Constraining the redshifted 21-cm signal with the unresolved soft X-ray background

We use the observed unresolved cosmic X-ray background (CXRB) in the 0.5-2 keV band and existing upper limits on the 21-cm power spectrum to constrain the high-redshift population of X-ray sources, focusing on their effect on the thermal history of the Universe and the cosmic 21-cm signal. Because the properties of these sources are poorly constrained, we consider hot gas, X-ray binaries and mini-quasars (i.e. sources with soft or hard X-ray spectra) as possible candidates. We find that (1) the soft-band CXRB sets an upper limit on the X-ray efficiency of sources that existed before the end of reionization, which is one-to-two orders of magnitude higher than typically assumed efficiencies, (2) hard sources are more effective in generating the CXRB than the soft ones, (3) the commonly assumed limit of saturated heating is not valid during the first half of reionization in the case of hard sources, with any allowed value of X-ray efficiency, (4) the maximal allowed X-ray efficiency sets a lower limit on the depth of the absorption trough in the global 21-cm signal and an upper limit on the height of the emission peak, while in the 21-cm power spectrum it sets a minimum amplitude and frequency for the high-redshift peaks, and (5) the existing upper limit on the 21-cm power spectrum sets a lower limit on the X-ray efficiency for each model. When combined with the 21-cm global signal, the CXRB will be useful for breaking degeneracies and helping constrain the nature of high-redshift heating sources.