Gravitational wave background from sub-luminous GRBs: prospects for second- and third-generation detectors

We assess the detection prospects of a gravitational wave background associated with sub-luminous gamma-ray bursts (SL-GRBs). We assume that the central engines of a significant proportion of these bursts are provided by newly born magnetars and consider two plausible GW emission mechanisms. First, the deformation-induced triaxial GW emission from a newly born magnetar. Secondly, the onset of a secular bar-mode instability, associated with the long-lived plateau observed in the X-ray afterglows of many gamma-ray bursts. With regards to detectability, we find that the onset of a secular instability is the most optimistic scenario: under the hypothesis that SL-GRBs associated with secularly unstable magnetars occur at a rate of (48-80) Gpc(-3) yr(-1) or greater, cross-correlation of data from two Einstein Telescopes (ETs) could detect the GW background associated to this signal with a signal-to-noise ratio of 3 or greater after 1 year of observation. Assuming neutron star spindown results purely from triaxial GW emissions, we find that rates of around (130-350) Gpc(-3) yr(-1) will be required by ET to detect the resulting GW background. We show that a background signal from secular instabilities could potentially mask a primordial GW background signal in the frequency range where ET is most sensitive. Finally, we show how accounting for cosmic metallicity evolution can increase the predicted signal-to-noise ratio for background signals associated with SL-GRBs.