New exact cosmological solutions to Einstein s gravity minimally coupled to a Quintessence field

Transcription

1 New exact cosmological solutions to Einstein s gravity minimally coupled to a Quintessence field Olga Arias, Tame Gonzalez and Israel Quiros Physics Department. Las Villas Central University. Santa Clara Villa Clara. Cuba israel@mfc.uclv.edu.cu Abstract A linear relationship between the Hubble expansion parameter and the time derivative of the scalar field is assumed in order to derive exact analytic cosmological solutions to Einstein s gravity with two fluids: a barotropic perfect fluid of ordinary matter, together with a self-interacting scalar field fluid accounting for the dark energy in the universe. A priori assumptions about the functional form of the self-interaction potential or about the scale factor behavior are not neccessary. These are obtained as outputs of the assumed linear relationship between the Hubble expansion parameter and the time derivative of the scalar field. As a consequence only a class of exponential potentials and their combinations can be treated. The relevance of the solutions found for the description of the cosmic evolution are discussed in some detail.

2 Dark energy or missing energy is one of the contemporary issues the physics community is more interested in due, mainly, to a relatively recent (revolutionary) discovery that our present universe is in a stage of accelerated expansion[1], that was preceded by an early period of decelerated expansion[]. This missing component of the material content of the universe is the responsible for the current stage of accelerated expansion and accounts for /3 of the total energy content of the universe, determining its destiny[]. This is a new form of energy with repulsive gravity and possible implications for quantum theory and supersymmetry breaking[]. A self-interacting, slowly varying scalar field, most often called quintessence, has been meant to account for the dark energy component. In any successful model of this class, the scalar field energy density should be subdominant at high redshift (in the past) and dominant at low redashift (at present and in the future) in order to agree with observations[, 3]. A variety of self-interaction potentials for the quintessence field has been studied. Among them, the simplest exponential potential (a single exponential) model is unacceptable because it can not produce the transition from subdominant to dominant energy density[3]. Other combinations of exponential potentials have been also studied[4, 5, 6, 7, 8]. Combinations of exponentials are interesting alternatives since these arise in more fundamental (particle) contexts: supergravity and superstring[9], where these types of potentials appear after dimensional reduction. In most cases the occurrence of a self-interaction potential for the scalar field makes difficult to solve analytically the field equations, although some techniques for deriving solutions have been developed. In Ref.[10], for instance, the form of the scale factor is given a priori and, consequently, the self-interaction potential can be found that obeys the field equations. This method has been repeatedly used[11, 1]. However there are cases when exact solutions can be found once the form of the potential is given[7, 8]. In other cases some suitable relationship between the self-interaction potential and the scalar field kinetic energy is assumed[13]. In this letter we explore a linear relationship between the Hubble expansion parameter and the time derivative of the scalar field to derive exact analytic cosmological solutions to Einstein s gravity with two fluids: a barotropic perfect fluid of ordinary matter, together with a self-interacting scalar field fluid accounting for the dark energy in the universe. 1 The assumed relationship between the Hubble parameter and the time derivative of the scalar field is suggested by an implicit symmetry of the field equations. We will be concerned with flat Friedmann-Robertson-Walker (FRW) cosmologies with the line element given by: ds = dt + a(t) δ ik dx i dx k, (1) where the indexes i, k =1,, 3anda(t) is the scale factor. We point out that it is not neccessary to make any a priori assumptions about the functional form of the self-interaction potential or about the scale factor behavior. These 1 This method has been already used in [14] to derive 4d Poincare invariant solutions in thick brane contexts. 1

