Realistic cyclic magnetic universe

(1)

c

World Scientiﬁc Publishing Company DOI:10.1142/S0218271812500733

REALISTIC CYCLIC MAGNETIC UNIVERSE

L. G. MEDEIROS Escola de Ciˆencia e Tecnologia, Universidade Federal do Rio Grande,

do Norte. Campus Universit´ario, s/n, CEP 59072-970, Natal, Brazil

leogmedeiros@gmail.com

Received 12 September 2011 Revised 10 August 2012 Accepted 10 August 2012 Published 25 September 2012

This work presents a complete cyclic cosmological scenario based on nonlinear magnetic field. It is constructed from a model composed of five fluids, namely baryonic matter, dark matter, radiation, neutrinos and a cosmological magnetic field. The first four fluids are treated in the standard way and the fifth fluid, the magnetic field, is described by a nonlinear electrodynamics. The free parameters are fitted by observational data (SNIa, CMB, extragalactic magnetic fields, etc.) and by simple theoretical considerations. As a result arises a cyclic cosmological model that preserves the main successes of the standard Big Bang model and solves some other problems such as the initial singularity, the present acceleration and the Big Rip.

Keywords: Cosmology; nonlinear electrodynamics. PACS Number(s): 98.80.-k, 98.62.En

1. Introduction

Although the inﬂationary scenario still maintains its status of the paradigm in cosmology, there is an increasing interest in the alternative proposal of models dis-playing a bounce.1–5 _{This means the possible existence of a collapsing era prior}

to the actual expanding phase. One of the main diﬀerences of these two proposals concerns the behavior of small perturbations and its evolution from a perfect spa-tially homogeneous phase to the production of inhomogeneities. There are hopes, due to a thorough analysis of these processes,6_{that this question will be settled in}

the near future. Another important point in favor of bouncing models is that they avoid the problem of initial singularity. The present paper examines one of these scenarios in which the dynamical history of the universe is partially controlled by a magnetic field. Its main effects and its compatibility with actual observations are analyzed. The main hypothesis of this model concerns the nonlinearity character of magnetic field.

Int. J. Mod. Phys. D 2012.21. Downloaded from www.worldscientific.com

(2)

The most general form for the dynamics of the electromagnetic (EM) ﬁeld, com-patible with covariance and gauge conservation principles yields for the Lagrangian the form L = L(F, G), where F ≡ Fµν_F

µν and G ≡ FµνFµν∗ . In magnetic uni-verse framework, the invariant G is irrelevant, since the average of the electric ﬁeld vanishes as it will be shown in the next sections. Thus, the Lagrangian appears as a regular function that can be developed as positive or negative powers of the invariant F. Positive powers dominate the dynamics of the gravitational ﬁeld in the neighborhood of its moment of extremely high curvatures.7 _{Negative powers}

control the other extreme, that is, in the case of very weak EM ﬁelds.8 _{In this case}

it modifies the evolution of the universe for large values of the scale factor, inducing the phenomenon of recent acceleration of the universe. The configuration of pure averaged magnetic field combined with the dynamic equations of general relativity received the generic name of magnetic universe.

The most remarkable property of a magnetic universe configuration is the fact that from the energy conservation law it follows that the dependence on time of the magnetic field B(t) is the same irrespective of the specific form of the Lagrangian. This property allows us to obtain the dependence of the magnetic field on the scale factor a(t), without knowing the particular form of the Lagrangian L(F ). This dependence is responsible for the property which states that strong magnetic fields dominate the geometry for small values of the scale factor; on the other hand, weak fields determine the evolution of the geometry for latter eras when the radius is big enough to excite these terms.

Recently, the both eﬀects were combined generating a cyclic cosmological toy model.9 _{The purpose of this paper is to improve this model and to investigate how}

realistic it can be. It is a well-known fact that the standard cosmological model unavoidably leads to a singular behavior of the curvature invariants in what has been termed the Big Bang. On the other hand, there are some evidences that point in the direction that the Universe is undergoing an accelerated expansion.10,11 In principle, these two problems have no simple connection and thus does not have a unique combined solution. It should be tempted to try to unify these two questions in a single model. It will be shown that in the framework of a magnetic universe, these two problems are solved at once.

In general, the cyclic cosmological models can be divided in two classes: those which generate the cyclic behavior with nonconventional matter ﬁelds,12,13 _and

those which produce the cyclicity through of extension of general relativity.14–16The model proposed here belongs to the ﬁrst class where the nonconventional matter is represented by a nonlinear magnetic ﬁeld.

The plan of this paper is as follows. In Sec. 2, it is made a review of Tolman process of average in order to conciliate the energy distribution of the EM ﬁeld with a spatially isotropic geometry. Section 3 presents the notion of the magnetic universe and its generic features concerning the dynamics of EM ﬁeld generated by a Lagrangian L = L(F ). Section 4 presents the conditions of bouncing and acceleration of a Friedmann–Robertson–Walker (FRW) universe. In Sec. 5, it is

(3)

fitted or constrained the five independent parameters presents in this models. Sec-tion 6 presents the complete scenario consisting of the five eras. The paper ends with some comments on this scenario. Moreover, a discussion about the model stability is presented in Appendix A.

2. The Average Procedure and the Fluid Representation

The eﬀects of a nonlinear EM theory in a cosmological setting have been studied in several papers.17–22

Given a generic gauge-independent Lagrangian L = L(F ), it follows that the associated energy–momentum tensor, deﬁned by

T_µν =√2 −γ δL√−γ δγµν , (1) reduces to T_µν =−4L_FF_µαF_αν− Lg_µν. (2)

The standard cosmological scenario, provided by the Friedmann–Lemaitre– Robertson–Walker (FLRW) geometry, obeys the cosmological principle, which means that the three-space is homogeneous and isotropic geometry. Thus, for com-patibility with the cosmological framework, it is necessary an average procedure in the EM ﬁeld.23_{As usual, we set the volumetric spatial average of a quantity X at}

the time t by X≡ lim V →V0 1 V X√−gd3x, (3) where V = √−gd3_{x and V}

0 is a suﬃciently large time-dependent three-volume. In this notation, it is necessary to impose that

E_i= 0, B_i= 0, E_iB_j = 0, (4) E_iE_j=−1 3E 2_g ij, BiBj =− 1 3B 2_g ij. (5)

With these conditions, the energy–momentum tensor of the EM ﬁeld associated with L = L(F ) can be written as a perfect ﬂuid,

T_µν= (ρ + p)v_µv_ν− pg_µν, (6) where ρ =−L − 4L_F¯E2, p = L−4 3(2B 2_{− E}2_)L ¯ F (7)

and L_F¯≡ dL/d ¯F . The bar above F points out that spatial average was performed.

