MUSE-DARK-II: 3D morpho-kinematic modelling of lensed galaxies. Tully-Fisher relation of $z \sim 1$ star-forming galaxies

Abstract

In a series of papers on lensed kinematics, we seek to combine the sensitivity of 3D forward modelling to low signal-to-noise ratio outskirts with the enhanced spatial resolution of cluster lensing. In this first paper, we (i) present and validate our methodology, which directly constrains the source parameters by incorporating lensing deflections into the $\texttt{GalPaK}^\texttt{3D}$ forward-modelling algorithm, and (ii) investigate the evolution of the stellar-mass and baryonic-mass Tully-Fisher relations (sTFR and bTFR) since $z \sim 1$. We define a robust sample of strongly lensed star-forming galaxies (SFGs) from the MUSE Lensing Cluster survey, spanning magnifications $μ= 1.4 - 12.4$ and stellar masses $M_\star = 10^{8.1} - 10^{10.3} M_\odot$. Using a series of mock galaxies, we find that our method is significantly more reliable at recovering morpho-kinematic properties than approaches that ignore differential magnification, even for relatively modest magnifications ($μ< 6$). Restricting the analysis to 95 rotationally supported SFGs with well-constrained velocities, we find a significant evolution of the sTFR zero-point ($Δb^\mathrm{sTFR} = -0.42^{+0.05}_{-0.05}~\mathrm{dex}$ in stellar mass) but no detectable evolution of the bTFR zero-point ($Δb^\mathrm{bTFR} = 0.00^{+0.06}_{-0.06}~\mathrm{dex}$ in baryonic mass) relative to $z \approx 0$. Our results are consistent with a mild evolution of the stellar-to-halo mass ratio and support the view that the sTFR has evolved only weakly over the past $\sim 8$ Gyr, aside from shifts driven by the redshift dependence of halo-defining quantities such as the critical density and overdensity. The absence of detectable evolution in the bTFR zero-point suggests that the increasing contribution of cold gas mass at higher redshift fully compensates the evolution observed in the stellar component alone. [abridged]

Figures (12)

• Figure 1: Schematic view of the lensing configuration and coordinate frames. The observed image-frame coordinates $\mathbf{r}^\mathrm{img} - \mathbf{r}_0^\mathrm{img}$ are mapped to source-frame coordinates $\mathbf{r}^\mathrm{src} - \mathbf{r}_0^\mathrm{src}$ via the lens equation $\mathbf{r}^\mathrm{src} = \mathbf{r}^\mathrm{img} - \bm{\alpha}(\mathbf{r}^\mathrm{img})$. The source-frame coordinates correspond to positions $\mathbf{r}$ in the physical galaxy frame, where parametric flux and velocity profiles are defined. These coordinate transformations do not alter the model grid, which remains aligned and regularly sampled along the IFS spatial axes and the LOS. This schematic is inspired by Bartelmann_01 and DiTeodoro_15.
• Figure 2: Comparison of the relative errors in the inferred properties, defined as $\delta p / p = (p^\mathrm{fit} - p^\mathrm{true}) / p^\mathrm{true}$, for a kinematic model fitted either directly in the image frame (left column) or in the source frame using the GalPaK$^\texttt{3D}$ Strong Lensing Extension (right column). Each row shows the error density distribution for a given property, restricted to the rotation-dominated subsample of mock galaxies ($v_{1.8}/\sigma_0 > 1$). The effective radius is corrected a posteriori by the magnification when fitted in the image plane. The PA error is relative to $\mathrm{PA}^\mathrm{true} = \qty{130}{\degree}$, as in Bouche_15b. Dashed lines show the empirical relation $\delta p / p = \pm S/N_\mathrm{max} \times (\sqrt{\mu} R_e /R_\mathrm{PSF})^{\alpha}$ based on the coefficients $\alpha$ from Bouche_15b, while solid vertical lines indicate $S/N_\mathrm{max} \times (\sqrt{\mu} R_e /R_\mathrm{PSF})^{\alpha} = 10$.
• Figure 3: Same as Fig. \ref{['fig:mock_relative_error']}, additionally restricted to the subsample of mock galaxies satisfying $\mu > 3$.
• Figure 4: S/N of the brightest [ O ii] spaxel against the apparent size-to-PSF ratio. The hatched grey region marks the parameter space excluded by our selection. Red symbols mark 6 galaxies chosen to showcase the range of apparent sizes and S/N in Fig.~\ref{['fig:highlights']}. Open symbols mark galaxies that are either dispersion-dominated ($v_{1.8}/\sigma_0 \leq 1$) or have poorly constrained velocities ($\Delta v_{1.8}/v_{1.8} \geq 30\%$), where $\Delta v_{1.8}$ is the posterior standard deviation. The background hexagonal bins illustrate the fraction of rotation-dominated ($v_{1.8}/\sigma_{0} > 1$) mock galaxies with a robust velocity recovery ($|v^\mathrm{fit}_{1.8} - v^\mathrm{true}_{1.8}| / v^\mathrm{true}_{1.8} < 30\%$). Bins with less than 20 mocks are not shown.
• Figure 5: General sample properties. The panels show the distributions of redshift (top left) and magnification (bottom left), together with the locations of the galaxies in the $M_\star$–SFR (middle) and $M_\star$–$R_e$ (right) planes. Effective radii are corrected to rest-frame 5000 Å following Eqs. 1 and 2 of Wel_14, expressed relative to the size–mass relation of Nedkova_21 (interpolated with redshift), and scaled to $z = 1$. Similarly, SFRs are expressed relative to the star-forming main sequence of Boogaard_18, adjusted to the redshift and stellar mass of each galaxy, and normalized to $z = 1$. The shaded regions indicate the $\pm1\sigma$ (dark grey) and $\pm2\sigma$ (light grey) scatter around the corresponding relations. Open symbols and histogram segments mark galaxies that are either dispersion-dominated ($v_{1.8}/\sigma_0 \leq 1$) or have poorly constrained velocities ($\Delta v_{1.8}/v_{1.8} \geq 30\%$), where $\Delta v_{1.8}$ is the posterior standard deviation.