Abstract
Unveiling the true nature of dark matter, which manifests itself only through gravity, is one of the principal quests in physics. Leading candidates for dark matter are weakly interacting massive particles or ultralight bosons (axions), at opposite extremes in mass scales, that have been postulated by competing theories to solve deficiencies in the Standard Model of particle physics. Whereas dark matter weakly interacting massive particles behave like discrete particles (ϱDM), quantum interference between dark matter axions is manifested as waves (ψDM). Here, we show that gravitational lensing leaves signatures in multiply lensed images of background galaxies that reveal whether the foreground lensing galaxy inhabits a ϱDM or ψDM halo. Whereas ϱDM lens models leave well documented anomalies between the predicted and observed brightnesses and positions of multiply lensed images, ψDM lens models correctly predict the level of anomalies remaining with ϱDM lens models. More challengingly, when subjected to a battery of tests for reproducing the quadruply lensed triplet images in the system HS 0810+2554, ψDM is able to reproduce all aspects of this system whereas ϱDM often fails. The ability of ψDM to resolve lensing anomalies even in demanding cases such as HS 0810+2554, together with its success in reproducing other astrophysical observations, tilt the balance toward new physics invoking axions.
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Acknowledgements
J.L. acknowledges a grant from the Research Grants Council of Hong Kong through General Research Fund 17304519, which also partially supports a graduate scholarship for A.A. T.B. is supported by the Spanish project grant PID2020-114035GB-100 (MINECO/AEI/FEDER, UE) as well as the General Research Fund 17304519. M.O. acknowledges support by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI grants JP20H04725, JP20H00181, JP20H05856 and JP18K03693. J.M.D. acknowledges the support of projects PGC2018-101814-B-100 and MDM-2017-0765. R.E. acknowledges support by the Institute for Theory and Computation at the Center for Astrophysics as well as grants 21-atp21-0077, NSF AST-1816420 and HST-GO-16173.001-A. H.-Y.S. acknowledges funding support from Jade Mountain Young Scholar Award NTU-111V1201-5, sponsored by the Ministry of Education, Taiwan, as well as by the National Science and Technology Council (NSTC) of Taiwan under grant NSTC 111-2628-M-002-005-MY4 and the NTU Academic Research-Career Development Project under grant NTU-CDP-111L7779. This work was carried out in part using the High Performance Computing (HPC) facilities at the University of Hong Kong.
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A.A. wrote original parallel computing code to perform ψDM lensing simulations, performed analysis of results and contributed to interpretation of results. T.B. and J. Lim supervised the project and contributed to interpretation of results. J. Lim, A.A. and T.B. wrote the paper. J.M.D. contributed to lensing simulations and interpretation of results. G.F.S. contributed to analysis of GRFs and interpretation of results. M.O., E.L. and R.E. contributed to constructing the mathematical framework of ψDM density fluctuations. T.C., H.-Y.S. and M.C.H.Y. contributed to methodology for generating GRFs. J. Li contributed to lens modelling using glafic. S.K.L. and J. Li carried out Markov chain Monte Carlo calculations.
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Extended data
Extended Data Fig. 1 Standard deviation of GRF.
Projected radial dependence in σκ(ξ) ≡ σΣ(ξ)/Σcr for the GRF relative to κsmooth ≡ Σsmooth(ξ)/Σcr for an NFW (blue) global profile computed according to Eq. 8, and for a PL (red) global profile computed in the same manner. Imprinting GRFs having an appropriate σκ onto these profiles to generate model ψDM lenses, fluctuations in the column mass density of these lenses diminish with increasing projected radius, ξ, from the halo centre. At the Einstein radius for these particular model lenses and the lensed source (see main text), the GRF has a standard deviation of ~ 10%-15% the local mean column mass density of both these profiles.
Extended Data Fig. 2 Cusp and fold configurations.
In the image plane, dots indicate multiply lensed image positions of a compact background source. Four images are distributed around an elliptical critical curve corresponding to the Einstein ring of the foreground lensing galaxy; a fifth image located near the lens centre as indicated by a × is not plotted as it is both demagnified and usually undetectable against the lensing galaxy. In the source plane, a solitary dot indicates the position of the source located near a diamond-shaped caustic, which encloses the region that gives rise to five lensed images; if located along the caustic, two of these lensed images merge to appear at the critical curve. a-b, Source located near a cusp of the caustic, giving rise to an image configuration whereby the three most closely separated images are the most highly magnified. c-d, Source located near a fold of the caustic, giving rise an image configuration whereby the two most closely separated images are the most highly magnified.