(4)

3. Magnetic Universe

To construct a realistic magnetic universe we start assuming that the universe is composed by the five fluids, namely baryonic matter, cold dark matter, neutrinos, radiation and a cosmological magnetic field. The first four fluids will be treated in the standard way. Thus baryonic and dark matters are modeled by nonrelativistic fluids (zero pressure) and neutrinos and radiation are featured as ultra-relativistic fluid (the mass of neutrinos is neglected). The fifth fluid, the magnetic field, will be described by the Lagrangian for the nonlinear electrodynamics given by

LNLED= α2F2− 1 4F − µ2 F + β2 F2, (8)

where the dimensional constants α, β and µ are to be determined by observation. The nonlinear terms can be interpreted as a phenomenological approach. But a more interesting and fundamental scenario is when these terms represent a classical interpretation of vacuum polarization. Indeed, Heisenberg, Schwinger and others showed that quantum corrections due to vacuum polarization change the classical Lagrangian introducing nonlinear terms.24,25 _{Another important point is that the}

existence of a large-scale magnetic field is a real possibility in cosmology, since in the early universe the electric field is screened by the charged primordial plasma, while the magnetic field lines are frozen.26

Let us suppose that the five fluids are independent. Moreover, the average of the electric part of the cosmological EM field vanishes by construction, i.e. E2_{= 0.} In this case, each term of LNLED becomes cosmological independent in the sense that each one behaves as a noninteracting perfect fluid. To prove this statement we start from the conservation of the energy–momentum tensor in the co-moving framework,

˙

ρ_i+ 3˙a

a(ρi+ pi) = 0, (9)

where i represents cold matter m (baryonic and dark), ultra-relativistic components UR (radiation and neutrinos) and the cosmological magnetic ﬁeld B.

In the context of this model, ρ_B and p_B are determined using (7) and (8),

ρ_B =−α2F¯2+1 4 ¯ F +µ 2 ¯ F − β2 ¯ F2, (10) p_B =−5α 2 3 ¯ F2+ 1 12 ¯ F−7µ 2 3 1 ¯ F + 11β2 3 1 ¯ F2, (11) where ¯ F = 2B2. (12)

Substituting these values in the conservation law, it follows

L_F¯ (B2) + 4B2˙a a = 0, (13)

(5)

where L_F¯ ≡ ∂L/∂ ¯F . The solution of this equation yields B =√2B0 a0 a 2 . (14)

The solution L_F¯ = 0 was neglected because in this case the magnetic ﬁeld becomes independent of any initial conditions, which is clearly a nonphysical result.

Although the last result was deduced from the specific Lagrangian LNLED, it is in fact independent of Lagrangian particular form. This property is easily understood from the structure of (13). Thus for each power ¯Fk it is possible to associate a specific component configuration with density of energy ρ_Bk and pressure p_Bk in such a way that the corresponding equation of state is given by

p_Bk= 4k 3 − 1 ρ_Bk. (15)

Treating baryonic matter and dark matter as a single ﬂuid and neutrinos and radiation as another single ﬂuid, the magnetic universe is completely featured for six independent components. In this context, the total energy density and the total pressure are given by

ρ_T = ρ_m+ ρUR+ ρ_Bi, p_T = p_m+ pUR+ p_Bi, (16) where ρ_m= ρ_m0 a0 a 3 , p_m= 0, ρUR= ρUR0 a0 a 4 , pUR= 1 3ρUR, ρ_B1=−16α2B₀4 a0 a 8 , p_B1= 5 3ρB1, ρ_B2= B₀2 a0 a 4 , p_B2= 1 3ρB2, ρ_B3= µ 2 4B2 0 a a0 4 , p_B3 =−7 3ρB3, ρ_B4=− β 2 16B4 0 a a0 8 , p_B4=−11 3 ρB4. (17)

The real evolution of the universe depends on the values of the six constants present in (17). However, at least in a qualitative manner, it is possible to distinguish ﬁve distinct cosmological stages by just looking at the form of the terms in (17). These ﬁve stages are:

• The bouncing era: This era is governed by ρB1 and it dominates in the earliest universe (before nucleosynthesis). The bouncing era is responsible for a collapsing phase, i.e. the scale factor attains a minimum value.

(6)

• The radiation era: This era is governed by ρUR. After the bounce, ρ + 3p changes the sign and the universe enters a deceleration stage. During this epoch occurs the primordial nucleosynthesis.

• The matter era: This era is dominated by ρm. It is expected that the structure formation occurs in this epoch.

• The acceleration era: In this period ρB3 governs the cosmic evolution. The term

ρ + 3p changes the sign again and the universe enters an accelerated regime. • The re-bouncing era: This era is dominated by ρB4. The acceleration changes

the sign once more and starts a phase in which ¨a < 0. The scale factor attains a

maximum and re-bounces.

After the re-bouncing era the universe starts a collapsing phase and passes through all the three intermediate stages until a new bouncing era. This unity of five stages constitutes an eternal cyclic configuration that repeats itself indefinitely. In Sec.6, it will be shown explicitly that the universe passes through all of the five stages.

Before to the end of this section, a last point concern to positivity of the total energy must be clariﬁed. A simple look at the components of the energy density shows that if the radius of the universe can attain arbitrary small and/or arbitrary big value (domain of ¯F2_{and 1/ ¯}_F2_{, respectively) then one should face the question} regarding the positivity of its energy content. To understand this question it is necessary to look the ﬁrst Friedmann equation:

˙a

a

2

= ρ_T. (18)

Observation: In this work it is considered only Euclidean spatial section, i.e.

κ = 0.

Based on (18), a simple analysis for the evolution of the scale factor (to the future or past) shows that ˙a becomes zero exactly when ρ_T = 0. In the mathematica point of view, ρ_T = 0 implies that the function a(t) is in a maximum or minimum. It will be shown in the next sections that the model has only two extreme points for the scale factor, and these points correspond to a minimum (domain of ¯F2₎ and a maximum (domain of 1/ ¯F2). The minimum (maximum) point is a bounce (re-bounce) point and ﬁxes the minimum (maximum) value for the scale factor. Therefore, before arriving at the undesirable values where the density of energy could attain negative values, the universe bounces and re-bounces avoiding this diﬃculty.

Let us now turn to the generic conditions needed for the universe to have a bounce and a phase of accelerated expansion.

4. Conditions for Acceleration and Bouncing

The conditions for bouncing and acceleration determine the general structure of the complete scenario of cyclic magnetic universe. Both of them are derived directly

(7)

from Friedmann’s equations: ¨ a a =− 1 2(ρT + 3pT), ˙a a 2 = ρ_T. (19)

The equation above determines if the universe is accelerating or decelerating. So, the necessary and suﬃcient condition for an accelerate phase is ρ_T + 3p_T < 0. Using

(17) in this equation it is quite simple to see that ρ_m, ρUR, ρB2and ρB4contribute for a deceleration stage and the other two terms — ρ_B1 and ρ_B3 — contribute for a acceleration stage. In this manner, there are three decelerate phases (radiation, matter and re-bouncing eras) and two accelerate phases (bouncing and acceleration eras).

A bouncing universe has as its main feature avoid the initial singularity. The condition for the bounce is the existence of a minimum for the scale factor:

˙aBou= 0 and ¨aBou> 0, which is equivalent to ˙a a 2 Bou = ρBou= 0 and ¨ a a Bou > 0. (20)

In the other extreme it has a re-bounce. The condition for the re-bounce is the existence of a maximum for the scale factor

˙a a 2 RBou = ρRBou = 0 and ¨ a a RBou < 0. (21)

The bouncing and re-bouncing are important pieces in the cyclic behavior, and both are present in this model. Indeed, if we compare ρURwith ρB1, it is easily seen that there is a value for the scale factor by which the condition (20) is satisﬁed. A similar analysis involving ρ_B3, ρ_B4and (21) reveals an analogous behavior in the re-bouncing sector.

