Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory

Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory

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July 23, 2019
Abstract

In this paper we investigate the evolution of a Jordan-Brans-Dicke scalar field, , with a power-law potential in the presence of a second scalar field, , with an exponential potential, in both the Jordan and the Einstein frames. We present the relation of our model with the induced gravity model with power-law potential and the integrability of this kind of models is discussed when the quintessence field is massless, and has a small velocity. The fact that for some fine-tuned values of the parameters we may get some integrable cosmological models, makes our choice of potentials very interesting. We prove that in Jordan-Brans-Dicke theory, the de Sitter solution is not a natural attractor. Instead, we show that the attractor in the Jordan frame corresponds to an “intermediate accelerated” solution of the form , as where and , for a wide range of parameters. Furthermore, when we work in the Einstein frame we get that the attractor is also an “intermediate accelerated” solution of the form as where and , for the same conditions on the parameter space as in the Jordan frame. In the special case of a quadratic potential in the Jordan frame, or for a constant potential in the Einstein’s frame, the above intermediate solutions are of saddle type. These results were proved using the center manifold theorem, which is not based on linear approximation. Finally, we present a specific elaboration of our extension of the induced gravity model in the Jordan frame, which corresponds to a particular choice of a linear potential of . The dynamical system is then reduced to a two dimensional one, and the late-time attractor is linked with the exact solution found for the induced gravity model. In this example the “intermediate accelerated” solution does not exist, and the attractor solution has an asymptotic de Sitter-like evolution law for the scale factor. Apart from some fine-tuned examples such as the linear, and quadratic potential in the Jordan frame, it is true that “intermediate accelerated” solutions are generic late-time attractors in a modified Jordan-Brans-Dicke theory.

a]Antonella Cid

b]Genly Leon

c]Yoelsy Leyva

Prepared for submission to JCAP

Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory

• Grupo de Cosmología y Gravitación GCG-UBB and Departamento de Física, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

• Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaíso, Chile

• Departamento de Física, Facultad de Ciencias, Universidad de Tarapacá, Casilla 7-D, Arica, Chile

Keywords: Modified Gravity, Jordan-Brans-Dicke, Dark Energy, Asymptotic Structure.

1 Introduction

A large amount of research has been devoted to the explanation of the late-time acceleration of the universe, either by introducing the concept of Dark Energy or by modifying the gravitational sector itself. Among the simplest candidates for Dark Energy, one can find canonical scalar fields, phantom fields or the combination of both fields in a unified quintom model, see [1, 2]; for the second approach there are several attempts reviewed in [3] (see references therein). Despite their interpretation, both approaches can be transformed one into the other, since the crucial issue is just the number of degrees of freedom beyond General Relativity and standard model particles (see [4] for a review on such a unified point of view). Finally, the above scenarios are well-suited not just for late-time implications, likewise for the description of an inflationary stage [5].

One example of modified gravitational theory is the so called scalar-tensor theory of gravity [6, 7, 8], in particular the Jordan-Brans-Dicke (JBD) theory [6, 7]. In this theory the effective gravitational coupling is time-dependent. The strength of this coupling is determined by a scalar field, the so-called JBD field, . In modern context, JBD theory appears naturally in supergravity models, Kaluza-Klein theories and in all known effective string actions [9, 10, 11, 12, 13, 14, 15, 16]. Furthermore, we can promote the Brans-Dicke (BD) parameter, , presents in the original theory to a non-constant BD parameter and to consider a non-zero self-interaction potential even surviving astrophysical tests [17, 18].

In [19] it was investigated the dynamics of the JBD scalar field with a quadratic potential and barotropic matter. The authors used the dynamical systems approach, revealing that the complexity of dynamical evolution, in homogeneous and isotropic cosmological models, depends on the BD parameter, , and the barotropic matter index, . The authors claim that the quadratic potential function leads naturally to a de Sitter state. The results in [19] were extended by [20] for an arbitrary potential function. In [21], it was investigated the observational constraints on the JBD cosmological model using observational data coming from distant supernovae type Ia, the Hubble function, measurements, information coming from the Alcock-Paczyński test, and baryon acoustic oscillations. However, the values found in [19, 20, 21] for the BD parameter are several orders of magnitude lower than the bound imposed by the Solar System tests [22, 23], and the bounds estimated on the basis of cosmological arguments [24] and [25]. This was the main objection to the models [19, 20, 21] in [26].

