Exact Solution of an Ekpyrotic Fluid and a Primordial Magnetic Field in an Anisotropic Cosmological Space-Time of Petrov D

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1 Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 12, HIKARI Ltd, Exact Solution of an Ekpyrotic Fluid and a Primordial Magnetic Field in an Anisotropic Cosmological Space-Time of Petrov D R. Alvarado CINESPA, Escuela de Fisica Universidad de Costa Rica, Costa Rica Copyright c 2017 Rodrigo Alvarado. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This document obtained and analyzed an exact solution of an ekpyrotic fluid P = 5 3 µ and a primordial magnetic field, so that the latter does not induce currents or electric fields. It is determined that the main role, in relation to the anisotropy in the solution, is played by the magnetic field. The solution is analyzed in points where, in appearance, some extent of singularity could exist, using the Kretschmann invariant and analyzing the metric s functions, and it is established that this is singular at t = 0. The value of the Kretschmann invariant, in these proximities, tends to be the same as the one obtained in the case of a free ekpyrotic fluid of the magnetic field; therefore, the magnetic field is left at the background when t 0. It is determined that the solution presents this behavior in small time values, and tends to be isotropic and equivalent to the solution of the ekpyrotic fluid obtained for the flat model of Friedmann, Robertson, Walker and Lemaitre FRWL; for big time values, it tends to behave as the Kasner LRS solution. Keywords: cosmology, Einstein, exact, magnetic field, solution 1 Introduction The dynamism of cosmology has become stronger due to the data of the microwave background radiation obtained by the satellites COBE, WMAP and

2 602 R. Alvarado PLANCK, and the discovery of the acceleration of the universe [1, 2]; there have existed some problems which have been discussed on [3]. The problem of the existence of cosmic magnetic fields in interstellar and intergalactic spaces is not new, it is actually a current issue [4], [5]. The review of these fields produces values of [4]1 3µG in structures of matter such as clusters and superclusters, and among those that appear to be a considerable value and a possible influence on several cosmic phenomena; for example, the radiation of the microwave background is of 3µG [6]. These and other problems are explained on [7]. In relation to the established on [3, 7], it is evident that it exists an interest to study cosmological models that represent the observations in a complete way; for example, homogeneous anisotropic models, heterogeneous isotropic models or heterogeneous anisotropic models. If the Universe was created with a primordial magnetic field, that nature forces to consider, at least, the anisotropic space in the process of determining a solution. Because of this, the current study analyzes a magnetic field and an ekpyrotic fluid with an anisotropic but homogeneous symmetry. One example of the primordial magnetic field in an ideal fluid of a dark energy model has been analyzed on [7], where is determined that the main role, in relation to the present anisotropy, in this solution is played by the magnetic field, and that the solution is free of singularities. Moreover, the solution, in long periods of time, presents a behavior which tends to be isotropic and equivalent to the solution of dark energy obtained in the flat model of Friedmann, Robertson, Walker y Lemaitre FRWL. Ekpyrotic models and their cyclical extensions resolve the standard issues of plane, horizon and cosmologic homogeneity by offering a slow phase of contraction of the universe before the Big Bang [8]. On the other hand, it has been obtained that the non-linear state equation in an anisotropic space-time of Petrov D can occur because of different thermodynamic processes which the particles can suffer. Besides, there exists the chance of matter change so that after some time, it gets back to its initial state without completely reaching it, and the implication that the effective entropy does not grow; however, leaving space for an almost oscillatory state in which the change in the matter happens in a gradual way [9]. The previous information, motivates to study other possible and similar scenarios with different fluid models. A possible scenario is the ekpyrotic fluid and a primordial magnetic field which does not induce to currents or electric fields; this scenario will be analyzed in the present study.

3 Cosmological exact solutions The Symmetry, Einstein Tensor, the Electromagnetic Field and an Ekpyrotic Fluid 2.1 The Symmetry and Einstein Tensor The anisotropic symmetry of the Petrov D has been considered on [3] as follows ds 2 = F dt 2 t 2/3 Kdx 2 + dy 2 t2/3 K 2 dz2, 1 where F and K are functions of t. The components of the Einstein s tensor [10] G β α = R β α 1 2 δβ αr different from zero, of 1, are G 0 0 = 4 K2 9 t 2 K 2, 2 12t 2 K 2 F 3Kt K 2F F t + 3F t 2K 2 K 5 K 2 + 4K 2 F t + F G 1 1 = 12t 2 K 2 F 2, 3 G 2 2 = G 1 1, 4 6Kt K 2F F t 3F t 4K 2 K K 2 + 4K 2 F t + F G 3 3 = 12t 2 K 2 F 2, 5 where the points over the functions represent derivatives of time. 2.2 The Magnetic Field The magnetic field is considered in this way because the only components of the tensor of the electromagnetic field [10] F µν different from zero are F 12 = F 21 = B 0z = const, where it can be noticed that the invariant F µν F µν = 2Bt 2 = 2B 0z 2 π/t 4/3 K 2, where Bt = B 0z π 1/2 /t 2/3 K is the magnitude of the effective magnetic field. The effective magnetic field does not generate currents or induced electric fields since the flow of the magnetic field Φ does not change with time dφ = BtdAt = Bt g 11 g 22 dxdy = B 0z π 1/2 dxdy. 6 The choice of the field, in the given way, allows the compliance of field equations F µν ;µ = 0, besides the equality to zero of the divergence of the energy momentum