5. Setting the Six Constants

In order to analyze the complete scenario it is necessary to determine the six con-stants wherein only ﬁve are independent.a Although ﬁve independent constants are a elevated number of free parameters, this model describes all the evolution for the universe, from the bounce until the re-bounce. It keeps untouchable the main successes of the standard Big Bang model (primordial nucleosynthesis, CMB gen-eration, etc.) and solves some other problems such as the initial singularity, the present acceleration and the Big Rip.

a_{The condition}_{k = 0 lays on a bond between the six parameters.}

(8)

The six parameters are divided into two distinct classes: The first one, repre-sented by α, B0 and ρUR0, acts on the primordial universe; and the second one, represented by µ, β and ρ_m0, affects only the recent universe. Because of this dif-ference, each class must be fixed by independent manner. Let us start with the primordial constants.

5.1. Constants for the primordial universe

In this subsection, it will be ﬁxed or constrained the three primordial, parameters, namely α, B0 and ρUR0. The most simple to determine is ρUR0. So let us start with it.

Nowadays, the constant ρUR0 represents the energy density for the all ultra-relativistic particles. It is basically composed of radiation ρ_γ0 and neutrinos ρ_ν0. For radiation, the precise CMB measurement27shows that

ρ_γ0= 4.6× 10−5ρ_c, with ρ_c≡ 8πG 3H2 0

= 1.9× 10−9h−2erg, where h 0.72.

Theoretical considerations for a massless neutrinos28_imply

ρ_ν0= 6× 7 16× 4 11 4/3 ρ_γ0= 3.1× 10−5ρ_c.

Therefore, the ultra-relativistic energy density is

ρUR0= 7.7× 10−5ρc. (22) The next parameter that should be ﬁxed is B0. As could be seen in (17), the constant B0acts directly only in the primordial universe through ρB2. Nevertheless, its numerical value is so necessary to determine the α, β and µ constants. Thus, B0 aﬀects not only the primordial universe but also, in the indirect way, the present universe.

Physically the parameter B0 is related to a cosmological magnetic field. Sup-posing the existence of a large-scale magnetic field, its background mean value B0 must be B0 < 10−9G.29,30 In the cgs system, the energy density related to this magnetic field is ρ_B0¯ = ¯ B₀2 8π = 4× 10 −20_b2_erg, ₍₂₃₎

where b is just a parametrization (b≤ 1).

On the other hand, the present energy density for the cosmological magnetic ﬁeld ρ_B20 is given by ρ_B20 = ¯ F 4 = B 2 0.

(9)

So, the comparison between the last equation and (23) results in

B02= 4× 10−20b 2

erg = 1.1× 10−11b2ρ_c. (24) Replacing (24) and (22) in (17), the first relevant result arises: ρ_B2 is always subdominant, i.e. the Maxwell term concerning to cosmological magnetic field is not important to the universe evolution. In fact, even the actual cosmological exper-iments including CMB anisotropy observation performed by WMAP satellite do not have accuracy to measure a cosmological magnetic field. Perhaps, the future experiments of CMB anisotropy (e.g. Planck satellite) are able to detect it.

The last parameter necessary to fix is α. This parameter rules the bouncing stage; hence, it is concerning at the very early universe. These features make it diffi-cult to precisely determine α. Therefore, instead of fixing it we just establish a range of validity for α. The maximum and minimum allowed values are obtained through two physical conditions: The bounce must occur considerably before primordial nucleosynthesis but it should not extrapolate the validity of classical gravitation. The first condition implies|ρUR(anuc)| ρUR(anuc), i.e.

α2 ργ0 64B4 0 anuc a0 4 .

Replacing (22) and (24) in the last equation it follows

α2 5.94× 10 15 b2 T_γ0 T_γnuc 4 ρ−1_c ,

where the scale factor is converted to temperature through the relation a∼ T_γ−1. Using T_γ0 2 × 10−10MeV and supposing that the nucleosynthesis took place around 1 MeV we ﬁnally obtain

α29.5× 10

−24

b2 ρ

−1

c . (25)

For consistence with the second condition the bounce should happen below the Planck scale, i.e. T_γBou 10−19GeV. In this model, the bouncing condition (20) is obtained when ρUR+ ρB1 0, which implies

aBou a0 4 = T_γ0 T_γBou 4 = 64B 4 0 ρ_γ0 α 2_. So, T_γ0 T_γBou 4  1.6 × 10−127 and hence α2 9.5× 10 −112 b2 ρ−1c . (26)

(10)

At last, using (25) and (26), it is obtained the range of validity for the parame-ter α: 9.5× 10−112 b2 ρ −1 c  α 2 9.5× 10−24 b2 ρ −1 c . (27)

The term α2_F2_{at the Lagrangian could be interpreted as an one-loop quantum} correction in the infrared regime. Indeed, its correction was obtained ﬁrst by Euler– Heisenberg.24 _{In this context, α}2 _{values are given by}

α2 = 1 90 1 137 2 m_ec 3 1 m_ec2 ⇒ α2 4.2 × 10−32cm 3 erg = 3.68× 10 −40_ρ−1 c . (28)

Comparing (28) with (27) it could be seen that both results are consistent. So, in this scenario, the bouncing stage was generated by quantum electrodynamics eﬀects.

5.2. Constants for the recent universe

In this subsection, we intend to determine the three remain constants namely µ,

β and ρ_m0. Diﬀerently for the three previous parameters, these constants are rel-evant at the same time (present universe). For this reason, they should be set simultaneously.

The procedure that was adopted is based on the ﬁt of the model using supernova data. This procedure concerns only to recent universe, so it could be neglected the terms ρ_B1, ρUR and ρB2. In this case, the luminosity distance dL is written as

d_L(z)≡ dh H0 = (1 + z) _z 0 d¯z H(¯z), (29)

where the Hubble function H(z) is given by

H(z) H0 = Ω_m0(1 + z)3_{+ Ω} µ0(1 + z)−4+ Ωβ0(1 + z)−8 (30) with the three Ω’s deﬁned as

Ω_m0≡ρm0 ρ_c , Ωµ0≡ µ2 4B2 0ρc and Ω_β0≡ − β 2 16B4 0ρc . (31)

For z = 0, (30) implies in Ω_m0+ Ω_µ0+ Ω_β0 = 1. Thus, for the present universe, the model has only two free parameters.

The set of SNIa which was used in the ﬁt is the Union set.31_{This set collects the}

main set of supernovas which was obtained during the last 12 years. It is constituted of 414 SN divided in 13 subsets where all of them were analyzed through the same procedure. Based on this procedure, the authors excluded 107 SN, so only 307 supernovas were regarded true standard candles. It is with these 307 SN which is ﬁtted the two parameters.