In this latter paper, the authors stated that the de Sitter solution is an attractor in the Jordan frame of the BD theory only for the quadratic potential . This result lead them to the claim that de BD cosmology does not have the CDM model as the universal attractor. Additionally, the authors showed that in the stable de Sitter critical point, as well as in the stiff-matter equilibrium configurations, the dilaton is necessarily massless. Due to the recent discussions in the literature concerning this topic, we consider it is worthy to investigate the subject further. In this paper we investigate a JBD scalar field, , with potential , (where we have chosen the positive square root by convention), in the presence of a second scalar field, , with exponential potential to the matter content. Because the addition of , we call this scenario a “modified JBD” theory. We assume that the BD parameter is finite, so the limiting case and (we use ), where GR is recovered, is excluded here.

Several gravity theories consider multiple scalar fields, e.g., assisted inflation scenarios [27, 28, 29, 30, 31, 32], quintom dark energy paradigm [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], among others [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. In particular there are some theories where the role of dark matter is played by a scalar field which dynamically behaves as dust during certain epoch in evolution [56, 57, 58]. The main motivation of this work is to analyse if the de Sitter solution represents a natural attractor in the modified JBD model presented. We investigate the Jordan and Einstein frames, and in both cases we prove that, under the parameter choices and , the late time attractor has an effective equation of state parameter . This region in the parameter space is compatible with the ranges described by observations [22, 23, 24, 25, 59]. We prove that in this modified JBD model, the de Sitter solution is not a natural attractor. Instead, we show that the attractor in the Jordan frame corresponds to an “intermediate accelerated” solution of the form as with . Furthermore, when we work in the Einstein frame we get that the attractor is, as well, an “intermediate accelerated” solution of the form as with .

A scale factor of the form where and was introduced in [60, 61, 62] in the context of inflation. Since the expansion of the universe with this scale factor is slower than the de Sitter inflation ( where is constant), but faster than the power-law inflation ( where ), it was called intermediate inflation. Intermediate inflationary models arise in the standard inflationary framework as exact cosmological solutions in the slow-roll approximation to potentials that decay with inverse power-law of the inflaton field [63]. These models have been studied in some warm inflationary scenarios [64, 65, 66, 67, 68, 69, 70, 71, 72, 73].

Intermediate inflation is also found in the context of scalar tensor theories with a variable BD field in the Jordan frame and different matter content. In Ref. [74] a fluid with constant state parameter is considered. In Ref. [75] the author takes into account a scalar field as matter source and intermediate inflation is found in the slow-roll approximation. In the reference [67, 68] it was investigated warm intermediate inflation in the JBD theory but formulated in the Einstein frame. Since this kind of solutions appear as late-time attractors in our context, we call them “intermediate accelerated” solutions.

We note that under the scalar field rescaling for and without the second scalar field , we obtain from our model a special case of the so-called induced gravity model, which is integrable for a power-law potential . The general solution of this class of models is known, see for example [76]. After conformal transformation, the induced gravity model with power-law potential becomes a General Relativity model with an exponential potential, that it is integrable as well [76, 77]. In this sense, our model can be considered a generalization of the induced gravity models described above since we have included a new scalar field as the matter source. So it would be interesting to see how the behavior of the solutions for the induced gravity model changes when a small scalar field is added. In the subsection 2.1 a discussion of this issue is presented. However, the main purpose of our investigation is not to find analytical solutions but to study the asymptotic behavior of the solutions space of this kind of scenarios without using fine-tuning of the parameters and initial conditions. Dynamical systems theory is a powerful tool for doing this research. Nevertheless, the fact that for some fine-tuned values of the parameters we may get some integrable cosmological models, makes our choice of potentials very interesting.

