**1.4. The evolution of primordial magnetic fields**

A crucial issue for the investigation of a possible primordial origin of present time galactic and intergalactic magnetic fields is that concerning the time evolution of the magnetic fields in the cosmic medium. Three conditions are needed for the persistence of large static fields:

- a) intrinsic stability of the field;
- b) the absence of free charges which could screen the field;
- c) to have a small diffusion time of the field with respect to the age of the Universe.

Condition a) does not hold for strong electric fields. It is a
firm prediction of QED that an electric field decays by converting
its energy in electron-positron pairs if *e* |*E*|
*m*_{e}^{2}
[37,
38].
This, however, is a purely
electric phenomenon. Although, at the end of the sixties, there
was a claim that strong magnetic fields may decay through a
similar phenomenon [39]
the argument was proved to be
incorrect. Only very strong fields may produce nontrivial
instabilities in the QCD (if *B* > 10^{17} G) and the
electroweak vacuum
(if *B* > 10^{23} G) which may give rise to a partial
screening of the field. These effects (see
Chap. 5)
may have some relevance for processes which occurred at very
early times and, perhaps, for the physics of very peculiar collapsed
objects like magnetars
[40].
They are, however, irrelevant for the evolution of cosmic magnetic
fields after BBN time.
The same conclusion holds for finite temperature and densities effects
which may induce screening of static magnetic fields (see e.g. Ref.
[41]).

Condition b) is probably trivially fulfilled for magnetic fields due to the
apparent absence of magnetic monopoles in nature.
It is interesting to observe that even a small
abundance of magnetic monopoles at present time would have
dramatic consequences for the survival of galactic and
intergalactic magnetic fields which would lose energy by
accelerating the monopoles. This argument was first used by Parker
[42]
to put a severe constraint on the present time
monopole flux, which is *F*_{M}
10^{-15}
cm^{-2} s^{-1}sr^{-1}. It was recently proposed
by Kephart and Weiler
[43]
that magnetic monopoles accelerated by galactic magnetic fields could give
rise to the highest energy cosmic rays (*E*
10^{19} eV)
and explain the violation of the famous Greisen-Zatsepin-Kuzmin
cut-off [44].

Also the condition c) does not represent a too serious problem for the survival of primordial magnetic fields. The time evolution law of a magnetic field in a conducting medium has been already written in Eq. (1.5).

Neglecting fluid velocity this equation reduces to the diffusion equation which implies that an initial magnetic configuration will decay away in a time

(1.17) |

where *L* is the characteristic length scale of the spatial
variation of **B**. In a cosmological framework, this means that
a magnetic configuration with coherence length *L*_{0} will
survive until the present time *t*_{0} (*t* = 0
corresponds to the big-bang time) only if
_{diff}(*L*_{0}) >
*t*_{0}. In our convention, *L*_{0}
corresponds to the present time length scale determined by the Hubble law

(1.18) |

where *a(t)* is the Universe scale factor and
*L*(*t*_{i}) is the
length scale at the time at which the magnetic configuration was
formed. Note that *L*_{0} may not coincide with the actual
size of the magnetic configuration since other effects (see below) may
come-in to change the comoving coherence length. As we see from
Eq. (1.17) the relevant quantity controlling the decay
time of a magnetic configuration is the electric conductivity of
the medium. This quantity changes in time depending on the varying
population of the available charge carriers and on their kinetics
energies. However, since most of the Universe evolution takes
place in a matter dominated regime, during which all charge
carriers are non-relativistic, our estimate of the magnetic
diffusion length is simpler. In general, electric
conductivity can be determined by comparing Ohm law **J** =
**E** with the
electric current density definition **J** = *n e* **v**,
where for simplicity we considered a single
charge carrier type with charge *e*, number density *n* and
velocity **v**. The mean drift velocity in the presence of the
electric field **E** is **v** ~ *e***E**
/ *m* where *m*
is the charge carrier mass and
is the average time between
particle collisions. Therefore the general expression for the
electron conductivity is
^{(2)}

(1.19) |

After recombination of electron and ions into stable atoms
Universe conductivity is dominated by residual free electrons.
Their relative abundance is roughly determined by the value that
this quantity took at the time when the rate of the reaction
*p + e* « *H* +
became
smaller than the Universe
expansion rate. In agreement with the results reported in
Ref. [46], we use

(1.20) |

where _{0}
is the present time density parameter and *h* is
Hubble parameter. Electron resistivity is dominated by Thomson
scattering off cosmic background photons. Therefore
1 /
*n*_{}
_{T}, where
_{T} =
[(*e*^{4}) /
(6 m_{e}^{2})]
is the Thomson cross section, and
*n*_{} = 4.2 × 10^{2} (1 +
*z*)^{3}. Substituting these expressions in
Eq. (1.19) we get

(1.21) |

It is noticeable that after recombination time Universe
conductivity is a constant. Finally, the cosmic diffusion length,
i.e. the minimal size of a magnetic configuration which can
survive diffusion during the Universe time-life *t*_{0}, is
found by substituting *t*_{0} = 2 ×
(_{0}
*h*^{2})^{-1/2} s^{-1} into Eq. (1.17)
which, adopting
_{0} = 1
and *h* = 0.6, gives

(1.22) |

It follows from this result that magnetic diffusion is negligible on galactic and cosmological scales.

