Lensing Statistics Research Articles

Galaxy-CMB and galaxy-galaxy lensing on large scales: Sensitivity to primordial non-Gaussianity

A convincing detection of primordial non-Gaussianity in the local form of the bispectrum, whose amplitude is given by the fNL parameter, offers a powerful test of inflation. In this paper we calculate the modification of two-point cross-correlation statistics of weak lensing - galaxy-galaxy lensing and galaxy-Cosmic Microwave Background (CMB) cross-correlation - due to fNL. We derive and calculate the covariance matrix of galaxy-galaxy lensing including cosmic variance terms. We focus on large scales (l<100) for which the shape noise of the shear measurement becomes irrelevant and cosmic variance dominates the error budget. For a modest degree of non-Gaussianity, fNL=+/-50, modifications of the galaxy-galaxy lensing signal at the 10% level are seen on scales R~300 Mpc, and grow rapidly toward larger scales as \propto R^2. We also see a clear signature of the baryonic acoustic oscillation feature in the matter power spectrum at ~150 Mpc, which can be measured by next-generation lensing experiments. In addition we can probe the local-form primordial non-Gaussianity in the galaxy-CMB lensing signal by correlating the lensing potential reconstructed from CMB with high-z galaxies. For example, for fNL=+/-50, we find that the galaxy-CMB lensing cross power spectrum is modified by ~10% at l~40, and by a factor of two at l~10, for a population of galaxies at z=2 with a bias of 2. The effect is greater for more highly biased populations at larger z; thus, high-z galaxy surveys cross-correlated with CMB offer a yet another probe of primordial non-Gaussianity.

Breaking the degeneracy: Optimal use of three-point weak lensing statistics

We study the optimal use of three-point statistics in the analysis of weak lensing by large-scale structure. The three-point statistics have long been advocated as a powerful tool to break measured degeneracies between cosmological parameters. Using ray-tracing simulations, incorporating important survey features such as a realistic depth-dependent redshift distribution, we find that a joint two- and three-point correlation function analysis is a much stronger probe of cosmology than the skewness statistic. We compare different observing strategies, showing that for a limited survey time there is an optimal depth for the measurement of three-point statistics, which balances statistical noise and cosmic variance against signal amplitude. We find that the chosen CFHTLS observing strategy was optimal and forecast that a joint two- and three-point analysis of the completed CFHTLS-Wide will constrain the amplitude of the matter power spectrum σ 8 to 10% and the matter density parameter Ω m to 17%, a factor of 2.5 improvement on the two-point analysis alone. Our error analysis includes all non-Gaussian terms, finding that the coupling between cosmic variance and shot noise is a non-negligible contribution which should be included in any future analytical error calculations.

Galaxy shear estimation from stacked images

Statistics of the weak lensing of galaxies can be used to constrain cosmology if the galaxy shear can be estimated accurately. In general this requires accurate modelling of unlensed galaxy shapes and the point spread function (PSF). I discuss suboptimal but potentially robust methods for estimating galaxy shear by stacking images such that the stacked image distribution is closely Gaussian by the central limit theorem. The shear can then be determined by radial fitting, requiring only an accurate model of the PSF rather than also needing to model each galaxy accurately. When noise is significant asymmetric errors in the centroid must be corrected, but the method may ultimately be able to give accurate un-biased results when there is a high galaxy density with constant shear. It provides a useful baseline for more optimal methods, and a test-case for estimating biases, though the method is not directly applicable to realistic data. I test stacking methods on the simple toy simulations with constant PSF and shear provided by the GREAT08 project, on which most other existing methods perform significantly more poorly, and briefly discuss generalizations to more realistic cases. In the appendix I discuss a simple analytic galaxy population model where stacking gives optimal errors in a perfect ideal case.

Sources of contamination to weak lensing tomography: redshift-dependent shear measurement bias

The current methods available to estimate gravitational shear from astronomical images of galaxies introduce systematic errors which can affect the accuracy of weak lensing cosmological constraints. We study the impact of KSB shape measurement bias on the cosmological interpretation of tomographic two-point weak lensing shear statistics. We use a set of realistic image simulations produced by the STEP collaboration to derive shape measurement bias as a function of redshift. We define biased two-point weak lensing statistics and perform a likelihood analysis for two fiducial surveys. We present a derivation of the covariance matrix for tomography in real space and a fitting formula to calibrate it for non-Gaussianity. We find the biased aperture mass dispersion is reduced by ~20% at redshift ~1, and has a shallower scaling with redshift. This effect, if ignored in data analyses, biases sigma_8 and w_0 estimates by a few percent. The power of tomography is significantly reduced when marginalising over a range of realistic shape measurement biases. For a CFHTLS-Wide-like survey, [Omega_m, sigma_8] confidence regions are degraded by a factor of 2, whereas for a KIDS-like survey the factor is 3.5. Our results are strictly valid only for KSB methods but they demonstrate the need to marginalise over a redshift-dependent shape measurement bias in all future cosmological analyses.