Extended Data Fig. 3 Caustics.
Close-up around a cusp of the caustic for a ϱDM versus ψDM halo that both have the same global profile (see Extended Data Table 1). Colours indicate lensing magnification (brighter for higher magnification) imposed onto the brightest of the multiply lensed images depending on where the source is located. a, The simple and smooth caustic (white loci) of a ϱDM halo, for which the fine striations are computational artefacts. b, The complex caustic of a ψDM halo with many branching microcaustics, for which the finest striations are again computational artefacts. Black rectangles bound the same region in the source plane near a cusp of the caustic, within which we placed a source at seventy-five different locations to mimic ψDM halos created using seventy-five different GRF realisations (to save computation effort and time) for computing the positional and brightness anomalies in Fig 3.
Extended Data Fig. 4 Position and brightness anomalies computed using two approaches.
Probability distribution in a, position and b, brightness anomalies generated by: (i) imprinting 75 different GRF realisations having λdB = 180 pc onto the ϱDM lens of Fig. 2 to create a suite of ψDM lenses, and then generating multiply lensed images for each ψDM lens by placing the source at a fixed location (dashed curves); versus (ii) imprinting a single GRF realisation having also λdB = 180 pc onto the ϱDM halo, and generating multiply lensed images by placing the source at seventy-five different locations within the black rectangle shown in Extended Data Fig. 3 (solid curves). Blue curves are for pure ψDM halos, whereas red curves are for ψDM halos having density modulations damped by 50% to include a baryonic component. Although having similar medians, the second approach yields slightly elevated tails toward higher values.
Extended Data Fig. 5 Predictions of bestfit ϱDM lens model for HS 0810+2554.
a, Image positions of the quadruply lensed pair of radio jets (red and blue crosses) and optical QSO (black crosses) predicted by our best-fit ϱDM lens model. Their observed image positions are indicated by the corresponding circles. Large black ellipse is the Einstein ring of our lens model at the redshift of the lensed source. b, Source positions for the pair of radio jets (labelled R1 and R2, respectively, for the blue and red crosses) and optical QSO (labelled C). The QSO is assumed to correspond to the core of the radio jets, and therefore lie along a line connecting these jets. Our best-fit lens model requires the core to be closer to radio jet R1 than R2, and all to be located near the caustic.
Extended Data Fig. 6 Posterior probability distribution function for the parameters of our best-fit ϱDM lens model for HS 0810+2554.
Histograms showing distribution in parameter values for each of the six parameters describing the PL model derived from a MCMC analysis. Each parameter exhibits a distribution in values that can be closely described by a Gaussian having a mean and ± 1σ ranges as indicated by the dashed lines. Contour plots with contour levels plotted at the 1σ, 2σ, and 3σ indicate the correlation between pairs of model parameters. Some of these parameters, like those between the ellipticity and y-position of the lens centre, exhibit strong correlations and therefore degeneracies. The distributions in parameter values were used to estimate uncertainties in the predicted positions of the multiply lensed images in HS 0810+2554 as indicated by the crosses in Fig. 4.
Extended Data Fig. 7 Best ψDM realization for overall agreement of lensed radio jets.
A single realization of our model ψDM lens that yields the best overall agreement between the predicted and observed positions of the multiply lensed radio images (indicated by blue and red symbols) among the seventy-five different realizations explored. We do not consider how well our model ψDM lens reproduces the HST positions of the multiply lensed QSO (indicated by black symbols) as there is an uncertainty of a few 10 mas between the registration of the radio and optical reference frames. As in Fig. 4, observed image positions are indicated by ellipses, each with a radius or semi-major and semi-minor axes of 3σ (where σ is the measurement uncertainty) so as to encompass 99.7% of all possible positions. Dots indicate the positions of the quadruply lensed images predicted by this particular realization of the model ψDM lens, with arm lengths corresponding to ± 3σtol (for which σtol reflects tolerances in the inferred parameters of the model ϱDM lens as described in Methods).
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Amruth, A., Broadhurst, T., Lim, J. et al. Einstein rings modulated by wavelike dark matter from anomalies in gravitationally lensed images. Nat Astron 7, 736–747 (2023). https://doi.org/10.1038/s41550-023-01943-9
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DOI: https://doi.org/10.1038/s41550-023-01943-9
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