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 ${rm d}ln Omega_{rm gw}/{rm d}ln flesssim 0.3$ 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.
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