Inflation Driven by Nonlinear Electrodynamics in Anisotropic Spacetime

The well-known Big Bang theory has explained how the universe came into being. This extraordinary event caused the later universe to be accelerated by a scale factor a(t). However, standard Big Bang theory has had some problems that can’t be explained, such as monopoles, horizons, and flatness. To solve this problem, a model of inflation in the early universe is required. Recent studies show that nonlinear electrodynamics coupled with general relativity can describe the inflation of the universe. In this work, we consider a model of nonlinear electrodynamics in anisotropic spacetime. We derive the dynamical equation from Einstein’s field equation and the law of conservation of energy-momentum tensor. Then, we use the perturbation method to solve the dynamical equation of the universe and obtain the evolution of the non-singular scale factor with anisotropy parameter ϵ. Using a phase-space analysis of the inflationary model, we obtain a phase portrait in the presence of fixed points. Our result shows that in the model of nonlinear electrodynamics coupled to gravity in anisotropic spacetime, the universe can undergo an inflationary mechanism if ϵ < 1. We also show the absence of singularity in density and pressure using this model.


Introduction
Einstein's theory of General Relativity (GR) is presently recognized as the best gravitational theory for describing our universe.However, the standard big bang cosmological model, which is founded on GR, is inevitable to have some basic problems: Why is our universe so flat (the flatness problem), and how two distinct spacetime patches in the universe can be causally connected after the big bang (the horizon problem)?.In order to solve this problem, a model of the inflationary scenario in the early universe has been proposed.Inflation is defined as the epoch in the early universe when the scale factor expands exponentially in only 10 −34 seconds.An inflated universe was first proposed in 1981 by Alan Guth as a solution to some important puzzles in standard cosmology such as monopoles, horizons, and flatness [1].Cosmic inflation later became one of the main paradigms in modern cosmology [2].It is very consistent with the results obtained from the cosmic microwave background (CMB), radiation detector observed by Wilkinson Microwave Anisotropy Probe (WMAP) and Planck [3,4,5].The standard inflation model is based on the assumption that spacetime at the beginning of the universe can be described by the Friedmann-Robertson-Walker (FRW) metric which is homogeneous and isotropic [6,7].The universe is isotropic on a multi-billion-light-year scale, meaning it must have the same appearance and behavior in all directions.Since the big bang nearly 14 billion years ago, the universe should have expanded identically everywhere.That expectation matches what astronomers see when they observe the CMB.However, recent studies suggest that the universe may be either closer or farther away than what would be expected in an isotropic model.This discovery could be a sign that the universe is actually anisotropic, i.e. expanding faster in some regions than in others [8,9].Furthermore, the standard cosmological model, which is based on the FRW geometry as the source of Maxwell's electrodynamics, leads to a cosmic singularity at some point in the past.To solve this puzzle, researchers have proposed numerous mechanisms such as modified gravity theories [10,11], scalar inflation fields [12,13], nonminimal coupling theories [14], nonlinear lagrangian with quadratic terms in the curvature [15], and nonlinear electrodynamics without modification of general relativity [16,17,18,19,20].
Using nonlinear electrodynamics without modifying general relativity is one approach to explain the inflationary universe without creating a cosmic singularity.The theory of nonlinear electrodynamics was first proposed by Max Born and Leopold Infeld in 1934 with the aim of eliminating the singularity of point charges in classical electrodynamics.Lagrangian Born-Infeld has an interesting feature of turning into Maxwell theory for low electromagnetic fields [21].Nonlinear electrodynamics coupled to gravity can explain the inflation of the universe [22,17,23,18].In the early epoch, the electromagnetic field was the source of the gravitational field and the inflation of the universe.Kruglov has pointed out in his article that the universe with a stochastic magnetic field can experience inflation [23].
Milne and McCrea had proposed the analogy between Newtonian mechanics and FRW cosmology within the framework of classical mechanics [36,38,37].The cosmological model is analogous to a one-dimensional Hamiltonian system of particles with all its dynamics described in terms of an effective potential function that can be obtained explicitly.These results demonstrate the advantages of using potential functions rather than equations of state to investigate the origins of inflation [29].Quantitative methods of dynamical systems have been applied to study cosmological models since the 1970s [30,31].The application of dynamical systems methods in the context of scalar field cosmology was investigated by Belinsky et al [32], also for various other models discussed in [29,33,34,35].
In this paper, we use FRW anisotropic spacetime to study the inflationary mechanism driven by nonlinear electrodynamics.We start our work from the action of general relativity coupled with nonlinear electrodynamics.We get the Einstein field equation by varying the action with respect to the metric.Then, we get the dynamical equation from Einstein's field equation and the conservation of energy-momentum tensor.Due to its complexity, the dynamical equations are difficult to solve.We use the perturbation method with anisotropy parameter, ϵ, to solve the equation.Finally, we obtain the evolution of the non-singular scale factor and show that the universe undergoes inflation driven by nonlinear electrodynamics in anisotropic spacetime.Note that in this model of nonlinear electrodynamics, we consider the case where the electric field vanishes, i.e.E 2 = 0, and a non-zero averaged magnetic field leads to a magnetic Universe.The inflation model is analyzed as a Newtonian dynamic system and its stability is examined.The main analysis relies on a one-dimensional form of the potential function that provides complete information about the dynamics of the inflation model.In conclusion, we can say that a very strong magnetic field in the early universe leads the universe to inflate.

