Isothermal Ellipsoids Research Articles

Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing

Abstract In Hezaveh et al. we showed that deep learning can be used for model parameter estimation and trained convolutional neural networks to determine the parameters of strong gravitational-lensing systems. Here we demonstrate a method for obtaining the uncertainties of these parameters. We review the framework of variational inference to obtain approximate posteriors of Bayesian neural networks and apply it to a network trained to estimate the parameters of the Singular Isothermal Ellipsoid plus external shear and total flux magnification. We show that the method can capture the uncertainties due to different levels of noise in the input data, as well as training and architecture-related errors made by the network. To evaluate the accuracy of the resulting uncertainties, we calculate the coverage probabilities of marginalized distributions for each lensing parameter. By tuning a single variational parameter, the dropout rate, we obtain coverage probabilities approximately equal to the confidence levels for which they were calculated, resulting in accurate and precise uncertainty estimates. Our results suggest that the application of approximate Bayesian neural networks to astrophysical modeling problems can be a fast alternative to Monte Carlo Markov Chains, allowing orders of magnitude improvement in speed.

The discovery of a five-image lensed quasar at z = 3.34 using PanSTARRS1 and Gaia

Abstract We report the discovery, spectroscopic confirmation and mass modelling of the gravitationally lensed quasar system PS J0630−1201. The lens was discovered by matching a photometric quasar catalogue compiled from Pan-STARRS1 and Wide-field Infrared Survey Explorer photometry to the Gaia data release 1 catalogue, exploiting the high spatial resolution of the latter (full width at half-maximum ∼0.1 arcsec) to identify the three brightest components of the lensed quasar system. Follow-up spectroscopic observations with the William Herschel Telescope confirm the multiple objects are quasars at redshift zq = 3.34. Further follow-up with Keck adaptive optics high-resolution imaging reveals that the system is composed of two lensing galaxies and the quasar is lensed into an ∼2.8 arcsec separation four-image cusp configuration with a fifth image clearly visible, and a 1.0 arcsec arc due to the lensed quasar host galaxy. The system is well modelled with two singular isothermal ellipsoids, reproducing the position of the fifth image. We discuss future prospects for measuring time delays between the images and constraining any offset between mass and light using the faintly detected Einstein arcs associated with the quasar host galaxy.

THE BOSS EMISSION-LINE LENS SURVEY. IV. SMOOTH LENS MODELS FOR THE BELLS GALLERY SAMPLE*

ABSTRACT We present Hubble Space Telescope F606W-band imaging observations of 21 galaxy-Lyα emitter lens candidates in the Baryon Oscillation Spectroscopic Survey Emission-Line Lens Survey (BELLS) for the GALaxy-Lyα EmitteR sYstems (BELLS GALLERY) survey. Seventeen systems are confirmed to be definite lenses with unambiguous evidence of multiple imaging. The lenses are primarily massive early-type galaxies (ETGs) at redshifts of approximately 0.55, while the lensed sources are Lyα emitters (LAEs) at redshifts from two to three. Although most of the lens systems are well fit by smooth lens models consisting of singular isothermal ellipsoids in an external shear field, a thorough exploration of dark substructures in the lens galaxies is required. The Einstein radii of the BELLS GALLERY lenses are, on average, 60% larger than those of the BELLS lenses because of the much higher source redshifts. This will allow for a detailed investigation of the radius evolution of the mass profile in ETGs. With the aid of the average ∼13× lensing magnification, the LAEs are frequently resolved into individual star-forming knots with a wide range of properties. They have characteristic sizes from less than 100 pc to several kiloparsecs, rest-frame far-UV apparent AB magnitudes from 29.6 to 24.2, and typical projected separations of 500 pc to 2 kpc.

The influence of weak lensing on measurements of the Hubble constant with quad-image gravitational lenses

We investigate the influence of matter along the line of sight and in the strong lens vicinity on the properties of quad image configurations and on the measurements of the Hubble constant (H0). We use simulations of light propagation in a nonuniform universe model with the distribution of matter in space based on the data from Millennium Simulation. For a given strong lens and haloes in its environment we model the matter distribution along the line of sight many times, using different combinations of precomputed deflection maps representing subsequent layers of matter on the path of rays. We fit the simulated quad image configurations with time delays using nonsingular isothermal ellipsoids (NSIE) with external shear as lens models, treating the Hubble constant as a free parameter. We get a large artificial catalog of lenses with derived values of the Hubble constant, Hfit. The average and median of Hfit differ from the true value used in simulations by < 0.5 km/s/Mpc which includes the influence of matter along the line of sight and in the lens vicinity, and uncertainty in lens parameters, except the slope of the matter distribution, which is fixed. The characteristic uncertainty of Hfit is ~3 km/s/Mpc. Substituting the lens shear parameters with values estimated from the simulations reduces the uncertainty to ~2 km/s/Mpc.