3 are obtained as outputs of the assumed linear relationship between the Hubble expansion parameter and the time derivative of the scalar field, once one integrates the field equations explicitely. As a consequence only a class of exponential potentials and their combinations can be treated. However, this is not a serious drawback of the method since, as pointed out above, exponential potentials are of prime importance in dark energy contexts accounted for by a quintessence field. The relevance of the solutions found for the description of the cosmic evolution will be discussed in some detail. We use the system of units in which 8πG = c =1. The field equations are: 3H = ρ m + 1 φ + V, () Ḣ +3H =(1 γ)ρ m 1 φ + V, (3) φ +3H φ = V, (4) where γ is the barotropic index of the fluid of ordinary matter, H = ȧ is the Hubble expansion a parameter and the dot accounts for derivative in respect to the cosmic time t. The energy density of the ordinary matter (cold dark matter plus baryons and/or radiation) is related with the scale factor through ρ m = ρ 0,γ a 3γ,whereρ 0,γ is an integration constant coming from integrating the conservation equation. Let us combine equations () and (3) to obtain Ḣ +3H = γ ρ m +V. (5) An implicit symmetry of the left hand side (LHS) of equations (4) and (5)is evident under the change H k φ. I. e., if one assumes a linear relationship between the Hubble expansion parameter and the time derivative of the scalar field; H = k φ, a = e kφ, (6) where k is a constant parameter, the LHS of equations (4) and (5) coincide up to the factor k. We obtain a differential equation for determining the functional form of the potential V : V + 1 k V = γ k ρ 0,γe 3kγφ. (7) Explicit integration yields the following potential which is a combination of exponentials: V = ξ 0 e 1 k φ + γ ρ 6k γ 0,γe 3kγφ,whereξ 0 is an integration constant. This potential can be given in terms of the scale factor if one considers equation (6): V = ξ 0 a 1 k + γ 6k γ ρ 0,γa 3γ. (8) An interesting feature of this potential is that it depends on the type of ordinary fluid which fills the universe. Otherwise, it depends on the barotropic index γ of the matter fluid.

4 This fact implies some kind of interaction between the ordinary matter and the quintessence field much like the interacting quintessence studied in Ref.[15]. By substituting (8) back into Eq. (), we can rewrite this last equation in the following form: (ȧ/a) = k γ 6k ρ 0,γa 3γ + k ξ 0 6k 1 a 1/k, or in the form it will be useful for the rest of this letter: da α γ k a 3γ + β k a 1/k = t + t 0, (9) where α γ k = k γ ρ 6k γ 0,γ, β k = k ξ 0,andt 6k 1 0 is an integration constant. For the purpose of observational testing of the solutions to the field equations (), (3) and (4), it is useful to look for the following magnitudes of astrophysical interest: the scalar field energy density, the scalar field state parameter, ρ φ = 1 φ + V =3H ρ m, (10) 1 ω φ = φ V 1 φ + V = (1 Ω m 1/3k ), (11) 1 Ω m where Ω m = ρ m /3H is the density parameter for ordinary matter that is related with the scalar field density paremeter Ω φ through (just another writing for the field equation ()), Ω φ = ρ φ 3H =1 Ω m, (1) and last but not the least, the deceleration parameter q = (1 + Ḣ/H ): q = 1+ 1 k + 3γ Ω m. (13) While deriving equations (10)-(13) we have used the field equations (), (3), (4) and their combinations. Assuming the linear relationship (6) between the Hubble parameter and the time derivative of the scalar field means that we are introducing a new free parameter, however, this parameter k can be assumed to be a known function of the other free parameters of the theory or of the barotropic index, etc. Another possibility, perhaps the most promising, is to choose a value for k according to the best fitting of the model to the observations. In this leter, for the sake of simplicity, we will study the limiting situations when k<<1and k>>1 respectively. In these cases one can easily find exact analytic solutions to the field equations. A) k<<1 In this case the potential (8) takes the following form: V = ξ 0 a 1/k + γ ρ 0,γ a 3γ, (14) 3