The paper is organized as follows. In section 2 the model is presented and the field equations in the Jordan’s frame are displayed. In the subsection 2.1 the relation of our model with the induced gravity model for power-law potential is presented and the integrability of this kind of models is discussed when the quintessence field is massless and has a small velocity, . In the subsection 2.2, the system is written as a dynamical system and the stability of the critical points is discussed. We separated the analysis in two parts: the analysis at the finite region, and the analysis at the infinite region, covering all the possibilities. In the subsection 2.3 we present the intermediate accelerated solution as a possible future attractor in the Jordan frame. In subsection 2.4 we investigate a linear potential of , which is equivalent to an extension of the so-called induced gravity model [76, 78]. The dynamical system is reduced to a two dimensional one, and the late-time attractor is linked with the solutions found in section 2.1. In section 3 the equations are written in the Einstein’s frame, through a conformal transformation. In the subsection 3.1 the system is written as a dynamical system and the stability of the critical points is discussed. Special emphasis is given to the possible late time attractors. In subsection 3.2 we present the intermediate accelerated solution as a possible future attractor in the Einstein frame. Concluding remarks are given in section 4.

2 Field Equations in the Jordan’s frame

Let us consider the action written in the Jordan frame as given by:

 SJF =∫√−g(ΦR2−ω02Φgμν∂μΦ∂νΦ−U(Φ)−12gμν∂μϕ∂νϕ−V(ϕ))d4x, (2.1)

where denotes the JBD scalar field, is the BD parameter and represents a quintessence scalar field. For the sake of simplicity we restrict our attention to the cases , with and , but the analysis can be extended to general potentials using similar techniques as in [19, 26]. and are constants. By construction we have assumed is positive and finite, it follows (the value gives infinity). The JBD scalar field plays the role of an effective Planck mass, consequently we assume . However, it can asymptotically evolves to its minimum value as we shall show in the following sections.

By considering a flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric:

 ds2=−dt2+a(t)2[dr2+r2(dθ2+sin2θdφ2)], (2.2)

the field equations become

 ¨Φ=(122ω0+3−3)H˙Φ+12H2Φ2ω0+3−2ΦU′(Φ)2ω0+3−2ω0˙Φ2(2ω0+3)Φ−3˙ϕ22ω0+3, (2.3a) ¨ϕ=−3H˙ϕ−V′(ϕ), (2.3b) 3H2Φ=ω0˙Φ22Φ+U(Φ)+12˙ϕ2+V(ϕ)−3H˙Φ, (2.3c) ˙H=4ω0H˙Φ(2ω0+3)Φ−6H22ω0+3+U′(Φ)2ω0+3−ω0(2ω0+1)˙Φ22(2ω0+3)Φ2−ω0˙ϕ2(2ω0+3)Φ, (2.3d)

where the Hubble expansion rate is given by and the dot denotes derivatives with respect to the cosmic time.

By defining the following effective energy densities and the effective pressuress :

 ρ1=3H2(1−Φ)+ω0˙Φ22Φ+U(Φ)−3H˙Φ, (2.4a) ρ2=12˙ϕ2+V(ϕ), (2.4b) p1=H(3−8ω0(2ω0+3)Φ)˙Φ+H2((6ω0+9)Φ−6ω0+3)2ω0+3−2U′(Φ)2ω0+3−U(Φ)+ +(2ω20+ω0(2ω0+3)Φ2−ω02Φ)˙Φ2+(2ω0(2ω0+3)Φ−1)˙ϕ2, (2.4c) p2=12˙ϕ2−V(ϕ), (2.4d)

the system (2.3) can be written as

 ˙ρ1+3H(ρ1+p1)=0, (2.5a) ˙ρ2+3H(ρ2+p2)=0, (2.5b) H2=13(ρ1+ρ2), (2.5c) ˙H=−12(ρ1+p1+ρ2+p2). (2.5d)

The above phenomenological definitions of the energy densities are not unique, specially if an interaction term between both fields is considered [79].

2.1 Relation with the induced gravity model

Let us observe that by setting , we obtain from (2.1) the so-called induced gravity model [76, 78]:

 SIG=∫√−g(W(σ)R−12gμν∂μσ∂νσ−U(σ2))d4x, (2.6)

under the choices and , given . This model admits exact solutions that we want to discuss in the following.

Starting with , (we have chosen the positive square root by convention) and choosing the parameters we obtain the potential

 U=γ2U0σ24−6γ2.

In order for to be real we have chosen which implies .

