Gravitational wave background from Population III binary black holes
  consistent with cosmic reionization
  The recent discovery of the gravitational wave source GW150914 has revealed a
coalescing binary black hole (BBH) with masses of $\sim 30~M_\odot$. Previous
proposals for the origin of such a massive binary include Population III
(PopIII) stars. PopIII stars are efficient producers of BBHs and of a
gravitational wave background (GWB) in the $10-100$ Hz band, and also of
ionizing radiation in the early Universe. We quantify the relation between the
amplitude of the GWB ($\Omega_{\rm gw}$) and the electron scattering optical
depth ($\tau_{\rm e}$), produced by PopIII stars, assuming that $f_{\rm
esc}\approx 10\%$ of their ionizing radiation escapes into the intergalactic
medium. We find that PopIII stars would produce a GWB that is detectable by the
future O5 LIGO/Virgo if $\tau_{\rm e} \gtrsim 0.07$, consistent with the recent
Planck measurement of $\tau_e=0.055 \pm 0.09$. Moreover, the spectral index of
the background from PopIII BBHs becomes as small as ${\rm d}\ln \Omega_{\rm
gw}/{\rm d}\ln f\lesssim 0.3$ at $f \gtrsim 30$ Hz, which is significantly
flatter than the value $\sim 2/3$ generically produced by lower-redshift and
less-massive BBHs. A detection of the unique flattening at such low frequencies
by the O5 LIGO/Virgo will indicate the existence of a high-chirp mass,
high-redshift BBH population, which is consistent with the PopIII origin. A
precise characterization of the spectral shape near $30-50$ Hz by the Einstein
Telescope could also constrain the PopIII initial mass function and star
formation rate.