4 604 R. Alvarado tensor of the electromagnetic field em T µν ;µ = 0, where the tensor em T µν is the energy momentum tensor of the electromagnetic field, whose only components, different from zero, in em T µ ν, are emt 0 0 = em T 1 1 = em T 2 2 = em T 3 3 = B 0z The Model of an Ekpyrotic Fluid 8 t 4/3 K 2. 7 The model of an ekpyrotic fluid that can be considered a perfect fluid, and its energy momentum tensor has the pattern [10] T αβ = µ + P u α u β g αβ P, 8 where T αβ is the energy momentum tensor of the perfect fluid, u α is the tetradimensional speed, g αβ the metric tensor, µ and P are respectively the energy density and the pressure of the fluid. It will be considered that u 1 = u 2 = u 3 = 0 and u 0 0. The ekpyrotic fluid model that will be studied has an state equation with the relation P = 5µ/3. The energy momentum tensor of the ekpyrotic fluid presents the following pattern ektν α = δνδ 0 0 α 5 δ 1 3 ν δ1 α + δνδ 2 2 α + δνδ 3 3 α µ, 9 where δν α is the delta of Kronecker. From the equality T ;µ µν = em T ;µ µν + ek T ;µ µν = 0, and considering that em T ;µ µν = 0, it is obtained that µ = α 3t 8/3. 3 Einstein s Equations and the Solution of the Model of the Magnetic Field and Ekpyrotic Fluid Einstein equations present the form [10] G β α = κt β α, where T β α = em T β α + ek T β α. From 2-5, 7, 9, it is obtained the following system of equations independent from each other 4 K 2 9 t 2 K 2 12t 2 K 2 F 3B 0z α t 4/3 K 2 24t 4/3 K 2 = 0, 10 3Kt K 2F F t + 3F t 2K 2 K 5 K 2 + 4K 2 F t + F 12t 2 K 2 F t2/3 B 0z α t 2/3 K 2 /3 24K 2 t 2 = 0,, 11

5 Cosmological exact solutions 605 6Kt K 2F F t 3F t 4K 2 K K 2 + 4K 2 F t + F 12t 2 K 2 F t2/3 B 0z α t 2/3 K 2 /3 24K 2 t 2 = 0, 12 From the equation 10, it is obtained that 2 4 K 2 9 t 2 K 2 F = t 2/3 3B 2 0z + 8 t 4/3 α K2. 13 Considering 13 in 11 and 12, it is obtained that both equations satisfy each other 27t 3 K K 2 t K + 36 KK 2 t 2 16K 3 B 2 0z + 8t 1/3 K 2 α 9 t 2 K tK K 2 12 K 14 K 2 t 8 K 2 K = 0. Before determining the solution of the equation 14, it is convenient to establish that if it is considered the magnetic field only, it does not have suitable solutions for the problem in investigation. If it is considered that 14 in α = 0, it is obtained that the equation solution is 2/3 K = t t + t2 C 4/3 t + t2 C 2/3 3 C + C 2/3 t + t2 C 2/ C 15 and from 13, F = 8K 2 3 C C 2/3 t + t 2 C 4/3 2 3 t + t 2 C, 16 4/3 t2 C t 2/3 2 B 0z From the solution 16, it is obtained that the function F is negative, void or complex; none of the previous possibilities correspond to a solution of a spacetime. Therefore, the solution with a magnetic field alone, with the form that has been considered, is not admissible in that symmetry a similar situation is presented when considering a Zeldovich fluid P = µ. The found solution of the equation 14 follows the pattern K = αt 4/3 B0z α From 17, the function F in 13 can be expressed as F = t2/3 3/2 A 1/2 + A 3/2 α A 2 A A

