Evolution of magnetic component in Yang-Mills condensate dark energy models

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1 Evolution of magnetic component in Yang-Mills condensate dark energy models arxiv:gr-qc/ v2 26 Jan 2007 Wen Zhao Department of Applied Physics Zhejiang University of Technology Hangzhou, Zhejiang, China Abstract The evolution of the effective Yang-Mills (YM) condensate dark energy model with electric and magnetic components is studied. In the case of electric field being dominant, the magnetic field disappears with the expansion of the universe. The total YM condensate tracked the evolution of the radiation in the early universe, but in the later stage, it becomes ω y 1, so the cosmic coincidence problem is also avoided. But in the case of magnetic field being dominant, ω y > 1/3 holds for all time. So the constraint of E 2 > B 2 must be satisfied for the YM condensate as a kind of candidate of dark energy. PACS numbers: k, Es, w, v wzhao7@mail.ustc.edu.cn 1. Introduction Recent observations on the Type Ia Supernova[1], Cosmic Microwave Background Radiation[2] and Large Scale Structure[3] all suggest a flat universe consisting of dark energy (73%), dark matter (23%) and baryon matter (4%). How to understand the physics of the dark energy is an important issue, having the equation of state (EoS) ω < 1/3 and causing the recent accelerating expansion of the universe. The simplest model is the cosmological constant Λ with ω Λ 1 (which can be viewed as the vacuum energy) and fits the observation fairly well. However, a number of evidences suggest that the EoS of the dark energy may be evolutive. This has stimulated a number of approaches to build the dark energy models with a dynamic field. One class of approaches to the dark energy is to introduce dynamic scalar fields. The most popular one is the quintessence models[4], which can naturally give a state of 1 < ω < 1. To get the state of ω < 1, phantom[5], k-essence[6], quintom[7] are also suggested. Different from the scalar field models, a number of authors have investigated the vector field models[8]. The effective YM field condensate model has also been introduced to describe the dark energy[9, 10, 11, 12, 13]. It has interesting feathers: the YM fields are the indispensable cornerstone to particle physics, gauge bosons have been observed. There is no room for adjusting the form of effective YM Lagrangian as it is predicted by quantum corrections according to field theory. In the previous works, we have investigated the simplest case with only electric component and found several interesting characters of this model: 1) this dark energy can naturally generate the EoS of ω y > 1 and ω y < 1[11], which is different from the scalar

2 quintessence models; 2) with the expansion of the universe, the EoS of the YM condensate naturally runs to the critical state of ω y = 1[11], which is consistent to the observations[14]; 3) the cosmic coincidence problem is naturally avoided in the YM condensate dark energy models[12, 13]; 4) the EoS of the dark energy can cross -1 in the double-field models or coupled models[11, 13]; 5) the big rip is naturally avoided in the YM dark energy models[13]. In this letter, we will discuss the general case of the YM condensate dark energy with both electric and magnetic components. We find that, if the magnetic component was subdominant in the initial condition, it will rapidly decrease to zero with the expansion of the universe. The states of ω y > 1 and ω y < 1 all can be realized in the models, and in the former case, the state of YM condensate was ω y 1/3 in the earlier stage, and later it turned into ω y 1, which is similar to the case with only electric component. So the cosmic coincidence problem is also naturally avoided in the models. But if the magnetic component was dominant in the initial condition, the state of YM condensate will keep ω y > 1/3, which can not be a kind of candidate of dark energy. So we get a new constraint of the YM condensate dark energy models. 2. The effective Yang-Mills field model The effective YM condensate cosmic model has been discussed in Ref.[9, 10, 11, 12, 13]. The effective lagrangian density up to 1-loop order is[15, 16] L eff = b 2 F ln F eκ 2. (1) where b = 11N/24π 2 for the generic gauge group SU(N) is the Callan-Symanzik coefficient[17] F = (1/2)Fµν a F aµν plays the role of the order parameter of the YM condensate, e 2.72, κ is the renormalization scale with the dimension of squared mass, the only model parameter. The attractive features of this effective YM lagrangian include the gauge invariance, the Lorentz invariance, the correct trace anomaly, and the asymptotic freedom[15]. With the logarithmic dependence on the field strength, L eff has a form similar to he Coleman-Weinberg scalar effective potential[18], and the Parker-Raval effective gravity lagrangian[19]. The effective YM condensate was firstly put into the expanding Friedmann-Robertson- Walker (FRW) spacetime to study inflationary expansion[9] and the dark energy[10]. We work in a spatially flat FRW spacetime with a metric ds 2 = a 2 (τ)(dτ 2 δ ij dx i dx j ), (2) where τ = (a 0 /a)dt is the conformal time. Assume that the universe is filled with the YM condensate. Here we study the SU(2) group. The energy density and pressure are given by ρ y = 1 2 ǫ(e2 + B 2 ) b(e2 B 2 ), (3) p y = 1 6 ǫ(e2 + B 2 ) 1 2 b(e2 B 2 ), (4) respectively, where dielectric constant is given by ǫ bln (E 2 B 2 )/κ 2. Here one can define two strength quantities F E 2 B 2, and Q E 2 + B 2. The corresponding dimensionless quantities are defined by f F/κ 2, q Q/κ 2, and ε ǫ/b = ln f. It is easily to find that q f must be satisfied in all discussion. Then the energy density and pressure are rewritten as ρ y = 1 2 bκ2 (εq + f), p y = 1 2 bκ2 ( 1 3 εq f ), (5)