5 meanwhile Eq. (9) can be written as, a 1/k da α γ 0a 1/k + β 0 = t + t 0, (15) where α γ 0 = k γρ 0,γ < 0andβ 0 = k ξ 0. If one introduces the new varaiable x = α 0a γ 1/k k then (15) can be put into the form: dx α γ. Explicit integration yields 0 x +β 0 a(t) ={ ξ 0 sin γρ0,γ [ γρ 0,γ k (t + t 0)]} k. (16) For this soltuion to make sense ξ 0 should be negative, implying that the potential (14) is negative and then the energy density is itself a negative quantity. Therefor, this case seems to be not of interest for the description of the cosmic evolution in any stage. Besides, in this case the deceleration parameter q =1/k +3γΩ m / is always positive, i. e., no inflation stage in the cosmic evolution is available. B) k>>1 This seems to be the most promising case regarding observational testing. The potential (8) can be written as follows: and the equation (9) is now V = ξ 0 a 1/k + γ 6k 6 γ ρ 0,γa 3γ, (17) da α γ a 3γ + β a = t + t 0, (18) where α γ = ρ 0,γ/3 andβ = ξ 0 /3. Under the change of variable x = β a 3γ/ the equation (18) can be transformed into dx = 3γ β (t + t α γ +x 0 ) so, after the integral in the LHS is explicitly taken and, in terms of the old variable, we obtain; a(t) ={ ρ 0,γ ξ 0 sinh [ 3γ 3 (t + t 0)]} 1/3γ. (19) This leads to the following expression for the Hubble expansion parameter: H(t) = coth[ 3γ (t + t 3 3 0)]. The evolution of the scalar field can be found with the help of the second equation in (6), φ(t) =φ 0 + 3kγ ln[sinh[3γ 3 (t + t 0)]], (0) where φ 0 =ln[ρ 0,γ /ξ 0 ] 1/3kγ. The deceleration parameter is given by the following expression: q = 1+ 3γ {cosh[3γ 3 (t + t 0)]}. (1) 4

6 meanwhile, for the other observationally relevant magnitudes: and, ρ m = ξ 0 {sinh[ 3γ 3 (t + t 0)]}, Ω m = {cosh[ 3γ 3 (t + t 0)]}, () ρ φ = ξ 0, Ω φ =tanh [ 3γ 3 (t + t 0)]. (3) The list of observationally relevant magnitudes ends up with the scalar field state parameter (most often called state equation) ω φ = 1. This last value is evident from Eq. (11) in the limit k>>1, and is consistent with present cosmological observations[18]. This fact, toguether with the result in (3) that the scalar field energy density is a constant magnitude over the cosmic evolution strongly supports the idea that the scalar field in our model plays the role of a cosmological constant (a vacuum energy) and, consequently, it may account for the dark energy in the universe. In fig.1 the evolution of the deceleration parameter is shown for γ = 1. We see that the potential (17) supports both an early inflation stage and a late time accelerated expansion through a stage of decelerated expansion (the period during which, in fig., q was positive) in agreement with recent observation of the SN1997ff at redshift z =1.7, confirming a decelerated phase when the universe was a few seconds old[19]. In fig. the evolution of the scale factor (19) is shown for γ = 1 (dust). A period of infinite contraction until the Big bang singularity is reached, is followed by a period of infinite expansion. This solution is in a sense similar to that of the Pre-Big bang scenario[16] or that of a geometrized instanton[17]. However, in those cases a vacuum solution (gravity coupled to a dilaton or to a geometrized instanton respectively) was explored. For γ = 4/3 a different configuration is obtained; the cosmic evolution begins with a Big bang and proceeds with an infinite expansion. Finally, from fig.3 where the evolution of both Ω m and Ω φ is shown, we see that the quintessence field φ was subdominant in the past as required by nucleosynthesis constraints[0], is comparable to the matter density parameter at present (according to present observations Ω m 0.3 andω φ 0.7[]) and will be dominant in the future. The free parameters ξ 0 and t 0 can be fixed once one tries to fit the model with the experimental observations of SN1a (we include the known bounds on Ω m, q 0, etc.[, 3]), including the recent observation of SN1997ff at redshift z =1.7. In fig.4 the evolution of the deceleration parameter q as function of the redshift z is shown for ξ 0 =1,t 0 = We see that the transition from decelerated to accelerated expansion occurs at z Although it is claimed that the transition should be at z 0.5[19], this result is not in contradiction with the SN1997ff observation at z =1.7. In fig.5 we show the distance modulus δ(z) as function of redshift (ξ 0 =1,t 0 = 1.35) both, computed according to the model (solid line) and the experimental curve (dots). Relative deviations are of about.4%. We recall that the best way to reconcile observations with the solution of (9) is by adjusting the free parameter k according to the best fitting of the model to the experimental data so, in principle, although k could be large enough, powers in 1/k expansion could be of interest. 5

