**8.2. Faraday rotation of CMB**

Large scale magnetic fields present at the decoupling epoch can also depolarize CMB [313]. The polarization of the CMB represents a very interesting observable which has been extensively investigated in the past both from the theoretical [314] and experimental points of view [315, 316]. Forthcoming satellite missions like PLANCK [317] seem to be able to achieve a level of sensitivity which will enrich decisively our experimental knowledge of the CMB polarization with new direct measurements.

If the background geometry of the universe is homogeneous but not isotropic the CMB is naturally polarized [314]. This phenomenon occurs, for example, in Bianchi-type I models [318]. On the other hand if the background geometry is homogeneous and isotropic (like in the Friedmann-Robertson-Walker case) it seems very reasonable that the CMB acquires a small degree of linear polarization provided the radiation field has a non-vanishing quadrupole component at the moment of last scattering [319, 320].

Before decoupling photons, baryons and electrons form a unique fluid which possesses only monopole and dipole moments, but not quadrupole. Needless to say, in a homogeneous and isotropic model of FRW type a possible source of linear polarization for the CMB becomes efficient only at the decoupling and therefore a small degree of linear polarization seems a firmly established theoretical option which will be (hopefully) subjected to direct tests in the near future. The linear polarization of the CMB is a very promising laboratory in order to directly probe the speculated existence of a large scale magnetic field (coherent over the horizon size at the decoupling) which might actually rotate (through the Faraday effect) the polarization plane of the CMB.

Consider, for
instance, a linearly polarized electromagnetic wave of physical
frequency
travelling along the
direction in a plasma of ions and electrons together with a magnetic
field ()
oriented along an arbitrary direction ( which might coincide with
in the simplest case).
If we let the polarization vector at the origin (*x* = *y* =
*z* = 0, *t* = 0) be directed along the
axis,
after the wave has traveled a length
*x*, the
corresponding angular shift
(
) in the polarization
plane will be :

(8.30) |

From Eq. (8.30)
by stochastically averaging over all the possible orientations
of and
by assuming that the last scattering surface is infinitely thin
(i.e. that
*xf*_{e}
*n*_{e}
_{T}^{-1}
where _{T} is the
Thompson cross section) we get an expression connecting the RMS of the
rotation angle to the magnitude of
at
*t*
*t*_{dec}

(8.31) |

(in the previous equation we implicitly assumed that the frequency of
the incident electro-magnetic radiation is centered around the maximum
of the CMB). We can easily argue from Eq. (8.31) that if
*B*(*t*_{dec})
*B*_{c} the
expected rotation in the polarization plane of the CMB is non negligible.
Even if we are not interested, at this level, in a precise estimate of
, we point out that more
refined determinations of the expected Faraday rotation signal (for an
incident frequency
_{M} ~ 30 GHz)
were recently carried out
[321,
322]
leading to a result fairly consistent with (8.30).

Then, the statement is the following. *If* the CMB is linearly
polarized and *if* a large scale magnetic field is
present at the decoupling epoch, *then* the polarization plane of
the CMB can be rotated
[313].
The predictions of different models can then be confronted
with the requirements coming from a possible detection of depolarization
of the CMB
[313].