal da Para´ıba 58051-970 Jo˜ao Pessoa, Para´ıba, Brasil
J.C.R.E. Oliveira
Centro de F´ısica do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal andDepartamento de Engenharia F´ısica da Faculdade de Engenharia da Universidade do Porto,Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
A.B. Pavan
Instituto de F´ısica, Universidade de S˜ao Paulo, Caixa Postal 66318, 05315-970 S˜ao Paulo SP, Brasil (Dated: November 9, 2018)In this work we investigate the duality linking standard and tachyon scalar field cosmologies.We determine the transformation between standard and tachyon scalar fields and between theirassociated potentials, corresponding to the same background evolution. We show that, in general,the duality is broken at a perturbative level, when deviations from a homogeneous and isotropicbackground are taken into account. However, we find that for slow-rolling fields the duality is stillpreserved at a linear level. We illustrate our results with specific examples of cosmological relevance,where the correspondence between scalar and tachyon scalar field models can be calculated explicitly.
PACS numbers: 98.80.Cq
I. INTRODUCTION
The study of cosmic duality has shown to be very useful in particular in the understanding of the correspondencebetween different families of inflationary and dark energy models. Also, in field theory and string theory dualitieshave been used to obtain quantitative predictions in various limits [1], in some cases with interesting implications tocosmology [2]. In a Friedmann-Robertson-Walker Universe (FRW) the equations of motion for the evolution of thetotal density field are invariant under the duality transformation linking a standard expanding cosmology dominatedby quintessence and a contracting cosmology with phantom behavior [3]. This duality has been generalized in orderto include more complex dark energy models, in particular those where an interaction between dark matter and darkenergy fluids is present. In all these models the duality may occur at the background level but is generally broken ata perturbative level.Cosmological scalar fields are expected to be relevant both at early and late times both in the context of primordialinflation and dark energy, respectively. In [4] Padmanabham noted a possible correspondence between a minimallycoupled quintessence scalar field and a tachyon field described by a specific Lagrangian, suggested by Sen [5, 6], in thecontext of string theory. Such correspondence was also explored by Gorini et al. in [7], which explicitly demonstratedthat distinct scalar field and tachyon models may indeed give rise to the very same cosmological evolution, providedsome general assumptions are satisfied [4, 7, 8]. In the present work we return to this issue, focusing on a large classof models for which the first-order formalism described in [12] can be applied. This framework has been very useful toinvestigate exact analytical solutions in the context of supergravity [9], branes [10, 11] and more recently in cosmology[12, 13].The outline of this paper is as follows. In Sec. II we consider FRW models with a generic scalar field and show thatthere is always an infinite set of scalar field models consistent with the same background evolution but, in general, withvery different sound speeds. In Sec. III we explore, at the background level, the duality between standard quintessenceand tachyon scalar field cosmologies. In Sec. IV we investigate the correspondence between standard and tachyonscalar field cosmologies in an inflationary phase in the early universe, considering not only the background evolutionbut also the spectra of scalar and tensor perturbations generated during inflation. In the end of the section we givetwo specific examples where the correspondence between scalar and tachyon scalar field models can be calculatedexplicitly. Throughout this work we use units in which 4 πG = 1. II. FRW MODELS WITH A GENERIC SCALAR FIELD
In this paper we shall consider models with a real scalar field, χ , minimally coupled to gravity with the action S = Z d x √− g (cid:18) − R + L ( χ, X ) (cid:19) , (1)where X = χ ,µ χ ,µ /
