A&A 433, L53-L56 (2005)

DOI: 10.1051/0004-6361:200500094

**V. B. Semikoz ^{1,2} - D. Sokoloff ^{3}**

1 - AHEP Group, Instituto de Fisica
Corpuscular; CSIC/Universitat de Valencia, Edificio de Paterna, Apartado
22085, 46071 Valencia, Spain

2 - IZMIRAN,
Troitsk, Moscow Region 142190, Russia

3 -
Department of Physics, Moscow State University, Moscow 119992,
Russia

Received 17 November 2004 / Accepted 1 March 2005

**Abstract**

The magnetic helicity has paramount significance in the
nonlinear saturation of the galactic dynamo. We argue that
magnetic helicity conservation is violated at the lepton stage in
the evolution of the early Universe. As a result, a cosmological
magnetic field which can be a seed for the galactic dynamo obtains
from the beginning a substantial magnetic helicity which has to be
taken into account in the magnetic helicity balance at a later
stage of the galactic dynamo.

**Key words: **magnetic fields - the early
Universe

Magnetic fields of galaxies are believed to be generated by a galactic dynamo based on the joint action of the so-called -effect and differential rotation. The -effect is connected with a violation of mirror symmetry in MHD-turbulence and therefore caused by rotation. For a weak galactic magnetic field, the mirror asymmetry is associated with helicity of the velocity field and is proportional to the linkage of vortex lines.

The magnetic helicity is an inviscid integral of motion and its
conservation strongly constrains the nonlinear evolution of the
galactic magnetic field. The helicity density
of
a galactic large-scale magnetic field is enhanced by galactic
dynamo (here
is a large-scale magnetic field, is its vector-potential). Because of magnetic helicity
conservation, this income must be compensated by magnetic helicity
of a small-scale magnetic field. Note that the magnetic helicity
density is bounded from above by *b*_{l}^{2} *l*, where *b*_{l} is the
magnetic field strength at the scale *l*. Hence the supply of a
small-scale magnetic helicity occurs to be insufficient for the
compensation required. This fact strongly constrains the galactic
dynamo action (see Brandenburg 2001a,b, and references therein).

On the other hand, a weak almost homogeneous cosmological magnetic field can introduce a new element in this scheme. If we adopt a thickness of the gaseous galactic disc as a typical scale for galactic dynamo action and G, then the magnetic helicity supplied via the cosmological magnetic field ( , where is the horizon size) can be of the same order as that one for the galactic magnetic field ( , ).

The cosmological magnetic field if it exists must be substantially weaker than the galactic magnetic field (see Beck et al. 1996, for review). According to the analysis of rotation measures of remote radio sources, . This estimate however is based on the assumption that a substantial part of charged particles in the Universe is in form of thermal electrons in the intergalactic medium. A more robust estimate, is based on the isotropy constrains.

We estimated above that if the cosmological magnetic field is of the order G its magnetic helicity density can be comparable with the magnetic helicity density of the galactic magnetic field. Of course, the estimate gives a much lower value for the magnetic helicity density. However it is more than natural to expect a magnetic helicity concentration in the processes of galactic formation. The question is whether a mechanism of magnetic helicity production can be suggested for physical processes in the early Universe.

Here we suggest a mechanism for magnetic helicity generation by a collective neutrino-plasma interactions in the early Universe after the electroweak phase transition.

Let us consider the electron-positron plasma as a two-component
medium, for which the strong correlation between opposite charges
due to Coulomb forces gives
.
Here
is the common fluid velocity of *electroneutral*
conducting gas while its positively and negatively charged
components have different velocities,
,
where a small difference
gives the separation of charges at small scales and enters the
electromagnetic current
*e*
obeying in MHD the Maxwell
equation,
(
.

The electric field
derived from Euler equations for
plasma components takes the form (Semikoz 2004) which includes
the contribution of weak interactions
taken
in the collisionless Vlasov approximation. We do not consider
other known terms which describe weak interaction collisions
(Dolgov & Grasso 2002), Biermann battery effects, etc., and
which do not play substantially in favor of helicity generation.
In turn, we keep in
only the axial vector
term which violates the parity:

Here is the Fermi constant, is the proton mass; is the axial weak coupling, upper (lower) sign is for electron (muon or tau) neutrinos, is the neutrino density asymmetry, is the neutrino current asymmetry; is the unit vector along the mean magnetic field;

Accounting for the first line in Eq. (1) we
obtain the axial vector term
where the helicity coefficient
is the *scalar *in the standard model (SM) with neutrinos instead of the
pseudoscalar
in standard MHD:

Here we substituted , and assumed a scale of neutrino fluid inhomogeneity , that is small compared with a large -scale of the mean magnetic field.

