of ionized bubbles. This is based on the assumption 5,6,7 that the cosmic gas was heated by stellar remnants -particularly X-ray binaries -to temperatures well above the cosmic microwave background at that time (∼30 K). Here we show instead that the hard spectra (that is, spectra with more high-energy photons than low-energy photons) of X-ray binaries 8,9 make such heating ineffective, resulting in a delayed and spatially uniform heating 1 that modifies the 21-cm signature of reionization. Rather than looking for a simple rise and fall of the large-scale fluctuations (peaking at several millikelvin), we must expect a more complex signal also featuring a distinct minimum (at less than a millikelvin) that marks the rise of the cosmic mean gas temperature above the microwave background.Observing this signal, possibly with radio telescopes in operation today, will demonstrate the presence of a cosmic background of hard X-rays at that early time.While stellar remnants at high redshift have been previously considered, a more reliable prediction of the radiative feedback from X-ray binaries (XRBs) is now possible due to a recent detailed population synthesis simulation of their evolution across cosmic time 8,9 . This simulation was calibrated to all available observations in the local and low redshift Universe, and it predicts the evolution of the luminosity and X-ray spectrum of XRBs with redshift. In particular, high-mass XRBs (especially black hole binaries) should dominate, with a ratio at high redshift of bolometric X-ray luminosity to star-formation rate (SFR) ofWe have allowed for an uncertainty in the X-ray efficiency with an extra parameter in eq. 1, where f X = 1 indicates our standard value. We focus on XRBs as the most natural heating source, since other observed sources should be sub-dominant at high redshift (see Methods section).Previous calculations of X-ray heating 6,10,11,12,13 have assumed power-law spectra that place most of the X-ray energy at the low-energy end, where the mean free path of the soft X-rays is relatively short. This means that most of the emitted X-rays are absorbed soon after they are emitted, before much energy is lost due to cosmological effects. The absorbed energy is then enough to heat the gas by the time of reionization to ∼ 10 times the temperature of the Cosmic Microwave Background (CMB; see Methods section). Thus, it is generally assumed 2 that reionization occurs when T gas ≫ T CMB , a limit referred to as saturated heating since the 21-cm intensity then becomes independent of T gas and mainly dependent on ionization and density. A different possibility whereby heating is delayed until reionization has only been previously considered as a fringe case of having an unusually low X-ray luminosity to SFR ratio 10,11 .However, the average radiation from XRBs is expected to have a much harder spectrum ( Fig. 1) whose energy content (per logarithmic frequency interval) peaks at ∼ 3 keV. Photons above a (roughly redshift-independent) critical energy of ∼ 1 keV have such a long me...
