  We study the primordial spectra and the gravitational-wave background (GWB)
of three models of semi-classical, quantum or string gravity where the big bang
is replaced by a bounce and the tensor spectrum is blue-tilted: ekpyrotic
universe with fast-rolling Galileons, string-gas cosmology with Atick-Witten
conjecture and pre-big-bang cosmology. We find that the ekpyrotic scenario does
not produce a GWB amplitude detectable by present or third-generation
interferometers, while the string-gas model is ruled out for producing too
large a signal. In contrast, the GWB of the pre-big-bang scenario falls within
the sensitivity window of both LISA and Einstein Telescope, where it takes the
form of a single or a broken power law depending on the choice of parameters.
The latter will be tightly constrained by both detectors.
