In this paper, we study the relationship between the 2-10 keV X-ray luminosity (L_X), assumed to originate from X-ray binaries (XRBs), and star formation rate (SFR) in UV-selected z<0.1 Lyman break analogs (LBAs). We present Chandra observations for four new GALEX-selected LBAs. Including previously studied LBAs, Haro 11 and VV 114, we find that LBAs demonstrate L_X/SFR ratios that are elevated by ~1.5sigma compared to local galaxies, similar to the ratios found for stacked LBGs in the early Universe (z>2). We show that these LBAs are unlikely to harbor AGN, based on their optical and X-ray spectra and the spatial distribution of the X-rays in three spatially extended cases. We expect that high-mass X-ray binaries (HMXBs) dominate the X-ray emission in these galaxies, based on their high specific SFRs (sSFRs=SFR/M* > 10^{-9}/yr), which suggest the prevalence of young stellar populations. Since both LBAs and LBGs have lower dust attenuations and metallicities compared to similar samples of more typical local galaxies, we investigate the effects of dust extinction and metallicity on the L_X/SFR for the broader population of galaxies with high sSFRs (>10^{-10}/yr). The estimated dust extinctions (corresponding to column densities of N_H<10^{22}/cm^2) are expected to have insignificant effects on observed L_X/SFR ratio for the majority of galaxy samples. We find that the observed relationship between L_X/SFR and metallicity appears consistent with theoretical expectations from X-ray binary population synthesis models. Therefore, we conclude that lower metallicities, related to more luminous HMXBs such as ultraluminous X-ray sources (ULXs), drive the elevated L_X/SFR observed in our sample of z<0.1 LBAs. The relatively metal-poor, active mode of star formation in LBAs and distant z>2 LBGs may yield higher total HMXB luminosity than found in typical galaxies in the local Universe.
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