2. The energy-momentum tensor of the real scalar field can be written in a perfect fluid form T µν = ( ρ + p ) u µ u ν − pg µν , (2)by means of the following identifications u µ = χ ,µ √ X , ρ = 2 X L ,X − L , p = L ( X, χ ) . (3)In Eq. (2), u µ is the 4-velocity field describing the motion of the fluid (for timelike χ ,µ ), while ρ and p are its properenergy density and pressure, respectively. The equation of state parameter, w is w ≡ pρ = L X L ,X − L , (4)and, if L ,X = 0, the sound speed squared is given by c s ≡ p ,X ρ ,X = L ,X L ,X + 2 X L ,XX . (5)If X is a small quantity, compared to the energy density associated with the scalar field potential, then a genericLagrangian is expected to admit an expansion of the form [14] L = − V ( χ ) + f ( χ ) X + g ( χ ) X + ... , (6)where f and g are functions of χ . Consequently, c s = 1 − gf X + ... , (7)where we are implicitly assuming that f = 0 for all relevant values of the scalar field χ . Hence, c s ∼ ds = dt − a ( t ) (cid:0) dx + dy + dz (cid:1) , (8)where t is the physical time and x , y and z are comoving spatial coordinates. Einstein’s equations then imply H = 23 ρ , (9a)˙ H = − ( ρ + p ) , (9b)where H = ˙ a/a is the Hubble parameter and a dot represents a derivative with respect to physical time. Energy-momentum conservation leads to ˙ ρ = − H ( ρ + p ) . (10)This is not an independent equation since it can be obtained using Eqs. (9a) and (9b). Given initial conditions for ρ the dynamics of the universe is completely determined by w ( a ). However, this is insufficient to determine L ( χ, X )since there is always an infinite set of scalar field models consistent with a given background evolution. This may beaccomplished for example by adjusting a potential for the scalar field, as long as the model allows for the given w ( a ).On the other hand, the background dynamics fully determines the equation of state parameter w = − HH . (11)The same applies to the sound speed, but only if the pressure is a function of the density alone ( p = p ( ρ )). In thatcase c s = − H ˙ HH . (12)For slow-rolling fields this would lead to c s ∼ − III. STANDARD AND TACHYON SCALAR FIELD COSMOLOGIES
In this paper we shall investigate the duality between two families of scalar field models described by the Lagrangians L = 12 φ ,µ φ ,µ − V ( φ ) , (13a) L = − U ( ψ ) p − ψ ,µ ψ ,µ , (13b)where V and U are the potentials for the standard and tachyonic real scalar fields, φ and ψ respectively. The equationof state parameters for the standard and tachyon fields are w φ = ˙ φ / − V ˙ φ / V , (14a) w ψ = − ψ , (14b)assuming they are homogeneous. On the other hand, the sound speed squared is c s = 1 for the standard scalarfield and c s = − w in the case of the tachyon field. This makes the tachyon field an interesting unified dark energycandidate since c s → w → U is a constant then thetachyon model reduces to the standard Chaplygin gas model [15]. Note that ˙ ψ ≤ − ≤ w ψ ≤ − ≤ w φ ≤
1. Hence, the duality between standard and tachyon scalar field cosmologies can only be effectivefor w ≤ H = 13 ˙ φ + 23 V , ˙ H = − ˙ φ , (15)for the standard scalar field, and H = 23 U q − ˙ ψ , ˙ H = − ˙ ψ q − ˙ ψ U , (16)for the tachyon field.Within the first-order formalism introduced in ref. [12], the correspondence may be obtained explicitly. The startingpoint is to assume H is a function of either φ or ψ alone, or equivalently that there is only one value of H correspondingto each value of φ (or ψ ). This assumption is satisfied in many situations of cosmological interest but is not valid ingeneral (for example if φ (or ψ ) is oscillating around a minimum of the potential). However, it will be satisfied in thecase of slow-rolling scalar fields. Using Eqs. (15) and (16) one obtains˙ φ = − H ,φ , (17a)˙ ψ = − H ,ψ H , (17b)and the potentials V ( φ ) = 32 H − ( H ,φ ) , (18a) U ( ψ ) = 32 H r −
49 ( H ,ψ ) H , (18b)for the standard and tachyon scalar fields, respectively.The correspondence between φ and ψ can be made explicitly by taking into account that Eqs. (17a) and (17b)imply that dφdψ = ± r H , (19)so that φ = ± r Z Hdψ , (20a) ψ = ± r Z dφH . (20b)Expanding Eq. (18b) up to first order terms in ( H ,ψ ) /H and using Eq. (19) one obtains U = 32 H −
29 ( H ,ψ ) H = 32 H − ( H ,φ ) V . (21)Consequently, for slow-rolling fields, the scalar field potentials V and U are approximately the same.Note that the knowledge of H ( φ ) ( or H ( ψ )) completely determines V ( φ ) ( or U ( ψ )) but the reverse is not true.This is a result of the freedom to fix the kinetic energy of the scalar field at a given initial time, for a given scalarfield potential. Given V ( φ ) and a particular H ( φ ), the solution ¯ H ( φ ) = H ( φ ) + ∆ H ( φ ) is also possible with the samepotential as long as 32 (∆ H ) + 3 H ∆ H −
12 (∆ H ,φ ) + H ,φ ∆ H ,φ = 0 . (22)However, in general, this change in H ( φ ) will lead to a modification of the corresponding potential. IV. DYNAMICS OF INFLATION
Up to this point the discussion has been fairly general with few assumptions being made about the dynamics of theuniverse. In this section we shall investigate the correspondence between standard and tachyon scalar field cosmologiesduring an inflationary phase in the early universe. This is essential for establishing a correspondence between the twotheories since the classical fluctuations we observe in the universe today are expected to have been generated duringinflation.
