Faint mm NIKA2 dusty star-forming galaxies. Finding the high-redshift population

Aims. High-redshift dusty star-forming galaxies (DSFGs) are proposed to be the progenitors of massive quiescent galaxies arising at cosmic noon, providing a crucial insight into the formation, assembly, and early quenching of massive galaxies in the early Universe. However, their high redshift combined with high dust obscuration adds significant difficulties to their redshift measurement, which is mandatory for detailed studies of their physical properties. Blind mm spectral scans are the most unbiased way in prinicple for obtaining accurate spectroscopic redshifts for these sources, but identifying faint molecular and atomic lines within limited telescope time for faint DSFGs is also difficult with these scans. Methods. We developed a new framework to constrain the source redshift. The method jointly accounts for the detection and/or nondetection of spectral lines and the prior information from the photometric redshift and total infrared luminosity from spectral energy distribution analysis. The method uses the estimated total infrared luminosity to predict the line fluxes at given redshifts and generates model spectra. The redshift-dependent spectral models were then compared with the observed spectra to determine the redshift. Results. We applied this joint redshift analysis method to four high-z dusty star-forming galaxy candidates selected from the NIKA2 observations of the HLSJ091828.6+514223 (HLS) field that were further observed by NOEMA with blind spectral scans. These sources only have Herschel SPIRE photometry as ancillary data. They were selected because SPIRE counterparts are faint or entirely lacking and thus favor to select the highest-redshift candidates. The method finds a spectroscopic redshift of 4 in the five NOEMA-counterpart detected sources, with z < 3. Based on these measurements, we derived the CO and [CI] lines and mm continuum fluxes from the NOEMA data and studied the properties of their interstellar medium and star formation. We find cold dust temperatures in some of the HLS sources compared to the general population of submm galaxies, which might be related to the bias introduced by the SPIRE-dropout selection. All sources except for one have a short gas-depletion time of a few hundred million years, which is typical of high-z submm galaxies. The only exception shows a longer gas-depletion time of up to a few billion years. This is comparable to the gas-depletion times of main-sequence galaxies at the same redshift. Furthermore, we identify a possible overdensity of dusty star-forming galaxies at z = 5.2 that is traced by two sources in our sample, as well as a lensed galaxy HLSJ091828.6+514223. Conclusions. We demonstrate that our method when applied to mm-selected DSFGs is able to determine the redshift accurately. This accuracy with only multiple emission lines with a low signal-to-noise ratio shows promising potential for the blind redshift search in large samples of high-z DSFGs, even in the absence of optical to near infrared photometric redshifts.