Heavily obscured ( N _ { H } \gtrsim 3 \times 10 ^ { 23 } cm ^ { -2 } ) Active Galactic Nuclei ( AGNs ) not detected in even the deepest X-ray surveys are often considered to be comparably numerous to the unobscured and moderately obscured AGNs . Such sources are required to fit the cosmic X-ray background ( XRB ) emission in the 10–30 keV band . We identify a numerically significant population of heavily obscured AGNs at z \approx 0.5 –1 in the Chandra Deep Field-South ( CDF-S ) and Extended Chandra Deep Field-South by selecting 242 X-ray undetected objects with infrared-based star formation rates ( SFRs ) substantially higher ( a factor of 3.2 or more ) than their SFRs determined from the UV after correcting for dust extinction . An X-ray stacking analysis of 23 candidates in the central CDF-S region using the 4 Ms Chandra data reveals a hard X-ray signal with an effective power-law photon index of \Gamma = 0.6 _ { -0.4 } ^ { +0.3 } , indicating a significant contribution from obscured AGNs . Based on Monte Carlo simulations , we conclude that 74 \pm 25 \% of the selected galaxies host obscured AGNs , within which \approx 95 \% are heavily obscured and \approx 80 \% are Compton-thick ( CT ; N _ { H } > 1.5 \times 10 ^ { 24 } cm ^ { -2 } ) . The heavily obscured objects in our sample are of moderate intrinsic X-ray luminosity [ \approx ( 0.9 – 4 ) \times 10 ^ { 42 } erg s ^ { -1 } in the 2–10 keV band ] . The space density of the CT AGNs is ( 1.6 \pm 0.5 ) \times 10 ^ { -4 } Mpc ^ { -3 } . The z \approx 0.5 –1 CT objects studied here are expected to contribute \approx 1 \% of the total XRB flux in the 10–30 keV band , and they account for \approx 5 –15 % of the emission in this energy band expected from all CT AGNs according to population-synthesis models . In the 6–8 keV band , the stacked signal of the 23 heavily obscured candidates accounts for < 5 \% of the unresolved XRB flux , while the unresolved \approx 25 \% of the XRB in this band can probably be explained by a stacking analysis of the X-ray undetected optical galaxies in the CDF-S ( a 2.5 \sigma stacked signal ) . We discuss prospects to identify such heavily obscured objects using future hard X-ray observatories .