Universe inflation and nonlinear electrodynamics

WE analyse the universe inflation when the source of gravity is electromagnetic fields obeying nonlinear electrodynamics with two parameters and without singularities. The cosmology of the universe with stochastic magnetic fields is considered. The condition for the universe inflation is obtained. It is demonstrated that singularities of the energy density and pressure are absent as the scale factor approaches to zero. When the scale factor goes to infinity one has equation of state for ultra-relativistic case. The curvature invariants do not possess singularities. The evolution of universe is described showing that at large time the scale factor corresponds to the radiation era. The duration of universe inflation is analysed. We study the classical stability and causality by computing the speed of the sound. Cosmological parameters such as the spectral index $n_s$, the tensor-to-scalar ratio $r$ and the running of the spectral index $\alpha_s$ are evaluated.


Introduction
Universe inflation may be justified by the modification of general relativity (F (R)-gravity) [1,2,3], insertion of cosmological constant Λ in Einstein-Hilbert action or by introduction of a scalar field (quintessence) [4].Another way to explain universe acceleration is to use the source of Einstein' gravity in the form of nonlinear electrodynamics (NED).First NED was proposed by Born and Infeld [5] to smooth a singularity of point-like charges and to have self-energy finite.It is worth noting that at strong electromagnetic fields classical Maxwell's electrodynamics becomes NED due to quantum corrections [6,7,8].Thus, we imply that in early time of the universe evolution electromagnetic fields were very strong and linear Maxwell's electrodynamics should be replaced by NED.Inflation can be described by Einstein' gravity coupled to some NED [9,10,11,12,13,14,15,16,17,18,19,20].We explore here NED with two parameters [21] which includes rational NED [16,18] for some parameter and for weak fields it becomes Maxwell's electrodynamics.We show that in the framework of our model the universe inflation takes place for the stochastic magnetic background field.
It was shown that the stochastic fluctuations of the electromagnetic field are present in a relativistic electron-positron plasma and, therefore, plasma fluctuations can generate a stochastic magnetic field [22,23].Thermal fluctuations in the pre-recombination plasma could be the source of a primordial magnetic field and magnetic fluctuations may be sustained by plasma before the epoch of Big Bang nucleosynthesis.Probably, there were strong low-frequency random magnetic fields in the early stage of the radiationdominated era.In our galaxy and other spiral galaxies magnetic fields of the order of B = 10 −6 G are present on scales of several Kpc [25].The fields B = 10 −6 G can have the primordial origin and are explained with the help of the galactic dynamo theory This theory explains a mechanism transferring angular momentum energy into magnetic energy and needs the existence of weak seed fields of the order of B = 10 −19 G at the epoch of the galaxy formation.The source of seed magnetic fields could be thermal fluctuations in the primordial plasma.Magnetic energy over larger scales can be due to long wavelength fluctuations which reconnect and redistribute magnetic fields [26].As the electric field is screened by the charged primordial plasma we consider the background magnetic field [23].We imply that directional effects are not present ( B = 0) according to the standard cosmological model.
In section 2 we consider Einstein's gravity coupled to NED with dimensional parameter β and dimensionless parameter γ.The source of gravity is a stochastic magnetic field.The range of magnetic fields, when an acceleration of universe occurs, is obtained.We show that singularities of the energy density and pressure are absent at γ ≥ 1.It is demonstrated that, when the scale factor approaches infinity, one has equation of state (EoS) for ultra-relativistic case.Singularities of curvature invariants are absent for γ ≥ 1.The evolution of the universe is investigated in section 3. We find the dependence of the scale factor on the time and show that when time goes to infinity one has the radiation era.In subsection 3.1, the deceleration parameter as a function of the scale factor is computed showing the evolution of universe.We analyse the amount of the inflation by calculating e-foldings N.For some parameters the reasonable e-foldings N ≈ 70 is realised.We found the duration of the universe inflation depending on parameters beta and gamma.In subsection 3.2 the causality and unitarity principles are studied by computing the speed of sound.In section 4 the cosmological parameters, the spectral index n s , the tensor-to-scalar ratio r, and the running of the spectral index α s , are evaluated.We show that they are in approximate agreement with the PLANK and WMAP data.Section 5 is a Summary.