Introducing the parametrization [76]

 a=σ−1exp(u+v), (2.7a) σ=exp(A(u−v)), (2.7b)

where is a constant to be specified, the Friedman equation (2.3c) for becomes

 (2A2−3γ2)˙u2−2(2A2+3γ2)˙u˙v+(2A2−3γ2)˙v2+γ2U0=0. (2.8)

Choosing the constant , (2.8) transforms to (see, e.g., similar equations (28) in [76] and (2.24) in [78]):

 ˙u˙v=U012. (2.9)

Substituting the expressions

 ¨v=−U0¨u12˙u2,˙v=U012˙u, (2.10)

the Raychaudhuri equation (2.3d) becomes

 3γ2(12˙u2+U0)2+4√6γ(12˙u2+U0)(−3¨u−12˙u2+U0) +6(U0−12˙u2)(−4¨u−12˙u2+U0)=0, (2.11)

and the equation of motion for the scalar field (2.3a), now reduces to

 −γ3(12˙u2+U0)e√6γ(u−v)(−4√6¨u+12(γ−√6)˙u2+(γ+√6)U0)96(3γ2−2)˙u2=0. (2.12)

Since is nonzero, it follows that both equations are simultaneously satisfied if and only if

 ¨u=12(γ−√6)˙u2+(γ+√6)U04√6. (2.13)

The equation (2.13) admits the general solution :

 (2.14a) Substituting the result for u on the equation (2.9), and integrating out the resulting equation we obtain (2.14b)

respectively, where and are integration constants.

Since we have chosen , given that , hereafter we use the branch given by hyperbolic functions:

 u(t)=c2−2ln(cosh(√6−γ2√U0(24c1+t)2√2))√6γ−6, (2.15a) v(t)=c3+2ln(sinh(√6−γ2√U0(24c1+t)2√2))√6γ+6. (2.15b)

Substituting (2.15) in (2.7) we obtain the solutions:

 σ(t)=e√32γ(c2−c3)sinh−√6γ√6γ+6(Δ(t))cosh−√6γ√6γ−6(Δ(t)), (2.16a) a(t)=e√32γ(c3−c2)+c2+c3sinh√6γ+2√6γ+6(Δ(τ))cosh√6γ−2√6γ−6(Δ(τ)), (2.16b) H(t)=√U0csch(2Δ(t))(2√6γ−3(γ2−2)cosh(2Δ(t)))3√2√6−γ2, (2.16c)

where

 Δ(t)=√6−γ2√U0(24c1+t)2√2. (2.17)

2.1.1 Including a massless scalar field

The equation of motion (2.3b) for a massless scalar field is given by

 ¨ϕ+3˙aa˙ϕ=0, (2.18)

and it admits the solution , where is an integration constant. Combining the parametrization (2.7) with the Raychaudhuri (2.3d) and the equation of motion for the scalar field (2.3a) we obtain

 ¨u=(√6γ+2)(3γ2−2)μ2exp(2√6γ(u−v)−6u−6v)4γ2 +(√32γ−3)˙u2+(3√32γ+3)˙u˙v−γU02√6, (2.19a) ¨v=−(√6γ−2)(3γ2−2)μ2exp(2√6γ(u−v)−6u−6v)4γ2 +(3−3√32γ)˙u˙v+(−√32γ−3)˙v2+γU02√6, (2.19b)

and the Friedmann equation (2.3c), assuming that , i.e., the massless case, now becomes

 ˙u˙v=G(u,v), (2.20a) G(u,v)=112⎛⎜ ⎜⎝(2−3γ2)μ2exp(2√6γ(u−v)−6u−6v)γ2+U0⎞⎟ ⎟⎠. (2.20b)

Combining the above equations we obtain

 3(√6γ+2)(3γ2−2)μ2exp(2√6γ(u−v)−6u−6v)γ −24γ¨u+12γ(√6γ−6)˙u2+γ(√6γ+6)U0=0, (2.21)

which reduces to (2.13) for .

