We present measurements of the evolution of normal-galaxy X-ray emission from $z approx$ 0-7 using local galaxies and galaxy samples in the 6 Ms Chandra Deep Field-South (CDF-S) survey. The majority of the CDF-S galaxies are observed at rest-frame energies above 2 keV, where the emission is expected to be dominated by X-ray binary (XRB) populations; however, hot gas is expected to provide small contributions to the observed- frame < 1 keV emission at $z < 1$. We show that a single scaling relation between X-ray luminosity ($L_{rm X}$) and star-formation rate (SFR) is insufficient for characterizing the average X-ray emission at all redshifts. We establish that scaling relations involving not only SFR, but also stellar mass ($M_star$) and redshift, provide significantly improved characterizations of the average X-ray emission from normal galaxy populations at $z approx$ 0-7. We further provide the first empirical constraints on the redshift evolution of X-ray emission from both low-mass XRB (LMXB) and high-mass XRB (HMXB) populations and their scalings with $M_star$ and SFR, respectively. We find $L_{rm 2-10 keV}({rm LMXB})/M_star propto (1 + z)^{2-3}$ and $L_{rm 2-10 keV}({rm HMXB})$/SFR $propto (1 + z)$, and show that these relations are consistent with XRB population-synthesis model predictions, which attribute the increase in LMXB and HMXB scaling relations with redshift as being due to declining host galaxy stellar ages and metallicities, respectively. We discuss how emission from XRBs could provide an important source of heating to the intergalactic medium in the early Universe, exceeding that of active galactic nuclei.
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