We use VANDELS spectroscopic data overlapping with the $simeq$7 Ms Chandra Deep Field South survey to extend studies of high-mass X-ray binary systems (XRBs) in 301 normal star-forming galaxies in the redshift range $3 < z < 5.5$. Our analysis evaluates correlations between X-ray luminosities ($L_X$), star formation rates (SFR) and stellar metallicities ($Z_star$) to higher redshifts and over a wider range in galaxy properties than hitherto. Using a stacking analysis performed in bins of both redshift and SFR for sources with robust spectroscopic redshifts without AGN signatures, we find convincing evolutionary trends in the ratio $L_X$/SFR to the highest redshifts probed, with a stronger trend for galaxies with lower SFRs. Combining our data with published samples at lower redshift, the evolution of $L_X$/SFR to $zsimeq5$ proceeds as $(1 + z)^{1.03 pm 0.02}$. Using stellar metallicities derived from photospheric absorption features in our spectroscopic data, we confirm indications at lower redshifts that $L_X$/SFR is stronger for metal-poor galaxies. We use semi-analytic models to show that metallicity dependence of $L_X$/SFR alone may not be sufficient to fully explain the observed redshift evolution of X-ray emission from high-mass XRBs, particularly for galaxies with SFR $<30$ $M_odot$ yr$^{-1}$. We speculate that the discrepancy may arise due to reduced overall stellar ages in the early Universe leading to higher $L_X$/SFR for the same metallicity. We use our data to define the redshift-dependent contribution of XRBs to the integrated X-ray luminosity density and, in comparison with models, find that the contribution of high-mass XRBs to the cosmic X-ray background at $z>6$ may be $gtrsim 0.25$ dex higher than previously estimated.
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