New gravitational wave probe of vector dark matter

The longitudinal components of massive vector fields generated during inflation constitute a well-motivated dark matter candidate, with interesting phenomenological implications. During the epoch of radiation domination following inflation, their spectrum exhibits a peak at small scales, whose amplitude and position are governed by the parameters of the dark matter model. We calculate the stochastic gravitational wave spectrum induced at second order in fluctuations by such a longitudinal vector peak. We demonstrate that the amplitude of the gravitational wave spectrum can, in principle, reach significant values at nano-Hertz frequencies or lower. This result suggests a novel gravitational wave probe to test inflationary vector dark matter scenarios, independent from assumptions on the coupling of dark vectors to the Standard Model. Additionally, we derive new analytical formulas for the longitudinal vector transfer functions during radiation domination, offering a valuable tool for characterising the convolution integrals that govern the properties of the induced gravitational waves.