Improving blazar redshift constraints with the edge of the Ly$\alpha$ forest: 1ES 1553+113 and implications for observations of the WHIM

Blazars are some of the brightest UV and X-ray sources in the sky and are valuable probes of the elusive warm-hot intergalactic medium (WHIM; $T{\simeq} 10^5-10^7$ K). However, many of the brightest blazars$-$called BL Lac objects such as 1ES1553+113$-$have quasi-featureless spectra and poorly constrained redshifts. Here, we significantly improve the precision of indirect redshift constraints for blazars based on the edge of the $\rm{H\,I}$ Ly$\alpha$ forest observed in their UV spectra. We develop a robust technique to constrain the redshift of a $z 12.6$. We use a large sample of 192 AGN/QSOs at $0.01\lesssim z\lesssim0.45$ that have high-quality COS FUV spectra to characterize the intrinsic scatter in the gap between the AGN redshift and the edge of their Ly$\alpha$ forest. We present new COS NUV data for 1ES1553+113 and confirm its redshift of $z=0.433$ using our technique. We apply our Ly$\alpha$-forest-based redshift estimation technique to nine additional blazars with archival ${\it HST}$ UV spectra, most of which are key targets for future X-ray missions. Our inferred redshift constraints improve estimates for two BL Lacs (1ES1118+424 and S50716+714) and are consistent with previous estimates for the rest. Our results emphasize the need to obtain further UV spectra of bright blazars, of which many have uncertain redshifts, in order to maximize the scientific value of future X-ray WHIM observations that will improve our understanding of galaxy evolution.