3 respectively. The energy density of YM condensate must be the positive value, which corresponds to a constraint of the YM condensate The EoS of YM condensate is εq + f > 0. (6) ω y = εq 3f 3εq + 3f. (7) At the critical point with the condensate order parameter F = κ 2, one has ε = 0 and ω y = 1, the universe is in exact de Sitter expansion[9]. Around this critical point, F < κ 2 gives ε < 0 and ε < 1, and F > κ 2 gives ε > 0 and ω y > 1. So in the YM field model, EoS of ω y > 1 and ω y < 1 all can be naturally realized. When ε 1, the YM field has a state of ω y = 1/3, becoming a radiation component. As is known, an effective theory is a simple representation for an interacting quantum system of many degrees of freedom at and around its respective low energies. Commonly, it applies only in low energies. However, it is interesting to note that the YM condensate model as an effective theory intrinsically incorporates the appropriate states for both high and low temperature. As has been shown above, the same expression in Eq.(7) simultaneously gives p y ρ y at low energies, and p y ρ y /3 at high energies. Therefore, our model of effective YM condensate can be used even at higher energies than the renormalization scale κ. These characters are exact same with the simple case with only electric field. In the following discussion, we first consider the case of E 2 > B 2, i.e. the electric field is dominant. So the dielectric constant becomes ε = ln f. The effective YM equations are[11, 12] τ (a 2 ǫe) = constant. (8) which reduce to q + f = c a 4 ε 2, (9) where c is the integral constant, and the quantities q and f are the variables. The energy conservation equation also should be considered, which is (ρη) = p, (10) where we have defined η a 3, and the prime denotes d/dη. Using the expresses of ρ and p, one can simplify this equation ( 1 + q ) f + εq = 4 f 3 εqη 1. (11) By the equations of (9) and (11), one can numerically solve the evolution of the EoS of the YM dark energy. From these two equations, one can easily find that, in the YM field condensate models, the cosmic time can be entirely replaced by the scale factor and the evolution of the YM condensate with the scale factor is independent of the other components in the universe, so we can randomly choose the initial condition at a = a i, where a i can be chosen at any time, and the initial condition is chosen as q = q i, f = f i. (12) The integral constant c is also fixed c = (q i + f i )(ln f i ) 2. (13)

4 First we consider the case of ω i > 1, which requires that q i > 0, f i > 1. (14) From the definition of q and f, one knows that q i > f i must be satisfied. The value of q being closer to f suggests that the density of electric field being much larger than which of magnetic field, and q = f suggests that the YM condensate only has electric field. On the contrary, the value of q being much larger than f suggests that the density of electric field being much closer to which of magnetic field, and q f suggests that E 2 B 2. When the values of q and f are all close to 1, one knows that E 2 κ 2 and B 2 0. Here we consider three different models: Mod.a1: f i = 50, q i = 100; Mod.a2: f i = 5, q i = 100; Mod.a3: f i = 5, q i = 10. Solving the evolutive Eqs.(9) and (11), we get the evolution of the EoS, which are plotted in Fig.1. One can find that in all these models, the values of EoS can not cross 1. With the expansion of the universe, the value of ω y will become very close the attractor solution ω y = 1, which is very similar to the cosmological constant. But in the early stage of the universe, the value of ω y was very close to 1/3, so it can track the evolution of the radiation. These features are all same with the case with only electric field in our previous works[11, 12]. So the cosmic coincidence problem is also naturally avoided in these models. In Fig.2, we have plotted the evolution of electric and magnetic fields with the scale factor in these three models. We found that with the expansion of the universe, the value of energy density of the electric field will approach to the renormalization scale κ 2, but which of the magnetic field will approach to zero. The total state of YM condensate will approach to the case with only electric field. In order to account to the present observational value of the dark energy, one also need to finely tune the value of the renormalization scale, so the fine-tuning problem also exists. Then we discuss the case of ω i < 1. The constraint (6) requires that f i < q i < f i /ln f i, f i < 1, (15) which leads to the constraint e 1 < f i < 1. Here we also consider three different models: Mod.b1: f i = 0.9, q i = 2.0; Mod.b2: f i = , q i = 10.0; Mod.b3: f i = 0.5, q i = 0.6. In Fig.3, we have plotted the evolution of EoS of the YM condensate in these three models, and Fig.4 plots the evolution of the electric and magnetic fields of YM condensate. Similar to the previous models, with the expansion of the universe, the EoS of YM field will run to the critical state of ω y = 1, the density of the electric field will approach to the value of κ 2, and the energy density of the magnetic field will approach to zero. In conclusion, in the case of electric field being dominant, the magnetic field decreases quickly with the expansion of the universe. Now we consider the case of magnetic field being dominant. First we consider the extreme case of E 2 = 0, and the energy density and pressure of the YM condensate are ρ y = 1 2 ǫb2 1 2 bb2, p y = 1 6 ǫb bb2, (16) respectively, where the dielectric constant ǫ = bln(b 2 /κ 2 ), which corresponds to the dimensionless quantity ε = ln(b 2 /κ 2 ). From the constraint ρ y > 0, one gets ε > 1. (17)