7 Finally, because of its simplicity and because k can be chosen arbitrary, we want to briefly explore the false vacuum case γ = 0. The self-interaction potential (8) is now the sum of an exponential plus a cosmological constant, V = ξ 0 a 1/k ρ 0,v, (4) where ρ 0,v is the constant vacuum energy density. The integral in the LHS of Eq. (9) can be explicitely taken to yield the known power-law behavior; where a 0 =(ξ 0 /k (6k 1)) k. For the scalar field one gets a(t) =a 0 (t + t 0 ) k, (5) φ(t) =φ 0 +k ln[t + t 0 ], (6) where φ 0 =lna 1/k 0. In this case (see equation (13) for γ =0)q = 1 +1/k so, for k>1/, one has inflation (q <0). The linear relationship explored in this letter allows, also, deriving solutions in Brans- Dicke and non-minimally coupled theories in general and this will be the subject of forthcoming papers. Summing up. We have derived exact analytic solutions to gravity theory minimally coupled to a self-interacting scalar field by assuming a linear relationship between the Hubble expansion parameter and the time derivative of the scalar field. This relationship is suggested by an implicit symmetry of the field equations. It induces a restriction upon the type of potentials one can deal with: a combination of exponentials that depends on the barotropic index of the matter fluid. We have derived solutions for the limiting cases when the constant parameter introduced in the assumed relationship k << 1andk >> 1 since, in these cases, finding solutions is a very simple task. It is noticed, however, that for the purpose of experimental testing, it could be better to choose k by the best fitting of the model with observations. Nevertheless, we have tested the solution found for k >> 1, γ = 1, andthe main features of present observational cosmology are satisfied within admissible accuracy levels. We acknowledge the MES of Cuba by financial support of this research. References [1] S. Perlmutter et al., Astrophys. J. 517 (1999) , astro-ph/981133; A. G. Riess et al., Astron. J. 116 (1998) , astro-ph/980501; Astrophys.J. 560 (001) 49-71, astro-ph/ [] M. S. Turner, astro-ph/00008 (To appear in the Proceedings of 001: A Spacetime Odyssey (U. Michigan, May 001, World Scientific)). [3] P. J. E. Peebles and Bharat Ratra, astro-ph/

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9 Figure 1: The evolution of the deceleration parameter in cosmological time for dust (γ = 1). The parameters have been chosen such that, at present (t =0),q 0 = 0.5. It is seen that there are both an early inflation and late inflation periods (q <0) and an intermediate stage of decelerated evolution (q >0). Figure : Evolution of the scale factor in cosmological time for γ = 1. The parameter ξ 0 =1. An infinite period of inflationary contraction, through an intermediate period of decelerated evolution (including the Big bang event at t = t 0 = 1.35) is followed by an infinite period of accelerated expansion. This picture is in some sense similar to the string inspired Pre-Big bang scenario. Figure 3: The matter density parameter Ω m (dotted line) and the scalar field density parameter Ω φ (solid line) as functions of the comoving time for γ =1(ξ 0 = 1). It is seen an early stage when the contribution from the quintessence field was subdominant. At present (t =0) both contributions from dust and from the scalar field are of the same order (Ω 0,m =1/3 while Ω 0,φ =/3). In the future the quintessence field will be dominant. Figure 4: Deceleration parameter q as function of the redshift z for values of the parameters ξ 0 =1andt 0 = 1.35 that agree with the known bounds on Ω 0,m,Ω 0,φ, q 0,etc. The transition from decelerated expansion into accelerated expansion occurs at t Figure 5: Modulus distance vs redshift (ξ 0 =1,t 0 = 1.35). The solid line represents the results of the theoretical model, meanwhile the dots account for the experimental data. A satisfactory agreement is achieved (the relative deviations are of approximately.4%). 8