The successive discoveries of binary merger events by Advanced LIGO-Virgo have been revealing the statistical properties of binary black hole (BBH) populations. A stochastic gravitational wave background (GWB) is a useful tool to probe the cosmological evolution of those compact mergers. In this paper, we study the upper bound on a GWB produced by BBH mergers, whose stellar progenitors dominate the reionization process at the cosmic dawn. Since early reionization by those progenitors yields a high optical depth of the universe inconsistent with the {it Planck} measurements, the cumulative mass density is limited to $rho_star lesssim 10^7~M_odot~{rm Mpc}^{-3}$. Even with this upper bound, the amplitude of a GWB owing to the high-$z$ BBH mergers is expected to be as high as $Omega_{rm gw}simeq 1.48_{-1.27}^{+1.80}times 10^{-9}$ at $fsimeq 25$ Hz, while their merger rate at the present-day is consistent or lower than the observed GW event rate. This level of GWB is detectable at the design sensitivity of Advanced LIGO-Virgo and would indicate a major contribution of the high-$z$ BBH population to the local GW events. The spectral index is expected to be substantially flatter than the canonical value of $simeq 2/3$ generically produced by lower-redshift and less massive BBHs. Moreover, if their mass function is more top-heavy than in the local universe, the GWB spectrum is even more skewed toward lower frequencies, which would allow us to extract information on the mass function of merging BBHs at high redshifts.
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