(11)

The supernova data are usually assumed as Gaussian, hence it could be used for the statistical likelihood method to determine the free parameters. The standard procedure consist in to build and minimize a χ2 _{through the ﬁt of the model’s} parameter. Following the steps point out in Ref.31, let us start the construction of

χ2_{regarding only the statistical errors:}

χ2= i [µ_r(m_i, s_i, c_i| M, α, β) − µ(z_i| Ω_m0, Ω_µ0)− M]2 jk C_jke_je_k+ σ2_tot+ σ2_int , (32) with µ(z_i| Ω_m0, Ω_µ0) = 5 log d_h(z_i| Ω_m0, Ω_µ0) and µ_r= m_i− M + α(s_i− 1) − βc_i, (33) where m_iis the B-band peak magnitude, M is the absolute magnitude, s_irepresents the stretch in supernova light-curve, c_i represents the color variation between B and V bands, α is the stretch parameter, β is the color parameter and M is an additive factor that includes constants such as H0. The Cjkmatrix is the statistical covariant matrix, σtot is related with astrophysical dispersion and σint represents the intrinsical dispersion.

The term _jkC_jke_je_k represents the uncertainties associated with m_i, s_i and

c_i, where e_j = (1, α,−β). In principle α and β are free parameters, which must be ﬁtted for each speciﬁc cosmological model. Nevertheless, the authors in Ref.31 found out that α and β are almost independent of cosmological parameters. So, it is assumed that α and β are constants,bthe uncertainties of m_i, s_iand c_icould be directly passed to µ_r_i. In this case, χ2_{is written as}

χ2(Ω_m0, Ω_µ0,M) = N i=1 (µ_r_i− µ(z_i| Ω_m0, Ω_µ0)− M)2 σ2 µri+ σ 2 tot+ σ2int , (34)

where the parameter M was embedded in M. In Ref.31, is available the list with

z, µ_r_i and σ_µ_ri for each one of the 307 SNIa.

Besides σ_µ_ri, there are other three Gaussian sources of uncertainties: The ﬁrst one is the dispersion due gravitational lensing, the second one is the uncertainty in the Milky-Way dust extinction correction and the last one is the uncertainty due to peculiar velocity in the host galaxy.

• Gravitational lensing decreases the mode of the brightness distribution and causes

increased dispersion in the Hubble diagram at high red-shift.31–33_{The both strong}

and weak lensing eﬀects engender a dispersion of 0.093× z in magnitude,33and

b_{This approximation engenders a new source of error, which will be regarded as a systematic}

error.

(12)

if the statistics of SNIa is large enough this uncertainty could be considered as a Gaussian error.

• All the SNIa light-curve data were corrected for extinction caused by our Galaxy.

Nevertheless, these corrections contain statistical and systematic errors, which act directly on the magnitude value. These statistical uncertainties will be neglected because of three reasons: They only concern the SN with z < 0.2, and these ones represent approximately 20% of total set; out of Galactic plane these uncertainties value as a rule∼ 0.01; the implementation of these uncertainties is very compli-cated because the extinction and its errors change with the Galactic coordinates. The systematic uncertainties are treated later on.

• As point out in several papers,10,11,31,34 _{the peculiar velocity in the host galaxy}

causes an error σ_vwhich varies between 100 and 500 km/s. In this paper, we adopt

σ_v = 300 km/s. So, the uncertainty propagated to magnitude due the peculiar velocities is σ_µ_v = 5 d_hln 10σdh 2.172 z σ_v c , where σ_d_h =σv c = 0.001.

These three sources of uncertainty are encapsulated in σtot. Thus, the discussion above results in σ_tot2 = σ2_les+ σ2_µ v = (0.093z) 2₊ 2.172 z σ_v c 2 . (35)

The last source of statistical error considered is the uncertainty produced by an intrinsical dispersion σint. This intrinsical dispersion is related about noncorrected and/or unknown errors.

To estimate σint, a first fit with a small initial value (∼0.01 mag) is performed. Then a new σintimposing χ2= 1 is determined. The last step consists in performing the same routine once more to obtain a more accurate and definitive σint. During this procedure, σtot is not used because the intrinsical dispersion is related only with the “main” data, namely{z, µ, σ_µ}.

5.2.1. Systematic uncertainties

The systematic uncertainties are introduced in the same way performed in Ref.31. The method consists of introducing a Gaussian distribution with a ∆M_sparameter for each present systematic uncertainty σ_∆M_s. The new parameters get in additively in the modulus of distance, i.e.

µ(z| Ω_m0, Ω_µ0)→ µ(z | Ω_m0, Ω_µ0) + s

∆M_s and the previous likelihood changes to

Lprev(Ωm0, Ωµ0)→ Lpost(Ωm0, Ωµ0, ∆Ms),

(13)

where, Lpost(Ωm0, Ωµ0, ∆Ms)≡ Lprev(Ωm0, Ωµ0) s e− ∆M2_s 2σ2_∆Ms_.

In this kind of construction, we are supposing that a speciﬁc source of systematic error leads to a “variation” in the modulus of distance. The importance of its vari-ation is determined through a Gaussian distribution where the standard devivari-ation is identiﬁed with the systematic uncertainty.

At the Union set, there are two kinds of systematic errors: One which depends on the specific subset (e.g. due to observational effects) and the other which con-cerns to all set of supernovas (e.g. due to astrophysical or fundamental calibration effects). The systematic uncertainties common to all samples can be absorbed in the definition of absolute magnitude. On the other hand, it is expected a differ-ence between the nearby (z∼ 0.05) and distant (z ∼ 0.5) supernovas. So, we can cast the common systematic uncertainties to a uncertainty in the difference ∆M between absolute magnitudes of close and distant SNs. Following the steps dis-cussed in Ref. 31, it is defined zdiv = 0.2 as the point which splits the SNs in nearby and farther objects. Beside ∆M , we introduce a set of 13 parameters ∆M_s which represents the systematic uncertainties for each one of the 13 subsets.

The complete likelihood for the model is given by

Lpost= e− χ2_post 2 , (36) where χ2post = ₁₃ s=1 _N s i=1 (µ_r_is− µ + ∆M∗+ ∆M_s− M)2 σ2 µris + σ 2 tot+ σ2int +∆M 2 s σ2 ∆Ms +∆M 2 σ2 ∆M (37) with ∆M∗= ∆M for z > 0.2, 0 for z≤ 0.2.

The Union set was split in the sum of the 13 subset where each subset has N_s samples. Thus, comparing (34) with (37) it follows that 13_s=1N_s= N .

The systematic errors arise from seven distinct sources:

• Errors due to the ﬁxation of α and β parameters. • Errors due to the possible samples contamination.

• Errors due to the models of supernovas light-curve and K-corrections. • Error due to photometric zero point calibration.

• Error due to a possible variation of Malmquist bias with the red-shift. • Systematic errors due to gravitational lensing eﬀects.

• Systematic error due to the normalization in the corrections for Galactic

extinc-tion maps.

(14)

Taking into account all these eﬀects, the authors of Ref.31reached the following uncertainties:

σ_∆M = 0.04 and σ_∆M_s = 0.033 for all s. So, the errors related to each one of the 13 subsets are the same.

5.2.2. Results

First, we take into account only the statistical uncertainties. Using the data present in Ref. 31, a numerical calculation is performed to determine the cosmological parameters. The procedure could be summarized in ﬁve steps:

(1) Construction ofL(Ω_m0, Ω_µ0,M) not taking into account σtot. (2) Marginalization ofM parameter.c

(3) Calculation of σint.