6 606 R. Alvarado where A = α t 4/3 B 0z 2. It is noteworthy that if in the solutions 17 and 18 it is considered α = 0, these are undefined as a result of what was determined before. 4 Analysis of the Solution In t = 0, the energy momentum tensor Tα β is singular, as it is noticed when observing the last terms on the left in 10, 11, 12. The prior can also be determined and analyzed thanks to the Kretschmann invariant, which is defined as Krets = R µναβ R µναβ. The condition Krets < is necessary and sufficient [11] for the finitude of all the invariants of algebraic curvature, and with the form, for the metric 1, K + 3 t K 2 K + 3 t K K + 3 t K t 4 K 4 F Krets = t 4 K 4 F 2 12 F Kt K + 18 F Kt 2 K + 4 F K 2 36 F t 2 K F tk 2 9 F t 2 K K 2 t 4 K 4 F 4 12 F Kt K + 18 F Kt 2 K 8 F K 2 6 F tk 2 9 F t 2 K K 2 9 F t 2 K 2. t 4 K 4 F 4 19 When considering 17 and 18 in 19, it is obtained that in the proximities with t 0, the invariant Krets 40α 2 / 27t 16/3, corresponds with the analogous case [3] when the magnetic field is not present; therefore, the influence of the magnetic field in processes near singularity is negligible. The metric in this case tends to ds 2 dη 2 η dx 2 + dy 2 ηdz 2, 20 where it is substituted t = η α 3/4 and x = x/ 3 1/4 2α 3/8, y = y/ 3 1/4 2α 3/8 and z = α 9 8 z, the prior agrees with the solution of the same fluid in a metric of a flat FRWL k=0 with no magnetic field. As t grows, the magnetic field s influence increases as well and in larger times t ; the invariant Krets 2 13 α 3 B 2 0z/ 27t 4 0. For this limit the metric tends to ds 2 dη 2 η 4/3 dx 2 + dy 2 dz2, 21 η2/3 where it was substituted t = 2 η 23/4 α 3/4 B 0z, Z = 2 17/12 α 1/4 z/b 2/3 0z X = C a x, Y = C a y, where C a = 2 5/12 α 1/4 /B 1/6 0z. The solution 21 tends to be the one of the Kasner vacuum LRS.

7 Cosmological exact solutions Conclusions In a mixture of an ekpyrotic fluid P = 5 µ and a magnetic field, for which 3 the magnetic flow remains constant, and does not induce currents or electric fields, the magnetic field is decisive in terms of the anisotropy presented in the space, so that the space expands faster over the perpendicular plane to the magnetic field direction. The presence of the magnetic field is not significant in relation to the singular point in the metric t = 0. The solution tends to become isotropic at the beginning t 0, and equivalent to the one an ekpyrotic fluid model of FRWL; therefore, the influence of the magnetic field is scarce at the beginning. When t, the metric tends to the one of a Kasner vacuum LRS, in an analogous manner to what was obtained in one of the two possible solutions of the ekpyrotic fluid model, with the same symmetry, analyzed in another study [3]. References [1] A. G. Riess, A. V. Filippenko, P. Challis, et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, The Astron. Journal, , no. 3, [2] S. Perlmutter, G. Aldering, G. Goldhaber, et al., Measurements of Ω and Λ from 42 High-Redshift Supernovae, The Astrophysical Journal, , no. 2, [3] R. Alvarado, Cosmological Exact Solutions Set of a Perfect Fluid in an Anisotropic Space-Time in Petrov Type D, Advanced Studies in Theoretical Physics, , [4] P.P. Kronberg, Extragalactic magnetic fields, Rep. on Prog. in Phys., , [5] F. A. Membiela, Primordial magnetic fields from a non-singular bouncing cosmology, Nucl. Phys. B, , [6] E. Battaner and E. Florido, Rotation Curve of Spiral Galaxies and Its Cosmological Implications, Fund. Cosmic Phys., , [7] R. Alvarado, Exact Solution of Dark Energy and a Primordial and Undisturbed Magnetic Field at an Anisotropic Cosmological Space-Time of Petrov Type D, Advanced Studies in Theoretical Physics, ,

8 608 R. Alvarado [8] Jean-Luc Lehners, Ekpyrotic Nongaussianity: A Review, Advances in Astronomy, , [9] R. Alvarado, Thermodynamics and Small Temporal Variations in the Equations of State of Anisotropic Cosmological Models of Petrov Type D, Advanced Studies in Theoretical Physics, , [10] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Addison- Wesley, Reading, Massachusetts, [11] K. A. Bronnikov, E. N. Chudayeva and G. N. Shikin, Magneto-dilatonic Bianchi-I cosmology: isotropization and singularity problems, Class. Quantum Grav., , Received: August 31, 2017; Published: November 16, 2017