The merger rate of extremely low mass white dwarf binaries: links to the formation of AM CVn stars and underluminous supernovae

We study a complete, colour-selected sample of double-degenerate binary systems containing extremely low mass (ELM) <= 0.25M(circle dot) white dwarfs (WDs). We show, for the first time, that Milky Way disc ELM WDs have a merger rate of approximately 4 x 10(-5) yr(-1) due to gravitational wave radiation. The merger end product depends on the mass ratio of the binary. The ELM WD systems that undergo stable mass transfer can account for greater than or similar to 3 per cent of AM Canum Venaticorum (AM CVn) stars. More importantly, the ELM WD systems that may detonate merge at a rate comparable to the estimated rate of underluminous supernovae (SNe), rare explosions estimated to produce only similar to 0.2M(circle dot) worth of ejecta. At least 25 per cent of our ELM WD sample belong to the old thick disc and halo components of the Milky Way. Thus, if merging ELM WD systems are the progenitors of underluminous SNe, transient surveys must find them in both elliptical and spiral galaxies.