A&A 471, 409417 (2007)
DOI: 10.1051/00046361:20077109
Excitation of MHD waves in magnetized anisotropic cosmologies
A. Kuiroukidis^{1,2} 
K. Kleidis^{1,3} 
D. B. Papadopoulos^{1} 
L. Vlahos^{1}
1  Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
2  Department of Informatics, Technological Education Institute of Serres, 64124 Serres, Greece
3  Department of Civil Engineering, Technological Education Institute of Serres, 64124 Serres,
Greece
Received 16 January 2007 / Accepted 16 April 2007
Abstract
The excitation of cosmological perturbations in an
anisotropic cosmological model and in the presence of a
homogeneous magnetic field was studied, using the resistive
magnetohydrodynamic (MHD) equations. We have shown that
fastmagnetosonic modes, propagating normal to the magnetic field,
grow exponentially and saturate at high values, due to the
resistivity. We also demonstrate that Jeanslike instabilities
can be enhanced inside a resistive fluid and that the formation of
condensations influence the growing magnetosonic waves.
Key words: magnetic fields  magnetohydrodynamics (MHD)  instabilities  waves  cosmology: early Universe  relativity
Magnetic fields are known to have a widespread presence in our
Universe, being a common property of the intergalactic medium in
galaxy clusters (Kronberg 1994). Reports on Faraday rotation imply
significant magnetic fields in condensations at high redshifts
(Kronberg et al. 1992). Largescale magnetic fields
and their potential implications for the formation and the
evolution of the observed structures have been the subject of
theoretical investigation (see Thorne 1967; Jacobs 1968;
Ruzmaikina & Ruzmaikin 1971; Wasserman 1978; Zel'dovich et al. 1983; Adams et al. 1996;
Barrow et al. 1997; Tsagas & Barrow 1997; Jedamzik et al. 1998; Barrow et al. 2006, etc.). Magnetic fields observed in galaxies and galaxy clusters are in
energy equipartition with the gas and cosmic rays (Wolfe et al. 1992). The origin of these fields, which can be astrophysical, cosmological or both, remains an unresolved issue.
If magnetism has a cosmological origin, as observations of G
fields in galaxy clusters and highredshift protogalaxies seem to
suggest, it could have affected the evolution of the Universe
(Giovannini 2004; Barrow et al. 2006). There are
several scenarios for the generation of primordial magnetic fields
(e.g. see Grasso & Rubinstein 2001). Most of the early treatments
were Newtonian, with relativistic studies making a recent
appearance in the literature. A common factor between almost all
the approaches is the use of the MHD approximation, namely the
assumption that the magnetic field is frozen into an effectively
infinitely conductive cosmic medium (i.e. a fluid of zero
resistivity). With a few exceptions (e.g. see Fennelly 1980;
Jedamzik et al. 2000; Vlahos et al. 2005), the role of kinetic viscosity and the possibility of
nonzero resistivity have been ignored. Nevertheless, these
aspects are essential for putting together a comprehensive picture
of the magnetic behavior, particularly as regards the nonlinear
regime. The electric fields associated with the resistivity can
become a source of particle acceleration, while the induced
nonlinear currents may react back on the magnetic field (Vlahos et al. 2005).
Many recent studies have used a Newtonian or a FriedmannRobertsonWalker (FRW) model to represent the evolving Universe
and superimposed a largescale ordered magnetic field. The
magnetic field is assumed to be too weak to destroy the FRW
isotropy and the anisotropy, induced by it, is treated as a
perturbation (Ruzmaikina & Ruzmaikin 1971; Tsagas & Barrow 1997;
Durrer et al. 1998). Current observations give a
strong motivation for the adoption of a FRW model but the
uncertainties on the cosmological Standard Model are
several. Therefore, the limits of the approximations and the
effects one may lose by neglecting the anisotropy of the
background magnetic field should be investigated. The formation of
smallscale structures and the excitation of resistive
instabilities in BianchiType models have been explored (Fennelly
1980). Nevertheless, the excitation of MHDwaves in curved
spacetime and their subsequent temporal evolution is far from
being clearly understood (Papadopoulos et al. 2001).
In the present article we explore the evolution of a magnetized
resistive plasma in an anisotropic cosmological model. We begin
with a uniform plasma driving the dynamics of the curved spacetime
(the socalled zerothorder solution). This dynamical system
is subsequently perturbed by smallscale fluctuations and we study
their interaction with the anisotropic background, searching for
imprints on the temporal evolution of the perturbations'
amplitude.
In Sect. 2, we present the system of the field equations
appropriate to describe the model under consideration. In Sect. 3, we solve this system analytically, to derive the zerothorder
solution. In Sect. 4, we extract the firstorder perturbed
equations. In Sect. 5, we derive the dispersion relation for the
magnetized cosmological perturbations and in Sect. 6, we perform
a numerical study of their evolution, using a fifthorder RungeKuttaFehlberg temporal integration scheme. In Sect. 7, a
perturbation analysis over purely gravitational fluctuations
reveals an inherent Jeanslike instability.
Our results suggest that in a resistive plasma, within an interval
of 10^{11} s after the beginning of the interaction process,
fastmagnetosonic modes are excited, growing exponentially in time
and saturate at high values. In this way, magnetic field
perturbations can be retained at large amplitudes, for
sufficiently long timeintervals
,
resulting in the enhancement of the ambient magnetic field (dynamo
effect). In addition, the resistive plasma enhances the
condensations that can be formed within the anisotropic fluid due
to gravitational instability, which, in turn, influences the
growth of the magnetosonic waves.
We consider an axisymmetric BianchiType I cosmological model,
driven by an anisotropic and resistive perfect fluid, in the
presence of a timedependent magnetic field,
.
The corresponding lineelement is written in the form