Extensive light profile fitting of galaxy-scale strong lenses

We investigate the merits of a massive forward modeling of ground-based optical imaging as a diagnostic for the strong lensing nature of Early-Type Galaxies, in the light of which blurred and faint Einstein rings can hide. We simulate several thousand mock strong lenses under ground- and space-based conditions as arising from the deflection of an exponential disk by a foreground de Vaucouleurs light profile whose lensing potential is described by a Singular Isothermal Ellipsoid. We then fit for the lensed light distribution with sl_fit after having subtracted the foreground light emission off (ideal case) and also after having fitted the deflector's light with galfit. By setting thresholds in the output parameter space, we can decide the lens/not-a-lens status of each system. We finally apply our strategy to a sample of 517 lens candidates present in the CFHTLS data to test the consistency of our selection approach. The efficiency of the fast modeling method at recovering the main lens parameters like Einstein radius, total magnification or total lensed flux, is quite comparable under CFHT and HST conditions when the deflector is perfectly subtracted off (only possible in simulations), fostering a sharp distinction between the good and the bad candidates. Conversely, for a more realistic subtraction, a substantial fraction of the lensed light is absorbed into the deflector's model, which biases the subsequent fitting of the rings and then disturbs the selection process. We quantify completeness and purity of the lens finding method in both cases. This suggests that the main limitation currently resides in the subtraction of the foreground light. Provided further enhancement of the latter, the direct forward modeling of large numbers of galaxy-galaxy strong lenses thus appears tractable and could constitute a competitive lens finder in the next generation of wide-field imaging surveys.

THE SPATIAL STRUCTURE OF YOUNG STELLAR CLUSTERS. I. SUBCLUSTERS

The clusters of young stars in massive star-forming regions show a wide range of sizes, morphologies, and numbers of stars. Their highly subclustered structures are revealed by the MYStIX project's sample of 31,754 young stars in nearby sites of star formation (regions at distances <3.6 kpc that contain at least one O-type star.) In 17 of the regions surveyed by MYStIX, we identify subclusters of young stars using finite mixture models -- collections of isothermal ellipsoids that model individual subclusters. Maximum likelihood estimation is used to estimate the model parameters, and the Akaike Information Criterion is used to determine the number of subclusters. This procedure often successfully finds famous subclusters, such as the BN/KL complex behind the Orion Nebula Cluster and the KW-object complex in M17. A catalog of 142 subclusters is presented, with 1 to 20 subclusters per region. The subcluster core radius distribution for this sample is peaked at 0.17 pc with a standard deviation of 0.43 dex, and subcluster core radius is negatively correlated with gas/dust absorption of the stars -- a possible age effect. Based on the morphological arrangements of subclusters, we identify four classes of spatial structure: long chains of subclusters, clumpy structures, isolated clusters with a core-halo structure, and isolated clusters well fit by a single isothermal ellipsoid.

Background, foreground and nearby matter influence on strong gravitational lenses

We investigate strong lensing by non-singular finite isothermal ellipsoids taking into account the influence of the matter along the line of sight and in the close lens vicinity. We compare three descriptions of light propagation: the full approach taking into account all matter inhomogeneities along the rays, the single plane approach, where we take into account the influence of the strong lens neighbours but neglect the foreground and background objects, and the single lens approach. In each case we simulate many strong lensing configurations placing a point source at the same redshift but in different locations inside the region surrounded by caustics. We further analyze configurations of four or five images. For every simulated strong lensing configuration we attempt to fit a simplified lens model using a single isothermal ellipsoid or a single isothermal ellipsoid with external shear. The single lens fits to configurations obtained in the full approach are rejected in majority of cases with 95% significance. For configurations obtained in the single plane approach the rejection rate is substantially lower. Also the inclusion of external shear in simplified modeling improves the chances of obtaining acceptable fits, but the problem is not solved completely. The quantitative estimates of the rates of rejection of simplified models depend on the required accuracy of the models, and we present few illustrative examples, which show that both matter close to the lens and matter along the rays do have important influence on lens modeling. We also estimate the typical value of the external shear and compare the fitted parameters of the simplified models with the parameters of the lenses used in the simulations.