Let us stress that instead of the *difference* of electron
and positron contributions in axial vector terms entering the pair
motion equation (Semikoz 2004) and given by the polarized density
asymmetries
(*n*_{0}^{(-)}-*n*_{0}^{(+)}) we obtained here the *sum
* of them
(*n*_{0}^{(-)}+*n*_{0}^{(+)}) that can lead to a
significant effect in a hot plasma.

The admixture of the pseudovector
to the pure
vector ,
e.g. for the constitutive relations
,
due to the same neutrino-plasma weak interaction
described by the constants ,
,
has been already
discussed in literature (see Nieves & Pal 1994, Eqs. (3.5), (3.6)). In a forthcoming paper we show that such unusual
coefficients appear in *chiral media* and are simply connected
with
given by Eq. (2),
,
where
is the dielectric
permittivity of plasma.

Thus, using Eq. (1) from the Maxwell equation
one obtains the Faraday equation
generalized in SM with neutrinos and antineutrinos:

where we omitted the weak vector contribution suggested by Brizard et al. (2000) since we neglect any neutrino flux vorticity in the hot plasma of early universe. Because the early universe is almost perfectly isotropisc and homogeneous, we ignore here any contribution from large-scale motions as well. In the relativistic plasma the diffusion coefficient takes the form

The first term in r.h.s. of Eq. (3), , is associated with the parity violation in weak interactions in the early universe plasma.

We stress that the Eq. (3) is the usual equation for
mean magnetic field evolution with -effect based on
particle effects rather on the averaging of turbulent pulsations.
It is well-known (see e.g. Ruzmaikin et al. 1988) that Eq. (3) describes a self-excitation of a magnetic field with
the spatial scale
and the growth
rate
.
Semikoz & Sokoloff (2004) estimated
these values for the early universe to get

Here

Thus, while in the temperature region
there are many small random magnetic field
domains, a weak mean magnetic field turns out to be developed into
the uniform *global* magnetic field at temperatures below *T*_{0} (see Eq. (4). The global magnetic field can be
small enough to preserve the observed isotropy of the cosmological
model (Zeldovich 1965) while being strong enough to be
interesting as a seed for galactic magnetic fields. This scenario
was extensively discussed by experts in galactic magnetism
(Kulsrud 1999), however until now no viable origin for the global
magnetic field has been suggested. We believe that the dynamo
based on the -effect induced by particle physics solves
this fundamental problem and opens a new and important option in
galactic magnetism.

Let us consider how the collisionless neutrino interaction with
charged leptons can produce the primordial magnetic helicity
,
where *v* is the volume that
encloses the magnetic field lines.

For that we should substitute into the derivative,

the electric field given by Eq. (1). Neglecting any rotation of primordial plasma given by the first dynamo term, or retaining the resistive term and the weak interaction term given by Eq. (1) that is the main one in the absence of any vorticities, one finds from Eq. (6)

Note that the second term in the r.h.s. violates parity: it is a pure

First term in the r.h.s. of Eq. (7) gives conventional ohmic losses for magnetic helicity and usually is neglected in helicity balance.

where is the initial helicity value at the moment if it exists, and we present the helicity density entering the integrand as . Here we use the dimensionless variable . The maximum value

It is worth noting that such WKB value of
which is
scaled being frozen-in as
does obey the BBN limit
at the temperature
,
i.e.
.
Note
also that the sign of the first term in Eq. (8) is not
well determined since it depends on the combined neutrino density
asymmetry (Semikoz & Sokoloff 2004),
,
where
the values of the dimensionless neutrino chemical potentials
are given by the BBN limit (Dolgov et al. 2002):
,
or by the CMBR/LSS
bound (Hansen et al. 2002):
,
< 2.6. One can use, e.g., the
conservation of the lepton number
that implies
,
however, this does not
guarantee the definite sign of the combined neutrino density
asymmetry
.
The definite sign of the
magnetic helicity (left-handed, *H*<0), arising during electroweak
baryogenesis (Vachaspati 2001) is another case connected with the
CP-violation.

