  By comparing the orbital period distributions of black hole and neutron star
low mass X-ray binaries (LMXBs) in the Ritter-Kolb catalogue we show that there
is statistical evidence for a dearth of black hole systems at short orbital
periods (P_orb < 4h). This could either be due to a true divergence in orbital
period distributions of these two types of system, or to black hole LMXBs being
preferentially hidden from view at short orbital periods. We explore the latter
possibility, by investigating whether black hole LMXBs could be concealed by a
switch to radiatively inefficient accretion at low luminosities. The peak
luminosity and the duration of X-ray binary outbursts are related to the disc
radius and, hence, the orbital period. At short periods, where the peak
outburst luminosity drops close to the threshold for radiatively inefficient
accretion, black hole LMXBs have lower outburst luminosities, shorter outburst
durations and lower X-ray duty cycles than comparable neutron star systems.
These factors can combine to severely reduce the detection probability of short
period black hole LMXBs relative to those containing neutron stars. We estimate
the outburst properties and orbital period distribution of black hole LMXBs
using two models of the transition to radiatively inefficient accretion: an
instantaneous drop in accretion efficiency (eta) to zero, at a fraction (f) of
the Eddington luminosity (L_Edd) and a power-law efficiency decrease, eta
\propto \dot{M}^n, for L < f*L_Edd. We show that a population of black hole
LMXBs at short orbital periods can only be hidden by a sharp drop in
efficiency, either instantaneous or for n >= 3. This could be achieved by a
genuine drop in luminosity or through abrupt spectral changes that shift the
accretion power out of a given X-ray band.
