A Fuzzy Situation Eased: Cold Dark Matter with Multipoles Can Explain The Double Radio Quad Lens HS 0810+2554

TL;DR

This work tests whether cold dark matter (CDM) mass models augmented with azimuthal multipole perturbations can explain the positional anomalies of the eight radio images in the HS 0810+2554 strong lens. By constructing four models of increasing complexity that combine an elliptical power-law frame, multipoles (m = 1, 3, 4), and external shear, and by exhaustively considering 16 image configurations (two possible time orderings) across 64 fits, the authors show that a complex CDM mass distribution with small multipole amplitudes can reproduce the image positions with χ as low as 1.59 in the best case (Mod4–P1′). The results suggest that multipole mass-structure within the lensing galaxy can account for the lensing observables without invoking fuzzy dark matter, though degeneracies and potential subhalo effects remain, underscoring the value of time-delay measurements to break degeneracies. Overall, the study highlights the importance of internal mass complexity in strong lensing analyses and provides a framework for incorporating multipole perturbations into CDM lens models.

Abstract

Originally observed in isophotal density contours of elliptical galaxies, higher order perturbations in the form of Fourier modes, or multipoles, are becoming increasingly recognized as necessary to account for angular mass complexity in strong lensing analyses. When smooth, elliptical CDM mass models fail, multipoles often emerge as solutions. With the discovery of two radio jets in the source quasar, the strong gravitational lens HS 0810+2554 can no longer be well fit by elliptical mass models, suggesting perturbations on small-scales. In this paper, we investigate the efficacy of multipoles $m=1$ (lopsidedness), $m=3$ (triangleness), and $m=4$ (boxiness and diskiness) in addressing the image positional anomalies of the two radio quads of HS 0810+2554. Due to the exact pairing and arrival sequence of the images being unknown, we consider all feasible image configurations. With 64 unique best-fit models, we achieve a fit of $χ=1.59$ ($χ^2=2.53$), with $m=1,3,4$ multipole strengths of 0.9%, 0.4%, and 0.6%, respectively, with images in the reverse time ordering. Elliptical+shear models from previous works find $χ\!\sim\!7\!-\!10$, for comparison. With the morphological (i.e., standard) arrival sequence, we achieve a fit of $χ=2.95$ with two images being assigned to opposite sources. Therefore, CDM mass models with mass complexity in the form of multipoles are able to adequately explain the positional anomalies in HS 0810+2554. Alternative dark matter theories, like fuzzy dark matter, need not be invoked.

Figures (8)

• Figure 1: The gravitational lens HS 0810+2554. The image positions for the two quadruply lensed radio jets (blue and red 'X's) and the quadruply lensed RQQ (large black '+'s) are reported in Table \ref{['tab:positions']}. A-D designates the relative brightness labelling scheme of the images. The images are centered on the CASTLES survey reported optical galaxy center 1998kochanek, assuming a nominal (+6 mas, +4 mas) offset for the radio images so that images A1 and AQ are not coincident. The gray circle has a radius of 0.5 arcsec and is included to show the near-circular nature of the system.
• Figure 2: A pictorial representation of the eight unique 2x4 configurations (or pairings; P$_{1-8}$) of the eight radio images of HS 0810+2554 as outlined in Table \ref{['tab:combs']}. The four images in a configuration belonging to the first source ($\theta_{1,1-4}$) are colored blue and the second source ($\theta_{2, 1-4}$) colored red. Similarly to how images are underlined in Table \ref{['tab:combs']}, gray boxes are drawn around images to indicate when they are switched between sources. The image positions are reported in Table \ref{['tab:positions']} and centered on the CASTLES derived galaxy center with a (+6 mas, +4 mas) offset. A-D designates the relative brightness labelling scheme of the images.
• Figure 3: A zoomed in view centered around each of the eight observed radio image positions, with 3$\sigma$ error ellipses and the predicted positions from eight of our best fitting models. The top figures are centered on the image positions of the 'first' P$_1$ quad (#=1) and the bottom is centered on the 'second' P$_1$ quad (#=2; see Table \ref{['tab:positions']} and P$_1$ in Table \ref{['tab:combs']}). The columns are ordered alphabetically (A-D), not based on arrival time. The eight best-fit models are Mod$_{3,4}$ for image configurations P$_{1,3}$ and P$_{1,3}^\prime$. The marker shape for each image configuration is the same for both time orderings, with the reverse time order being unfilled, i.e., Mod$_3$--P$_1$ (filled circle) and Mod$_3$--P$_1^\prime$ (unfilled circle). Model images belonging to the 'first' quad within its image configuration are colored blue and 'second' quad colored red. The images of A1 and A2 are red and blue, respectively, for P$_3$ because those images are switched when compared to P$_1$ (see Table \ref{['tab:combs']} and Figure \ref{['fig:combs']}). The image positions from the best fitting SIE+EX from 2019hartley ($\chi=7.05$) are included as H's for comparison and the best fitting EPL model from 2023amruth ($\chi=9.84$) as A's. Images C1 and B2 from 2023amruth fall outside the window.
• Figure 4: Convergence, or $\kappa$, maps for the four best-fitting solutions of the two models Mod$_{3-4}$ and image configurations P$_1$ and P$_3$. Column 1 displays the $\kappa$ maps with the lowest $\chi$ values, whose positional errors can be seen in Figure \ref{['fig:best']}. Columns 2-4 show the three next best solutions. The fit parameters for all of these models are tabulated in Table \ref{['tab:params']}. The reverse time ordering $\kappa$ maps are displayed in Figure \ref{['fig:kappas2']}. The $\kappa$ map highlighted with the red rectangle ($\chi$=2.95) is the best-fit model with the morphological time ordering. Additionally, we consider this model to be reasonably physical, whereas other models displayed are less likely to be physical.
• Figure 5: Convergence, or $\kappa$, maps for the four best-fitting solutions of the two models Mod$_{3-4}$ and image configurations P$_1^\prime$ and P$_3^\prime$. Column 1 displays the $\kappa$ maps with the lowest $\chi$ values, whose positional errors can be seen in Figure \ref{['fig:best']}. Columns 2-4 show the three next best solutions. The fit parameters for all of these models are tabulated in Table \ref{['tab:params']}. The standard time ordering $\kappa$ maps are displayed in Figure \ref{['fig:kappas1']}. The $\kappa$ map highlighted with the red rectangle ($\chi$=1.59) is the best-fit model with the reverse time ordering. Additionally, we consider this model to be reasonably physical, whereas other models displayed are less likely to be physical.