**2.3. Dissipative effects on the MHD modes**

In the previous section we neglected any dissipative effect which may
possibly affect the evolution of the MHD modes. However, similar to the
damping
of baryon-photon sound waves by photon shear viscosity and heat
conductivity,
damping of MHD perturbations may also occur. This issue was studied in
detail by Jedamzik, Katalinic and Olinto
[72]
who first
determined the damping rates of fast and slow magnetosonic waves as well as
of Alfvén waves. Furthermore, it was shown in Refs.
[72,
73]
that dissipation of MHD modes produce an effective damping of inhomogeneous
magnetic fields. The dissipation process goes as follows. A spatially
tangled magnetic field produces Lorentz forces which accelerate the
plasma and set up
oscillations. Since the radiation-baryon pressure is much larger than the
magnetic pressure, as long as the photon mean-free-path is smaller than
the scale of the magnetic tangle, the motions can be
considered as being largely incompressible. In this situation mainly
Alfvén waves, which do not involve density fluctuations, are excited.
In the absence of dissipation, this process will continue until, for all
scales with magnetic
field relaxation time
~
/
*v*_{A} shorter than the Hubble time *t*_{H}, an
approximate equipartition between magnetic and kinetic energies is
produced. If the fluid is non-ideal, however, shear viscosity
will induce dissipation of kinetic energy, hence also of magnetic
energy, into
heat. In this case dissipation will end only when the magnetic field
reaches a force-free state.

In the absence of magnetic fields it is known that in the diffusive regime
(i.e. when the perturbation wavelength is much larger than the mean free
path
of photon or neutrinos) acoustic density fluctuations are effectively
damped because of the finite viscosity and heat conductivity
(Silk damping [74]).
At recombination time, dissipation occurs for modes smaller
than the approximate photon diffusion length,
d_{}
~ (*l*_{} *t*_{H})^{1/2}, where
*l*_{} is photon mean free path.
The dissipation of fast magnetosonic waves proceeds in a quite similar way.
Indeed, it was showed in Ref.
[72]
that the dissipation length scale
of these kind of waves coincide with the Silk damping scale.
More interesting is the result found in Refs.
[72,
73] which
shows that damping of Alfvén and slow magnetosonic waves is
significantly different from damping of sound and fast magnetosonic waves.
The reason for such a different behavior is that, for a small background
magnetic field *v*_{A} << 1 so that the oscillation
frequency of an Alfvén
mode (*v*_{A} *k / a*) is much smaller than the
oscillation frequency of a fast
magnetosonic mode with the same wavelength (*v*_{sound}
*k* / *a*). While all magnetosonic modes of
interest satisfy the condition for damping in the oscillatory regime
(*v*_{sound} <<
*l*_{} *k / a*), an Alfvén mode can become
*overdamped* when the photon (or neutrino) mean-free-path becomes
large enough for dissipative effects to overcome the oscillations
(*v*_{A} cos
*l*_{}(*T*) *k / a*, where
is the angle between
the background magnetic field and the wave vector). Because of the
strong viscosity, that
prevent fluid acceleration by the magnetic forces, damping is quite
inefficient for non-oscillating overdamped Alfvén modes with

(2.43) |

As a result, the damping scale of overdamped Alfvén modes at the end of
the diffusion regime is smaller than the damping scale of sound and
fast magnetosonic modes (Silk damping scale) by a factor which depends
on the strength of the background magnetic field and the
angle,
*L*_{A} ~ *v*_{A}
cos
*d*_{}.

From the previous considerations it follows that
the results discussed in the previous section hold only under the
assumption that the magnetic field coherence length is not much smaller
than the comoving Silk damping scale (*L*_{S} ~ 10 Mpc), in
the case of fast magnetosonic waves, and not smaller than
*L*_{A} for Alfvén waves.

Some other interesting work has been recently done by Jedamzik, Katalinic and Olinto [75] concerning the effects of dissipation of small-scale magnetic fields on the CMBR. The main idea developed in the paper by Jedamzik et al. is that the dissipation of tangled magnetic fields before the recombination epoch should give rise to a nonthermal injection of energy into the heat-bath which may distort the thermal spectrum of CMBR. It was showed by the authors of Ref. [75] that once photon equilibration has occurred, mainly via photon-electron scattering and double-Compton scattering, the resultant distribution should be of Bose-Einstein type with a non-vanishing chemical potential. The evolution of the chemical potential distortions at large frequencies may be well approximated by [76]

(2.44) |

where, in our case, *Q*_{B} =
*d*_{B} / *dt* is the dissipation rate of the
magnetic field and *t*_{DC} = 2.06 × 10^{33} s
(_{b}
*h*^{2})^{-1} *z*^{-9/2}
is a characteristic time scale for double-Compton scattering.
Jedamzik et al. assumed a statistically isotropic magnetic field
configuration with the following power spectrum

(2.45) |

and zero otherwise, normalized such that
<^{2}>
= *B*_{0}^{2}.
The energy dissipation rate was determined by substituting this spectrum in
the following Fourier integral

(2.46) |

together with the mode frequencies for Alfvén and slow magnetosonic waves determined in Ref. ([72])

(2.47) |

where 3(_{}
+ *p*_{})'
= , and
is the shear
viscosity. For *k*_{N} >>
*k*_{D}^{0}
*z*_{µ}^{3/2}, where
*k*_{D}^{0} = (15*n*_{e}^{0}
_{Th} /
2.39 × 10^{1}9 s)^{1/2},
an analytic solution of Eq. (2.44) was then found to be

(2.48) |

In the above *K* is a numerical factor of order 1, the precise
value depending
on the spectral index *n* and *z*_{µ} is
the characteristic redshift for
"freze-out" from double-Compton scattering. This redshift equals
*z*_{µ} = 2.5 × 10^{6} for
typical values
_{b}
*h*^{2} = 0.0125, and *Y*_{p} = 0.24.
The scale *k*_{D}^{0}
*z*_{µ}^{3/2} has a simple
interpretation. It is the
scale which at redshift *z*_{µ} is damped by
one e-fold. For the above values of
_{b}
*h*^{2} and Y_{p} the corresponding comoving
wavelength is
_{D} =
(2) /
(*k*_{D}^{0}
*z*_{µ}^{3/2}) = 395 pc.

The present upper limit on chemical potential distortion of the CMBR come
from the COBE/FIRAS data: |*µ*| < 9 × 10^{-5} at
95% confidence level
[77].
Comparing this limit with the prediction of Eq. (2.48)
it follows that primordial magnetic fields of strength
3 ×
10^{-8} G, and comoving
coherence length 400
pc are probably excluded.
On slightly larger scales, dissipation of spatially tangled magnetic
fields may
give to a different kind of CMBR distortion which may be described by a
superposition of blackbodies of different temperature, i.e. a Compton
*y* distortion
[78].
The absence of this kind of distortions in the
observed CMBR thermal spectrum disallow magnetic fields of
3 × 10^{-8} Gauss on scales ~ 0.6 Mpc.