Effects of supermassive binary black holes on gravitational lenses

Recent observations indicate that many if not all galaxies host massive central black holes (BHs). In this paper we explore the influence of supermassive binary black holes (SMBBHs) on their actions as gravitational lenses. When lenses are modelled as singular isothermal ellipsoids, binary black holes change the critical curves and caustics differently as a function of distance. Each black hole can in principle create at least one additional image, which, if observed, provides evidence of black holes. By studying how SMBBHs affect the cumulative distribution of magnification for images created by black holes, we find that the cross section for at least one such additional image to have a magnification larger than $10^{-5}$ is comparable to the cross section for producing multiple-images in singular isothermal lenses. Such additional images may be detectable with high-resolution and large dynamic range maps of multiply-imaged systems from future facilities, such as the SKA. The probability of detecting at least one image (two images) with magnification above $10^{-3}$ is $\sim 0.2 \fBH$ ($\sim 0.05 \fBH$) in a multiply-imaged lens system, where $\fBH$ is the fraction of galaxies housing binary black holes. We also study the effects of SMBBHs on the core images when galaxies have shallower central density profiles (modelled as non-singular isothermal ellipsoids). We find that the cross section of the usually faint core images is further suppressed by SMBBHs. Thus their presence should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.

Parametric strong gravitational lensing analysis of Abell 1689

We have derived the mass distribution of galaxy cluster Abell 1689 within 0.3 h−170 Mpc of the cluster centre using its strong lensing (SL) effect on 32 background galaxies, which are mapped in altogether 107 multiple images. The multiple images are based on some from the literature with modifications to both include new and exclude some of the original image systems. The cluster profile is explored further out to ∼2.5 h−170 Mpc with weak lensing (WL) shear measurements from the literature. The masses of ∼200 cluster galaxies are measured with the Fundamental Plane (FP) in order to model accurately the small-scale mass structure in the cluster. The cluster galaxies are modelled as elliptical truncated isothermal spheres. The scalings of the truncation radii with the velocity dispersions of galaxies are assumed to match those of: (i) field galaxies; and (ii) theoretical expectations for galaxies in dense environments. The dark matter (DM) component of the cluster is described by either non-singular isothermal ellipsoids (NSIE) or elliptical versions of the universal DM profile (elliptical Navarro, Frenk & White, ENFW). To account for substructure in the DM we allow for two DM haloes. The fitting of a non-singular isothermal sphere (NSIS) to the smooth DM component results in a velocity dispersion of 1450+39−31 km s−1 and a core radius of 77+10−8 h−170 kpc, while a Navarro, Frenk & White (NFW) profile has an r200 of 2.86 ± 0.16 h−170 Mpc(M200= 3.2 × 1015 M⊙ h70) and a concentration of 4.7+0.6−0.5. The total mass profile is well described by either a NSIS profile with σ= 1514+18−17 km s−1 and a core radius of rc= 71 ± 5 h−170 kpc, or an NFW profile with C= 6.0 ± 0.5 and r200= 2.82 ± 0.11 h−170 Mpc(M200= 3.0 × 1015 M⊙ h70). The errors are assumed to be due to the error in assigning masses to the individual galaxies in the galaxy component. Their small size is due to the very strong constraints imposed by multiple images and the ability of the smooth DM component to adjust to uncertainties in the galaxy masses. The agreement in the total mass profile between this work and that of the literature is better than 1σ at all radii, despite the considerable differences in the methodology used. Using the same image configuration as used in the literature, we obtain a SL model that is superior to some in the literature (rms of 2.7 compared to 3.2 arcsec). This is very surprising considering the larger freedom in the surface mass profile in their grid modelling. The difference is most likely a result of the careful inclusion of the cluster galaxies. Using also WL shear measurements from the literature, we can constrain the profile further out to r∼ 2.5 h−170 Mpc. The best-fitting parameters change to σ= 1499 ± 15 km s−1 and rc= 66 ± 5 h−170 kpc for the NSIS profile and C= 7.6 ± 0.5 and r200= 2.55 ± 0.07 h−170 Mpc(M200= 2.3 × 1015 M⊙ h70) for the NFW profile.

Smooth Particle Lensing

We present a numerical technique to compute the gravitational lensing induced by simulated haloes. It relies on a 2D-Tree domain decomposition in the lens plane combined with a description of N-Body particles as extended clouds with a non-singular density. This technique is made fully adaptive by the use of a density-dependent smoothing which allows one to probe the lensing properties of haloes from the densest regions in the center or in substructures to the low-density regions in the outskirts. 'Smooth Particle Lensing' has some promising features. First, the deflection potential, the deflection angles, the convergence and the shear are direct and separate end-products of the SPL calculation and can be computed at an arbitrary distribution of points on the lens plane. Second, this flexibility avoids the use of interpolation or a finite differentiation procedure on a grid, does not require padding the region with zeros and focuses the computing power on relevant regions. The SPL algorithm is tested by populating isothermal spheres and ellipsoids with particles and then comparing the lensing calculations to the classical FFT-based technique and analytic solutions. We assess issues related to the resolution of the lensing code and the limitations set by the simulations themselves. We conclude by discussing how SPL can be used to predict the impact of substructures on strong lensing and how it can be generalized to weak lensing and cosmic shear simulations.

