  We study detection prospects of a gravitational-wave background (GWB) sourced
by SU(2) gauge fields considering all possible observational constraints. More
precisely, we consider bounds set by cosmic microwave background measurements,
primordial black hole overproduction, as well as backreaction of the gauge
fields on the background evolution. Gravitational-waves data from the first
three observing runs of the LIGO-Virgo-KAGRA Collaboration show no evidence for
a GWB contribution from axion inflation. However, we are able to place
conservative constraints on the parameters of the SU(2) inflation with current
data. We investigate conditions on the inflationary potential that would lead
to a detectable signal that evades astrophysical and cosmological constraints
and discuss detection prospects for third generation networks.