Nonlinear Electrodynamics in Anisotropic Spacetime
We consider an anisotropic universe which belongs to Bianchi Type-I universe which can be written in the natural units as with a(t) is scale factor in the x directions, and b(t) is scale factor in the y and z directions.
The action of general relativity coupled with nonlinear electrodynamics given by where κ −1 = M P l , M P l is the reduced Planck mass, R is the Ricci scalar, and L is the Lagrangian density of the nonlinear electrodynamics [23], where β is a real parameter, F = 1 4 F µν F µν , is Maxwell's invariant, and the field strength of the gauge field A µ .The energy-momentum tensor of the nonlinear electrodynamics is given by and has non-vanishing trace ( From equation ( 2), the Einstein field equation is given by where G µν ≡ R µν − 1 2 g µν R, is the Einstein tensor, and R µν is the Ricci tensor.In this paper, we assume that the electromagnetic fields has a stochastic background since the curvature is considerably larger than the wavelength of the electromagnetic waves.For anisotropic universe, we modify the result of [24] for the averaged magnetic fields obey equations as follows: In the next calculation, for simplicity we omit the averaging brackets ⟨⟩.
The energy-momentum tensor (4) can be written as that of a perfect fluid energy-momentum tensor, 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012073 and using conditions (7), we have With the help of metric in equation ( 1) and Einstein field equation, we obtain modified Friedmann equation as follows where the dots over the letter denotes the derivatives with respect to cosmic time.The conservation of the energy-momentum tensor, ∇ µ T µν = 0, gives the relation The direct solution of the above equations is very difficult to solve due to their highly non-linear nature.we consider a small anisotropic, i.e. ϵ << 1, and then use a perturbation to find an approximate solution.We will show that the zeroth-order equation returns to the solution of the isotropic universe calculated in ref [23].

Small Anisotropic Solution
The equations of a vast majority of dynamical systems cannot be solved exactly, therefore it is important to develop methods for obtaining approximate solutions.Perturbation theory tends to be the only analytic method and one of the most effective approaches.In this method, a solvable system that differs from the insoluble system in minor ways is identified, and then the solvable system is perturbed to approximate the insoluble system [25].We use the perturbation method with the following form: By using the perturbation method we obtain zeroth and first-order equation of the energy density and pressure as follows Using the same technique applies to equations ( 12), (13), and ( 14) we obtain zeroth-order equation as follows and first-order equation as follows Ḃ1 Integrating equations ( 26) and ( 29) lead to where C 0 is the magnetic field corresponding to the scale factor b(0) = 1 and C 1 is an integration constant.From equation (26) and with the help of equations ( 33) and ( 34) we obtain the evolution of the magnetic field in anisotropic spacetime under the change of the scale factor b(t).
It shows that due to inflation, the scale factor increases, and the magnetic field decreases.16), ( 21), (22) and making used of equations ( 33) and ( 29), we attain and for the pressure, we get According to equations ( 35) and ( 37), there is no singularity in the energy density and pressure at b(t) → 0 and b(t) → ∞.Thus, The absence of singularities is an interesting feature of the cosmological model with nonlinear electrodynamics.This characteristic is also shown in Refs.[18,26,27] by using a different model of nonlinear electrodynamics.

