Observational Study of Higher Dimensional Magnetic Universe in Nonlinear Electrodynamics
Abstract
In this work, we have considered the flat FRW model of the universe in dimensions filled with the dark matter and the magnetic field. We present the Hubble parameter in terms of the observable parameters and with the redshift and the other parameters like . The natures of magnetic field , deceleration parameter and diagnostic have also been analyzed for accelerating expansion of the universe. From Stern data set (12 points), we have obtained the bounds of the arbitrary parameters by minimizing the test. The bestfit values of the parameters are obtained by 66%, 90% and 99% confidence levels. Now to find the bounds of the parameters () and to draw the statistical confidence contour, we fixed four parameters . Here the parameter determines the higher dimensions and we perform comparative study between three cases : 4D , 5D and 6D respectively. Next due to joint analysis with BAO observation, we have also obtained the bounds of the parameters () by fixing other parameters for 4D, 5D and 6D. The best fit of distance modulus for our theoretical model and the Supernova Type Ia Union2 sample are drawn for different dimensions.
1 Introduction
The origin of the classical Einstein field equations are Maxwell’s
electrodynamics which leads to the singular isotropic Friedmann
solutions. Over the last few years the standard cosmological
model, based on FriedmannRobertsonWalker (FRW) geometry with
Maxwell’s electrodynamics has got sufficient amount of interest
and many significant result are obtained ?; Murphy (1973); De Sitter (1917); Novello et al (1979, 1993); Breton et al (2005, 2000); Novello (2005); Klippert et al (2000). Recently the nonlinear electrodynamics
(NLED) is a very interesting subject of research in general
relativity. In 1934, Born and Infield Bron et al (1934) wanted to
modify the standard Maxwell theory in order to eliminate the
problem of infinite energy of electron. In present time a new
approach De Lorenci et al (2002) has been taken to avoid the cosmic
singularity through a nonlinear extension of the Maxwell’s
electromagnetic theory and black hole solution can be obtained
Kats et al (2007); Anninos et al (2009); Cai et al (2008). Another interesting feature
can be viewed that for construction of regular black hole
solutions Beato et al (1998, 1999); Salazar et al (1987),
nonlinear electrodynamics theories are most powerful tool. The
solution of the Einstein field equations together with NLED
signifies the nonlinear effects in strong gravitational and
magnetic fields. In the standard Maxwell Lagrangian, the nonlinear
terms can be added by imposing the existence of symmetries such as
parity conservation, gauge invariance, Lorentz invariance, etc.
Novello et al (1996); Munoz (1996), as well as by the introduction of first
order quantum corrections invariance to the Maxwell
electrodynamics Heisenberg et al (1936); Schwinger (1951).
Our theoretical models are continuously testing by the different
observational data. Currently the universe is expanding with
acceleration which is verified by different observations of the
SNeIa Perlmutter et al (1998, 1999); Riess et al (1998, 2004),
large scale redshift surveys Bachall et al (1999); Tedmark et al (2004), the
measurements of the cosmic microwave background (CMB)
Miller et al (1999); Bennet et al (2000) and WMAP Briddle et al (2003); Spergel et al (2003). The
observational facts are not clearly described by the standard big
bang cosmology with perfect fluid. Recently several interesting
mechanisms such as loop quantum cosmology Asthekar et al (2011),
modified gravity Cognola et al (2009), higher dimensional phenomena
Chakraborty et al (2010); Ranjit et al (2012), BransDicke theory Brans et al (1961),
braneworld model Gergely et al (2002) and so on, suggested that some
unknown matters are responsible for accelerating scenario of the
universe which has positive energy density and sufficient negative
pressure, known as dark energy Sahni et al (2000); Padmanabhan (2003). The most
suitable type of dark energy for that scenario is the scalar field
or quintessence Pee bles et al (1988) in which the potential dominates
over the kinetic term. In the present time several cosmological
models have been constructed by introducing dark energies such as
phantom Caldwell (2002); Bronnikov et al (2012); Chang et al (2012), tachyon scalar field
Sen (2002); Balart et al (2007); Farajollahi et al (2011); Del Campo et al (2009), hessence Wei et al (2005), dilaton
scalar field Morris (2012); Marcus (1990), Kessence scalar field
ArmendarizPicon et al (2001); Bouhmadi et al (2010); Malquarti et al (2003), DBI essence scalar field
Spalinski (2007); Martin et al (2008) and many others. Recent observational
evidence suggests that the present Universe is formed of
26% matter (baryonic + dark matter) and 74% of a smooth
vacuum energy component and about 0.01% of the thermal CMB
component. The information about the structure formation process
and other important cosmic observable are obtained by the angular
power spectrum of CMB components in anisotropic.
