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The inversion of gravitational lens systems is hindered by the fact that multiple mass distributions are often equally compatible with the observed properties of the images. Besides using clear examples to illustrate the effect of the so-called monopole and mass sheet degeneracies, this article introduces the most general form of said mass sheet degeneracy. While the well known version of this degeneracy rescales a single source plane, this generalization allows any number of sources to be rescaled. Furthermore, it shows how it is possible to rescale each of those sources with a different scale factor. Apart from illustrating that the mass sheet degeneracy is not broken by the presence of multiple sources at different redshifts, it will become apparent that the newly constructed mass distribution necessarily alters the existing mass density precisely at the locations of the images in the lens system, and that this change in mass density is linked to the factors with which the sources were rescaled. Combined with the fact that the monopole degeneracy introduces a large amount of uncertainty about the density in between the images, this means that both degeneracies are in fact closely related to substructure in the mass distribution. An example simulated lensing situation based on an elliptical version of a Navarro-Frenk-White profile explicitly shows that such degeneracies are not easily broken by observational constraints, even when multiple sources are present. Instead, the fact that each lens inversion method makes certain assumptions, implicit or explicit, about the smoothness of the mass distribution means that in practice the degeneracies are broken in an artificial manner rather than by observed properties of the lens system.
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Gravitational and plasma lensing share the same mathematical formalism in the limit of geometrical optics. Both phenomena can be effectively described by a projected, two-dimensional deflection potential whose gradient causes an instantaneous light deflection in a single, thin lens plane. We highlight the differences in the time-delay and lensing equations that occur because plasma lensing is caused by a potential directly proportional to the deflecting electron number density and gravitational lensing is caused by a potential related to the deflecting mass density by a Poisson equation. Since we treat plasma and gravitational lensing as thin-screen effective theories, their degeneracies are both caused by the unknown distribution of deflecting objects. Deriving the formalism-intrinsic degeneracies for plasma lensing, we find that they are analogous to those occurring in gravitational lensing. To break the degeneracies, galaxies and galaxy-cluster scale strong gravitational lenses must rely on additional assumptions or complementary observations. Physically realistic assumptions to arrive at self-consistent lens and source reconstructions can be provided by simulations and analytical effective theories. In plasma lensing, a deeper understanding of the deflecting electron density distributions is still under development, so that a model-based comprehensive lens reconstruction is not yet possible. However, we show that transient lenses and multi-wavelength observations help to break the arising degeneracies. We conclude that the development of an observation-based inference of local lens properties seems currently the best way to further probe the morphologies of plasma electron densities. Due to the simpler evidence-based breaking of the lensing degeneracies, we expect to obtain tighter constraints on the local plasma electron densities than on the gravitationally deflecting masses.
In this work we investigate the gravitationally lensed system B1422+231. High--quality VLBI image positions, fluxes and shapes as well as an optical HST lens galaxy position are used. First, two simple and smooth models for the lens galaxy are applied to fit observed image positions and fluxes; no even remotely acceptable model was found. Such models also do not accurately reproduce the image shapes. In order to fit the data successfully, mass substructure has to be added to the lens, and its level is estimated. To explore expectations about the level of substructure in galaxies and its influence on strong lensing, N-body simulation results of a model galaxy are employed. By using the mass distribution of this model galaxy as a lens, synthetic data sets of different four image system configurations are generated and simple lens models are again applied to fit them. The difficulties in fitting these lens systems turn out to be similar to the case of some real gravitationally lensed systems, thus possibly providing evidence for the presence and strong influence of substructure in the primary lens galaxy.
We study the abundance of substructure in the matter density near galaxies using ALMA Science Verification observations of the strong lensing system SDP.81. We present a method to measure the abundance of subhalos around galaxies using interferometric observations of gravitational lenses. Using simulated ALMA observations, we explore the effects of various systematics, including antenna phase errors and source priors, and show how such errors may be measured or marginalized. We apply our formalism to ALMA observations of SDP.81. We find evidence for the presence of a $M=10^{8.96pm 0.12} M_{odot}$ subhalo near one of the images, with a significance of $6.9sigma$ in a joint fit to data from bands 6 and 7; the effect of the subhalo is also detected in both bands individually. We also derive constraints on the abundance of dark matter subhalos down to $Msim 2times 10^7 M_{odot}$, pushing down to the mass regime of the smallest detected satellites in the Local Group, where there are significant discrepancies between the observed population of luminous galaxies and predicted dark matter subhalos. We find hints of additional substructure, warranting further study using the full SDP.81 dataset (including, for example, the spectroscopic imaging of the lensed carbon monoxide emission). We compare the results of this search to the predictions of $Lambda$CDM halos, and find that given current uncertainties in the host halo properties of SDP.81, our measurements of substructure are consistent with theoretical expectations. Observations of larger samples of gravitational lenses with ALMA should be able to improve the constraints on the abundance of galactic substructure.
In recent years, gravitational lensing has been used as a means to detect substructure in galaxy-sized halos, via anomalous flux ratios in quadruply-imaged lenses. In addition to causing anomalous flux ratios, substructure may also perturb the positions of lensed images at observable levels. In this paper, we numerically investigate the scale of such astrometric perturbations using realistic models of substructure distributions. Substructure distributions that project clumps near the Einstein radius of the lens result in perturbations that are the least degenerate with the best-fit smooth macromodel, with residuals at the milliarcsecond scale. Degeneracies between the center of the lens potential and astrometric perturbations suggest that milliarcsecond constraints on the center of the lensing potential boost the observed astrometric perturbations by an order of magnitude compared to leaving the center of the lens as a free parameter. In addition, we discuss methods of substructure detection via astrometric perturbations that avoid full lens modeling in favor of local image observables and also discuss modeling of systems with luminous satellites to constrain the masses of those satellites.
The statistics of peaks in weak lensing convergence maps is a promising tool to investigate both the properties of dark matter haloes and constrain the cosmological parameters. We study how the number of detectable peaks and its scaling with redshift depend upon the cluster dark matter halo profiles and use peak statistics to constrain the parameters of the mass - concentration (MC) relation. We investigate which constraints the Euclid mission can set on the MC coefficients also taking into account degeneracies with the cosmological parameters. To this end, we first estimate the number of peaks and its redshift distribution for different MC relations. We find that the steeper the mass dependence and the larger the normalisation, the higher is the number of detectable clusters, with the total number of peaks changing up to $40%$ depending on the MC relation. We then perform a Fisher matrix forecast of the errors on the MC relation parameters as well as cosmological parameters. We find that peak number counts detected by Euclid can determine the normalization $A_v$, the mass $B_v$ and redshift $C_v$ slopes and intrinsic scatter $sigma_v$ of the MC relation to an unprecedented accuracy being $sigma(A_v)/A_v = 1%$, $sigma(B_v)/B_v = 4%$, $sigma(C_v)/C_v = 9%$, $sigma(sigma_v)/sigma_v = 1%$ if all cosmological parameters are assumed to be known. Should we relax this severe assumption, constraints are degraded, but remarkably good results can be restored setting only some of the parameters or combining peak counts with Planck data. This precision can give insight on competing scenarios of structure formation and evolution and on the role of baryons in cluster assembling. Alternatively, for a fixed MC relation, future peaks counts can perform as well as current BAO and SNeIa when combined with Planck.
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