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, (t₀) ~ 10⁸ _ew. Suppose now that T_c ~ 100 GeV; than we will have that _ew(t₀) ~ 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 10³ 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 h₀² _GW 10^-8) is automatically satisfied since our hypermagnetic background is defined for 10^-5 Hz 10³ Hz. The large scale bounds would imply h₀² _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^-3Hz, 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].

Figure 13. The stochastic background of GW produced by inflationary models with flat logarithmic energy spectrum, illustrated together with the GW background of hypermagnetic origin. The frequencies marked with dashed lines correspond to the electroweak frequency and to the hypermagnetic diffusivity frequency.