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Astron. Astrophys. 327, 1-7 (1997)

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5. Conclusions

Primordial magnetic flux tubes are able to produce filamentary inhomogeneities of density which grow more or less linearly throughout the epoch considered in this paper. This epoch has been restricted from annihilation to just before the acoustic epoch, from [FORMULA] to [FORMULA], although the calculations may account for the evolution in epochs earlier than this time interval.

We have restricted ourselves to a particular case: that of a flux tube, or more specifically to the field configuration given by Eq. (1). This choice was in part due to the frequent observation of flux tubes in other cosmic ionized systems and to the fact that they are suggested by the material filaments often observed in the large scale structure. It is also a very simple symmetric structure defined with only one coordinate. However this particular choice restricts the generality of our results and other field configurations are possible.

However, flux tubes, or at least structures as defined by Eq. (1), constitute a rather general case if we consider a universe with no mean magnetic field and in which the fluctuating field is made up of characteristic cells with a coherent internal field orientation, but having no "a priori" relation with the orientation of the field in adjacent cells. Suppose a coherence cell in which the field can be represented by (0,0, [FORMULA]), with z clearly being the direction of the field within the cell. We look for a function [FORMULA], i.e. when we leave the cell following a perpendicular direction to the field in the cell. Before encountering another coherence cell, [FORMULA] would vanish. In the opposite direction (-x) we would have the same function. At the centre, we would have a maximum of [FORMULA], with [FORMULA], avoiding a discontinuity in the second derivative. If we adopt axisymmetry, we conclude that [FORMULA] can reasonably be assumed to be a Gaussian, as specified by our Eq. (1). It is true that flux tubes are long structures and the above argument does not consider the length of the structure defined by (1). Along z, the field can be independent of z and therefore represented by (0,0, [FORMULA]) within a z-length, which is considered long in this paper, but with an unimportant and unspecified value. Therefore, even if we consider a particular magnetic field configuration as our basic structure, our results remain rather general.

After the epoch considered in this paper, other effects, such as damping of small diameter filamentary structures prior to recombination, non-linear growth after recombination and mechanisms amplifying magnetic fields in recent pregalactic and galactic epochs, will complicate this simple picture, but this is beyond our objectives.

Our work is restricted to an unobservable epoch, which means that our results cannot strictly be compared with observations. Its objective is to provide initial conditions for other models devoted to more recent epochs, the whole history of cosmological magnetism being a huge task, which cannot be undertaken by a single model. Nevertheless, it is unlikely that large filamentary structures have disappeared in more recent epochs, so that our results may provide an explanation for presently observed large scale structures. The implications of this model in their interpretation are in part, considered in Paper III. It is really to be expected that large structures remain unaffected by complex processes after equality. This is justified as follows:

Damping of cosmic magnetic fields has been considered by Jedamzik, Katalinic and Olinto (1997) introducing viscosity, bulk viscosity and heat conduction. After neutrino decoupling the main damping mechanism is photon diffusion, which only affects structures up to the Silk mass, about [FORMULA].

When considering the growth of unmagnetized structures, it is assumed that non-linear effects are important only at smaller scales, up to [FORMULA] 10 Mpc, and this limit probably remains valid when magnetic fields are taken into account. In practice, this limit corresponds to the scale at which the rms galaxy fluctuations are unity, and is therefore independent of the involved forces. This issue has been considered by Kim, Olinto and Rosner (1996).

Finally, we must take into account specific mechanisms of magnetic field creation and amplification in recent times. A large variety have been proposed (Zweibel, 1988; Pudritz and Silk, 1989; Harrison, 1973; Tajima et al., 1992; Lech and Chiba, 1995, and others); See also the reviews by Rees (1987) and Kronberg (1994). The important fact is that these mechanisms induce small scale magnetic fields, smaller than a few Mpc . As far as we are aware, no mechanism for producing magnetic fields at scales larger than a few Mpc have been proposed for post-Recombination mechanisms.

After Equality, some mechanisms erase pre-existing magnetic fields, and others amplify them in a complicated way, thus modifying the pre-Equality magnetic fields considered in this paper, although these mechanisms only affect the small scale structures. The evolution of large scale structures could be described by the formulae in Appendix B into Paper I, i.e., the structures are maintained, simply growing with the expansion. Our model therefore, constitutes a tool to interpret present large scale structures.

It is a well established observational fact that the large-scale structure of the Universe is very rich in filaments (Gregory and Thomson, 1978; Oort, 1983; de Lapparent et al. 1986 and others. See for instance the review by Einasto, 1992) being more abundant than two-dimensional sheets (Shaty, Sahni and Shandarin, Einasto, 1992). They can play an important role in the formation of clusters (West, Jones and Forman, 1995).

The existence of large-scale filaments is currently accounted for by other hypotheses, but it is here suggested that primordial magnetic flux tubes constitute an additional alternative, or at least, a mechanism reinforcing other gravitational effects. Filaments are associated with magnetic fields in many astrophysical systems, such as the Sun and the interstellar medium, and we now see that this association can be extended to large-scale filamentary structures in the Universe.

The best studied large-scale filamentary structure is the Coma-A1367 supercluster, which is itself elongate and extended towards the Hercules supercluster. Its diameter is about 10 Mpc, thus constituting a large scale inhomogeneity in the sense considered here (i.e. [FORMULA] 0.28 Mpc; [FORMULA] 1 in the units defined above). Its length can be very large (Batuski and Burns, 1985). The distribution of early type galaxies is particularly thick (Doi et al., 1995).

As observed random velocities of groups and clusters with respect to the filament structure are relatively small ([FORMULA]; Tully, 1982) the observed distribution of galaxies reflects its distribution when the whole structure was formed. The evolution of the filament and of the network of filaments it belongs to, has evolved very little.

Rather interestingly, the magnetic field strength has been measured in this supercluster. In a region well outside the coma cluster in the direction toward A1367, Kim et al. (1989) observed a bridge of synchrotron emission with the same direction, of about 0.3-0.6 [FORMULA], a large value for an extracluster region. In the Coma cluster core region it is even larger, of the order of 1.7 [FORMULA] (Kim et al. 1990). Radio observations of the Coma cluster and its vicinity at different frequencies have been reported by Kim et al (1994) and Kim (1994). It would be very interesting to determine whether the direction of the magnetic field coincides with the NE-SW direction, which is that of the huge filament. This coincidence would be in noticeable agreement with the model here suggested.

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Online publication: April 8, 1998