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The first population of X-ray binaries (XRBs) is expected to affect the thermal and ionization states of the gas in the early Universe. Although these X-ray sources are predicted to have important implications for high-redshift observable signals, such as the hydrogen 21-cm signal from cosmic dawn and the cosmic X-ray background, their properties are poorly explored, leaving theoretical models largely uninformed. In this paper we model a population of X-ray binaries arising from zero metallicity stars. We explore how their properties depend on the adopted initial mass function (IMF) of primordial stars, finding a strong effect on their number and X-ray production efficiency. We also present scaling relations between XRBs and their X-ray emission with the local star formation rate, which can be used in sub-grid models in numerical simulations to improve the X-ray feedback prescriptions. Specifically, we find that the uniformity and strength of the X-ray feedback in the intergalactic medium is strongly dependant on the IMF. Bottom-heavy IMFs result in a smoother distribution of XRBs, but have a luminosity orders of magnitude lower than more top-heavy IMFs. Top-heavy IMFs lead to more spatially uneven, albeit strong, X-ray emission. An intermediate IMF has a strong X-ray feedback while sustaining an even emission across the intergalactic medium. These differences in X-ray feedback could be probed in the future with measurements of the cosmic dawn 21-cm line of neutral hydrogen, which offers us a new way of constraining population III IMF. [ABSTRACT FROM AUTHOR]

Properties of the first generation of stars [referred to as the Population III (Pop III) stars], such as their initial mass function (IMF), are poorly constrained by observations and have yet to converge between simulations. The cosmological 21-cm signal of neutral hydrogen is predicted to be sensitive to Lyman-band photons produced by these stars, thus providing a unique way to probe the first stellar population. In this paper, we investigate the impacts of the Pop III IMF on the cosmic-dawn 21-cm signal via the Wouthuysen–Field effect, Lyman–Werner feedback, Ly α heating, and cosmic microwave background heating. We calculate the emission spectra of star-forming haloes for different IMFs by integrating over individual metal-free stellar spectra, computed from a set of stellar evolution histories and stellar atmospheres, and taking into account variability of the spectra with stellar age. Through this study, we therefore relax two common assumptions: that the zero-age main-sequence emission rate of a Pop III star is representative of its lifetime mean emission rate, and that Pop III emission can be treated as instantaneous. Exploring bottom-heavy, top-heavy, and intermediate IMFs, we show that variations in the 21-cm signal are driven by stars lighter than 20 M⊙. For the explored models, we find maximum relative differences of 59 per cent in the cosmic-dawn global 21-cm signal, and 131 per cent between power spectra. Although this impact is modest, precise modelling of the first stars and their evolution is necessary for accurate prediction and interpretation of the 21-cm signal. [ABSTRACT FROM AUTHOR]