A. Slow-roll parameters and sound speed
We start by defining a set of first order slow-roll parameters, expressed as a function of H and its time derivatives,as [16] ǫ n ≡ ( − n H d n Hdt n × (cid:18) d n − Hdt n − (cid:19) − , (23)where n ≥ d H/dt = H . These parameters can be written either as a function of the standard or the tachyonicscalar field, namely ǫ = − ˙ HH = H ,φ H = 23 H ,ψ H (24) ǫ = ¨ HH ˙ H = − H ,φφ H = − " H ,ψψ H − H ,ψ H , (25) ǫ = − ... HH ¨ H = 2 H ,φφ H + H ,φ H ,φφφ HH ,φφ = 4 H H ,ψψ + 2 H H ,ψ H ,ψψψ − HH ,ψ H ,ψψ − H ,ψ H h HH ,ψψ − H ,ψ i . (26)The sound speed can be expressed in terms of the slow-roll parameter ǫ . Using Eqs. (3), (6), (9b) and (24) one canshow that ˙ H = − H ǫ = − X L ,X ∼ − Xf , (27)where the approximation is valid up to first order in X . Then, using Eq.(7), one obtains c s ∼ − gH f ǫ ∼ − c ǫ , (28)again up to first order in X . Here, c is a constant which is equal to 0 for a standard scalar field and 2 / B. Scalar and tensor perturbations
The spectra of scalar and tensor fluctuations at horizon crossing ( P R and P g , respectively) produced during inflationhas been calculated in [19] (see also [20]). Up to first-order in the slow-roll parameter ǫ P R ( k ) = (cid:2) α + 1 − c ) ǫ + α ǫ (cid:3) × H π ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = aH , (29) P g ( k ) = [1 − α + 1) ǫ ] × H π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = aH , (30)where α = 2 − ln 2 − γ ≃ − . γ is the Euler constant .The scalar spectral index n , the tensor spectral index n T , and the tensor-scalar ratio r , defined by n ≡ d ln P s ( k ) d ln k , n T ≡ d ln P g ( k ) d ln k , r = P g P R , (31)were also calculated in [19] up to second order in the slow-roll parameters: n = 1 − ǫ − ǫ − [2 ǫ + (2 α + 3 − c ) ǫ ǫ + αǫ ǫ ] , (32a) n T = − ǫ [1 + ǫ + ( α + 1) ǫ ] , (32b) r = 16 ǫ [1 + αǫ − c ǫ ] . (32c)This shows that, up to first-order in the slow-roll parameters, the spectral indexes are not sensitive to sound speed.The difference between two dual theories (at the background level), (a) and (b), only appears at second order in theslow-roll parameters: n a − n b = ( c a − c b ) ǫ ǫ , n a T − n b T = 0, r a − r b = 16( c b − c a ) ǫ . If (a) and (b) represent standardand tachyon scalar field inflationary cosmologies then c a = 0 and c b = 2 / n a − n b = − ǫ ǫ / n a T − n b T = 0, r a − r b = 32 ǫ / V. EXAMPLES
In this section we shall compute the correspondence, at the background level, between tachyon and standard scalarfield cosmologies for two specific classes of models of cosmological interest for the dynamics of inflation.