(1) 
The evolution of a curved spacetime in the presence of matter and an
e/m field is determined by the gravitational field equations

(2) 
(in the system of units where
), together with the energymomentum conservation law

(3) 
and Maxwell's equations



(4) 



(5) 
In Eqs. (2)(5), Greek indices refer to the fourdimensional
spacetime (Latin indices refer to the threedimensional spatial
section) and the semicolon denotes covariant derivative.
and
are the Ricci tensor and the scalar
curvature with respect to the background metric
,
while G is Newton's gravitational constant.
is the
antisymmetric tensor of the e/m field and
is the
corresponding current density.
The energymomentum tensor consists of two parts

(6) 
The first part is due to an anisotropic perfect fluid source of
the form

(7) 
where
is the energy density, p_{i}(t)are the components of the anisotropic pressure and the axial
symmetry of the metric (1) implies that
p_{2} (t) = p_{3} (t).
is the fluid's fourvelocity, satisfying
the conditions
and
,
with
being the
projection tensor.
The second part is due to the ambient e/m field

(8) 
where for
and
// ,
the nonzero
components of the Faraday tensor in the curved spacetime (1) read
(see Appendix A)

(9) 
The current density
may be determined by the invariant form
of Ohm's law

(10) 
where
is the locally measured charge
density and
is the (finite) electric resistivity, in units
of time. As a consequence of the Maxwell equations, we
obtain
.
Assuming that that the fluid has
zero netcharge, i.e.
,
Eq. (10) reduces to
.
A vanishing net charge indicates
that the perfect fluid consists of at least two components.
We look for axisymmetric BianchiType I cosmological solutions to
the EinsteinMaxwell equations (Appendices A and B), representing
the background metric of our problem. In this case, Eqs. (2) reduce to

(11) 
(the dot denotes timederivative) and Eqs. (4), (5) yield

(12) 
Equation (12) has a clear physical interpretation: the magnetic flux through a comoving
surface normal to the direction of the magnetic field is conserved.
On the other hand, the continuity Eq. (3) results in (Appendix C)