(4) Construction of the new likelihood taking into account σtot and the σint. (5) Finally, the minimization of χ2_(Ω

m0, Ωµ0). The results are shown in Fig.1.

The best ﬁt for Ω_m0 or Ω_µ0 could be established marginalizing all the other parameters. Performing these marginalization we get the following results:

Ω_m0= 0.44+0.03_−0.03 and Ω_µ0= 1.20+0.35_−0.33. (38) 0.3 0.35 0.4 0.45 0.5 0.55 0.6 m0 0.5 1 1.5 2 2.5 µ 0

Fig. 1. Parametric contour plot for Ωm0× Ωµ0 taking into account only the statistical

uncer-tainties. From inside to outside the contour lines correspond respectively to 68%, 95% and 99%

conﬁdence levels. The dashed line represents Ω_β0= 0.

c_{We are interested only in cosmological parameters.}

(15)

Finally, Ω_β0 is determined through the equation Ω_m0+ Ω_µ0+ Ω_β0= 1. Thus,

Ω_β0 =−0.64−0.35_+0.33. (39)

The next step is to include the systematic uncertainties in the procedure. Anal-ogously to the ﬁrst case, it is performed an analytic marginalization at the “non-cosmological parameters” namelyM, ∆M and the set of ∆M_s. Again, the method is straightforward but the calculation is much more cumbersome than the previous case.

Using the same steps described earlier, the following result is obtained. For this case, the best ﬁts for Ω_m0and Ω_µ0 are

Ω_m0= 0.448+0.042_−0.039 and Ω_µ0= 1.17+0.39_−0.37 (40) and the calculated value for Ω_β0 is

Ω_β0 =−0.62−0.39_+0.37. (41)

As expected, the inclusion of systematic uncertainties increases the uncertainties at the cosmological parameters.

5.2.3. Including a prior in Ω_m0

Measurements of X-ray gas mass fraction fgas done in galaxy clusters allow to establish excellent values to the ratio Ω_b0/Ω_m0.35,36_{On the other hand, primordial}

nucleosynthesis determines with good accuracy the actual baryon density Ω_b0. So,

0.3 0.35 0.4 0.45 0.5 0.55 0.6 m0 µ 0 0.5 1 1.5 2 2.5

Fig. 2. Parametric contour plot for Ωm0× Ωµ0taking into account the statistical and systematic

uncertainties. From inside to outside the contour lines correspond respectively to 68%, 95% and

99% conﬁdence levels. The dashed line represents Ω_β0= 0.

(16)

when we join these two facts it is possible to obtain an excellent estimation for Ω_m0. Based on this estimation, we intend to build a Gaussian prior to Ω_m0as follows:

P = Exp − Ω_m0− ¯Ω_m02 2σ_Ω2¯_m0 ,

where ¯Ω_m0and its uncertainty are obtained through gas mass fraction procedure. According to Ref.36, the fgasis linked with Ωb0/ ¯Ωm0as shown in Eq. (42):

f_gasΛCDM(z) =KAγb(z) 1 + s(z) Ω_b0 ¯ Ω_m0 dΛCDM_A (z) d_A(z) 3 2 , (42)

where d_Ais the angular diameter distance given by

d_A(z) = 1 (1 + z) _z 0 dz H(z). (43)

The f_gasΛCDM is the gas mass fraction for ΛCDM model (the reference model) with the following parameters: Ω_m0= 0.3, ΩΛ= 0.7 and h = 0.7. The other factors that appear in (42) are described below:

• K is a constant that parametrizes uncertainties in the accuracy of the

instru-ment calibration and X-ray modeling. Its value is K = 1± 0.1.

• γ is related with the nonthermic pressure support in the clusters. Its value is

1.0 < γ < 1.1.

• b(z) is the depletion factor, i.e. the ratio by which the baryon fraction is depleted

with respect to the universal mean. It is modeled by b(z) = b0(1 + αbz) where

b0= 0.83± 0.04 and −0.1 < αb< 0.1.

• The function s(z) = s0(1 + αsz) describes the baryonic mass fraction in stars where s0= (0.16± 0.05)h1/270 and−0.2 < αs< 0.2.

• The A factor takes into account the change in the subtended angle of the

clusters due to the diﬀerence between ΛCDM model and the others models. According to authors of Ref.36, this eﬀect could be neglected for all red-shifts of interest, i.e. A = 1.

To estimate ¯Ω_m0, a set of six clusters with low red-shifts is used (see Table1).

Table 1. The red-shift andfgasfor the six galaxy clusters.

The data were taken from Table 3 of Ref.36.

Cluster classiﬁcation z fgasΛCDMh1.570

Abell 1795 0.063 0.1074 (0.0075) Abell 2029 0.078 0.1117 (0.0042) Abell 478 0.088 0.1211 (0.0053) PKS0745-191 0.103 0.1079 (0.0124) Abell 1413 0.143 0.1082 (0.0058) Abell 2204 0.152 0.1213 (0.0116)

(17)

As we consider only clusters at low red-shift, the dependence on z of the func-tions b(z) and s(z) can be neglected. Thus, inverting (42) it follows:

¯ Ω_m0 Kb0 (1 + s0)fgasΛCDM dΛCDM A (z) d_A(z) 3 2 Ω_b0. (44)

Face on the other uncertainties, the variation of gamma could be neglected, i.e.

γ = 1. Using primordial nucleosynthesis models and data of deuterium abundance,

it is possible to derive the value for Ω_b0 with good accuracy37:

Ω_b0h2= 0.0214± 0.002 with h = 0.7 ± 0.1, (45) where the value of h was determined in Ref.38.

The procedure to determine ¯Ω_m0could be summarized in the following steps: (1) Fit of parameters using the set of SNIa without priors. The steps were discussed

previously.

(2) Calculation of d_A(z) using the previous ﬁt.

(3) Determination of ¯Ω_m0and its corresponding uncertainty for each one of the six clusters.

(4) Calculation of weighted mean and the mean uncertainty for ¯Ω_m0. (5) Finally, it repeats the previous step but now using the prior for Ω_m0.

These steps were performed few times until ¯Ω_m0stabilizes as ¯

Ω_m0= 0.278± 0.038. (46)

The complete result including the statistical and systematic uncertainties and the prior in Ω_m0is shown in Fig. 3. The best ﬁts for Ω_m0and Ω_µ0 are

Ω_m0= 0.367+0.026_−0.023 and Ω_µ0= 1.47+0.37_−0.34 (47) and the calculated value for Ω_β0 is

Ω_β0 =−0.84−0.37_+0.34. (48)

It is interesting to determine explicitly the values of µ and β. Thus, replacing (47), (48) and (24) in (31) follows: µ = 4B2 0ρcΩµ0= 8.0× 10−6bρc, β = −16B4 0ρcΩβ0= 3.98× 10−11b2ρ3/2c .

