Tensor microwave anisotropies from a stochastic magnetic field. DURRER, Ruth, FERREIRA, Pedro G., KAHNIASHVILI, Tina. Abstract

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1 Article Tensor microwave anisotropies from a stochastic magnetic field DURRER, Ruth, FERREIRA, Pedro G., KAHNIASHVILI, Tina Abstract We derive an epression for the angular power spectrum of cosmic microwave background anisotropies due to gravity waves generated by a stochastic magnetic field and compare the result with current observations; we take into account the non-linear nature of the stress energy tensor of the magnetic field. For almost scale invariant spectra, the amplitude of the magnetic field at galactic scales is constrained to be of order 1-9 G. If we assume that the magnetic field is damped below the Alfvén damping scale, we find that its amplitude at.1h-1 Mpc,?, is constrained to be? Reference DURRER, Ruth, FERREIRA, Pedro G., KAHNIASHVILI, Tina. Tensor microwave anisotropies from a stochastic magnetic field. Physical Review. D,, vol. 61, no. 31, p. 6 DOI : 1.113/PhysRevD ariv : astro-ph/9911 Available at: Disclaimer: layout of this document may differ from the published version.

2 Tensor Microwave Anisotropies from a Stochastic Magnetic Field R. Durrer 1, P.G. Ferreira 1;;3 and T. Kahniashvili 1 Departement de Physique Theorique, Universite de Geneve, quai Ernest Ansermet, CH-111 Geneve, Switzerland CERN Theory Division, CH-111, Geneve 3, Switzerland 3 CENTRA, Instituto Superior Tecnico, Av. Rovisco Pais, 1, 196 Lisboa Code, Portugal Department of Astrophysics, Abastumani Astrophysical Observatory, Kazbegi Ave. a, 386 Tbilisi, Georgia We derive an epression for the angular power spectrum of cosmic microwave background anisotropies due to gravity waves generated by a stochastic magnetic eld and compare the result with current observations; we take into account the non-linear nature of the stress energy tensor of the magnetic eld. For almost scale invariant spectra, the amplitude of the magnetic eld at galactic scales is constrained to be of order 1?9 Gauss. If we assume that the magnetic eld is damped below the Alfven damping scale, we nd that its amplitude at :1h?1 Mpc,, is constrained to be < 7:9 1?6 e 3n Gauss, for n <?3=, and < 9:5 1?8 e :37n Gauss, for n >?3=, where n is the spectral inde of the magnetic eld and H = 1hkm s?1 Mpc?1 is the Hubble constant today. PACS Numbers : 98.8.Cq, 98.7.Vc, 98.8.Hw I. INTRODUCTION The past few years have seen a tremendous surge of interest in the origin and evolution of galactic magnetic elds [1]. A number of mechanisms have been proposed for the origin of the seed elds, ranging from inationary mechanisms [], cosmological phase transitions [3] to astrophysical processes []. Much progress has been made in trying to disentangle the various non-linear processes which may be responsible for the growth of such a seed eld in the very early universe in particular the interplay between the magnetic eld and the primordial plasma [5,6] and the importance of turbulence [7]. Given a small seed eld at late times, two dierent mechanisms can cause its amplication to magnetic elds of order 1?6 Gauss observed in galaies: adiabatic compression of magnetic u lines can amplify a seed eld of order 1?9 Gauss to the present, observable values; the far more ecient (and controversial) galactic dynamo mechanism may be able to amplify seed elds as small as 1? Gauss [] or even 1?3 Gauss in universe with low mass density [8]. Clearly, to make some progress in identifying which one of these mechanisms is responsible for galactic magnetic elds, one would like to nd a constraint for the seed eld before it has been processed by local, galactic dynamics. The obvious observable for such a constraint is the cosmic microwave background (CM). It is interesting to note that a eld strength of 1?8 Gauss provides an energy density of = =(8 c ) 1?5, where is the density parameter in photons. We naively epect a magnetic eld of this amplitude to induce perturbations in the CM on the order of 1?5, which are just on the level of the observed CM anisotropies. It is thus justied to wonder to what etent the isotropy of the CM may constrain primordial magnetic elds. Our order of magnitude estimate makes clear that we shall never be able to constrain tiny seed elds on the order of 1?13 Gauss or less