The Sloan Lens ACS Survey. II. Stellar Populations and Internal Structure of Early‐Type Lens Galaxies

We use HST images to derive effective radii and effective surface brightnesses of 15 early-type (E+S0) lens galaxies identified by the SLACS Survey. Our measurements are combined with stellar velocity dispersions from the SDSS database to investigate for the first time the distribution of lens galaxies in the fundamental plane (FP) space. Accounting for selection effects (top priority to the largest Einstein radii and thus approximately to the largest velocity dispersions, σ ≳ 240 km s-1) and for passive evolution, the distribution of the lens galaxies inside the FP is indistinguishable from that of the parent sample of SDSS galaxies. We conclude that SLACS lenses are a fair sample of high velocity dispersion E+S0s. By comparing the central stellar velocity dispersion (σ) with the velocity dispersion that best fits the lensing models (σSIE) we find ⟨fSIE⟩ ≡ ⟨σ/σSIE⟩ = 1.01 ± 0.02 with 0.065 rms scatter. We infer that within the Einstein radii (typically Re/2) the SLACS lenses are very well approximated by isothermal ellipsoids, requiring a fine tuning of the stellar and dark matter distribution (the bulge-halo ``conspiracy''). Interpreting the offset from the local FP in terms of evolution of the stellar mass-to-light ratio, we find d log(M/LB)/dz = -0.69 ± 0.08 (rms 0.11) consistent with the rate found for field E+S0s and with most of the stars being old (zf > 2) and less than ~10% of the stellar mass having formed below z = 1. We discuss our results in the context of formation mechanisms such as collisionless (``dry'') mergers.

The Cosmic Lens All-Sky Survey: statistical strong lensing, cosmological parameters, and global properties of galaxy populations

Extensive analyses of statistical strong gravitational lensing are performed based on the final Cosmic Lens All-Sky Survey (CLASS) well-defined statistical sample of flat-spectrum radio sources and current estimates of galaxy luminosity functions per morphological type. The analyses are carried out under the assumption that galactic lenses are well-approximated by singular isothermal ellipsoids and early-type galaxies evolved passively since redshift z ∼ 1. Two goals of the analyses are: (i) to constrain cosmological parameters independently of other techniques (e.g. Type Ia supernovae magnitude-redshift relation, cosmic microwave background anisotropies, galaxy matter power spectra); and (ii) to constrain the characteristic line-of-sight velocity dispersion and the mean projected mass ellipticity for the early-type galaxy population. Depending on how the late-type galaxy population is treated (i.e. whether its characteristic velocity dispersion is constrained or not), we find for a flat universe with a classical cosmological constant that the matter fraction of the present critical density Ω m = 0.31 + 0 . 2 7 -0.14 (68 per cent) for the unconstrained case or 0.40 + 0 . 2 8 -0.16 (68 per cent) for the constrained case, with an additional systematic uncertainty of 0.11 arising from the present uncertainty in the distribution of CLASS sources in redshift and flux density. For a flat universe with a constant equation of state for dark energy w = p x (pressure)/ρ x (energy density) and the prior constraint ω ≥ -1, we find that -1 ≤ ω < -0.55 + 0 . 1 8 -0.11 (68 per cent) for the unconstrained case or -1 ≤ w < -0.41 + 0 . 2 8 -0.16 (68 per cent) for the constrained case, where w = -1 corresponds to a classical cosmological constant. The determined value of the early-type characteristic velocity dispersion [σ ( e ) *] depends on the faint-end slope of the early-type luminosity function [α ( 3 ) ] and the intrinsic shape distribution of galaxies; for equal frequencies of oblates and prolates, we find that σ ( e ) *(0.3 ≤ z ≤ 1) = 198 + 2 2 -18 km s - 1 (68 per cent) for a 'steep' α ( 3 ) = -1 or σ ( e ) *(0.3 ≤ z ≤ 1) = 181 + 1 8 -15 km s - 1 (68 per cent) for a 'shallow' α ( e ) = -0.54. Finally, from the relative frequencies of doubly imaged sources and quadruply imaged sources, we find that a mean projected mass ellipticity of early-type galaxies ∈ m a s s = 0.42 with a 68 per cent lower limit of 0.28 assuming equal frequencies of oblates and prolates.