Since ρ_c∼ 10−9erg, the values of µ and β are extremely small and undetectable in the classical earth experiments. The other interesting feature of this model is that the value of Ω_m0 is greater than the majority models present in literature. Even with the inclusion of the prior in Ω_m0the diﬀerence still remains. It is possible that this feature is related to the eminent deceleration provided by the β2/F2term. An important last point to comment is that with the inclusion of this prior, the region where Ω_β0 > 0 (below to the dashed line) is excluded in 2σ. It is an important

(18)

0.3 0.35 0.4 0.45 m0 0.5 1 1.5 2 2.5 µ 0

Fig. 3. Parametric contour plot for Ωm0× Ωµ0taking into account the statistical and systematic

uncertainties and a prior in Ωm0. From inside to outside the contour lines correspond respectively

to 68%, 95% and 99% conﬁdence levels. The dashed line represents Ω_β0= 0.

feature because with positive Ω_β0 the model does not engender the re-bounce [see condition (21)].

6. The Complete Scenario

Before discussing the complete scenario, it is necessary, based on the previous sec-tion, to choose the values for the six constants of this model. For the three constants associated with the primordial universe were selected (22) for ρUR0, (24) with b = 1 for B₀2 and α2 = 10−64ρ−1_c suitable with (27). For the three constants related to the recent universe were chosen the best fit for Ω_m0, Ω_µ0 and Ω_β0 given by (47) and (48). This best fit is the most convenient choice to characterize the complete scenario because it incorporates more observational cosmological features than the other two. Nevertheless, using another best fit the qualitative results discussed below remain the same.

6.1. The bouncing period and the primordial acceleration

Near to bounce, the Friedmann equations could be approximated by ˙a a 2 H2 0 ΩUR0 _a₀ a 4 − Ωα0 _a₀ a 8 and ¨ a a −H 2 0 ΩUR0 _a 0 a 4 − 3Ωα0 _a 0 a 8 ,

(19)

where ΩUR0 = ρUR0 ρ_c and Ωα0= 16α2_B4 0 ρ_c .

The ﬁrst condition for bouncing (20) implies aBou a0 4 = Ωα0 ΩUR0 ,

which allows to rewrite the Friedmann equations as ˙a a 2 H2 0 Ω2_UR0 Ω_α0 _a_Bou a 4 −aBou a 8 (49) and ¨ a a −H 2 0 Ω2 UR0 Ω_α0 _a Bou a 4 − 3aBou a 8 . (50)

The above two equations clearly show the existence of bounce. Using the numerical values for α2_{, B}2

0 and ρUR0, it is possible to determine the minimum for the scale factor: aBou Ω_α0 ΩUR0 1/4 a0 7.08 × 10−21a0, (51) where in this calculation the contribution of nonrelativistic matter was neglected.

The solution for (49) was studied in Ref.9 and has the following form:

a(t) = aBou4

(2Qt)2_{+ 1.} ₍₅₂₎

Here the initial condition a(0) = aBou was used. The time scale Q is deﬁned as

Q≡H√0ΩUR0

Ω_α0 = 4.1× 10 20_s−1_,

with H0= 72 km/s· Mpc (see Ref.38). This solution is valid throughout the range in which the nonrelativistic matter is negligible, e.g. from bounce to primeval nucle-osynthesis.

The bouncing period plot is shown in Fig.4.

Just looking for this plot, it is possible to realize that the primordial acceleration (inflation) engendered by the model is restricted. In fact, Eq. (50) states that the acceleration occurs only between aBou< a(t) < 1.32aBou, which in usual inflation-ary language corresponds just to 0.3 e-folds. It rules out the standard mechanism to solve the horizon problem. Nevertheless, in this model the universe is cyclic and eternal, so in fact this problem does not exist. Much more instigate (and com-plicated) is the question of evolution of energy density fluctuations. This issue is

(20)

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 1.0 1.2 1.4 1.6 1.8 2.0 _a(t)/a_Bou t (10-20_s)

Fig. 4. Plot for the scale factora in function of time t. This ﬁgure shows that the inﬂationary

phase is very restrained.

directly related with model stability (see Appendix A) and it will be the subject of future investigations.

6.2. The re-bouncing period and the recent acceleration

For the present, the Friedmann equations could be approximated by ˙a a 2 H2 0 Ω_m0 _a 0 a 3 + Ω_µ0 a a0 4 + Ω_β0 a a0 8 (53) and ¨ a a − H2 0 2 Ω_m0 _a 0 a 3 − 6Ωµ0 a a0 4 − 10Ωβ0 a a0 8 , (54)

where Ω’s are deﬁned in (31).

Unlike the equations of bounce, the equations above could not be solved ana-lytically. Even the aRBou could not be expressed from Ω’s in a simple manner. Nevertheless, numerical approaches allow us to study many features of this period. Figure5shows the numerical solution for (53).

From the numerical calculations involving (53) the maximum size for the uni-verse can be determined,

aRBou= 1.173a0, (55)

and the time remaining to reach it is

tRBou− t0= 3.37 Gyr.

(21)

0 2 4 6 8 10 12 14 16 18 0.6 0.7 0.8 0.9 1.0 1.1 1.2 t₀ a/a0 t(Gyr)

Fig. 5. Plot for the scale factora/a0 in function of timet in Giga years. t = 0 is chosen as the

instant when the actual acceleration begins.t0corresponds to interval of time between the start

of acceleration and the present time.

The period of acceleration ∆tacelis obtained from (54): ∆tacel= 5.33 Gyr.

It started at 5.31 billions of years ago and in terms of red-shift it began at

zacel= 0.533.

These results show that the acceleration era is almost ﬁnished, so spending another 20 million years we are entering into the re-bouncing period.

6.3. The main features for the complete scenario

To obtain global features for this model it is necessary to solve the ﬁrst Friedmann equation with all terms. It is given by

˙a a 2 = H₀2 Ω_B0 a0 a 4 + ΩUR0 a0 a 4 − Ωα0 a0 a 8 + Ω_m0 a0 a 3 + Ω_µ0 a a0 4 + Ω_β0 a a0 8 , (56) where Ω_B0 ≡B02 ρc is always negligible.

Again, we must use numerical methods to solve (56). The result is shown in Fig.6.

The plot above shows the complete evolution for one and a half cycle. This model is a eternal cosmological model, so it is meaningless to talk about the initial time and the age of the universe. Nevertheless, one could determine the time passed

(22)

-20 -10 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t₃ t₂ t₁ a/a0 t(Gyr)

Fig. 6. Plot for the scale factora/a0 in function of timet in Giga years. t = 0 is chosen as the

instant of bounce. The three timest1,t2 andt3 split the expansion period in three eras: Matter

era — 10−4 t < t1, acceleration era — t1 < t < t2 and re-bouncing era —t2< t < t3. The

other two eras, radiation and bouncing eras, are too close to origin, thus it is not possible to distinguish them, at least in these scales.

of bounce until today ∆ttoday and a period to complete a cycle ∆tcycle. Using the solution for (56) it follows that ∆ttoday= 13.0 Gyr and ∆tcycle= 32.6 Gyr.

The solution of (56) allows to establish the values for the scale factor at the bouncing and re-bouncing instants. In terms of a0we obtain aBou= 7.08× 10−21a0 and aRBou = 1.173a0, where the maximum variation for the scale factor is 1.66× 1020_{. Comparing these results with those obtained previously — Eqs. (51) and} (55) — it veriﬁes that both have the same numerical value. This conﬁrms the validity of analysis made in Secs.6.1 and6.2.