(13) 
and the particles' number conservation law reads

(14) 
The system of Eqs. (11)(14) admits the exact solution
where the index "0'' stands for the
corresponding values at t = t_{0} and t_{0} marks the beginning of
the interaction between magnetized plasma and curved spacetime.
Solution (15) represents an anisotropic cosmological model in
which the largescale anisotropy along the axis is due
to the presence of an ambient magnetic field. The combination of
Eqs. (11) and (15) indicates that, initially, the total energy
density is given by

(16) 
and the difference between fluid's pressure and the pressure of the magnetic field along the two anisotropic spatial directions is equal







(17) 
Equations (17) lead us to identify
p_{10} = p_{0} = p_{20}

(18) 
i.e. initially, when
R(t_{0}) = S(t_{0}), the two components of the
anisotropic pressure were equal in absolute value, something that
is confirmed also by Eq. (13). Furthermore, with Eqs. (16) and (17), we obtain

(19) 
which, according to Eq. (18), results in the equation of state for the matterenergy content at t = t_{0}

(20) 
For B_{0} = 0, i.e. as regards the perfect fluid itself, we obtain that,
initially,
.
Since
,
our model corresponds to a semirealistic
cosmological model of Bianchi Type I. These models are crude,
first order approximations to the actual Universe when we use
currently available theories and observations (Jacobs 1969).
For any dynamical system, much can be learnt by investigating the
possible modes of smallamplitude oscillations or waves. A plasma
is physically much more complicated than an ideal gas, especially
when there is an externally applied magnetic field. As a result, a
variety of smallscale perturbations may appear. We first assume a
uniform magnetized plasma in curved spacetime as background, which
is perturbed by small scale fluctuations. In this article, the
evolution of the background is described by the solution (15).
Accordingly, we introduce firstorder perturbations in the
EinsteinMaxwell equations, by decomposing the physical variables
of the fluid as



(21) 


p_{x} (t,z) = p_{1} (t) 




(22) 




and we insert the perturbed values (21) and (22) into Eqs. (11)(14), neglecting
all terms higher than or equal to the second order. The pressure
perturbation
introduces a longitudinal acoustic
mode, propagating along the direction and therefore

(23) 
where
is the speed of sound. The fourvelocity of the plasma fluid is perturbed
around its comoving value,
,
as

(24) 
Then, the condition
,
to the first leading order,
implies

(25) 
and, therefore,
.
Accordingly,
.
As regards the perturbations of the e/m field, we consider that
they correspond to a transverse e/m wave, propagating along the
axis



(26) 



(27) 
Therefore, the nonzero components of the Faraday tensor in curved spacetime
are modified as follows







(28) 
In what follows, we take into account the socalled Cowling approximation (Cowling 1941), admitting that
.
Therefore, the evolution of the perturbed quantities is
governed only by the energymomentum tensor conservation, together
with Maxwell's equations.
To begin with, we perturb the particles' number conservation law:
accordingly, Eq. (C.3) yields

(29) 
We continue with Maxwell's equations. Then, from Eq. (B.2), using Eqs. (21), (22), (24) and (25), we obtain



(30) 
Now, Eq. (B.3) becomes

(31) 
The conservation Eq. (C.2) results in



(32) 
while, to the first leading order, Eq. (C.1) collapses to an identity. Equations (29)(32) are the linearly independent first order perturbed
EinsteinMaxwell equations in the curved background (1). In the
flat spacetime  zero resistivity limit, they reduce to Eqs. (10.53a), (10.9) and (10.53c) of Jackson (1975), respectively.
To develop the theory of smallamplitude waves in curved
spacetime, we search for solutions to the linearized Eqs. (29)(32) in which all perturbation quantities are proportional to the exponential

(33) 
following the socalled adiabatic approximation (Zel'dovich
1979; Birrell & Davies 1982; Padmanabhan 1993). In this context,
the (slowly varying) timedependent frequency of the wave is
defined by the eikonal