Brief review of Maxwell’s electrodynamics and nonlinear
electrodynamics are presented in section II. The basic equations
in dimensional FRW universe and their solutions are given
in section III for interacting model. The nature of Omdiagnostic
are studied also. The observational data analysis mechanism for
nonlinear electrodynamic are described in section IV. The
test for best fit values of the observational
parameters are investigated with Stern and then Stern+BAO joint
data analysis. The best fit of distance modulus for our
theoretical model and the Supernova Type Ia Union2 sample are
drawn for different dimensions.
Finally, some observational conclusions are drawn in section V.
2 Nonlinear Electrodynamics
The Lagrangian density in Maxwell’s electrodynamics can be written as Camara et al (2004)
(1) 
where is the electromagnetic field strength tensor, is the electromagnetic field and is the magnetic permeability. The canonical energymomentum tensor is then given by
(2) 
The general class of Lagrangian for nonlinear electromagnetic field Novello et al (2004) can be written in the form
(3) 
where the sum may involve both positive and negative powers of
.
Here we consider the generalization of Maxwell’s electromagnetic Lagrangian up to the second order terms of the fields as in the form Camara et al (2004)
(4) 
where, and are arbitrary constants. Here
(5) 
where, is the dual of . Now, the electromagnetic field has the expression in terms of electric field and magnetic field as in the form . So the corresponding energymomentum tensor for nonlinear electromagnetic theory has the form
(6) 
Now we consider the homogeneous electric field in plasma gives rise to an electric current of charged particles and then rapidly decays. So the squared magnetic field dominates over , i.e., in this case, . So is now only the function of magnetic field (vanishing electric component) and hence the FRW universe may be called Magnetic Universe. Now from equation (6), we obtain the expressions of magnetic density and pressure as
(7) 
and
(8) 
It is to be noted that the density of the magnetic
field must be positive, so the magnetic field must satisfy
with and
. If
,
the strong energy condition is violated i.e.,
and in this case, the magnetic field generates dark energy which
drives acceleration of the universe.
3 Field Equations and Solutions of Higher Dimensional FRW Model
We consider the dimensional flat homogeneous and isotropic universe described by FRW metric which is given by Chatterjee (); Mukhopadhyay et al ()
(9) 
where is the scale factor and
(10) 
Now assume that the universe is filled with dark matter and magnetic field type dark energy, so the Einstein’s field equations in higher dimension are given by
(11) 
and
(12) 
where
(13) 
and
(14) 
where . Also and are the energy density and pressure of the dark matter with the equation of state given by , and and are the energy density and pressure due to magnetic field. The energy conservation equation in higher dimensional cosmology is given by
(15) 
where is the Hubble parameter defined as . According to the recent Supernovae and CMB data, the energy transfer decay rate should be proportional to the present value of the Hubble parameter. Then we consider the model of interaction between dark matter and dark energy governed by the magnetic field, through a phenomenological interaction term . Hence the energyconservation equation (15) becomes
(16) 
and
(17) 
For simplicity of the calculation, we take the interaction component as Bandyopadhyay et al (2011)
(18) 
where, is a small positive quantity, termed as interaction parameter. Using the above expressions of , and solving the equations (16)(18) we obtain Bandyopadhyay et al (2011)
(19) 
and
(20) 
where is a constant and redshift .
From the above solutions, it may be concluded that the interaction
term always decays with the evolution of the universe. If
, we get the noninteracting solutions, i.e., . When (i.e., for 4D), we can recover
the result of Ref Bandyopadhyay et al (2011) (i.e., ) and
also when , and we drop the matter density term
(i.e., ), then we can verify the result of Ref.
Camara et al (2004) for. Otherwise, our solutions are
not similar with their solutions.
The Hubble parameter and Deceleration parameter are given by
(21) 
and
(22) 
where and are given by (19) and (20). The
expression of is very complicated in terms of . So the
variation of deceleration parameter against redshift is
plotted in figure 1 for 4D , 5D and 6D .
From figure, we see that decreases from some positive value to
as decreases. So the model generates first deceleration
and then acceleration as universe expands.