Images for an isothermal ellipsoidal gravitational lens from a single real algebraic equation

We present explicit expressions for the lens equation for a cored isothermal ellipsoidal gravitational lens as a single real sixth-order algebraic equation in two approaches; 2-dimensional Cartesian coordinates and 3-dimensional polar ones. We find a condition for physical solutions which correspond to at most five images. For a singular isothermal ellipsoid, the sixth-order equation is reduced to fourth-order one for which analytic solutions are well-known. Furthermore, we derive analytic criteria for determining the number of images for the singular lens, which give us simple expressions for the caustics and critical curves. The present formulation offers a useful way for studying galaxy lenses frequently modeled as isothermal ellipsoids.

Flux Ratios as a Probe of Dark Substructures in Quadruple-Image Gravitational Lenses

We demonstrate that the flux ratios of quadruple-image lensed quasars provide a powerful means of probing the small-scale structure of dark matter halos. A family of smooth lens models can precisely predict certain combinations of flux ratios using only the positions of the images and the lens as inputs. Using five observed lens systems, we show that real galaxies cannot be described by smooth singular isothermal ellipsoids, nor by the more general elliptical power-law potentials. Large-scale distortions from the elliptical models cannot yet be ruled out. Nevertheless, we find by comparing with simulations that the data can be accounted for by a significant (≳5%) amount of dark substructures within a projected distance of several kiloparsecs from the center of the lenses. This interpretation favors the cold dark matter model over the warm or self-interacting dark matter models.

Why Is the Fraction of Four‐Image Radio Lens Systems So High?

We investigate the frequency of two- and four-image gravitational lens systems in the Jodrell-VLA Astrometric Survey (JVAS) and Cosmic Lens All-Sky Survey (CLASS), and the possible implications for dark matter halo properties. A simple lensing statistics model, which describes lens galaxies as singular isothermal ellipsoids with a projected axis ratio distribution derived from the surface brightness ellipticities of early-type galaxies in the Coma cluster, is ruled out at the 98% level since it predicts too few four-image lenses (quads). We consider a range of factors that may be increasing the frequency of radio quads, including external shear fields, mass distributions flatter than the light, shallow lensing mass profiles, finite core radii, satellite galaxies, and alterations to the luminosity function for faint flat-spectrum radio sources. We find that none of these mechanisms provide a compelling solution to the quad problem on their own while remaining consistent with other observational constraints.

The Gravitational Lens System B1030+074. Discovery and Follow-up

We report the discovery of a new double image gravitational lens system B1030+074 which was found during the Jodrell Bank - VLA Astrometric Survey (JVAS). We have collected extensive radio data on the system using the VLA, MERLIN, the EVN and the VLBA as well as HST WFPC2 and NICMOS observations. The lensed images are separated by 1.56 arcseconds and their flux density ratio at centimetric wavelengths is approximately 14:1 although the ratio is slightly frequency dependent and the images appear to be time variable. The HST pictures show both the lensed images and the lensing galaxy close to the weaker image. The lensing galaxy has substructure which could be either part of the galaxy or a companion object. We have modeled B1030+074 using a Singular Isothermal Ellipsoid that yielded a time delay of 156/h50 days. This lens is likely to be suitable for the measurement of the Hubble constant.

Shear and Ellipticity in Gravitational Lenses

Galaxies modeled as singular isothermal ellipsoids with an axis ratio distribution similar to the observed axis ratio distribution of E and S0 galaxies are statistically consistent with both the observed numbers of two-image and four-image lenses and the inferred ellipticities of individual lenses. However, no four-image lens is well fit by the model (typical $\chi^2/N_{dof} \sim 20$), the axis ratio of the model can be significantly different from that of the observed lens galaxy, and the major axes of the model and the galaxy may be slightly misaligned. We found that models with a second, independent, external shear axis could fit the data well (typical $\chi^2/N_{dof} \sim 1$), while adding the same number of extra parameters to the radial mass distribution does not produce such a dramatic improvement in the fit. An independent shear axis can be produced by misalignments between the luminous galaxy and its dark matter halo, or by external shear perturbations due to galaxies and clusters correlated with the primary lens or along the line of sight. We estimate that the external shear perturbations have no significant effect on the expected numbers of two-image and four-image lenses, but that they can be important perturbations in individual lens models. However, the amplitudes of the external shears required to produce the good fits are larger than our estimates for typical external shear perturbations (10-15% shear instead of 1-3% shear) suggesting that the origin of the extra angular structure must be intrinsic to the primary lens galaxy in most cases.