(34) 
through the relation

(35) 
Notice that, in Eq. (33), z is the comoving coordinate along the axis and
k is the comoving wavenumber. In an expanding Universe, the
corresponding physical quantities are defined as
and
,
so that
.
Before discussing the temporal evolution of the perturbation
quantities, it is important to trace what kind of waveforms are
admitted by this system. We have to derive their dispersion
relation,
,
at t = t_{0}. Provided that certain
kinds of modes (such as acoustic, magnetosonic etc.) do exist,
they can be excited through their interaction with the anisotropic
spacetime. An additional excitation, due to the nonzero
resistivity, is also possible (Fennelly 1980).
Accordingly, we assume a wavelike expansion for the perturbation
quantities of the form
Although the background quantities
depend on time, in the search for a dispersion relation at t =
t_{0}, we treat the perturbation amplitudes (A_{i}s) as constants. In this way, our search for potential waveforms at t
= t_{0}, is not disturbed by the inherent nonlinearity introduced
for t > t_{0}. Nevertheless, once the potential waveforms are
determined, their interaction with the curved spacetime in the
presence of an external magnetic field implies that for t > t_{0}the timedependence of their amplitudes is a priori
expected. Using Eqs. (36), Eq. (29) is written in the form

(39) 
where we have set

(40) 
Furthermore, using Eqs. (37), Eq. (30) reduce to

(41) 
while Eq. (31) becomes

(42) 
Finally, Eq. (32) yields







(43) 
With the aid of Eqs. (12) and (14), the combination of Eqs. (39)(43) results in
where
is the (dimensionless) Alfvén velocity. Equation (44) is the dispersion relation which determines the possible
waveforms admitted by this dynamical system for all
.
,
as defined by Eqs. (34) and (35), has the usual meaning of
the angular frequency of an oscillating process only in the
shortwavelength (highfrequency) regime of the mode k(Mukhanov et al. 1992). In other words, the
wave description in curved spacetime makes sense only when the
physical wavelength along the direction of propagation
is much smaller than the corresponding horizon length
,
i.e.

(45) 
Equation (45) implies that, in the anisotropic background (1), the
wave description makes sense as long as

(46) 
for all
.
In this limit, Eq. (44) becomes surprisingly transparent, namely

(47) 
The vanishing of the real part results in acoustic
and e/m
waves, while the vanishing of the
imaginary part results in fastmagnetosonic waves

(48) 
In the zeroresistivity limit (ideal plasma), the obvious modes expected are the
magnetosonic modes, which we recover. On the other hand, in most
astrophysical situations we have (Jackson 1975)

(49) 
In this case, Eq. (47) reads

(50) 
According to Eq. (50), in the very high frequency limit where no acoustic waves
are admitted, we are left with a waveform governed by the
dispersion relation

(51) 
which yields

(52) 
This result has a clear physical interpretation:
all the veryhighfrequency perturbations of the dynamical system
are suppressed due to the finite resistivity. Therefore, the only
modes that survive in a resistive cosmological model are the
(lowfrequency) MHD modes. In the next section, we discuss the
evolution of these modes.
In order to study the temporal evolution of the magnetosonic modes
for
,
we assume that their amplitudes are no longer
timeindependent
In Eqs. (53)(55), the wavenumber k is related to the frequency
through Eq. (48)
and, once again, we have taken into account the equation of state
for the perfect fluid.
We decompose the timedependent amplitude of the perturbations (53)(55) into a real and an imaginary part, as
that reduces Eqs. (29)(32) to the following first order system



(58) 



(59) 







(60) 







(61) 



(62) 



(63) 
We integrate numerically the system (58)(65), using a fifth order RungeKuttaFehleberg scheme with variable integration step. The time is measured in units of t_{0} and,
therefore,
.
In terms of ,
the
physical wavenumber reads
and the Hubble
parameter along the yzplane is written in the form
.
According to Eq. (45), for a certain value of
,
a wave is well inside the horizon as long as

