**8.3. Relic gravitational waves**

Magnetic fields can source the evolution equations of the fluctuations of the geometry also over length-scales much smaller than the ones where CMB anisotropy experiments are conduced. This suggests that magnetic inhomogeneities may leave an imprint on the relc background of gravitational waves.

If a hypermagnetic background is present for *T* >
*T*_{c}, then, as discussed
in Section 5 and 6,
the energy momentum tensor will acquire a small anisotropic component
which will source the evolution equation of the tensor fluctuations of
the metric. Suppose now, that
|_{Y}|
has constant amplitude and that it is also homogeneous. Then as argued in
[323]
we can easily deduce the critical fraction of energy density present
today in relic gravitons of EW origin

(8.32) |

(*z*_{eq} is the redshift from the time of matter-radiation,
equality). Because of the structure of the AMHD equations, stable
hypermagnetic fields will be present not only for
_{ew} ~
*k*_{ew} / *a* but for all the range
_{ew} <
<
_{} where
_{} is the diffusivity
frequency. Let us assume, for instance, that
*T*_{c} ~ 100 GeV and *g*_{*} = 106.75.
Then, the (present) values of
_{ew} is

(8.33) |

Thus, _{}(*t*_{0})
~ 10^{8}
_{ew}. Suppose
now that *T*_{c} ~ 100 GeV; than we will have that
_{ew}(*t*_{0}) ~ 10^{-5} Hz.
Suppose now, that

(8.34) |

as, for instance, implied by the analysis of the electroweak
phase diagram in the presence of a magnetized background.
This requirement imposes
*r* 0.1-0.001 and,
consequently,

(8.35) |

Notice that this signal would occurr in a (present) frequency
range between 10^{-5} and 10^{3} Hz. This signal
satisfies the presently available phenomenological
bounds on the graviton backgrounds of primordial origin.
The pulsar timing bound ( which applies for present frequencies
_{P} ~
10^{-8} Hz and implies *h*_{0}^{2}
_{GW}
10^{-8}) is
automatically satisfied since our hypermagnetic background is defined
for 10^{-5} Hz
10^{3} Hz. The
large scale bounds would imply *h*_{0}^{2}
_{GW} < 7
× 10^{-11} but a at much lower frequency
(i.e. 10^{-18} Hz). The signal discussed here is completely
absent for frequencies <
_{ew}. Notice that
this signal is clearly distinguishable from other stochastic
backgrounds occurring at much higher frequencies (GHz region)
like the ones predicted by quintessential inflation
[324,
325,
326].
It is equally distinguishable from signals due to
pre-big-bang cosmology (mainly in the window of
ground based interferometers
[327]).
The frequency of operation of the interferometric devices
(VIRGO/LIGO) is located between few Hz and 10 kHz
[327].
The frequency of operation of LISA is well below the Hz
(i.e. 10^{-3}Hz, approximately). In this model the signal
can be located both in the LISA window and in the VIRGO/LIGO window
due to the hierarchy between the hypermagnetic diffusivity scale and the
horizon scale at the phase transition.

In Fig. 13 the full thick line illustrates the
spectrum of relic gravitational waves
produced in a conventional model for the evolution of the universe. The
flat plateau corresponds to modes which left the horizon during the
inflationary stage of expansion and re-entered duriing the radiation
dominated phase. The
decreasing slope between 10^{-16} and 10^{-18} Hz is
due to modes leaving the horizon during inflation and re-entering
during the maatter dominated stage of expansion. Clearly, the signal
provided by a background of hypermagnetic fields can be even 7 order of
magnitude larger than the inflationary prediction.
The interplay between gravitational waves and large-scale
magnetic fields has been also the subject of recent interesting
investigations
[329,
330,
331].