• Black holes perturbation theory
• cosmology
• Gravitational waves
• stochastic background of gravitational waves

The detection of gravitational waves by the interferometers LIGO and Virgo, has opened a golden age of the physics of gravitational waves (GW).
The next challenge, for which the space interferometer LISA is designed, is the detection of the cosmological stochastic background of GW.
The cosmological background of GW is generated by the amplification of quantum fluctuations of the metric during the epoch of the inflation.
Primordial GW are not expected in non-inflationary models, making them a smoking-gun probe of Inflation.
Moreover, since the inflationary GW background is generated at energy scales
which are many orders of magnitude above those achievable by particle accelerators, its detection could
open a window on new high energy physics.
Black holes (BHs) interact with GW either absorbing or reflecting depending on the ratio between the incoming wave frequency and the intrinsic characteristic frequencies that depend on the BH mass and spin.
It is thus plausible that a cosmological background could interact with the astrophysical
population of BHs and thus its spectral characteristics of the would be modified in a way
that reflects the cosmological population of BHs.

In the first part of this thesis, we investigate the problem of the scattering of a plane and monochromatic GW on a non-rotating BH.
Since the Schwarzchild BH has spherical symmetry, the natural way to study its perturbation is to decompose the wave in eigenstates
of the angular momentum: the tensorial spherical harmonics. Using a fourth order Runge-Kutta numerical method, we find that the BH absorbs a portion of the GW at fixed angular number l with an absorption coefficient that depends on the frequency of GW.

In the second part of this thesis, we use our results to find out how the GW background is modified by the interaction with the astrophysical population of BHs. We conjecture a population of BHs constant in the comoving volume and with various ranges of mass chosen:
stellar of 10 solar mass, supermassive of 10^6 solar mass and primordial black holes of 10 solar mass.
We further estimate the interaction cross section equal to the area of the light ring r=3M, and
we compute the fraction of GW absorbed by BH at each l.

∆ Ω_gw/Ω_gw ~ 10^{-24} for stellar black holes
∆ Ω_gw/Ω_gw ~ 10^{-14} for supermassive black holes
∆ Ω_gw/Ω_gw ~ 10^{-18} for stellar black holes
We conclude by noting that, even if the effect of the astrophysical BH population is very small,
because the absorption coefficients depends on the angular momentum, we expect the emergence of correlations in the cosmological GW spectrum
coming from different positions in the sky.