A Salpeter IMF and an NFW halo: Disentangling the dark and stellar mass through precise lens modelling of a double-source-plane system reinforces the canonical model of elliptical galaxies

Abstract

We present a strong lensing analysis of the double source plane lens J0946+1006 (colloquially "Jackpot" lens) to measure the inner dark matter density profile, the stellar-to-halo mass ratio, and the stellar initial mass function normalisation using a two component stellar plus dark matter mass model. The stellar mass follows a multi-Gaussian expansion light model with a free global mass-to-light ratio and an allowed radial $M/L$ gradient, while the dark matter is described by an elliptical generalised NFW halo. The double-source-plane geometry provides additional leverage against the mass-sheet transformation and helps constrain the radial mass profile. Despite allowing both a radial stellar $M/L$ gradient and a generalised NFW halo, the data prefer the canonical picture: an approximately constant stellar mass-to-light ratio with a Salpeter-like IMF normalisation, and a dark matter halo consistent with NFW. We infer $M_{\star} = 4.4^{+0.25}_{-0.39}\times 10^{11}\,M_{\odot}$ and an inner halo slope $γ_{\rm in}^{\rm halo} = 1.04^{+0.10}_{-0.14}$. The halo mass is $M_{200}^{\rm halo} = 1.11^{+0.37}_{-0.32}\times 10^{13}\,M_{\odot}$, implying $\log_{10}(M_{200}/M_{\star})=1.41^{+0.13}_{-0.14}$. At fixed halo mass, the inferred stellar mass lies $\sim0.1$ dex above typical literature stellar halo mass relations at similar redshift, which is comparable to the intrinsic scatter of these relations. We expect this approach to provide a practical template for future dark matter studies with the large double-source-plane lens samples from Euclid.

Figures (10)

• Figure 1: The false-colour image of the Jackpot lens (left), and its $5^{\prime\prime}$ cutout (right). The colour composite is constructed from HST bands with F475W mapped to blue (B), F606W to green (G), and F814W to red (R).
• Figure 2: The lens model of star + gNFW mass model. Top-left: observed image. Top-middle: best-fit model image (posterior-median). Top-right: normalized residual $(\mathrm{data}-\mathrm{model})/\sigma$, clipped to $\pm 3\sigma$. Bottom-left: reconstructed inner source $s_1$ on its source plane; orange dashed contours show $\log\kappa$ of the $s_1$ mass component. Bottom-middle: reconstructed outer source $s_2$ on its source plane. Bottom-right: lens-light subtracted image with the same log stretch as the top-left panel. Axes are in arcsecond.
• Figure 3: Posterior distributions for the composite star + gNFW halo model under different combinations of external information. Diagonal panels show the 1D marginalised posteriors; off-diagonal panels show the 2D posteriors with 68% and 95% credible contours. The parameters shown are $M_{\mathrm{200}}^{\mathrm{halo}} / 10^{12} M_\odot$ (halo mass), $\gamma_{\mathrm{in}}^{\mathrm{halo}}$ (inner logarithmic density slope), $\nabla(M/L)$ (logarithmic radial gradient of the stellar mass-to-light ratio), $q_{\mathrm{halo}}$ (projected axis ratio), and $M_\star / 10^{11} M_\odot$ (total stellar mass). Filled blue contours show the lensing-only inference; orange dashed lines show the posterior reweighted by the kinematic likelihood; green dashed lines include the concentration-mass prior; and solid black lines show the combined lensing + kinematic + concentration-mass constraints. The shaded horizontal band marks the stellar-mass range corresponding to a Salpeter IMF which comes from Sonnenfeld2012.
• Figure 4: Aperture-averaged line-of-sight stellar velocity dispersion fit with pPXF. The observed MUSE spectrum and the pPXF best-fit model (normalised to a median flux of 1) are marked in black and red, respectively. The residuals of the data with respect to the model are shown in green. The regions masked, to avoid the impact of the lensed source emission lines or of the subtraction of sky, laser, or telluric lines, are shaded in grey. The data minus model residuals in these regions are marked in blue. In the top-left corner, we show a cutout of the summed white image of the MUSE cube, marking the circular aperture used to extract the lens spectrum in red.
• Figure 5: Mass decomposition of the lens model. Top-left: Two-dimensional convergence map of the median lens model, where the total mass is shown in colour scale and the stellar (blue) and dark matter (red) components are shown as overlaid contours. Top-right: Contours of the total convergence alone. Bottom-left: Radial mass profiles of the stellar, dark matter, and total components. The solid curves show the median profiles, while the shaded bands indicate the 68% and 95% credible regions derived from 500 posterior samples. Bottom-right: Cumulative dark matter fraction $f_{\rm DM}(<R)$ as a function of radius, with median (solid) and 68%/95% credible regions (shaded). The vertical dashed line marks the inner arc, while the solid black and orange lines indicate the outer arc and $R=5\,\mathrm{kpc}$, respectively.