2 Cosmology with a stochastic magnetic field of background The Einstein-Hilbert action coupled to the matter source is given by where R is the Ricci scalar, κ 2 = 8πG and G is Newton's constant with the dimension [L 2 ].We consider the source of gravitational field in the form of NED with the Lagrangian [21] Here, F = F µν F µν /4 = (B 2 − E 2 )/2 is the field invariant and E and B are the electric and magnetic fields, correspondingly.We introduce the parameter ǫ = ±1, dimensional parameter β > 0 with the dimension [L 4 ], and dimensionless parameter γ > 0. If B > E one uses ǫ = 1 and for B < E we should set ǫ = −1 to have the real Lagrangian.It is worth noting that as βF → 0 Lagrangian (2) approaches to Maxwell's Lagrangian.It was shown in Ref. [24] that there are no singularities in the electric field of point-like charges and electric self-energy in the model with the Lagrangian (2) for ǫ = −1.Here, we consider inflation of the universe filled by stochastic magnetic fields (ǫ = 1).After varying action (1), one obtains the Einstein and electromagnetic field equations where and the stress-energy tensor is given by We use the line element of homogeneous and isotropic cosmological spacetime as follows: with a scale factor a(t).Let us consider the cosmic background in the form of stochastic magnetic fields.As the electric field is zero we use ǫ = 1.We have the isotropy of the Friedman-Robertson-Walker space-time after averaging the magnetic fields [27].It is implied that the wavelength of electromagnetic waves is smaller that the curvature.Therefore, one should impose equations as B = 0, with the brackets denoting an average over a volume.For simplicity we will omit the brackets in the following.Note that the NED stress-energy tensor can be represented in the form of a perfect fluid [13].The Friedmann's equation for three dimensional flat universe is where ä = ∂ 2 a/dt 2 .When ρ + 3p < 0 the universe acceleration takes place.Making use of Eq. ( 6) we obtain By virtue of Eq. ( 10) one finds From Eq. ( 11) and the condition ρ + 3p < 0 for the acceleration of the universe, we obtain with γ > 1/2.The plot of the function √ βB versus γ is given in Fig. 1.The  where W (z) is the Lambert function.Thus, the strong magnetic fields drives the universe inflation.The stress-energy tensor conservation, ∇ µ T µν = 0, leads to the equation Making use of Eq. ( 10), one finds After integrating Eq. ( 13), with the help of Eq. ( 14), we obtain were B 0 is the magnetic field corresponding to the value a(t) = 1.It is worth mentioning that Eq. ( 15) holds for any Lagrangians [13].Because the scale factor increases during the inflation, the magnetic field decreases.By virtue of Eqs. ( 10) and ( 15) one finds lim As a result, singularities of the energy density and pressure as a(t) → 0 are absent at γ ≥ 1. Equation (16) shows that at the beginning of the universe evolution (a ≈ 0), the model gives ρ = −p at γ = 1 for the rational NED, corresponding to de Sitter space-time, i.e. we have the same property as in the ΛCDM model.Making use of Eq. ( 10) we obtain EoS The function w versus (βB 2 ) γ is depicted in Fig. 2 for γ = 0.75, 1, 1.5.Making use of Eqs. ( 17)one finds As a(t) → ∞ (B → 0) one has the EoS for ultra-relativistic case [28].For γ > 1 and (βB 2 ) γ = 1/(γ − 1), one has de Sitter space-time, w = −1.Thus, for an example, at γ = 1.5 and (βB 2 ) 1.5 = 2 one has w = −1 (see Fig. 2).From Eqs. ( 3) and ( 6), we obtain the Ricci scalar The function βR/κ 2 versus (βB 2 ) γ is depicted in Fig. 3. Making use of Eq. ( 19) we find lim Note that when the scale factor a(t) → 0 one has B → ∞ and the singularity of the Ricci scalar is absent for γ ≥ 1.The Ricci tensor squared R µν R µν and the Kretschmann scalar R µναβ R µναβ may be expressed as linear combinations of κ 4 ρ 2 , κ 4 ρp, and κ 4 p 2 [16] and, according to Eq. ( 16), they are finite as a(t) → 0 and a(t) → ∞ for γ ≥ 1.As t → ∞ the scale factor increases and spacetime approaches to the Minkowski space-time.From Eqs. ( 12) and ( 15) we find that the universe accelerates at a(t) < β 1/4 √ B 0 (2γ − 1) 1/(4γ) and the universe inflation occurs.