5 The EoS of the YM condensate is ω y = ε + 3 3ε 3. (18) So the constraint in Eq.(17) can be rewritten as ω y > 1 3, (19) which means that this kind of YM condensate can not give a negative pressure, and can not be a candidate of dark energy. Then we discuss the general case with the initial condition B 2 > E 2. The energy density, pressure and EoS of YM condensate are in the Eqs.(5) and (7). The constraint of ρ y > 0 yields 0 > f > εq. (20) So one can easily get ω y > 1/3, i.e. the YM condensate can not have a negative EoS. However if it is possible for the YM condensate to evolute from the state of B 2 > E 2 to the state of B 2 < E 2, and give the negative pressure? If it is possible, then in the turning point, the YM condensate must have the state of B 2 = E 2, where ǫ =. But from the effective YM equation (9), one knows that either the conditions a = 0 or E 2 = B 2 = 0 must be satisfied, when ǫ =, which all can not be realized. So this kind of transform can not occur. In conclusion, the YM condensate with the state of B 2 > E 2 can not give a negative pressure, so it can not be considered as a kind of candidate of dark energy. 3. Summary The effective YM condensate has the advantageous characters: the YM fields are indispensable to particle physics, there is no room for adjusting the functional form of the lagrangian as it is predicted by quantum field theory. As a model for the cosmic dark energy, it can naturally solve the coincidence problem. In this study, we have investigated the evolution of the magnetic component in the YM condensate, which shows that the models can be separated into two cases: one is E 2 > B 2, the electric field dominant; and the other is E 2 < B 2, the magnetic field dominant. In the former case, the energy density of magnetic field decreases to zero with the expansion of the universe, but which of the electric field run to the state of E 2 = κ 2. In the early stage of the universe, ω y 1/3, which tracked the evolution of the radiation, and in the later stage, ω y 1. So the comic coincidence problem is also avoided. But in the latter case, ω y > 1/3 is satisfied for all time, which can not be considered as a kind of candidate of dark energy. ACKNOWLEDGMENT: The author thanks Yang Zhang for helpful discussion. References [1] A.G.Riess et al., Astron.J. 116, 1009 (1998); S.Perlmutter et al., Astrophys.J. 517, 565 (1999); J.L.Tonry et al., Astrophys.J. 594, 1 (2003); R.A.Knop et al., Astrophys.J. 598, 102 (2003); [2] C.L.Bennett et al., Astrophys.J.Suppl. 148, 1 (2003); D.N.Spergel et al., Astrophys.J.Suppl. 148, 175 (2003); D.N.Spergel et al., arxiv:astro-ph/ ; [3] M.Tegmark et al., Astrophys.J. 606, 702 (2004), Phys.Rev.D 69, (2004); A.C.Pope et al., Astrophys.J. 607, 655 (2004); W.J.Percival et al., MNRAS 327, 1297 (2001); [4] C.Wetterich, Nucl.Phys.B 302, 668 (1988); Astron.Astrophys. 301, 321 (1995); B.Ratra and P.J.E.Peebles, Phys.Rev.D 37, 3406 (1988); R.R.Caldwell, R.Dave and P.J.Steinhardt, Phys.Rev.Lett. 80, 1582 (1998);

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7 Figure 1: In models a1, a2, a3, the evolution of EoS of YM condensate with the scale factor. Figure 2: In models a1, a2, a3, the evolution of electric and magnetic fields with the scale factor. Figure 3: In models b1, b2, b3, the evolution of EoS of YM condensate with the scale factor. Figure 4: In models b1, b2, b3, the evolution of electric and magnetic fields with the scale factor.

8 Mod.a2 Mod.a1: f i =50, q i =100 Mod.a2: f i =5, q i =100 Mod.a3: f i =5, q i = y Mod.a3 Mod.a log 10 ( a / a i )

9 10 8 E 2 / ( B 2 / +1 ) Mod.a2 Mod.a3 The upper lines denote: E 2 / 2 ; and the lower lines denote: B 2 / 2 +1 Mod.a log 10 ( a / a i )

10 0-1 Mod.b2 Mod.b1-2 y -3 Mod.b3 Mod.b1: f i =0.9, q i =2.0 Mod.b2: f i =0.9999, q i =10.0 Mod.b3: f i =0.5, q i = log 10 ( a / a i )

11 E 2 / ( B 2 / +1 ) Solid lines denote: E 2 / 2 Dot lines denote: B 2 / Black lines: Mod.b1 Red lines: Mod.b2 Blue lines: Mod.b log 10 ( a / a i )