EFFECTS ON THE COSMIC MICROWAVE BACKGROUND

**2.1. The effect of a homogeneous magnetic field**

It is well know from General Relativity that electromagnetic fields can affect the geometry of the Universe. The energy momentum tensor

(2.1) |

where
*F*^{µ}
is the electromagnetic field tensor, acts as a
source term in the Einstein equations. In the case of a
homogeneous magnetic field directed along the *z*-axis

(2.2) |

Clearly, the energy-momentum tensor becomes anisotropic due to the presence
of the magnetic field. There is a
positive pressure term along the *x* and *y*-axes but a "negative
pressure" along the field direction. It is known that an isotropic
positive pressure increases the deceleration of the universe
expansion while a negative pressure tends to produce an
acceleration. As a consequence, an anisotropic pressure must give
rise to an anisotropy expansion law
[53].

Cosmological models with a homogeneous magnetic field have been considered by several authors (see e.g. [54]). To discuss such models it is beyond the purposes of this review. Rather, we are more interested here in the possible signature that the peculiar properties of the space-time in the presence of a cosmic magnetic field may leave on the Cosmic Microwave Background Radiation (CMBR).

Following Zeldovich and Novikov [53] we shall consider the most general axially symmetric model with the metric

(2.3) |

It is convenient to define
=
/ *a*;
=
/ *b*; and

(2.4) |

Then, assuming *r*,
< 1, the Einstein
equations are well approximated by

(2.5) | |

(2.6) |

where *H* = (2 +
) and
are
defined by the equation of state *p* =
( - 1)
. It is easy to
infer from
the first of the previous equations that the magnetic field acts so as to
conserve the anisotropy that would otherwise decay with time
in the case *r* = 0. By substituting the asymptotic value of the
anisotropy, i.e.
6*r*,
into the evolution equation for *r* in the RD era
one finds

(2.7) |

where *q* is a constant.
Therefore, in the case the cosmic magnetic field is homogeneous, the
ratio of the magnetic and blackbody radiation densities is not a constant,
but falls logarithmically during the radiation era.

In order to determine the temperature anisotropy of the CMBR we
assume that at the recombination time *t*_{rec} the
temperature is everywhere *T*_{rec}. Then, at the present
time, *t*_{0}, the temperature of relic photons coming
from the *x* (or *y*) and *z* directions will be respectively

(2.8) |

Consequently, the expected temperature anisotropy is

(2.9) |

By using this expression, Zeldovich and Novikov estimated that a
cosmological magnetic field having today the strength of
10^{-9} ÷ 10^{-10} Gauss would produce a
temperature anisotropy
*T / T*
10^{-6}.

The previous analysis has been recently updated by Barrow,
Ferreira and Silk [55].
In that work the authors derived an upper
limit on the strength of a homogeneous magnetic field at the recombination
time on the basis of the 4-year Cosmic Background Explorer (COBE)
microwave background isotropy measurements
[56]. As it is
well know COBE detected quadrupole anisotropies at a level
*T / T* ~
10^{-5} at an angular scale of few degrees.
By performing a suitable statistical average of the data and
assuming that the field keeps frozen-in since the recombination
till today, Barrow at al. obtained the limit

(2.10) |

In the above *f* is a *O*(1) shape factor accounting for possible
non-Gaussian characteristics of the COBE data set.

From this results we see that COBE data are not incompatible with a primordial origin of the galactic magnetic field even without invoking a dynamo amplification.