Om DIAGNOSTIC:
Recently, Sahni et al Sahni et al (2003, 2008) proposed a new cosmological parameter named Om which is a combination of Hubble parameter and the cosmological redshift and provides a null test of dark energy. Om diagnostic has been discussed together with statefinder for generalized Chaplygin gas model from cosmic observations in Tong et al (2009); Lu et al (2009). Generally, it was introduced to differentiate CDM from other dark energy models. For CDM model, is a constant, independent of redshift . Also it helps to distinguish the present matter density constant in different models more effectively. The main utility for Om diagnostic is that the quantity of Om can distinguish dark energy models with less dependence on matter density relative to the EoS of dark energy. Our starting point for Om diagnostics in the Hubble parameter and it is defined as:
(23) 
where and and is the present value of the Hubble parameter. Now in our interacting magnetic field model, we obtain
(24) 
where . We draw the
diagnostic against redshift in figure 2 for 4D, 5D and
6D. The diagnostic always increases as decreases (universe expands).
4 Observational Constraints
In this section, we investigate the observational constraints of the higher dimensional FRW model of the magnetic universe. We shall determine the expected bounds of the theoretical parameters by statistical best fit test with the basis of  (Stern) Stern et al (2010) and Stern+BAO Wu et al (2007); Thakur et al (2009); Paul et al (); Paul et al (); Ghose et al (); Chakraborty et al (2012) joint data analysis. We also determine the statistical confidence contours between two parameters and in different dimensions. The best fit of distance modulus for our theoretical model and the Supernova Type Ia Union2 sample are analyzed for different dimensions. To investigate the bounds of model parameters here we consider Stern () data set with 12 data of  (Stern) in the following Table 1 Stern et al (2010).

Table 1: and
for different values of .
Defining and using the expression of and , the expression of becomes
(25) 
This equation can be written in the form , where
known as normalized Hubble parameter contains six model
parameters beside the
redshift parameter . Now to find the bounds of the parameters
and to draw the statistical confidence contour (66%, 90% and
99% confidence levels) we fixed four parameters . In the this case we find the bounds of
and draw the contours between them. Here the
parameter determines the higher dimensions and we perform
comparative study between three
cases : 4D , 5D and 6D respectively.
4.1 Analysis with Stern () Data Set
Here we analyze the model parameters using twelve data Stern et al (2010) of Hubble parameter for different redshift given by Table 1. We first form the Chi square statistic (with 11 degree of freedom) as a sum of standard normal distribution as follows:
(26) 
where and are theoretical and observed values of Hubble parameter at different redshifts respectively and is the corresponding error. Here, is a nuisance parameter and can be safely marginalized. We consider the observed parameters , = 72 8 Kms Mpc and a fixed prior distribution. Here we shall determine the model parameters by minimizing the statistic. The reduced chi square can be written as
(27) 
where is the prior distribution. We now plot the graphs
for different confidence levels (i.e., 66%, 90% and 99%
confidence levels) and for three different dimensions (4D, 5D and
6D). Now our best fit analysis with Stern observational data
support the theoretical range of the parameters. When we fix the
parameters , the 66% (solid,
blue), 90% (dashed, red) and 99% (dashed, black) contours for
are plotted in figures 3, 4 and 5 for 4D ,
5D and 6D respectively. The best fit values of
and minimum values of for different
values of (i.e., different dimensions) are tabulated in
Table 2. For each dimension, we compare the model parameters
through the values of the parameters and by the statistical
contours. From this comparative study, one can understand the
convergence of theoretical values of the parameters to the values
of the parameters obtained from the observational data set and how
it changes from normal four dimension to higher
dimension (6D).

Table 2:  (Stern): The best fit values of , and the minimum values of for different dimensions.
4.2 Joint Analysis with Stern BAO Data Sets
The Baryon Acoustic Oscillations (BAO) in the primordial baryonphoton fluid, leave a characteristic signal on the galaxy correlation function, a bump at a scale 100 Mpc, as observed by Eisenstein et al Eisenstein et al (2005). We shall investigate the two parameters and for our model using the BAO peak joint analysis for low redshift (with range ) using standard distribution. The BAO peak parameter may be defined by
(28) 
where
(29) 
Here, is the normalized Hubble parameter and is the typical redshift of the SDSS data sample. This quantity can be used even for more general models which do not present a large contribution of dark energy at early times. Now the function for the BAO measurement can be written as in the following form
(30) 
where the value of the parameter for the flat model () of the FRW universe is obtained by using SDSS data set Eisenstein et al (2005) from luminous red galaxies survey. Now the total joint data analysis (Stern+BAO) for the function defined by
(31) 
Now our best fit analysis with SternBAO observational data support the theoretical range of the parameters. In figures 68, we plot the graphs of for different confidence levels 66% (solid, blue), 90% (dashed, red) and 99% (dashed, black) contours for 4D, 5D and 6D respectively and by fixing the other parameters . The best fit values of and minimum values of for different values of (i.e., different dimensions) are tabulated in table 3.