Evolution of the universe
We will use the Friedmann equation for three dimensional flat universe to describe evolution of the universe which is given by Making use of Eqs. ( 10) and ( 21), one obtains With the help of the unitless variables x ≡ a/(β 1/4 √ B 0 ), y ≡ √ 6β ẋ/κ Eq. ( 22) becomes The function y versus x is depicted in Fig. 4. According to Fig. 4, the  inflation lasts from Big Bang till the graceful exit, x end = (2γ − 1) 1/(4γ) , and then the universe decelerates.After integrating Eq. ( 22) we obtain From Eq. ( 24) we arrive at the equation where F (a, b; c; z) is the hypergeometric function and △t is the duration of the inflation.It is worth noting that the hypergeometric function entering Eq. ( 25) has the property c−b = 1, and therefore, it can be expressed through the incomplete B-function by the relation B z (p, q) = p −1 z p F (1−q, p; p+1; z) [29].Equation ( 25) allows us to study the evolution of the universe inflation.

Universe inflation
The expansion of the universe can be described by the deceleration parameter.From Eqs. ( 9), ( 10), ( 15) and ( 21) we obtain the deceleration parameter The plot of the deceleration parameter q versus x = a/(βB 2 0 ) 1/4 is depicted in Fig. 5.The inflation, q < 0, takes place till the graceful exit q = 0 (x end = (2γ − 1) 1/(4γ) ).When x = (2γ − 1) 1/(4γ) the acceleration stops, the deceleration parameter is zero, and then one has the deceleration phase (q > 0).Singularities at the early epoch are absent.Now, we evaluate the amount of the inflation with the help of e-foldings [30] where t end corresponds to the final time of the inflation and t in is an initial time.The graceful exit point is x end = (2γ − 1) 1/(4γ) and one finds a(t end ) = (2γ − 1) 1/(4γ) b (b = β 1/4 √ B 0 ).It is known that the horizon and flatness problems can be solved when e-foldings N ≈ 70 [30].From Eq. ( 27) we find the scale factor corresponding to the inflation initial time One can analyze different scenarios of the universe inflation by varying γ, b, and N.This model possesses phases of the universe inflation, the graceful exit and deceleration.Making use of Eq. ( 25) and condition a(t end ) ≫ a(t in ) (a(t end ) = 10 31 a(t in )), we obtain To study the scale factor at a(t end ) ≫ 1, we use the Pfaff transformation [29] By virtue of Eq. ( 30), at a(t end ) ≫ 1, we obtain where Γ(x) is Gamma-function.From Eqs. ( 29) and ( 31), at a(t end ) ≫ 1, we obtain in the leading order the equation as follows: Making use of Eq. ( 32), at a(t end ) ≫ 1, we arrive at which corresponds to the radiation era.With relation Γ( Taking the duration of the inflation to be △t ≈ 10 −32 s and the values of κ = √ 8πG and a(t end ) = (2γ − 1) 1/(4γ) β 1/4 √ B 0 , one can find from Eq. ( 33) the coupling Figure 6 shows the dependance of y = β/(κ 2 ∆t 2 ) versus γ, where we use a(t end ) = (2γ − 1) 1/(4γ) β 1/4 √ B 0 .According to Fig. 6 when parameter γ increases the coupling β decreases to have the same time of the universe inflation.

Cosmological parameters
By virtue of Eqs.(10) one obtains By solving Eq. (37) for a particular γ, one can express βB 2 as a function of ρ.Then Eq. ( 36) may be represented as EoS for the perfect fluid where z = βB 2 is the solution to Eq. (37) (2βρ[z γ + 1] − z = 0) and z is a function of ρ.Analytical solution to Eq. ( 37) is possible only for several values of γ.The case γ = 1 was studied in [17].Here, we will investigate the case γ = 2.For this case the solution to Eq. ( 37) is given by z  gravity were implied.The range of magnetic fields, when the inflation of the universe takes place, has been obtained.It was shown that there are not singularities of the energy density and pressure at γ ≥ 1.We have demonstrated that as the scale factor goes to infinity, EoS of ultra-relativistic case takes place.It was shown that singularities of curvature invariants are absent for γ ≥ 1.We have investigated the evolution of the universe.The dependence of the scale factor on the time has been obtained.To study the evolution of universe, the deceleration parameter as a function of the scale factor has been calculated.To analyse the amount of the inflation we have calculated e-foldings.It was shown that for some parameters the reasonable e-foldings N ≈ 70 holds.The duration of the universe inflation as the function of parameters β and γ has been obtained.We computed the speed of sound to study the causality and unitarity principles in our model.It was found the range of background magnetic fields when causality and unitarity hold.The cosmological parameters, the spectral index n s , the tensor-to-scalar ratio r, and the running of the spectral index α s , were computed.We showed that n s , r, and α s are in agreement with the PLANK and WMAP data.
Let us discuss the similarities and differences of our model of the universe inflation with the model in Ref. [36].Similarities: The Ricci scalar curvature, the Ricci tensor squared and the Kretschmann scalar have no singularities at early/late stages, as a → 0. In the model [36] and in our model w = 1/3, q = 1 as a → ∞.The early universe did not have Big-Bang singularity and accelerated in the past.The slow-roll parameters such as spectral index n s , and tensor-to-scalar ratio r were analysed and compared with results of observational data.The classical stability and causality are studied.Differences: The behaviour of EoS parameter w and deceleration parameter q as a → 0 are different.In our model w = (1 − 4γ)/3, q = 1 − 2γ but in the model of [36], w = −1, q = −1 for any α as a → 0. The duration of the universe inflation as the function of parameters β and γ was computed in our model.Phase space analysis was performed in [36].Corrections of NED Lagrangian to Maxwell's Lagrangian are different at weak-field limit.