FAst STatistics for weak Lensing (FASTLens): fast method for weak lensing statistics and map making

With increasingly large data sets, weak lensing measurements are able to measure cosmological parameters with ever greater precision. However this increased accuracy also places greater demands on the statistical tools used to extract the available information. To date, the majority of lensing analyses use the two point-statistics of the cosmic shear field. These can either be studied directly using the two-point correlation function, or in Fourier space, using the power spectrum. But analyzing weak lensing data inevitably involves the masking out of regions or example to remove bright stars from the field. Masking out the stars is common practice but the gaps in the data need proper handling. In this paper, we show how an inpainting technique allows us to properly fill in these gaps with only $N \log N$ operations, leading to a new image from which we can compute straight forwardly and with a very good accuracy both the pow er spectrum and the bispectrum. We propose then a new method to compute the bispectrum with a polar FFT algorithm, which has the main advantage of avoiding any interpolation in the Fourier domain. Finally we propose a new method for dark matter mass map reconstruction from shear observations which integrates this new inpainting concept. A range of examples based on 3D N-body simulations illustrates the results.

THE SURVIVAL OF DARK MATTER HALOS IN THE CLUSTER Cl 0024+16

Theories of structure formation in a cold dark matter dominated universe predict that massive clusters of galaxies assemble from the hierarchical merging of lower mass subhalos. Exploiting strong and weak gravitational lensing signals inferred from panoramic Hubble Space Telescope imaging data, we present a high-resolution reconstruction of the mass distribution in the massive, lensing cluster Cl 0024+16 at z = 0.39. Applying galaxy–galaxy lensing techniques we track the fate of dark matter subhalos as a function of projected cluster-centric radius out to 5 Mpc, well beyond the virial radius. We report the first detection of the statistical lensing signal of dark matter subhalos associated with late-type galaxies in clusters. The mass of a fiducial dark matter halo that hosts an early-type L* galaxy varies from M = 6.3+2.7−2.0 × 1011 M☉ within r < 0.6 Mpc, 1.3+0.8−0.6 × 1012 M☉ within r < 2.9 Mpc, and increases further to M = 3.7+1.4−1.1 × 1012 M☉ in the outskirts. The mass of a typical dark matter subhalo that hosts an L* galaxy increases with projected cluster-centric radius in line with expectations from the tidal stripping hypothesis. The mass of a dark matter subhalo that hosts a late-type L* galaxy is 1.06+0.52−0.41 × 1012 M☉. Early-type galaxies appear to be hosted on average in more massive dark matter subhalos compared to late-type galaxies. Early-type galaxies also trace the overall mass distribution of the cluster whereas late-type galaxies are biased tracers. We interpret our findings as evidence for the active assembly of mass via tidal stripping in galaxy clusters. The mass function of dark matter subhalos as a function of projected cluster-centric radius is compared with an equivalent mass function derived from clusters in the Millennium Run simulation populated with galaxies using semianalytic models. The shape of the observationally determined mass functions based on an I-band-selected sample of cluster members and lensing data are in agreement with the shapes of the subhalo mass functions derived from the Millennium Run simulation. However, simulated subhalos appear to be more efficiently stripped than lensing observations suggest. This is likely an artifact of comparison with a dark matter only simulation. Future simulations that simultaneously follow the detailed evolution of the baryonic component during cluster assembly will be needed for a more detailed comparison.

Sources of contamination to weak lensing three-point statistics: constraints fromN-body simulations

We investigate the impact of the observed correlation between a galaxies shape and its surrounding density field on the measurement of third order weak lensing shear statistics. Using numerical simulations, we estimate the systematic error contribution to a measurement of the third order moment of the aperture mass statistic (GGG) from three-point intrinsic ellipticity correlations (III), and the three-point coupling between the weak lensing shear experienced by distant galaxies and the shape of foreground galaxies (GGI and GII). We find that third-order weak lensing statistics are typically more strongly contaminated by these physical systematics compared to second-order shear measurements, contaminating the measured three-point signal for moderately deep surveys with a median redshift z_m ~ 0.7 by ~ 15%. It has been shown that accurate photometric redshifts will be crucial to correct for this effect, once a model and the redshift dependence of the effect can be accurately constrained. To this end we provide redshift-dependent fitting functions to our results and propose a new tool for the observational study of intrinsic galaxy alignments. For a shallow survey with z_m ~ 0.4 we find III to be an order of magnitude larger than the expected cosmological GGG shear signal. Compared to the two-point intrinsic ellipticity correlation which is similar in amplitude to the two-point shear signal at these survey depths, third order statistics therefore offer a promising new way to constrain models of intrinsic galaxy alignments. Early shallow data from the next generation of very wide weak lensing surveys will be optimal for this type of study.