in this way, but primordial elds of 1?9 Gauss may have left their traces in the CM. A number of methods have been proposed in the past few years for measuring a cosmological magnetic eld using the CM: the eect on the acoustic peaks [9], Faraday rotation on small [1] and large [11] scales and vorticity [1,13] can all lead to observable anisotropies in the CM if the primordial magnetic eld strength is of the order of 1?9 to 1?8 Gauss. The most stringent bound from the CM presented thus far was for the case of a homogeneous magnetic eld [1]; the authors use the COE data to nd the constraint < 6:8 1?9 ( h ) 1= Gauss where the Hubble constant is H = 1hkm s?1 Mpc?1 and is the energy density in units of the critical value. Although there is no fundamental reason to discard the possibility of a homogeneous magnetic eld, all physical mechanisms proposed to date lead to the presence of stochastic magnetic elds with no homogeneous term; in this paper we consider such elds. For these types of conguration one is allowed to have uctuations on a wide range of scales and the magnetic eld will serve as a non-linear driving force to the metric uctuations; in the parlance of cosmological perturbation theory, the magnetic eld evolves as a sti source, without being aected by the uid perturbations (back reaction) [15] which may be induced. Stochastic magnetic elds have also been considered in [1], where the CM anisotropies due to the induced uid vorticity has been analyzed. Here we determine gravitational eects of the magnetic eld. For simplicity, and to allow for a purely analytical analysis, we constrain ourselves to tensor perturbations. Similar contributions are also epected from vector and scalar perturbations which then would add to the nal result. In this sense the anisotropies computed here are a strict lower bound (underestimating the true eect probably by about a factor of three). The main result of this work is that one can obtain 1

3 reasonably tight constrains for scale invariant magnetic elds; For causally generated magnetic elds the constraints are weaker and are strongly dependent on the evolution of the magnetic eld in the radiation era on small scales. For simplicity we concentrate on the case = 1. Througout, we use conformal time which we denote by. Greek indices run from to 3, Latin ones from 1 to 3. We denote spatial (3d) vectors with bold face symbols. The value of the scale factor today is a( ) = 1. II. THE STRESS TENSOR OF THE MAGNETIC FIELD During the evolution of the universe, the conductivity of the inter galactic medium is eectively innite. In this regime we can decouple the time evolution from the spatial structure: scales like (; ) = ()=a on suciently large scales. On smaller scales the interaction of the magnetic eld with the cosmic plasma becomes important leading mainly to two eects: on intermediate scales, the oscillates like cos(v A k), where v A = =(( + p)) 1= is the Alfven velocity and on small scales, the eld is eponentially damped due to shear viscosity [6]. We will model () as a statistically homogeneous and isotropic random eld. The transversal nature of leads us to h i (k) j (q)i = 3 (k? q)( ij? ^k i^kj ) (k) : (1) where we use the Fourier transform conventions j (k) = d 3 ep(i k) j () ; j () = 1 d 3 k () 3 ep(?i k) j (k) : The Alfven oscillations modulate the initial power spectrum by a factor (k)! (k) cos (v A k) : This can be approimated by a reduction of a factor in the power spectrum on scales with v A k > 1. ut as we shall see, our most stringent constraints will come either from very small scales where the spectrum is eponentially damped or from much larger scales where oscillations can be ignored. We will incorporate the eponential damping by a cuto in the power spectrum at the damping scale. Let us investigate the consequence of causality for the spectrum (k). If is generated by some causal mechanism, it is uncorrelated on super horizon scales, h i (; ) j ( ; )i = for j? j > : () Here it is important that the universe is in a stage of standard Friedmann epansion, so that the causal horizon size is about. During an inationay phase the causal horizon diverges and our subsequent argument does not apply. In this somewhat misleading sense, one calles in- ationary perturbations 'a-causal'. According to Eq. (), h i (; ) j ( ; )i is a function with compact support