(66) 
The validity of Eq. (66) for long intervals determines the appropriate values of the comoving
wavenumber. Now, the analysis depends on where do we place the
initial time, t_{0}.
According to the Standard Model (Kolb & Turner 1990), after
nucleosynthesis the Universe goes on expanding and cooling until
s. At that time, the temperature drops to the
point where electrons and nuclei can form stable atoms
(recombination). Before that time, during the socalled radiation epoch, photons couple strongly with matter, the main
constituent of which is in the form of plasma. Therefore, the
latest time at which plasma could play a role of cosmological
significance is the recombination time (t_{R} = 1.2
10^{13} s). In the limiting case where t_{0} = t_{R}, the condition (66) reads
s^{1} and, therefore, an appropriate choice for k would be
k
= 10^{12} s^{1}.
In order to decide on the initial values of the unperturbed
quantities, we write Eq. (16) in ordinary units, namely

(67) 
We adopt a typical behavior for the energydensity, valid at the late
stages of the radiation epoch (see Weinberg 1972, Eq. (15.6.42))

(68) 
where T is the temperature and
is the blackbody constant. At the time of recombination (t_{0} = 1.2
10^{13} s, T = 4000 ), we obtain
,
which, through Eq. (67), is effectively a choice on
B_{0}, namely
Gauss. Notice that this value
lies barely within limits of the constraint

(69) 
a necessary condition to retain the anisotropy of the metric (Thorne 1967). Extrapolation of this
result along the lines of Eq. (15) to the present epoch (
10^{9} y) suggests that, today, the
corresponding magnetic field should be
10^{10} Gauss. This value lies within limits of the upper
bound for the presentday magnetic field strength, arising from
the largeangular scale anisotropy of the microwave radiation
background (MRB) at last scattering (Barrow et al. 1997, 2006)

(70) 
We estimate the amount of distortion which the
expansion anisotropy along the xaxis (caused by the unperturbed
magnetic field) induces on the microwave pattern at the present
epoch. The contribution of a largescale coherent magnetic field
to the microwave quadrupole anisotropy is given by (Madsen 1989)

(71) 
where
and
denote the present
values of the magnetic field and the background radiation
energydensity, z is the redshift at which the anisotropy begins
to grow (in our case, at the recombination time where
). The present value of the microwave background temperature
is
,
corresponding to an energydensity
of
for the
radiation field. Accordingly, our analysis suggests that the
presentday quadrupole anisotropy along the xaxis should be

(72) 
i.e. almost four times larger than the corresponding COBE result.
Taking into account that, initially, the unperturbed quantities
are of the order of unity, we normalize all the perturbation
quantities at t = t_{0}, to 0.01 in cgs units. On the
other hand, initially, the equation of state for the perfect fluid
admits
,
while, for the resistivity we adopt the
Spitzer relation (Krall & Trivielpiece 1973)

(73) 
In a radiationdominated background, we have (Kolb & Turner 1990)

(74) 
and therefore, during recombination, Eq. (64) results in
.
In order
to demonstrate how
may trigger instabilities, we consider
three cases, namely
,
and
.
The output of the numerical integration consists of the electric
and the magnetic field perturbations' amplitude



(75) 



(76) 
and illustrates their temporal evolution. In Fig. 1, we present the magnetic field perturbation
versus time. We consider two cases:
 For
(ideal plasma), the magnetic field
perturbation grows steeply at early times. It appears that the
interaction of the perturbed quantities with the anisotropic
spacetime results in the amplification of the convective
field
,
which is the only one to
survive in the idealplasmalimit [e.g. see Eq. (30)]. Through
Faraday's law, any amplification in the convective field leads to
an analogous growth in
,
at the expense of the
cosmological expansion. Accordingly, after exhausting any available energy, the magnetic field perturbation reaches a
maximum value before it is suppressed due to the cosmological
redshift.
 On the other hand, for
,
the magnetic field
perturbation also increases rapidly at early times after t_{0}(
), reaching values up to 3 times its
initial one. However, in this case, the perturbation's amplitude
is saturated, acquiring sufficiently large values for long enough
time intervals
.
This is due to the
fact that, besides the convective field
,
a nonzero
resistivity also favors convective currents
.
For
,
the lhs of Eq. (30) corresponds,
through Ampere's law, to an electric current. Accordingly, now,
the energy available to be absorbed by the perturbed quantities is
larger and therefore the magnetic perturbation remains at high
levels for longer time intervals.
As a result, after saturation, the magnitude of constitutes a fraction of
of the unperturbed
value of the magnetic strength. In this case, the quadrupole
anisotropy induced in the MRB along the xaxis reads

(77) 
resulting in

(78) 
i.e. the corresponding value is enlarged by 1%.