Testing the DGP model with gravitational lensing statistics

The self-accelerating braneworld model (DGP) seems to provide a simple alternative to the the standard $\Lambda$CDM cosmology to explain the current cosmic acceleration, which is strongly indicated by measurements of Type Ia supernovae, as well as other concordant observations. In this work, we investigate observational constraints on this scenario from gravitational lensing statistics using the Cosmic Lens All-Sky Survey (CLASS) lensing sample. We show that a large parameter space of the DGP model is in good agreement with this radio source gravitational lensing sample. In the flat case, $\Omega_\mathrm{K}=0$, the likelihood is maximized, ${\cal L}={\cal L_\mathrm{max}}$, for $\1 = 0.30_{-0.11}^{+0.19}$. If we relax the prior on $\Omega_\mathrm{K}$, the likelihood peaks at $\{\1,\2 \} \simeq \{0.29, 0.12\}$, just slightly in the region of open models. However the confidence contours are pretty elongated so that we can not discard either close or flat or open models

Strong lensing statistics and the power spectrum normalisation

We use semi-analytic modelling of the galaxy-cluster population and its strong lensing efficiency to explore how the expected abundance of large gravitational arcs on the sky depends on $\sigma_8$. Our models take all effects into account that have been shown to affect strong cluster lensing substantially, in particular cluster asymmetry, substructure, merging, and variations in the central density concentrations. We show that the optical depth for long and thin arcs increases by approximately one order of magnitude when $\sigma_8$ increases from 0.7 to 0.9, owing to a constructive combination of several effects. Models with high $\sigma_8$ are also several orders of magnitude more efficient in producing arcs at intermediate and high redshifts. Finally, we use realistic source number counts to quantitatively predict the total number of arcs brighter than several magnitude limits in the R and I bands. We confirm that, while $\sigma_8\sim0.9$ may come close to the known abundance of arcs, even $\sigma_8\sim0.8$ falls short by almost an order of magnitude in reproducing known counts. We conclude that, should $\sigma_8\sim0.8$ be confirmed, we would fail to understand the strong-lensing efficiency of the galaxy cluster population, and in particular the abundance of arcs in high-redshift clusters. We argue that early-dark energy or non-Gaussian density fluctuations may indicate one way out of this problem.

Validity of strong lensing statistics for constraints on the galaxy evolution model

We examine the usefulness of the strong lensing statistics to constrain the evolution of the number density of lensing galaxies by adopting the values of the cosmological parameters determined by recent Wilkinson Microwave Anisotropy Probe observation. For this purpose, we employ the lens-redshift test proposed by Kochanek and constrain the parameters in two evolution models, simple power-law model characterized by the power-law indexes νn and νv, and the evolution model by Mitchell et al. based on cold dark matter structure formation scenario. We use the well-defined lens sample from the Sloan Digital Sky Survey (SDSS) and this is similarly sized samples used in the previous studies. Furthermore, we adopt the velocity dispersion function of early-type galaxies based on SDSS DR1 and DR5. It turns out that the indexes of power-law model are consistent with the previous studies, thus our results indicate the mild evolution in the number and velocity dispersion of early-type galaxies out to z= 1. However, we found that the values for p and q used by Mitchell et al. are inconsistent with the presently available observational data. More complete sample is necessary to withdraw more realistic determination on these parameters.

A NEW SURVEY FOR GIANT ARCS

We report on the first results of an imaging survey to detect strong gravitational lensing targeting the richest clusters selected from the photometric data of the Sloan Digital Sky Survey (SDSS) with follow-up deep imaging observations from the Wisconsin–Indiana–Yale NOAO (WIYN) 3.5 m telescope and the University of Hawaii 88 inch telescope (UH88). The clusters are selected from an area of 8000 deg^2 using the red cluster sequence technique and span the redshift range 0.1 ≾ z ≾ 0.6, corresponding to a comoving cosmological volume of ~2 Gpc^3. Our imaging survey thus targets a volume more than an order of magnitude larger than any previous search. A total of 240 clusters were imaged of which 141 had sub-arcsecond image quality. Our survey has uncovered 16 new lensing clusters with definite giant arcs, an additional 12 systems for which the lensing interpretation is very likely, and 9 possible lenses which contain shorter arclets or candidate arcs which are less certain and will require further observations to confirm their lensing origin. Among these new systems are several of the most dramatic examples of strong gravitational lensing ever discovered, with multiple bright arcs at large angular separation. These will likely become “poster-child” gravitational lenses similar to Abell 1689 and CL0024+1654. The new lenses discovered in this survey will enable future systematic studies of the statistics of strong lensing and their implications for cosmology and our structure formation paradigm.