Table 3:  (Stern)+BAO: The best fit values of , and the minimum values of for different dimensions.
4.3 Current Supernovae Type Ia Data
In this section, we use Supernova Type Ia data at high redshifts
Perlmutter et al (1998, 1999); Riess et al (1998, 2004) and
Baryonic Acoustic Oscillation (BAO) Eisenstein et al (2005) to restrict
the parameters of the model for different dimensions. The
observations directly measure the distance modulus of a Supernovae
and its redshift Riess et al (); Kowalaski et al (2008). Now, take recent
observational data, including SNe Ia which consists of 557 data
points and belongs to the Union2 sample Amanullah et al (2010).
From the observations, the luminosity distance determines the dark energy density and is defined by
(32) 
and the distance modulus (distance between absolute and apparent luminosity of a distance object) for Supernovas is given by
(33) 
The best fit of distance modulus as a function of redshift for our theoretical model and the Supernova Type Ia Union2 sample are drawn for different dimensions (4D, 5D and 6D) in figure 9 for our best fit values of and by fixing the other parameters .
5 Discussions
In this work, we have considered the flat FRW model of the
universe in dimensions filled with the dark matter and the
magnetic field. We present the Hubble parameter in terms of the
observable parameters and with the redshift
and the other parameters like . The magnetic field and
deceleration parameter have been calculated. The magnetic
field follows the power law form of redshift . Now the
variation of deceleration parameter against redshift has
been plotted in figure 1 for 4D , 5D and 6D
. From figure, we see that decreases from some positive
value to as decreases. So the model generates first
deceleration and then acceleration as universe expands. Recently
proposed diagnostic has also been discussed for our model.
We draw the diagnostic against redshift in figure 2 for
4D, 5D and 6D. The diagnostic always increases as decreases
(universe expands).
We have investigated the observational constraints of the higher
dimensional FRW model of the magnetic universe. Here we have
chosen the observed values of ,
and = 72 Kms Mpc. From
Stern data set (12 points), we have obtained the bounds of the
arbitrary parameters by minimizing the test. The
bestfit values of the parameters are obtained by 66%, 90% and
99% confidence levels. Now to find the bounds of of the
parameters () and to draw the statistical confidence
contour, we fixed four parameters
and . Here the
parameter determines the higher dimensions and we perform
comparative study between three cases : 4D , 5D and
6D respectively. We have plotted the graphs for different
confidence levels i.e., 66%, 90% and 99% confidence levels and
for three different dimensions (4D, 5D and 6D) in figures 35. Now
our best fit analysis with Stern observational data support the
theoretical range of the parameters. The best fit values of
and minimum values of for different
dimensions are tabulated in Table 2. For each dimension, we
compare the model parameters through the values of the parameters
and by the statistical contours. From this comparative study, one
can understand the convergence of theoretical values of the
parameters to the values of the parameters obtained from the
observational data set and how it changes from normal four
dimension to higher dimension (6D). Next due to joint analysis
with Stern+BAO observational data, we have also obtained the
bounds of the parameters by fixing some other
parameters for 4D, 5D and 6D.
In figures 68, we have plotted the graphs of for
different confidence levels 66% (solid, blue), 90% (dashed, red)
and 99% (dashed, black) contours for 4D, 5D and 6D respectively.
The best fit values of and minimum values of
for different dimensions are tabulated in Table 3. The
best fit of distance modulus as a function of redshift
for our theoretical model and the Supernova Type Ia Union2
sample are drawn for different dimensions (4D, 5D and 6D) in
figure 9 for our best fit values of and by
fixing the other parameters .
Acknowledgement:
The authors are thankful to IUCAA, Pune, India for warm
hospitality where part of the work was carried out. Also UD is
thankful to CSIR, Govt. of India for providing research project
grant (No. 03(1206)/12/EMRII).
Footnotes
 affiliation: Department of Mathematics, Seacom Engineering College, Howrah  711 302, India. Email: chayanranjit@gmail.com
 affiliation: Department of Mathematics, Seacom Engineering College, Howrah  711 302, India. Email:shuvendu.chakraborty@gmail.com
 affiliation: Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah711 103, India. Email: ujjaldebnath@yahoo.com
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