Another interesting analysis that can be done is the evolution of total equation of state ω_T. Using the relation

ω_T =pT

ρ_T,

we can study its evolution in terms of time. In most of the time during a com-plete cycle|ω_T| ≤ 1. This range include two long periods of deceleration (radiation and matter eras) and two periods of acceleration (primordial and recent). However, because of features of bouncing and re-bouncing, ω_T must eventually assume val-ues less than −1 and values greater than 1. Indeed, near to bouncing in a range

|t − tBou| < 1.1 × 10−21s we have ωT <−1, which means a phantom equation of state. Near to re-bouncing in a range|t − tRBou| < 2.3 Gyr we have ωT > 1, i.e. an Ekpyrotic equation of state. It is noteworthy that during all period of recent accel-eration ω_T >−1. This analysis conﬁrms that this model is an explicit realization

of a quintom cyclic scenario.12,39

(23)

7. Conclusion

In this paper, a cyclic cosmological model with five fluids namely baryonic and nonbaryonic matter, radiation, neutrinos and a cosmic magnetic field described by a nonlinear electrodynamics was constructed. With five independent parameters it reproduces correctly the three expansion phases — radiation, matter and accelerate phases — and produces bouncing and re-bouncing stages. Another important point is that the main features for the model could be analyzed by splitting it into two independent phases (primordial and actual) with a linking phase to connect both. In the primordial phase the relevant components are the bouncing and ultra-relativistic components characterized by the constants α and ΩUR0. The ultra-relativistic constant was fixed through CMB measurement and theoretical con-sideration about the neutrinos background. On the other hand, the constant α remains widely free. It happens because there are not much information about the pre-nucleosynthesis universe. Another way to determine α is to interpret the term

α2_F2_{as an one-loop quantum correction due to vacuum polarization in the infrared} regime.24,25 In this case, α2 10−40ρ−1_c , and this value is in complete agreement with the cosmological constraints (see Sec.5.1).

Nowadays, the relevant components are the nonrelativistic matter, the accel-eration and the re-bouncing terms. The three constants — Ω_m0, Ω_µ0 and Ω_β0 — associated with each term were fixed using data of SNIa. Figures 1–3 show that the re-bounce probably occurs. For the complete fit, represented by Fig. 3, the nonexistence of re-bouncing is excluded at 95% confidence level. Another impor-tant feature for this model is the high value obtained for Ω_m0 compared with the standard ΛCDM model. Even with a robust prior for nonrelativistic matter this feature remains. I believe that it happens due to the eminent deceleration engen-dered by the re-bouncing term. To better clarify this point, it would be important to use other cosmological tests. This possibility will be investigated in the future.

An important question concerns about the evolution of energy density fluctu-ations. As pointed out in Sec. 6.1, the model does not engender an inflationary period. Nevertheless, it is not an essential problem for eternal cyclic universe since the causal connection could be established in the previous cycle. Still, the sub-ject about formation and/or dissociation of structure (linked with the evolution of energy density fluctuations) is an important topic and should be investigated in the future.

Another question concerns about the features of NLED used. As mentioned in Sec.3, the NLED was only used to describe the cosmological magnetic field while the radiation was retracted by the standard electrodynamics. So, the nonlinear effects act only in the magnetic field and the description of a free radiation remains as in Maxwell theory. The main reason to adopt this approach is because quantum corrections due to vacuum polarization change the classical Lagrangian introduc-ing nonlinear terms.24 _{The new terms appear only when we are treating with a}

quasistatic EM ﬁeld.25So, the fundamental Lagrangian is the Maxwell one and the

(24)

extra terms are just eﬀects of vacuum polarization. This approach is directly related to the model stability. Indeed, if we consider the LNLEDas fundamental Lagrangian instead of Maxwell Lagrangian the model necessarily will present instabilities near to bounce (see Appendix A).

For a wide point of view, we can speculate that the EM phenomenons in extremum conditions (e.g. very large and very short scale) are described by a nonlin-ear generalization of Maxwell electrodynamics. Assuming that this generalization in a given scale could be expanded in positive and negative powers of F ≡ F_µνFµν, it is necessary for convergence which α_kFk _{> α}

k+1Fk+1and βkF−k > βk+1F−(k+1) for k > 0. So, it can be argued that in this model the maximum and the mini-mum values for the scale factor limit the inﬂuence of Fk _{terms with} _{|k| > 2, i.e.} terms of kind F3_{, F}4_{, . . . and F}−3_{, F}−4_{, . . . are negligible in all cosmological time.} In this context, the proposal Lagrangian is an approximation for a generic nonlinear electrodynamics theory.

Acknowledgments

I would like to acknowledge M. Novello, J. Salim and A. N. Ara´ujo for their very useful discussion and comments. I also acknowledge the support of FAPERJ. Appendix A: Model Stability

The purpose of this appendix is to discuss the model stability. As it is well known, all flat cosmological models with bounce violate the null energy condition (NEC). On the other hand, in Ref.40it was shown that under fairly general conditions the NEC violation necessarily implies instability (ghosts). More specifically, the authors showed that when all null vectors violate NEC (as in isotropic systems) or when superluminal excitations are not present, the NEC violation will result in insta-bilities. These statements extend in a straightforward manner for isotropic fluids through the Lagrangian formulation for fluid dynamics.41 Thus, it is reasonable to ask if the proposed model is stable?

To answer this question we will restrict the analysis to a region near the bounce. In this case the relevant components are represented by the following Lagrangian:

L α2F2−1

4F− 1

4FUR, (A.1)

where F is associated with the magnetic ﬁeld and FUR with the ultra-relativistic constituents.

To analyze the stability of a model it is necessary to perturb this model and ana-lyze the behavior of these perturbations. Thus, in the context of magnetic universe model, this analysis should be separated in two cases: perturbations taken before the achievement of spatial averages and perturbations taken after this achievement. Let us discuss the ﬁrst case (perturbations taken before). The term F2 arises when quantum corrections from vacuum polarization are interpreted classically.24,25

Indeed, Schwinger showed in Ref. 25 that for quasistatic EM ﬁelds the eﬀects

(25)

of vacuum polarization induce corrections in the classical Lagrangian and the ﬁrst of these corrections is given by α2_F2_{. It is worth noting that the complete} results obtained in Refs.24 and25are nonperturbative. So, the term α2_F2 repre-sents the inclusion (classical) of eﬀects of vacuum polarization in the cosmological background. Therefore, the quantum perturbations should be done in the usual Lagrangian of electrodynamics 1₄F which does not present any instability

prob-lem. In this context, the quantization of L α2_F2₋ 1

4F or any other LNLED is completely meaningless.

For the second case, the perturbations must be performed in energy–momentum tensor of perfect ﬂuid (isotropic background). So, the conclusions found in Ref.40 can be directly applied, and therefore the violation of NEC will imply in an unstable model. Note that these perturbations should necessarily be of classical type (such as gravitational perturbations) since the spatial average has already been taken. In the magnetic universe model, this period of instability occurs only near to bounce and it quickly ends (see Sec. 6.1).

