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

TL;DR

This paper tackles the challenge of achieving sub-percent accuracy in stellar velocity dispersion measurements, a critical ingredient for cosmography via gravitational time delays, by systematically identifying and mitigating dominant systematics, especially template mismatch. It introduces a reproducible recipe that marginalizes over template libraries using a Bayesian information criterion (BIC) weighting scheme, combines multiple high-quality empirical libraries, and validates the approach with MUSE, KCWI, JWST NIRSpec, and SDSS data. The study demonstrates that, with cleaned templates and appropriate wavelength ranges and polynomials, template-induced systematics can be reduced to well below 1%, with covariances between galaxies also suppressed to the sub-percent level, enabling robust H0 inferences. The authors provide public software to implement the method and plan to apply it across future TDCOSMO work, marking a significant step toward standardized, precision stellar kinematics for cosmography.

Abstract

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 H$_0$ 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$\sim$30--160 Å$^{-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.

Figures (18)

• Figure 1: Example of rejected templates from each of the three stellar libraries. For each spectrum, we state the library name, object name, $T_{\rm eff}$, and metallicity above the respective panel. All three templates were removed because they were too noisy in the wavelength region of interest.
• Figure 2: Example spectra of stars in common between the three libraries. For each example, the spectra are shown in the top panel of the two, and ratios with respect to the MILES spectrum are shown in the bottom one. SSP spectra with solar metallicity and an age of 10 Gyr are also shown for comparison. The fourth template shown (HD118100) was removed from all three clean libraries, owing to the strong emission lines in the CaHK absorption features. It was removed from MILES by author selection and was flagged independently by visual inspection in the other two libraries.
• Figure 3: Test of the robustness of the spectral fit to the template distribution, as described in Section \ref{['ssec:template_robustness']}. The left panel shows a representative pPXF fit to the mock spectrum, showing the data (black), best-fit model (red), and residuals (green diamonds). The fits for the two biased library cases are indistinguishable, except for different random noise realizations. The middle and right panels show the input SFH (orange) and the averaged recovered pPXF SFH (blue) of 100 Monte Carlo realizations for the cases with template libraries overpopulated at young and old ages, respectively. The recovered SFHs are statistically indistinguishable, demonstrating the robustness of the method.
• Figure 4: 1D spectra extracted from central spaxels of three galaxies from the MUSE sample at different redshifts. In each panel, the black line is the observed spectrum and the red line is the best-fit model from pPXF. The gray regions mark the excluded wavelength range from fitting, and the green markers are residuals between data and the fitted model at each wavelength. The blue pixels, generally emission lines or sky residuals, have been excluded by sigma-clipping.
• Figure 5: Comparison between the KCWI (red) and SDSS (blue) spectra of the same objects in the wavelength range used for the KCWI fit. The KCWI spectra shown here are integrated over spaxels within a radius of $1.5\arcsec$, the same size as the SDSS fiber. SDSS spectra (blue) shown here have S/N $<15\ \text{\AA}^{-1}$.