  We calculate the gravitational waves (GW) spectrum produced in various Early
Universe scenarios from gauge field sources, thus generalizing earlier
inflationary calculations to bouncing cosmologies. We consider generic
couplings between the gauge fields and the scalar field dominating the energy
density of the Universe. We analyze the requirements needed to avoid a
backreaction that will spoil the background evolution. When the scalar is
coupled only to $F \tilde F$ term, the sourced GW spectrum is exponentially
enhanced and parametrically the square of the vacuum fluctuations spectrum,
${\cal P}^s_T\sim ({\cal P}^v_T)^2$, giving an even bluer spectrum than the
standard vacuum one. When the scalar field is also coupled to $F^2$ term, the
amplitude is still exponentially enhanced, but the spectrum can be arbitrarily
close to scale invariant (still slightly blue), $n_T\gtrsim 0$, that is
distinguishable form the slightly red inflationary one. Hence, we have a proof
of concept of observable GW on CMB scales in a bouncing cosmology.
