  Spinning bosonic stars (SBSs) can form from the gravitational collapse of a
dilute cloud of scalar/Proca particles with non-zero angular momentum. In a
recent work we found that the scalar stars are transient due to a
non-axisymmetric instability which triggers the loss of angular momentum. We
further study the dynamical formation of SBSs using 3-dimensional
numerical-relativity simulations of the Einstein-(massive, complex)Klein-Gordon
system and of the Einstein-(complex)Proca system. We incorporate a quartic
self-interaction potential in the scalar case to gauge its effect on the
instability; we investigate (m=2) Proca stars to assess their stability; we
attempt to relate the instability of SBSs to the growth rate of azimuthal
density modes and the existence of a corotation point. We show that: the
self-interaction potential can only delay the instability in scalar SBSs; m=2
Proca stars always migrate to the stable m=1 spheroidal family; unstable m=2
Proca stars and m=1 scalar boson stars exhibit a corotation point. This
establishes a parallelism with rotating neutron stars affected by dynamical
bar-mode instabilities. We compute the gravitational waves (GWs) emitted and
investigate the detectability of the waveforms comparing the characteristic
strain of the signal with the sensitivity curves of a variety of detectors,
computing the signal-to-noise ratio. By assuming that the characteristic
damping timescale of the bar-like deformation in SBSs is only set by GWs
emission and not by viscosity (unlike in neutron stars), we find that the
post-collapse emission could be orders of magnitude more energetic than that of
the bar-mode instability itself. Our results indicate that GW observations of
SBSs might be within the reach of future experiments, offering a potential
means to establish the existence of such stars and to place tight constraints
on the mass of the bosonic particle.