A. Example 1
Consider H ( φ ) = Ae − Bφ + C , where A > B > C ≥ V ( φ ) = 12 A (3 − B ) e − Bφ + 3 ACe − Bφ + 32 C , (33)with φ = 1 B ln (cid:0) AB ( t − t ) + e Bφ (cid:1) , (34)which simplifies to φ = B − ln( AB t ) if we choose, without loss of generality, initial conditions such that φ → −∞ when t →
0. In this case, we have H = 1 B t + C , (35)and we obtain the following power law solution for the evolution of the scale factor with the time a ( t ) = ( t/t ) /B e C ( t − t ) , (36)where the scalar factor has been normalized to unity at the time t >
0. Hence, the universe is always accelerated(¨ a >
0) for t ≥ C = 0 and B <
C > B ≤
1. If
C >
B > t > t i = ( B − /CB .If C = 0 and B = √
3, the dual tachyon field is ψ = ± r
23 1
AB e Bφ , (37)and the corresponding potential is given by U ( ψ ) = r − B B ψ . (38)In this case the potential (33) reduces to the simple exponential form considered in [22], which corresponds to thetachyon potential (38) studied in [23]. On the other hand, if C = 0 and B = √
3, the first term of the potential givenby Eq. (33) vanishes. In this case the dual tachyon field is given by ψ = ± r
23 1 BC ln (cid:0) A + Ce Bφ (cid:1) , (39)and, for A <<
1, the corresponding potential becomes U ( ψ ) = 32 C (cid:0) Ae − Dψ (cid:1) , (40)with D = 3 / √ C .The slow-roll parameters for the first example are ǫ = B / (1 + CA e Bφ ) , (41) ǫ = − ǫ and ǫ = 3 ǫ . The beginning and the end of inflation is defined by the condition ¨ a = 0 (or alternatively ǫ =1). For C >
B >
1, inflation starts when t ≥ t i = ( B − /CB , which corresponds to φ i = ln ( A ( B − /C ) /B .The number of e-foldings since the beginning of inflation is given by N ( φ ) = Z tt i Hdt = − Z φφ i HH, φ dφ = φ − φ i B + C (cid:0) e Bφ − e Bφ i (cid:1) AB , (42)with t ≥ t i and φ ≥ φ i . B. Example 2
Here we consider a model with H ( φ ) = Aφ n + B , (43)where
A > B ≥ n are real constants. We shall assume that the model is valid only for φ >
0. From (18a) weobtain the potential V ( φ ) = 32 (cid:0) Aφ n + B (cid:1) − n A φ n − , (44)If B = 0 and n = 1 / B = 0 and n = − / φ with physical time is given by φ = (cid:16) n ( n − At + φ − n )0 (cid:17) / (2(1 − n )) , (45)where we take t = 0 and φ >
0. Using (43) and normalizing the scale factor at t = 0 so that a = 1, one obtains a ( t ) = exp (cid:18) − (cid:16) n ( n − At + φ − n )0 (cid:17) / (1 − n ) / (4 nA ) + Bt (cid:19) . (46)If n > t → t ∗ = φ − n )0 / (4 n (1 − n ) A ) < φ → ∞ with H → ∞ and V → ∞ . If n < t = t ∗ = φ − n )0 / (4 n (1 − n ) A ) < φ = 0 and ˙ φ = − H ,φ = 0. At this point the first order formalism describedin Sect. III ceases to be valid. In both cases inflation may only occur at late times, when V → constant. If 0 < n < t = t ∗ = φ − n )0 / (4 n (1 − n ) A ) > φ = 0 and ˙ φ = − H ,φ = 0, if n > /