From a diﬀerent point of view, we can look at (A.1) as a fundamental Lagrangian. Thus, it is necessary to analyze the stability of this Lagrangian. The term−₄1FUR will be neglected because it does not generate instabilities. So,

L_B α2F2−1

4F. (A.2)

The energy–momentum tensor of L_Bis calculated with the assistance of (2) and results in T_µν= (1− 8α2F )F_µαF_αν− α2F2−1 4F g_µν,

remembering that F_µν is determined only by magnetic ﬁeld B.

To establish that there is a violation of NEC it is necessary to analyze the quantity T_µνnµnν where nµ= (ω, n) is a null vector. Perfoming this calculation in FLRW background we obtain the following equation:

T_µνnµnν= [1− 8α2F ] n× B a 2 , (A.3)

where F is function of B2. Thus, for NEC violation (T_µνnµnν < 0) to occur the

ﬁrst bracket should be negative. Another important point is that if T_µνnµ_nν _{< 0} then all null vectors violate NEC.

In general, the above equation is not spatially isotropic since B may depend on points in space. Indeed, it is assumed that B has a random spatial distribution whose spatial average is determined by (5). Therefore, it is possible that there are regions of space with instability (where NEC is violated for all null vectors) and other regions which are perfectly stable. Note that the instabilities only appear at many diﬀerent points of space if, on average, the term 8α2_{F is of the order (or} larger) of unity, i.e. 8α2F¯ 1, and it is exactly what happens near to bounce. So,

(26)

using the bouncing condition (20) and (A.2) follows ρBou 1 4 ¯ FBou− α2F¯Bou2 = 0⇒ 8α 2_F_¯ Bou= 2, where ¯F = 2B2 — Eq. (5).

Since, on average, the signal of (A.3) is determined by [1− 8α2F ]¯ or [1− 16α2B2],

it is possible to conclude that near to bounce most regions will present unstable perturbations. However, as the universe expands, the factor B2_{∼ a}−4 _decreases, causing a decrease in the number of regions with unstable perturbations. Note that with the increase of a in only one order of magnitude the probability of having a region with unstable perturbations is negligible. One interesting possibility for future work is to study how the perturbations generated near to bounce evolve.

References

1. M. Gasperini and G. Veneziano, Phys. Rep.373 (2003) 1, arXiv:hep-th/0207130.

2. J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, Phys. Rev. D 64 (2001)

123522, arXiv:hep-th/0103239.

3. M. Novello and S. E. Perez Bergliaﬀa, Phys. Rep.463 (2008) 127.

4. R. Aldrovandi, R. R. Cuzinatto and L. G. Medeiros, Eur. Phys. J. C 58 (2008) 483,

arXiv:0801.0705.

5. C. Corda and H. J. M. Cuesta, Astropart. Phys.34 (2011) 587, arXiv:1011.4801.

6. P. Peter and N. Pinto-Neto, Phys. Rev. D 78 (2008) 063506.

7. V. De Lorenci, R. Klippert, M. Novello and J. M. Salim, Phys. Rev. D 65 (2002)

063501.

8. M. Novello, S. E. Perez Bergliaﬀa and J. Salim, Phys. Rev. D 69 (2004) 127301,

arXiv:astro-ph/0312093.

9. M. Novello, A. N. Araujo and J. Salim, Int. J. Mod. Phys. A 24 (2009) 5639,

arXiv:0802.1875.

10. A. G. Riess et al., Astron. J.116 (1998) 1009, arXiv:astro-ph/9805201.

11. S. Permutter et al., Nature 391 (1998) 51, arXiv:astro-ph/9712212.

12. H. H. Xiong, Y. F. Cai, T. Qiu, Y. S. Piao and X. Zhang, Phys. Lett. B 666 (2008)

212, arXiv:0805.0413.

13. J. V. Narlikar, G. Burbidge and R. G. Vishwakarma, J. Astrophys. Astron.28 (2007)

67, arXiv:0801.2965.

14. Y. F. Cai and E. N. Saridakis, J. Cosmol. Astropart. Phys. 10 (2009) 20,

arXiv:0906.1789.

15. Y. F. Cai and E. N. Saridakis, Class. Quantum Grav. 28 (2011) 35010,

arXiv:1007.3204.

16. Y. F. Cai and E. N. Saridakis, J. Cosmol.17 (2011) 7238, arXiv:1108.6052.

17. P. Vargas Moniz, Phys. Rev. D 66 (2002) 103501.

18. V. Dyadichev, D. Gal’tsov, A. Zorin and M. Zotov, Phys. Rev. D 65 (2002) 084007,

arXiv:hep-th/0111099.

19. R. Garcia-Salcedo and N. Breton, Int. J. Mod. Phys. A15 (2000) 4341,

arXiv:gr-qc/0004017.

20. M. Sami, N. Dadhich and T. Shiromizu, Phys. Lett. B 568 (2003) 118,

(27)

21. E. Elizalde, J. Lidsey, S. Nojiri and S. Odintsov, Phys. Lett. B 574 (2003) 1, arXiv:hep-th/0307177.

22. R. Garcia-Salcedo and N. Breton, Class. Quantum Grav.20 (2003) 5425,

23. R. Tolman and P. Ehrenfest, Phys. Rev.36 (1930) 1791.

24. W. Heisenberg and H. Euler, Z. Phys.98 (1936) 714.

25. J. Schwinger, Phys. Rev. 82 (1951) 664.

26. D. Lemoine and M. Lemoine, Phys. Rev. D 52 (1995) 1955.

27. Particle Data Group, http://pdg.lbl.gov/ (2008).

28. S. Weinberg, Cosmology (Oxford University Press, New York, 2008).

29. S. Bertone et al., Mon. Not. R. Astron. Soc.370 (2006) 319, arXiv:astro-ph/0604462.

30. F. Govone and L. Feretti, Int. J. Mod. Phys. D 13 (2004) 1549,

arXiv:astro-ph/0410182.

31. M. Kowalski et al., Astrophys. J.686 (2008) 749, arXiv:0804.4142.

32. M. Sasaki, Mon. Not. R. Astron. Soc.228 (1987) 653.

33. D. E. Holz and E. V. Linder, Astrophys. J.631 (2005) 678.

34. P. Astier et al., Astron. Astrophys. 447 (2006) 31, arXiv:astro-ph/0510447.

35. S. Shindler, Space Sci. Rev.100 (2002) 299, arXiv:astro-ph/0107028.

36. S. W. Allen et al., Mon. Not. R. Astron. Soc.383 (2008) 879, arXiv:0706.0033.

37. D. Kirkman et al., Astrophys. J. Suppl.149 (2003) 1, arXiv:astro-ph/0302006.

38. W. L. Freedman et al., Astrophys. J.553 (2001) 47, arXiv:astro-ph/0012376.

39. Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia, Phys. Rep. 493 (2010) 1,

arXiv:0909.2776.

40. S. Dubovsky et al., J. High Energy Phys.603 (2006) 25, arXiv:hep-th/0512260v2.

41. R. Jackiw et al., J. Phys. A37 (2004) 327, arXiv:hep-ph/0407101.