The high conductivity of the cosmic medium has other relevant consequences for the evolution of magnetic fields. Indeed, as we already mentioned in the Introduction, the magnetic flux through any loop moving with fluid is a conserved quantity in the limit . More precisely, it follows from the diffusion equation (1.5) and few vector algebra operations (see Ref. [15]) that

(1.23) |

where *S* is any surface delimited by the loop. On scale where
diffusion can be neglected the field is said to be *frozen-in*, in
the sense that lines of force move together with
the fluid. Assuming that the Universe expands
isotropically ^{(3)},
and no other effects come-in, magnetic flux conservation implies

(1.24) |

This will be one of the most relevant equations in our review. It
should be noted by the reader that *B(t)* represents the local
strength of the magnetic field obtained by disregarding any effect that may
be produced by spatial variations in its intensity and direction.
Eq. (1.24) is only slightly modified in the case the a
uniform magnetic field produce a significative anisotropic
component in the Universe expansion (see
Sec. 2.1).

Another quantity which is almost conserved due to the
high conductivity of the cosmic medium is the, so-called, *
magnetic helicity*, defined by

(1.25) |

Helicity is closely analogous to vorticity in fluid dynamics. In a
field theory language, *H* can be
identified with the
Chern-Simon number which is known to be related to the topological
properties of the field. Indeed, it is known from
Magnetohydrodynamics (MHD) that *H*
is proportional to the
sum of the number of links and twists of the magnetic field lines
[47].
As it follows from Eq. (1.5), the time
evolution of the magnetic helicity is determined by

(1.26) |

As we shall show in Chap. 4, several models proposed to explain the origin of primordial magnetic fields predict these fields to have some relevant amount of helicity.

In the previous section we have already mentioned the important
role played by magnetic helicity in some MHD dynamo mechanisms
driving an instability of small scale magnetic fields into large scale
fields. A similar effect may take place at a cosmological level
leading to significative corrections to the simple scaling laws
expressed by the Eqs. (1.18, 1.24). Generally, these
kind of MHD effects occur in the presence of some, turbulent
motion of the conductive medium (note that
Eqs. (1.18, 1.24) has been derived under the assumption
of vanishing velocity of the fluid *v* = 0). Hydrodynamic
turbulence is generally parameterized in terms of the Reynolds
number, defined by

(1.27) |

where is the kinematic
viscosity. Fluid motion is said to be
turbulent if *Re* >> 1. In the presence of a magnetic field
another parameter turns-out to be quite useful. This is the *magnetic
Reynolds number* defined by

(1.28) |

where =
[1 / (4
)]. When
*Re*_{M} >> 1 transport of the magnetic lines with
the fluid dominates
over diffusion. In this case hydrodynamic turbulence in a
conducting medium give rise to *magnetic turbulence*. It is
often assumed in MHD that a fully developed magnetic turbulence
give rise to equipartion between the kinetic and the magnetic
energy of the fluid. Whether equipartition hypothesis is valid or not
is a controversial issue.

Both the hydrodynamic and magnetic Reynolds numbers can be very
large in the early Universe. This is a consequence of the
high electric conductivity and low viscosity of the medium
and, especially, of the large scales which are involved. The
electric conductivity of the early Universe has been computed by
several authors. A first simple estimation of
in the
radiation dominated era was performed by Turner and Widrow
[45].
In terms of the resistivity
their result is
~
/ *T* . A more
exact series of calculations can be found in Ref.
[48] which include
logarithmic corrections due to Debye and dynamical screening. As
a result a more correct expression for
is

(1.29) |

Other detailed computations of the Universe conductivity close to the QCD and the electroweak phase transitions are available in Refs. [49]. The kinematic viscosity follows the behavior [50]

(1.30) |

In the early Universe
>> , i.e.
*Re*_{M} >> *Re*.
Concerning the absolute value of these parameters, using the
previous expressions it is easy to verify that for a reasonable
choice of the velocity field that may be produced by a phase
transition, *v*
10^{-3},
both *Re* and *Re*_{M} are much
larger than unity by several orders of magnitude, even for very
small scales (for more details see Chap. 4).
It seems that the early Universe was a quite "turbulent child" !
Turbulence is expected to cease after *e*^{+}
*e*^{-} annihilation since
this process reduces the plasma electron population and therefore
increases the photon diffusion length hence also the kinematic
viscosity. This should happen at a temperature around 1 MeV.