2, or ˙ φ = − H ,φ = ∞ , if n < / n = 1 then φ = φ e − At (47)and a ( t ) = exp (cid:18) φ A (cid:0) − e − A t (cid:1) + B t (cid:19) . (48)The case with n = 0 is trivial, with φ = φ , H = H and V = V at all times.The slow-roll parameters are given by ǫ = 4 n A ( Aφ + Bφ − n ) , (49a) ǫ = − n (2 n − AAφ + Bφ − n ) , (49b) ǫ = 4 n (3 n − AAφ + Bφ − n ) , (49c)and a necessary and sufficient condition for inflation to occur is ǫ < Aφ + Bφ − n < | n | A .The relation for the dual tachyon field, in terms of the standard scalar field, depends of n and B. For n = 1 / B = 0, the dual tachyon field is ψ = ± r φ − n A (1 − n ) , (50)and the corresponding potential is U ( ψ ) = 32 A ( Dψ ) n/ (1 − n ) (cid:18) − n ( Dψ ) / (2 n − (cid:19) / , (51)where D = p / A (1 − n ), which, recently [25] was considered to describe intermediate inflation, for 0 < / (1 − n ) < n = 1 /
2, the dual tachyon field is ψ = ± r
23 1 A ln( Aφ + B ) , (52)and the corresponding potential is U ( ψ ) = 32 e Dψ (cid:18) − A e − Dψ (cid:19) / , (53)where D = ± p / A , which for D > ψ . For n = 1 and B = 0, the dualtachyon field is ψ = ± r AB arctan r AB φ ! (54)and, for | B | ≥ /
3, the corresponding potential is U ( ψ ) = 32 B sec ( Dψ ) (cid:18) − AB sin (2 Dψ ) (cid:19) / , (55)where D = p AB/ VI. ENDING COMMENTS
In this paper we have shown that, as long as w ≤
0, for each tachyon scalar field cosmology there is a standardscalar field model which leads to the same background evolution, and we have explicitly determined the associatedtransformation between the models. This correspondence, is broken at a perturbative level for w = − c s = 1 and c s = − w for standard and tachyon models, respectively). Still, wehave demonstrated that for w ∼ −
1, the correspondence remains valid, even at a perturbative level, up to first orderin the slow-roll parameters. In particular, in the inflationary regime the two models generate identical spectra ofscalar and tensor perturbations, up to first order in the slow-roll parameters, thus leading to very similar cosmologicalsignatures. We have explicitly determined the correspondence, at the background level, between tachyon and standardscalar field cosmologies for two specific examples of cosmological interest.The authors would like to thank CAPES, CNPq, FAPESP, Brasil and FCT, Portugal for partial support. [1] J. E. Lidsey, D. Wands and E. J. Copeland, Phys. Rept. 337, 343 (2000).[2] G. Veneziano, Phys. Lett. B , 287 (1991).[3] L. P. Chimento and R. Lazkoz, Phys. Rev. Lett. 91, 211301 (2003).[4] T. Padmanabhan, Phys. Rev. D , 021301(R) (2002).[5] A. Sen, JHEP , 048 (2002); Mod. Phys. Lett. A , 1797 (2002).[6] A. Sen, JHEP , 065 (2002); Int. J. Mod. Phys. A , 5513 (2005).[7] V. Gorini, A. Yu. Kamenshchik, U. Moschella and V. Pasquier, Phys. Rev. D , 111 (2006).[9] M. Cvetic and H. H. Soleng, Phys. Rev. D 51, 5768 (1995); Phys. Rep. , 159 (1997).[10] K. Skenderis and P.K. Townsend, Phys. Lett. B , 46 (1999); O. DeWolfe, D.Z. Freedman, S.S. Gubser and A. Karch,Phys. Rev. D , 046008 (2000); D.Z. Freedman, C. Nunez, M. Schnabl and K. Skenderis, Phys. Rev. D , 104027 (2004).[11] A. Celi, A. Ceresole, G. Dall’Agata, A. Van Proeyen and M. Zagermann, Phys. Rev. D , 045009 (2005); V.I. Afonso, D.Bazeia and L. Losano, Phys. Lett. B , 526 (2006); A. Celi, JHEP , 078 (2007); A. Ceresole and G. Dall’Agata,JHEP , 110 (2007); W. Chemissany, A. Ploegh and T. Van Riet, Class. Quantum Grav. , 4679 (2007).[12] D. Bazeia, C.B. Gomes, L. Losano and R. Menezes, Phys. Lett. B
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