Efficient computation of the gravitational wave spectrum emitted by eccentric massive black hole binaries in stellar environments

We present a fast and versatile method to calculate the characteristic spectrum hc of the gravitational wave background (GWB) emitted by a population of eccentric massive black hole binaries (MBHBs). We fit the spectrum of a reference MBHB with a simple analytic function and show that the spectrum of any other MBHB can be derived from this reference spectrum via simple scalings of mass, redshift and frequency. We then apply our calculation to a realistic population of MBHBs evolving via three-body scattering of stars in galactic nuclei. We demonstrate that our analytic prescription satisfactorily describes the signal in the frequency band relevant to pulsar timing array (PTA) observations. Finally, we model the high-frequency steepening of the GWB to provide a complete description of the features characterizing the spectrum. For typical stellar distributions observed in massive galaxies, our calculation shows that three-body scattering alone is unlikely to affect the GWB in the PTA band and a low-frequency turnover in the spectrum is caused primarily by high eccentricities.