Let us emphasize that cosmological helicity production via the
collective neutrino interaction with hot plasma ceases if the
neutrino chemical potential
(hence the
neutrino density asymmetry
)
vanishes,
,
.
Solely the inequality
is known from the BBN bound on light elements
abundance at
(Dolgov et al. 2002), thereby
substituting for a rough estimate
we
estimate the integral for *h*(*x*) as
.

Thus, collecting numbers for
,
*J*(*x*) and using the
electron Compton length
,
one finds the huge value of *cosmological helicity density *
that could seed galactic magnetic helicity

Traditional galactic dynamo considered galactic magnetic field produced from a very weak seed field. This implies that the magnetic helicity of the seed field is weak. We argue that the applicability of this viewpoint is limited. The seed field for galactic dynamo can be a field of substantial strength and substantial helicity. The first part of this statement is already quite well-accepted in modern galactic dynamo (see e.g. Beck et al. 1996) while the second one is new. In this letter we suggest a physical mechanism for the magnetic helicity production for the seed field of galactic dynamo. As far as we know, such mechanisms was not considered previously. Note that Field & Carroll (2000) pointed out in a general form the importance of the electroweak phase transition for magnetic helicity generation.

We stress that the epoch just after the electro-weak phase transition and that one of galaxy formation are quite remote in respect to their physical properties. We appreciate that the magnetic helicity evolution in the time interval between these epoches has to be addressed separately. In particular, large-scale magnetic helicity produced by galactic dynamo is antisymmetric in respect to the galactic equator while the magnetic helicity from any cosmological sources is obviously independent on the galactic equator position. It is far from clear how important this asymmetry is for nonlinear galactic dynamos and for the observed asymmetry of magnetic field in Milky Way. Note also that the strong cosmological magnetic field could prevent the inverse MHD cascade on the scale of galaxies (Milano et al. 2003; Brandenburg & Matthaeus 2004).

We should remark that the huge helicity value (9)
exists only in hot ultrarelativistic (
)
early
universe plasma where
is sufficiently large. The
evolution of magnetic helicity *H*(*t*), or how cosmological
magnetic helicity feeds protogalactic fields is a complicated
task. In the nonrelativistic plasma, first, positrons vanish, then
with the cooling for the frozen-in magnetic field
the electron density at the main Landau level drops,
,
resulting in
,
and the magnetic helicity
production becomes impossible.

Let us note that the neutrino collision mechanism (Dolgov &
Grasso 2002) can not produce magnetic helicity unlike in our
collisionless mechanism. This immediately comes after the
substitution of the electric field term stipulated by weak
collisions
(is the electric conductivity in the ultrarelativistic plasma) and
taken from Eq. (5) in (Dolgov & Grasso 2002), where the electric
current
is caused by the friction
force due to the difference of weak cross-sections for neutrino
scattering off electrons and positrons. This current is directed
along the fluid velocity. The generalized momentum
,
is the enthalpy,
is the -factor in the
ultrarelativistic plasma, obeys Euler equation (Semikoz 2004)

from which retaining the standard MHD terms only (the first and the second ones in the r.h.s. of Euler equation) one can obtain the velocity that does not contribute (in the lowest approximation over ) to the helicity change.

Let us note that we rely here on homogeneous magnetic fields with a scale which is less (however comparable) than the horizon , hence the magnetic force lines are closed within the integration volume , or applying the Gauss theorem one can show that the contribution of the first term in the r.h.s. of Eq. (10) to the helicity production (6) vanishes. This is exactly like for the gauge transformation of the vector potential in the helicity , . On the other hand, there remains an open question how to define the gauge invariant helicity for superhorizon scales.

There are other astrophysical objects for which axial vector weak forces acting on electric charges and driven by neutrinos can lead to the amplification of mean magnetic field and its helicity as given in Eq. (7). For instance, the neutrino flux vorticity which is proportional to can vanish for isotropic neutrino emission from supernovas in the diffusion approximation when neutrino flux is parallel to the radius . In such case the mechanism of collective neutrino-plasma interactions originated by the axial vector weak currents becomes more efficient to amplify magnetic field than the analogous mechanism based on weak vector currents (Brizard et al. 2000).

This work was supported by the RFFI grants 04-02-16094 and 04-02-16386. Helpful discussions with A. Brandenburg and T. Vachaspati are acknowledged.

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