Gravitational wave background from binary systems
  Basic aspects of the background of gravitational waves and its mathematical
characterization are reviewed. The spectral energy density parameter
$\Omega(f)$, commonly used as a quantifier of the background, is derived for an
ensemble of many identical sources emitting at different times and locations.
For such an ensemble, $\Omega(f)$ is generalized to account for the duration of
the signals and of the observation, so that one can distinguish the resolvable
and unresolvable parts of the background. The unresolvable part, often called
confusion noise or stochastic background, is made by signals that cannot be
either individually identified or subtracted out of the data. To account for
the resolvability of the background, the overlap function is introduced. This
function is a generalization of the duty cycle, which has been commonly used in
the literature, in some cases leading to incorrect results. The spectra
produced by binary systems (stellar binaries and massive black hole binaries)
are presented over the frequencies of all existing and planned detectors. A
semi-analytical formula for $\Omega(f)$ is derived in the case of stellar
binaries (containing white dwarfs, neutron stars or stellar-mass black holes).
Besides a realistic expectation of the level of background, upper and lower
limits are given, to account for the uncertainties in some astrophysical
parameters such as binary coalescence rates. One interesting result concerns
all current and planned ground-based detectors (including the Einstein
Telescope). In their frequency range, the background of binaries is resolvable
and only sporadically present. In other words, there is no stochastic
background of binaries for ground-based detectors.
