Cosmological 21cm line observations to test scenarios of super-Eddington accretion on to black holes being seeds of high-redshifted supermassive black holes

In this paper, we study scenarios of the super-Eddington accretion onto black holes at high redshifts $z > 10$, which are expected to be seeds to evolve to supermassive black holes until redshift $z \sim 7$. For an initial mass, $M_{\rm BH, ini} \lesssim 2 \times 10^{3} M_{\odot}$ of a seed BH, we definitely need the super-Eddington accretion, which can be applicable to both astrophysical and primordial origins. Such an accretion disk inevitably emitted high-energy photons which had heated the cosmological plasma of the inter-galactic medium continuously from high redshifts. In this case, the cosmic history of cosmological gas temperature is modified, by which the absorption feature of the cosmological 21 cm lines is suppressed. By comparing theoretical predictions of the 21cm line absorption with the observational data at $z\sim17$, we obtain a cosmological upper bound on the mass-accretion rate as a function of the seed BH masses. In order to realize $M_{\rm BH} \sim 10^9 M_{\odot}$ at $z \sim 7$ by a continuous mass-accretion on to a seed BH, to be consistent with the cosmological 21cm line absorption at $z \sim 17$, we obtained an severe upper bound on the initial mass of the seed BH to be $M_{\rm BH, ini} \lesssim 10^2 M_{\odot}$ ($M_{\rm BH, ini} \lesssim 10^6 M_{\odot}$) when we assume a seed BH with its comoving number density $n_{\rm seed,0} \sim 10^{-3} {\rm Mpc}^{-3}$ ($n_{\rm seed,0} \sim 10^{-7} {\rm Mpc}^{-3}$). We also discuss some implications for application to primordial black holes as the seed black holes.