Evolution of the Scale Factor
In this section, we study the dynamics of the universe by using Einstein's equations and the energy density in equation (35).The zeroth order equation is given by ( 27) and using equation ( 33), we get where γ ≡ √ βC 0 / √ 2. Integrating equation ( 40) yields the quadratic equation as follow where the integration constant is chosen such that b(0) = √ γ.The real solution to equation ( 41) is given by We obtain the scale factor for the isotropic universe in equation ( 43).For the anisotropic solution, we solve equation (30) with help of equations ( 33), ( 34) and ( 40), as a result Integration of equation ( 44) yields where the integration constant is chosen such that a 1 = 0 when b = 1.From equation ( 19) we obtain the scale factor for the anisotropic universe as a function of b, as stated below Evolution of the scale factor for isotropic and anisotropic universe depicts in Fig. (1).We plot graph a vs x = κγt √ 3β with C 1 = 1.We define the anisotropy parameter as ϵ and take γ = 0.5.The value of γ is based on the best value given by Kruglov in his paper to plot the scale factor in isotropic spacetime with the same nonlinear electrodynamics model [23].We can see there are no singularities in this model.The scale factor a(t) approaches zero only at t → −∞.It states at the early epoch universe undergoes inflation due to a strong magnetic field.
Figure 1.Evolution of the scale factor with γ = 0.5 and anisotropiy parameter ϵ = 0, 0.01, 0.1, 0.29, and 1.We state the graph with ϵ = 0 (blue line) as the isotropic universe.The blue line coincides with the green line with ϵ = 0.01.The graph shows that the universe undergoes inflation if the anisotropy parameter remains small, i.e. ϵ < 1.
The values ϵ = 0.01 from Fig. (1) produce outcomes that are very similar to the isotropic spacetime.ϵ = 0.1 also produces a model with an inflation mechanism, even if we choose the number ϵ = 0.29 based on the anisotropy parameter from WMAP [28], we still get an inflated universe.The line slows down briefly before rising at the very beginning (x = 0) if we choose ϵ = 1.The model of nonlinear electrodynamics can cause universe inflation in anisotropic spacetime if the anisotropy parameter remains small, i.e. ϵ < 1.

Phase Space Analysis
The dynamics of the inflationary model are determined by the equation (12) and can be reduced to the dynamics of fictitious particles of the unit mass moving under the influence of the effective potential U ef f .Therefore the inflationary model can be simply represented in terms of a dynamical system of a Newtonian type: The perturbation of the conservation of stress energy-momentum tensor yields B 0 = C 0 /b 2 and B 1 = C 1 − 2a 1 /3.Using equation ( 21), we can write the effective potential as Furthermore, the scale factor can be expressed as X ≡ b/b(0) which measures the value of b compared to the current value of the scale factor b(0), and we rescale the time variable as t → τ : κdt |C 0 | = dτ , we get, and the system is then equivalent to Equation (50) gives the possibility to investigate the evolution of inflationary model for all possible initial conditions.The effective potential is then expressed in terms of new parameters Since the scale factor has a positive definite value, X ≥ 0, then, it is convenient to define a new variables (u, v), which is a real number.We also redefined time variable to describe evolution of systems So, we get a system in terms of coordinates (u, v) where W (u) ≡ is the effective potential which can be written as The fixed point at the finite region of the phase space is We do compactification in order to have the general bounded system by using U = u √ 1+u 2 +v 2 and V = v √ 1+u 2 +v 2 .Thus, we have the equivalent flow At a point around fixed point, the dynamics of the system at that point is equivalent to the linear part of the fixed point.The linearization matrix near the origin of coordinates is We can obtain that the system has eigenvalues .Thus, the fixed point will be a saddle if −(14 + 47ϵC 1 ) > 0, and a center if −(14 + 47ϵC 1 ) < 0.
A global phase portrait is represented in Fig. 2 (a) shows the dynamics of the system for the choice ϵ = 0. shows the acceleration of the universe is given in the form of a potential function.The potential function shows a large negative gradient when the scale factor approaches zero, this can be interpreted as an inflationary era.In this case, there is one minimum point of the potential function, there are two eras of evolution of the universe i.e inflation and decelerating era.

Conclusion
A model of nonlinear electrodynamics in anisotropic spacetime has been studied to explain the inflation of the universe.In general, the dynamical equations in the anisotropic universe are more complicated than in the isotropic FRW universe.We use the perturbation method to solve the dynamical equations of the universe and obtain the evolution of the non-singular scale factor.It shows that nonlinear electrodynamics with anisotropic spacetime allow inflation of the universe with the anisotropy parameter ϵ < 1.By analyzing a global phase portrait, we show that there is one fixed point in the form of center.In conclusion, the strong magnetic field in the early epoch caused the universe to inflate.We also showed that, at the time of the Big Bang, there was no singularity in the energy density and pressure.

Figure 2 .
Figure 2. (a) A global phase portraits that represents the dynamics of the equation (57) and (58) for the choice γ = 0.80059, ϵ = 0.29 and C 1 = 1.(b) The function of effective potential W (b) vs. scale factor b.