Turbulence is expected to produce substantial modification in the
scaling laws of cosmological magnetic fields. This issue has been
considered by several authors. Brandenburg, Enqvist and Olesen
[51]
first consider MHD in an expanding Universe in the
presence of hydro-magnetic turbulence. MHD equations were written
in a covariant form and solved numerically under some simplifying
assumptions. The magnetic field was assumed to be distributed
randomly either in two or three spatial dimensions. In the latter
case a cascade (shell) model was used to reduce the number of
degree of freedom. In both cases a transfer of magnetic energy
from small to large magnetic configurations
was observed in the simulations.
In hydrodynamics this phenomenon is know as an *inverse cascade*.
Cascade processes are know to be related to certain conservation
properties that the basic equations obey
[52]. In the
two-dimensional inverse cascade, the relevant conserved quantity
is the volume integral of the vector potential squared,
*d*^{2} *x* **A**^{2}, whereas in the
three-dimensional cases it is the
magnetic helicity. It was recently showed by Son
[59] that
no inverse cascade can develop in 3d if the mean value of
*H*
vanishes. If this is the case, i.e. in the presence of non-helical
MHD turbulence, there is still an anomalous grow of the magnetic
correlation length with respect to the scaling given in
Eq. (1.18) but this is just an effect of a *selective
decay* mechanism: modes with larger wavenumbers decay faster than
those whose wavenumbers are smaller. Assuming Universe expansion
is negligible with respect to the decay time, which is given by
the eddy turnover time
_{L} ~ *L* /
*v*_{L}, and the decay of the
large wavenumber modes does not affect those with smaller
wavenumbers, Son found that the correlation length scale with time as

(1.31) |

where _{i} =
*L*_{i} / *v*_{i} is the eddy turnover time at
*t* = 0.
Assuming equipartition of the kinetic and magnetic energies, that
is *v*_{L} ~ *B*_{L}, it follows that the
energy decay with time like
*t*^{-6/5}. When the Universe expansion becomes not negligible,
i.e. when *t* > *t*_{0}, one has to take into
account that the correlation length grows as
^{2/5}, where
is the
conformal time. Since ~
*T*^{-1}, it follows the
*T*^{-2/5} law. In the real situation, the final correlation
length at the present epoch is,

(1.32) |

In the above, the first factor comes from the growth of the
correlation length in the time interval 0 < *t* <
*t*_{0} when eddy
decay is faster than Universe expansion; the second factor comes
from the growth of *L* in the *t* > *t*_{0}
period; the last factor
comes from trivial redshift due to the expansion of the Universe.
T_{nt} is the temperature of the Universe when the fluid
becomes non-turbulent. As we discussed *T*_{nt} ~ 1 MeV.
If, for example, we assume that turbulence was produced at the
electroweak phase transition, so that *T*_{i} =
*T*_{EW} ~ 100 GeV,
that *v*_{i} ~ 0.1 and *L*_{i} ~
10^{-2} *r*_{H}(*T*_{EW}) ~
10^{-2} cm, one find *L*_{0} ~ 100 AU. This result
has to be
compared with the scale one would have if the only mechanism of
dissipation of magnetic energy is resistive diffusion which, as we
got in Eq. (1.22) is ~ 1 AU.

A larger coherence length can be obtained by accounting for
the magnetic helicity which is probably produced during a
primordial phase transition. The conservation of
*H* has an
important consequence for the evolution of the magnetic field.
When *H* is non-vanishing, the
short-scale modes are not
simply washed out during the decay: their magnetic helicity must
be transferred to the long-scale ones. Along with the magnetic
helicity, some magnetic energy is also saved from turbulent decay.
In other words, an inverse cascade is taking place. Assuming
maximal helicity, i.e. that **B** ·
( × **B**) ~
*L B*^{2}, the conservation of this quantity during the
decay of turbulence implies the scaling law

This corresponds to "line averaging", which gives a much larger amplitude of the magnetic field than the usual "volume averaging". Equipartition between magnetic and kinetic energy implies

This relation together with the expression for the eddy decay
time, _{L} =
*L* / *v*_{L}, leads to the following scaling law for the
correlation length of helical magnetic structures

(1.33) |

Comparing this result with Eq. (1.31), we see that in
the helical case the correlation length grow faster than it does
in the turbulent non-helical case. The complete expression for the
scaling of *L* is finally obtained by including trivial redshift
into Eq. (1.33). Since in the radiation dominated era
*T* ^{-1} ~ *a* ~ *t*^{1/2}, we have
[50]

(1.34) |

and

(1.35) |

According to Son,
[50]
helical hydromagnetic turbulence
survives longer than non-helical turbulence allowing *T*_{nt}
to be as low as 100 eV. If again we assume that helical magnetic
turbulence is generated at the electroweak phase transition
(which will be justified in Chap. 4) we find

(1.36) |

which is much larger than the result obtained in the non-helical case. It is worthwhile to observe that,as the scale derived in the previous expression is also considerably larger than the cosmological magnetic diffusion length scale given in Eq. (1.22), that magnetic field fields produced by the EW phase transition may indeed survive until present.

^{2} In the case the average
collision time of the charge carrier is larger than the Universe
age _{U}, the latter
has to be used in place of
in Eq. 1.19
[45].
Back.

^{3} In the
Sec. 2.1 we shall discuss under
which hypothesis such an assumption is consistent with the
presence of a cosmic magnetic field.
Back.