Figure 1:
The timeevolution of the magnetic field perturbation, for B_{0} = 7 Gauss and for
several values of the resistivity
(s). Notice that, for
,
the perturbation's amplitude is saturated, acquiring
large values for long enough time intervals. 
Open with DEXTER 
The numerical results indicate a completely different behavior for
the electric field perturbation (Fig. 2). Not only are the growth
rate and the highest value of
slightly smaller than the
corresponding values of
,
but, also, the suppression rate
of the perturbation's amplitude is much larger than that of
,
resulting in a rapid decrease of the electric field at late
times. It appears that the expanding Universe disfavors strong
electric fields.

Figure 2:
The timeevolution of the electric field perturbation, for B_{0} = 7 Gauss and for
several values of the resistivity
(s). Notice that, in this case, there is no saturation. 
Open with DEXTER 
We conclude that, for reasonable values of the resistivity, the
magnetic field perturbations lead to a real instability, acquiring
large values for sufficiently long timeintervals. The influence
of resistivity in triggering instabilities in anisotropic
cosmological models has been the subject of research in the past
(Fennelly 1980). To the best of our knowledge, however, this is
the first time that a direct connection between the resistivity
and the saturation of the perturbations' amplitude at high values
for long time intervals is suggested and discussed.
The question that arises now is whether the cosmological model
under consideration admits other kinds of instability and what
their role is in connection to the resistive one. To answer this
question, we study the evolution of purely gravitational
perturbations, examining whether they admit a growing behavior
(Jeans instability) or not.
In the absence of e/m fields (and their fluctuations), one is left
with the system of perturbation equations



(79) 



(80) 
the combination of which yields

(81) 
Taking into account the particle number conservation law, Eq. (81) results in

(82) 
describing damped acoustic waves. With respect to ,
Eq. (82) is a second
order algebraic equation with roots

(83) 
provided that

(84) 
In this case, the energydensity perturbations (36) reduce to

(85) 
where
is given by

(86) 
Equation (86) represents the dispersion relation for the propagation of the energydensity
fluctuations. In the isotropic case, where
H_{R} = H = H_{S}, it yields

(87) 
which, with the aid of the corresponding Friedmann equation
,
reads

(88) 
Equation (88) is identical to the isotropic (FRW)
result, predicted by Weinberg (1972), in the relativistic theory
of small fluctuations.
In contrast to the high frequency e/m waves (52), as regards the
corresponding energydensity perturbations, propagation is
possible only when their physical wavenumber is larger than a
characteristic value arising from condition (84) otherwise,
after some time they become unstable and grow exponentially
with time (Jeanslike instabilities).
Taking into account the background solution (15), Eq. (84) at t = t_{0} reads

(89) 
and the corresponding Jeans length is given by

(90) 
Propagation of density perturbations with
is not possible, for all
and we are lead to a gravitational instability. The larger the
coordinate wavelength is, the more prominent the unstable
behavior will be (Fig. 3).

Figure 3:
The timeevolution of the energy density perturbation for several values of the coordinate
wavelength
in terms of
. 
Open with DEXTER 
Furthermore, one may ask whether the waves with wavenumber around
in a nonideal plasma may grow faster than those in an ideal
plasma. For every t > t_{0}, the physical Jeans length along
the axis
is larger than the
corresponding length along the other two axes
,
due to the background anisotropy, suggesting formation
of "cigarlike'' condensations within the anisotropic fluid.
Since this fluid is conductive, these condensations act in favor
of electric currents which may lead to a further amplification of
the e/m perturbations, fortifying any preexisting resistive
instability (Fig. 4). Therefore, a Jeanslike instability enhances
the phenomena related to the resistivity.