and hence its Fourier transform is analytic. The function h i (k) j (k)i ( ij? ^k i^kj ) (k) (3) is analytic in k. If we in addition assume that (k) can be approimated by a simple power law, we must conclude that (k) / k n, where n is a even integer. (A white noise spectrum, n = does not work because of the transversality condition which has led to the nonanalytic pre-factor ij? ^k i^kj.) y causality, there can be no deviations of this law on scales larger than the horizon size at formation, in. We assume that the probability distribution function of is Gaussian; although this is not the most general random eld, it greatly simplies calculations and gives us a good idea of what to epect in a more general case. The anisotropic stresses induced are given by the convolution of the magnetic eld, () ij (k) = 1 d 3 q i (q) j (k? q)? 1 l(q) l (k? q) ij : () With the use of the projection operator, P ij = ij? ^k i^kj we can etract the tensor component of Eq. (), () ij = (P a i P b j? (1=)P ij P ab ) ab ; (5) tracelessness, orthogonality and symmetry force the correlation function to be of the form h () ij (k; t) () lm (k ; t)i = j (k; t)j M ijlm (k? k ) h () ij (k; t) () ij (k ; t)i = j (k; t)j (k? k ); (6) were we make use of the tensor basis, M: The correlator on an isotropic tensor component has always the following tensorial structure, M ijlm = il jm + im jl? ij lm + k? ( ij k l k m + lm k i k j? il k j k m? im k l k j? jl k i k m? jm k l k i ) + k? k i k j k l k m : (7) We now determine the function j (k; t)j in terms of the magnetic eld. Using Wick's theorem we have h i (k) j (q) n (s) m(p)i = h i (k) j (q)ih n (s) m(p)i + h i (k) n(s)ih j (q) m(p)i + h i (k) m(p)ih n (s) j (q)i : (8)

4 The problem reduces itself to calculating self convolutions of the magnetic eld. The power spectrum of Eq. () is hij (k; ) 1 d 3 q (8) lm (k ; )i = d 3 ph i (q) j ((k? q)) l (p) m ((p? k ))i = (k? k ) d 3 q (q) (jk? qj) [( il? ^q i ^q l )( jm? (k d? q) j(k d? q) m ) + ( im? ^q i ^q m )( jl? (k d? q) j(k d? q) l )] : (9) Using Eqs. (8,5) and (6), this leads to j j = f(k) =a 8, where f(k) = 1 d 3 q (q) (8) (jk? qj)(1 + + ) ; (1) with = ^k ^q and = ^k k d? q. It remains to dene (k) from Eq. (1). We shall parameterize it in terms of an amplitude and a scale dependence through (k) = ( () 5 n+3?[ n+3 ] kn for k < k c otherwise (11) n >?3=, the spectrum of the energy momentum tensor becomes white noise, independent of n. Only the amplitude which is proportional to (k c ) n depends on the spectral inde. This is due to the fact that the integral (1) is dominated by the contributions at very small scales, k c k. The induced C` spectrum from gravity wave will therefore be independent of n for n >?3=, and obey the well known behavior C` / ` of a white noise source. To simplify, we just consider the dominant term and, in order not to articially produce a singularity at n =?3=, we drop the factor 1=(n + 3). Given the intent of this paper (to constrain the amplitude of the magnetic eld) we will include a factor of 1?1 in our nal result, guaranteeing that we are not overestimating CM anisotropies. The singularity at n =?3 is real. It is the usual logarithmic singularity of the scale invariant spectrum. III. THE CM ANISOTROPIES Armed with the structure and evolution of the stochastic magnetic eld we can now proceed to calculate its effects on tensor CM anisotropies. The metric element of the perturbed Friedman universe is given by ds = a ()(?d + ( ij + h ij )d i d j ) ; The normalization is such that h() i ()ij i = where the quantity in brackets represents the averaged magnetic eld smoothed over a comoving length scale. Note that we have assumed that the cuto scale today is smaller than. We require n >?3 so as not to over-produce long range coherent elds; we shall see that for n =?3 we obtain a scale invariant spectrum of CM anisotropies. We have included a short wavelength cuto to take into account the eponential damping due to shear viscosity in the cosmic plasma [6]. The mean energy density due to such a magnetic eld, which is an appropriately weighted integral of Eq. (11), will be strongly dependent on the cut o when n >?3. Using Eqs. (11) and (1) we can calculate f. The integral cannot be computed analytically, but the following result is a good approimation for all wave numbers k f (k) ' ()9 16 n+6? [ n+3](n + 