  Rapidly rotating Neutron Stars (NSs) in Low Mass X-ray Binaries (LMXBs) are
thought to be interesting sources of Gravitational Waves (GWs) for current and
next generation ground based detectors, such as Advanced LIGO and the Einstein
Telescope. The main reason is that many of the NS in these systems appear to be
spinning well below their Keplerian breakup frequency, and it has been
suggested that torques associated with GW emission may be setting the observed
spin period. This assumption has been used extensively in the literature to
assess the strength of the likely gravitational wave signal. There is now,
however, a significant amount of theoretical and observation work that suggests
that this may not be the case, and that GW emission is unlikely to be setting
the spin equilibrium period in many systems. In this paper we take a different
starting point and predict the GW signal strength for two physical mechanisms
that are likely to be at work in LMXBs: crustal mountains due to thermal
asymmetries and magnetically confined mountains. We find that thermal crustal
mountains in transient LMXBs are unlikely to lead to detectable GW emission,
while persistent systems are good candidates for detection by Advanced LIGO and
by the Einstein Telescope. Detection prospects are pessimistic for the magnetic
mountain case, unless the NS has a buried magnetic field of $B\approx 10^{12}$
G, well above the typically inferred exterior dipole fields of these objects.
Nevertheless, if a system were to be detected by a GW observatory, cyclotron
resonant scattering features in the X-ray emission could be used to distinguish
between the two different scenarios.