Figure 4:
The timeevolution of the magnetic field perturbation for
s, B_{0} =
7 Gauss and for several values of the coordinate wavenumber in terms of . 
Open with DEXTER 
On the other hand, numerical results indicate that waves with
wavelength around
become more prominent as the resistivity
grows (Fig. 5). This result also has a clear physical
interpretation: as we have already seen, any increase in the
resistivity fortifies the surrounding magnetic field. A strong
magnetic field organizes plasma along its lines, favoring any
preexisting condensations. Hence, resistive instabilities act in
favor of the corresponding gravitational ones and vice versa.

Figure 5:
The timeevolution of the energy density perturbation, for
and for several values of the resistivity (s). We observe that the Jeans instability becomes more prominent as the resistivity grows. 
Open with DEXTER 
We study the evolution of the magnetosonic waves in a magnetized,
resistive plasma, which governs the dynamics of an anisotropic
cosmological model. After constructing the general set of MHD and
Einstein equations for the anisotropic cosmological model (see the
Appendices), we solve the field equations to obtain the
zerothorder solution. In order to determine the waveforms
admitted by this system in the first place, we introduce wavelike
perturbations and, neglecting all terms higher or equal than the
second order, we extract the dispersion relation at t = t_{0},
i.e. at the beginning of the interaction between magnetized plasma
and curved spacetime. It appears that magnetosonic modes can be
excited due to the anisotropy and the resistivity. For
,
we integrate numerically the perturbed equations, using the
dispersion relation for the fastmagnetosonic waves.
We find that, at early times, both the electric and the magnetic
field perturbations grow exponentially, at least in the regime
where the linear analysis holds. However there is a major
difference in their behavior in the presence of a nonzero
resistivity. For
,
the magnetic field perturbation
after increasing to reach values up to 3 times its initial one, is
subsequently saturated, remaining at high levels for sufficiently
long time intervals
.
The situation is completely different to the electric field
perturbation. Not only the growth rate and the highest value of
are slightly smaller than the corresponding values of
,
but, also, the suppression rate of the perturbation's
amplitude is much larger than that of
.
Accordingly, the
electric field decreases rapidly at late times. It appears that
the expanding Universe disfavors strong electric fields.
We have shown that waves with wavenumber around
are
enhanced in non ideal plasmas.
Acknowledgements
The authors would like to thank Dr. Heinz Ishliker and Dr. Christos
Tsagas for helpful discussions. Financial support from the Greek
Ministry of Education under the Pythagoras Program, is gratefully
acknowledged.
We present the closed set of MHD and Einstein equations (in the
system of units where
)
for the anisotropic
cosmological models of BianchiType I

(A.1) 
in the presence of an anisotropic perfect fluid, which allows
for acoustic waves along the direction

(A.2) 
and an e/m field of the form

(A.3) 
where Greek indices refer to the fourdimensional spacetime and
Latin indices refer to the threedimensional spatial section. In
Eq. (A.3),
is the Faraday tensor in flat
spacetime. The components of the e/m field in curved spacetime are
defined by

(A.4) 
where the nonzero components of the orthonormal tetrad
of the local Lorentz frame for the metric (A.1) are given by







(A.5) 
Therefore, in the curved spacetime (A.1), the nonzero components of the Faraday tensor are







(A.6) 
In what follows, the dot denotes the
timederivative, while the prime denotes differentiation with
respect to z. The Einstein equations,
,
result in:
The (tt)component is given by







(A.7) 
The (xx)component is given by



(A.8) 
The (yy)component is given by



(A.9) 
The (zz)component is given by



(A.10) 
The (tz)component is given by



(A.11) 
The Maxwell equations in curved spacetime are written in the form







(B.1) 
where
is the
current density and
is the electric resistivity of the
fluid. Accordingly, we obtain



(B.2) 
and

(B.3) 
Taking the time and space component of
,
we obtain the required equations of motion in
a covariant form, namely
and
In addition, the particles' number conservation law
,
results in



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