3) kc n+3 + n n + 3 kn+3 : (1) This result seems to have a singularity at n =?3= which is however removable. The rst term dominates if n >?3= and while second term dominates if the opposite inequality is satised. For n >?3=, the gravity wave source is therefore white noise and its amplitude is determined by the upper cuto, k c. Note that if where h i i = and hj i ki = for tensor perturbations. The magnetic eld will source the evolution equation for h ij through h ij + _a a _ h ij + k h ij = 8G () ij : (13) Such a gravity wave induces temperature uctuations in the CM due to the fact that the photons move along the perturbed geodesics [15] T T ( ; ; n) = _ hij ((); )n i n j d : (1) Here denotes the (conformal) time of decoupling of matter and radiation due to recombination. We want to compute the angular power spectrum of T T, the C`, dened by h T T (n)t T (n )i nn = = 1 X l (` + 1)C`P`() : The C`s are solely determined by the power spectrum of metric uctuations. Dening h_ h (T ) ij (k ; )_ h (T ) lm (k; )i = j _H(k; )j M ijlm (k? k ) one can derive a closed form epression for C` (see [17]): 3

5 C` = 1 dkk ji(`; k)j `(`? 1)(` + 1)(` + ) ; (15) I = d _H(; k) j`(k(? )) (k(? )) (16) where j` denotes the spherical essel function of order `. We solve equation (13) using the Wronskian method; in terms of the dimensionless variable = k. The homogeneous solutions are the spherical essel functions j ; y in the radiation dominated era, and j 1 = ; y 1 = in the matter dominated era respectively. We assume that the magnetic elds were created in the radiation dominated epoch, at redshift z in. We then match the general inhomogeneous solutions of Eq. (13) at the time of equal matter and radiation, eq. Due to the rapid fallo of the source term in the matter dominated era, the perturbations created after eq are sub-dominant, and we nd for the dominant contribution at > eq _H(k; t) ' Gz zin eq ln kf(k) j (k) z eq k Inserting this result in Eq. (16), we obtain I = G z eq ln(z in =z eq )f(k) d j () : (17) j`(? ) (? ) ; (18) where = k, = k and = k. For wave numbers which are super-horizon at decoupling, <, the lower boundary in Eq. (18) can be set to. The remaining integral cannot be epressed in closed form, but is well approimated by []: :7 = 7 5 d j () j`(? ) (? ) = d J 5=() J`+1= (? ) ' 3= (? ) 5= d J 5=() J`+1= (? ) (? ) 3 p` J`+3 ( )) : (19) 3 The third integral above can be epressed in closed form ( [1], number ), and is reasonably well approimated by the last epression, we have checked the approimation numerically for l and varying. We can now do the integrals in Eq. (15) analytically to obtain `C` ' A for n >?3=, and `C` ' A n+6 3 (k c ) n+3`3 n+6 ()?n?[1? n] (n + 3)? [1? `6+n (1) n] (1?n) for?3 < n <?3=, where A = 5 1? () 9 z eq ln zin = 3 1?8 G z eq? [ n+3] ln 1 (z in=z eq) 1?9 Gauss? [ n+3] : () IV. RESULTS Eqs. () and (1) are our main result. They allow us to limit a possible primordial magnetic eld by requiring it not to over produce uctuations in the CM. Since the uctuations induced grow with ` for all values of the spectral inde?3 < n, we obtain the best limits for large values of `. We shall be conservative and assume an upper bound of `C` j`=5 < 8:5 1?9 [18]. Given that we are interested in galactic and cluster scales we = :1 h?1 Mpc for the remainder of this paper. In Fig. 1 we show the limit on a stochastic magnetic eld as a function of the spectral inde n, using the damping scale given below as cuto. We now focus on a few particular cases of interest and in doing so we will derive an analytic epression which approimates the upper bound of over the whole range of n. Scale invariant magnetic eld: From Eq (1) we see that the result is independent of the cuto. In the limit where n!?3 we nd that < 1?9 Gauss (3) i.e. of the same level as other constraints [9{1]. Causal magnetic eld: For this scenario we have, as eplained above, n ; we shall consider the case of n =. For instructive purposes let us rst consider a k c which is independent of the magnetic eld. The constraint is then < ln? 1 zin z eq (k c )? 7 Gauss : () The cuto k c will depend on the plasma properties and evolution; even though the conductivity of the cosmic plasma is very large, it is nevertheless nite. One actually nds [19] that = T, where the parameter 1 < < 7 is slowly temperature dependent. y Ohm's law, magnetic elds on small enough scales are eponentially suppressed, / ep??ak =, leading to a damping scale, k d () = (a=) 1= = (1?3 cm)?1=. This scale is smaller than the comoving horizon scale for all temperatures below the Planck scale. On scales smaller than 1=k d ( eq ), the induced gravity waves have damped away even before matter and radiation equality. Since the sourcing of gravity waves after equality is negligible, the damping scale relevant in our problem is k c = k d ( eq ),

