TDCOSMO. XIX. Measuring stellar velocity dispersion with sub-percent accuracy for cosmography

The stellar velocity dispersion (σ) of massive elliptical galaxies is a key ingredient in breaking the mass-sheet degeneracy and obtaining precise and accurate cosmography from gravitational time delays. The relative uncertainty on the Hubble constant H₀ is double the relative error on σ. Therefore, time-delay cosmography imposes much more demanding requirements on the precision and accuracy of σ than galaxy studies. While precision can be achieved with an adequate signal-to-noise ratio (S/N), the accuracy critically depends on key factors such as the elemental abundance and temperature of stellar templates, flux calibration, and wavelength ranges. We carried out a detailed study of the problem using multiple sets of galaxy spectra of massive elliptical galaxies with S/N∼30--160 Å^-1, along with state-of-the-art empirical and semi-empirical stellar libraries and stellar population synthesis templates. We show that the choice of stellar library is generally the dominant source of residual systematic errors. We propose a general recipe for mitigating and accounting for residual uncertainties. We show that a sub-percent level of accuracy can be achieved on individual spectra with our data quality, which we subsequently validated with simulated mock datasets. The covariance between velocity dispersions measured for a sample of spectra can also be reduced to sub-percent levels. We recommend this recipe for all applications that require high precision and accurate stellar kinematics. Thus, we have made all the software publicly available to facilitate its implementation. This recipe will also be used in future TDCOSMO collaboration papers.