Now we want to simplify further the equations, choosing a new time parameter such that

 u′v′˙τ2=G(u,v), (2.22)

where the comma denotes derivative with respect the new time . Thus, choosing we obtain

 u′v′=1. (2.23)

The second derivatives with respect to are given by

 ¨u=˙τ2u′′+¨τu′, (2.24a) ¨v=˙τ2v′′+¨τv′, (2.24b)

where

 ˙τ=√(2−3γ2)μ2e2√6γ(u−v)−6(u+v)γ2+U02√3, (2.25a) ¨τ=−(3γ2−2)μ2((√6γ−3)u′−(√6γ+3)v′)e2√6γ(u−v)−6(u+v)12γ2. (2.25b)

Finally, the equation (2.1.1) transforms to

 u′′(2(3γ2−2)μ2e2√6γu−2γ2U0e6u+2(√6γ+3)v) +γ(√6(3γ2−2)μ2e2√6γu(u′2+1)+γU0(√6γ+(√6γ−6)u′2+6)e6u+2(√6γ+3)v)=0. (2.26)

As in the previous case, since we are interested in the range of parameters , we omit the solutions of (2.1.1) given in terms of trigonometric functions and we use hyperbolic ones instead. Thus, for the unmodified case we recover the exact solution (2.15)

 u(τ)=c2−2ln(cosh(Δ(τ)))√6γ−6, (2.27a) v(τ)=c3+2ln(sinh(Δ(τ)))√6γ+6, (2.27b)

where , defined for .

We can use the solution (2.29) for constructing an approximated solution for the system when is a small parameter, i.e., assuming that the scalar field is massless and has a small velocity .

Assuming , we define

 u(τ)=U(τ)+μd2(τ)+O(μ2), (2.28a) v(τ)=V(τ)+μd3(τ)+O(μ2), (2.28b)

where are the seed solutions when given by

 U(τ)=c2−2ln(cosh(Δ(τ)))√6γ−6, (2.29a) V(τ)=c3+2ln(sinh(Δ(τ)))√6γ+6, (2.29b)

and are functions to be specified. Substituting in (2.28) and in (2.1.1), expanding in Taylor’s series with respect to the parameter near , we obtain respectively:

 E11+μE12+O(μ2)=0, (2.30) E21+μE22+O(μ2)=0, (2.31)

where the equations must be satisfied.

 E11=0⟹U′(τ)V′(τ)−1=0, (2.32a) E12=0⟹U′(τ)d3′(τ)+V′(τ)d2′(τ)=0, (2.32b) E21=0⟹√6γ−2U′′(τ)+(√6γ−6)U′(τ)2+6=0, (2.32c) E22=0⟹−2d2′′(τ)+2(√6γ−6)U′(τ)d2′(τ) +(2(√6γ+3)d3(τ)+6d2(τ))(√6γ−2U′′(τ)+(√6γ−6)U′(τ)2+6)=0. (2.32d)

Substituting (2.29) in the above equations makes the equations and trivially satisfied, and the equations and simplify now to

 d2′′(τ)=−√6√6−γ2d2′(τ)tanh(Δ(τ)), (2.33a) d3′(τ)=−(√6−γ)d2′(τ)coth2(Δ(τ))γ+√6, (2.33b)

where , with solutions

 d2(τ)=√23f1tanh(Δ(τ))√6−γ2+f2, (2.34a) d3(τ)=√23√√6−γf1coth(Δ(τ))(γ+√6)3/2+f3, (2.34b) where f1,f2,f3 are integration constants.

Henceforth, we obtain the first order (in the parameter ) solution

 u(τ)=c2−2ln(cosh(Δ(τ)))√6γ−6+μ⎡⎢ ⎢⎣√23f1tanh(Δ(τ))√6−γ2+f2⎤⎥ ⎥⎦+O(μ2), (2.35a) v(τ)=c3+2ln(sinh(Δ(τ)))√6γ+6+μ⎡⎢ ⎢ ⎢⎣√23√√6−γf1coth(Δ(τ))(γ+√6)3/2+f3⎤⎥ ⎥ ⎥⎦+O(μ2). (2.35b)

The relative errors in the approximation of (2.35) by (2.29) are:

 Er(u(τ)):=u(τ)−U(τ)u(τ)=μ(√23f1tanh(Δ(τ))√6−γ2+f2)c2−2ln(cosh(Δ(τ)))√6γ−6+O(μ2), (2.36a) Er(v(τ)):=v(τ)−V(τ)v(τ)=μ⎛⎝√23√√6−γf1coth(Δ(τ))(γ+√6)3/2+f3⎞⎠2ln(sinh(Δ(τ)))√6γ+6+c3+O(μ2). (2.36b)

Taking the limit it follows that the above relative errors tend to zero. Thus, the linear terms in in the equation (2.35) can be made a small percent of the contribution of the zeroth-solutions (2.29) taking large enough. Henceforth, this shows that the behavior of the solutions for the induced gravity model does not change abruptly when a small massless scalar field, , is added to the setup.