6 k c ' 1 13 h Mpc?1 : (5) If we insert this this damping scale in Eq. (), we obtain < 1?9 Gauss. A more realistic scenario is to assume that the magnetic eld will be damped by electron viscosity. To proceed with the analysis we shall split the stochastic magnetic eld into a high-frequency component and a low-frequency component; the scale which separates the two is the Alfven scale at equality, A = V A eq where V A is the Alfven velocity, VA = h i=(( + p)). From equation of [1] we see that the inhomogeneous magnetic eld will obey a damped harmonic oscillator equation, with a time dependent damping coecient, D = :k l (1 + z) (l is the physical photon mean free path) and frequency!() = VA k? ( _D=)? (D=). Within this setting we can estimate the damping scale of the magnetic eld in the oscillatory regime of this system; the amplitude of the eective homogeneous magnetic eld, A, which is responsible for the Alfven waves is related to through A ' 1?9 n+3 Gauss 3:8 1? A eq which leads to n+5 A = (13h?1 ) n+3 1?9 n+5 1?9 Gauss (6) Gauss We shall dene the damping scale to be the scale at which one e-fold of damping has occured by equality. From d = 1 one nds R eq D k c = :5Mpc?1 (7) For this estimate to be valid, the system must be in the damped oscillatory regime (as opposed to overdamping regime), i.e.!( eq ) > ; this condition is satised if A > 5:5h? 1?9 Gauss. We nd that indeed this is the case in the range of interest. Combining Equations (6),(), (1) and (7) and assuming a formation redshift of z in = 1 15 (although the nal result is very weakly dependent z in ) we nd that an an approimation to the bound is < 7:9 1?6 e :99n Gauss; for n <?3=; < 9:5 1?8 e :37n Gauss; for n >?3=; (8) The upper bounds corresponding to Eq. 8 represent a reasonable t to Figure 1. As one can see, the constraint on a causal magnetic eld is well above 1?9 Gauss. Throughout this derivation we have assumed that we can estimate the damping scale of the magnetic eld by looking solely at the Alfven modes. A linear analysis of the remaining degrees of freedom also indicate that the magnetic eld will be damped at the same scale as in Equation (7). It is possible that non-linear eects may FIG. 1. The upper bound as a function of spectral inde, n. We assume z in=z eq = 1 9 and = :1h?1 Mpc. prevent the tangled magnetic eld from damping at this scale but an accurate quantitative analysis is still lacking. Inationary magnetic elds: roken conformal invariance allied with the an inationary period will create large scale magnetic elds. ertolami and Mota [] estimate the spectral inde in such a mechanism to lie in the range around. We are then clearly in the regime where the cuto is important. Using the Alfven damping scale at equality and assuming magnetic eld generation at 1 1 GeV, we nd < (1?7?1?8 )Gauss for n varying from?:5 to :5. A similar result can be obtained for the model of Gasperini et al []. V. DISCUSSION Our calculation diers from most of the recent work on the impact of primordial magnetic elds on structure formation: In estimating the CM anisotropies we do not split the magnetic eld into a 'large' homogenous mode and a 'small' uctuation. The magnetic eld then aects metric perturbations quadratically. This has two eects. Firstly it allows us to consider the magnetic eld as a sti source, and discard (within the MHD approimation) the backreaction of the perturbations in the cosmological uid. Indeed if we were to consider backreaction then we would know a priori that we would be generating unacceptable perturbations in the cosmological uid. Another way of phrasing this is that the magnetic eld itself is 1= order perturbation theory, while its energy momentum tensor and consequently the induced metric perturbations are rst order perturbations. The MHD backreactions on would be 3= order and may thus be neglected in linear perturbation theory. We point out, 5

