Population synthesis of ultracompact X-ray binaries in the Galactic Bulge

[abridged] Aims. We model the number and properties of ultracompact X-ray binaries (UCXBs) in the Galactic Bulge. The objective is to learn about the formation of UCXBs and their evolution, such as the onset of mass transfer and late-time behavior. Methods. The binary population synthesis code SeBa and stellar evolutionary tracks are used to model the UCXB population. The luminosity behavior of UCXBs is predicted using long-term X-ray observations of the known UCXBs and the thermal-viscous disk instability model. Results. In our model, the majority of UCXBs initially have a helium burning star donor. In the absence of a mechanism that destroys old UCXBs, we predict (0.2 - 1.9) x 10^5 UCXBs in the Galactic Bulge, mostly at orbital periods longer than 60 min (a large number of long-period systems also follows from the observed short-period UCXB population). About 5 - 50 UCXBs should be brighter than 10^35 erg/s, mostly persistent sources with orbital periods shorter than 30 min and with degenerate helium and carbon-oxygen donors. This is one order of magnitude more than the observed number of (probably) three. Conclusions. This overprediction of short-period UCXBs implies that fewer systems are formed, or that a super-Eddington mass transfer rate is more difficult to survive than assumed. The very small number of observed long-period UCXBs with respect to short-period UCXBs, the surprisingly high luminosity of the observed UCXBs with orbital periods around 50 min, and the properties of the PSR J1719-1438 system all point to much faster UCXB evolution than expected from angular momentum loss via gravitational wave radiation alone. Old UCXBs, if they still exist, probably have orbital periods longer than 2 h and have become very faint due to either reduced accretion or quiescence, or have become detached. UCXBs are promising candidate progenitors of isolated millisecond radio pulsars.