Finally, going back to the original variables, we obtain the solutions:

 σ(τ)=e√32γ(c2−c3)sinh−√6γ√6γ+6(Δ(τ))cosh−√6γ√6γ−6(Δ(τ))× ⎡⎢ ⎢⎣1+12γμ⎛⎜ ⎜⎝√6(f2−f3)−4f1csch(2Δ(τ))(√6−γcosh(2Δ(τ)))√√6−γ(γ+√6)3/2⎞⎟ ⎟⎠⎤⎥ ⎥⎦+O(μ2), (2.37a) a(τ)=sinh√6γ+2√6γ+6(Δ(τ))cosh√6γ−2√6γ−6(Δ(τ))K1/31 −f2μ(γ(γ(√6γ+10)+2√6)−12)sinh(2Δ(τ))sinh−4√6γ+6(Δ(τ))cosh4√6γ−6(Δ(τ))4(γ+√6)2K1/31 +f3μ(γ(γ(√6γ+14)+10√6)+12)sinh(2Δ(τ))sinh−4√6γ+6(Δ(τ))cosh4√6γ−6(Δ(τ))4(γ+√6)2K1/31+O(μ2), (2.37b) H(τ)=˙τa′(τ)a(τ)=√G(u(τ),v(τ))a′(τ)a(τ) =√F(τ)csch(2Δ(τ))(4√3γ−3√2(γ2−2)cosh(2Δ(τ)))6√6−γ2 +2γf1μcoth(2Δ(τ))csch(2Δ(τ))((30−9γ2)F(τ)+(7γ2−18)U0)√3(γ+√6)(γ2−6)√F(τ) +√2f1μcsch2(2Δ(τ))((2γ4−9γ2+18)F(τ)+(−γ4+γ2−6)U0)(γ+√6)(γ2−6)√F(τ) −√2(γ4−5γ2+6)f1μcosh(4Δ(τ))csch2(2Δ(τ))(U0−F(τ))(γ+√6)(γ2−6)√F(τ) +α2μcsch(2Δ(τ))(F(τ)−U0)(√6(γ2−2)cosh(2Δ(τ))−4γ)2γ√F(τ) +α3μ(F(τ)−U0)(√6(γ2−2)coth(2Δ(τ))−4γcsch(2Δ(τ)))4γ√F(τ), (2.37c) ϕ =ϕ0+∫ττ0μ√G(u(τ′),v(τ′))a(τ′)3dτ′ =ϕ0+∫ττ0⎛⎜ ⎜⎝2√3μK1sinh12√6γ+6−3(Δ(τ′))cosh−12√6γ−6−3(Δ(τ′))√F(τ′)⎞⎟ ⎟⎠dτ′+O(μ2), (2.37d) t=t0+∫ττ0dτ′√G(u(τ′),v(τ′))=t0+∫ττ02√3√F(τ′)dτ′ +∫ττ02f1μcsch(Δ(τ′))sech(Δ(τ′))(U0−F(τ′))(2√3(γ2−3)cosh(2Δ(τ′))−3√2γ)√√6−γ(γ+√6)3/2F(τ′)3/2dτ′ (2.37e)

which generalize solutions (2.16).

In (2.37) we have introduced the expressions

 K1=e3√32γ(c2−c3)−3(c2+c3), K3=(c1γ)2e2√6γ(c2−c3)−6c2−6c3, α2=√√6−γγ(√2γ3+3√3γ2−6√3)f2(γ+√6)3/2(γ2−6), α3=−2√√6−γγ(√2γ3+5√3γ2+12√2γ+6√3)f3(γ+√6)3/2(γ2−6),

and

 F(τ)=U0