7 however that, to obtain an estimate of the damping scale due to the viscosity in the MHD we had to consider a split between long wave length and short wavelength uctuations in. Secondly, the stress energy tensor being quadratic in the magnetic eld, leads to a 'sweeping' of modes: large wavelength modes in T will in general be aected by all scales of the spectrum of [1]. As we have seen in the causal case, the small wavelength behaviour of the magnetic eld totally dominates the large wavelength pertubations. In [5] the magnetic eld is modeled as = + (1) () where is a homogeneous term; the stress energy tensor is then given by terms of the form i (1) j, which are linear in the stochastic component. A few comments are in order with regards to our result. Note that we are considering a specic class of models, where the magnetic eld seed is created at some well dened moment in the early universe and then evolves according to the MHD equations. If the magnetic eld is being constantly sourced throughout the radiation era, then our calculation is not valid. An eample of such a scenario was proposed by Vachaspati [3] where magentic elds are sourced by vortical imprints from an evolving network of cosmic strings; although the scaling behaviour of source may lead to / a?, the eective damping scale will be of order the horizon much larger than the Alfven damping scale. Another possibility has been put forward in [7], where the onset of turbulence induces an amplication of power on large scales but a supression of power on small scales. This would further increase k c but the results are still too qualitative to be properly included in an analysis such as ours. Acknowledgments: We thank John arrow, Orfeu ertolami, Kari Enqvist, Karsten Jedamzik, Jo~ao Magueijo, Evan Scannapiecco and Misha Shaposhnikov for useful discussions. We thank one of the referees for pointing out the importance of Equation (6) and another for alerting us to the importance of reference [1]. RD acknowledges the hospitality of the CfPA at U.C. erkeley, where this work was initiated. TK acknowledges the hospitality of Geneva University. New York, (1983); E.N. Parker, Cosmological Magnetic Fields, Oford University Press, (1979). [5] E. Kim, A. Olinto and R. Rosner, Ap. J. 68, 8 (1996); K.Jedamzik, V.Katalinic and A.Olinto Phys. Rev. D57, 36 (1998). [6] K.Subramanian and J. arrow Phys. Rev. D (1998). [7] A.randenburg, K.Enqvist and P.Oleson, Phys. Rev. D5, 191 (1996). [8] A.Davis, M.Lilley and O, Tornqvist astro-ph/99 [9] J. Adams, U.H. Danielsson, D. Grasso and H. Rubinstein, Phys. Lett. 388, 53 (1996). [1] A. Kosowsky and A. Loeb, Ap. J. 69, 1 (1996). [11] E.Scannapieco and P.G. Ferreira, Phys. Rev. D56, R793, (1997). [1] K.Subramanian and J. arrow Phys. Rev. Lett. 81, 3575 (1998). [13] R. Durrer, T. Kahniashvili and A. Yates, Phys. Rev. D58, 13 (1998). [1] J. arrow, P. Ferreira and J. Silk Phys. Rev. Lett. 78, 361 (1997). [15] R. Durrer, Fund. Cosmic Phys. 15, 9 (199). [16] P.J.Peebles, The Large Scale Structure of the Universe, Princeton, PUP (198). [17] F.F. Abott and M.. Wise, Nucl. Phys., 51 (198). [18] Ruhl, J. et al Ap. J. Lett 53 1, (1995). [19] This formula for the conductivity has been derived in: J.Ahonen and K.Enqvist Phys. Lett. 38, (1996), for T > 1MeV. For T < 1MeV, the scattering processes which determine the conductivity are Compton scattering and Rutherford scattering. Applying a simple formula for the conductivity in a non-relativistic plasma, one obtains the same result with. [] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York (197). [1] Gradzthein and Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, (199). [1] P.P. Kronberg, Rep. Prog. Phys. 57, 57 (199). [] M.S.Turner and L.M. Widrow, Phys. Rev. D37, 73 (1988);. Ratra, Ap. J. Lett. 391 L1 (199); W.D.Garretson, G.. Field and S.M.Carroll, Phys. Rev. D6 536 (199); M. Gasperini, M. Giovannini and G. Veneziano, Phys. Rev. Lett. 75, 3796 (1995); D. Lemoine and M. Lemoine Phys. Rev. D5, 1955 (1995); O.ertolami and D.F.Mota, gr-qc/9911. [3] T.W. Kibble and A. Vilenkin Phys. Rev. D5 679 (1995); J.T. Ahonen and K. Enqvist, Phys. Rev. D57 66 (1998); T. Vachaspati, Phys. Lett , (1991); M.Joyce and M.E.Shaposhnikov Phys. Rev. Lett. 79, 1193 (1997). [] Ya.. eldovich, A.A. Ruzmaikin and D.D. Sokolo, Magnetic Fields in Astrophysics, Gordon and reach, 6