Is modified gravity required by observations? An empirical consistency test of dark energy models

We apply the technique of parameter splitting to existing cosmological data sets, to check for a generic failure of dark energy models. Given a dark energy parameter, such as the energy density ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ or equation of state $w$, we split it into two meta-parameters with one controlling geometrical distances, and the other controlling the growth of structure. Observational data spanning Type Ia Supernovae, the cosmic microwave background (CMB), galaxy clustering, and weak gravitational lensing statistics are fit without requiring the two meta-parameters to be equal. This technique checks for inconsistency between different data sets, as well as for internal inconsistency within any one data set (e.g., CMB or lensing statistics) that is sensitive to both geometry and growth. We find that the cosmological constant model is consistent with current data. Theories of modified gravity generally predict a relation between growth and geometry that is different from that of general relativity. Parameter splitting can be viewed as a crude way to parametrize the space of such theories. Our analysis of current data already appears to put sharp limits on these theories: assuming a flat universe, current data constrain the difference $\ensuremath{\Delta}{\ensuremath{\Omega}}_{\ensuremath{\Lambda}}={\ensuremath{\Omega}}_{\ensuremath{\Lambda}}(\mathrm{geom})\ensuremath{-}{\ensuremath{\Omega}}_{\ensuremath{\Lambda}}(\mathrm{grow})$ to be $\ensuremath{-}{0.0044}_{\ensuremath{-}0.0057\ensuremath{-}0.0119}^{+0.0058+0.0108}$ (68% and 95% C.L. respectively); allowing the equation of state $w$ to vary, the difference $\ensuremath{\Delta}w=w(\mathrm{geom})\ensuremath{-}w(\mathrm{grow})$ is constrained to be ${0.37}_{\ensuremath{-}0.36\ensuremath{-}0.53}^{+0.37+1.09}$. Interestingly, the region $w(\mathrm{grow})&gt;w(\mathrm{geom})$, which should be generically favored by theories that slow structure formation relative to general relativity, is quite restricted by data already. We find $w(\mathrm{grow})&lt;\ensuremath{-}0.80$ at $2\ensuremath{\sigma}$. As an example, the best-fit flat Dvali-Gabadadze-Porrati model approximated by our parametrization lies beyond the $3\ensuremath{\sigma}$ contour for constraints from all the data sets.

Strong lensing statistics in large, zlsim 0.2, surveys: bias in the lens galaxy population

Large current and future surveys, like the Two-degree Field Survey (2dF), the Sloan Digital Sky Survey (SDSS) and the proposed Kilo-Degree Survey (KIDS), are likely to provide us with many new strong gravitational lenses. Taking cosmological parameters as known, we calculate the expected lensing statistics of the galaxy population in large, low-redshift surveys. Galaxies are modelled using realistic, multiple components: a dark matter halo, a bulge component and disc. We use semi-analytic models of galaxies coupled with dark matter haloes in the Millennium Run to model the lens galaxy population. Replicating the selection criteria of the 2dF, we create a mock galaxy catalogue. We predict that a fraction of 7.5 ± 0.2 x 10 -5 of radio sources will be lensed by galaxies within a survey like the 2dF below z < 0.2. We find that proper inclusion of the baryonic component is crucial for calculating lensing statistics - pure dark matter haloes produce lensing cross-sections several orders of magnitude lower. With a simulated sample of lensed radio sources, the predicted lensing galaxy population consists mainly of ellipticals (∼80 per cent) with an average lens velocity dispersion of 191 ± 3 km s -1 , producing typical image separations of ∼1.5 arcsec. The lens galaxy population lies on the fundamental plane but its velocity dispersion distribution is shifted to higher values compared to all early-type galaxies. We show that magnification bias affects lens statistics very strongly and increases the 4:2 image ratio drastically. Taking this effect into account, we predict that the ratio of 4:2 image systems is 18 ± 5 per cent, marginally consistent with the observed ratio found in the Cosmic Lens All-Sky Survey (CLASS). We also find that the population of four-image lens galaxies differs markedly from the population of lens galaxies in two-image systems. We find that while most lenses tend to be ellipticals, galaxies that produce four-image systems preferentially tend to be lower velocity dispersion systems with more pronounced disc components. Our key result is the explicit demonstration that the population of lens galaxies differs markedly from the galaxy population as a whole: lens galaxies have a higher average luminosity and, for a given luminosity, they reside in more massive haloes than the overall sample of ellipticals. This bias restricts our ability to infer galaxy evolution parameters from a sample of lensing galaxies.

The Cosmic Lens All-Sky Survey parent population - I. Sample selection and number counts

We present the selection of the Jodrell Bank Flat-spectrum (JBF) radio source sample, which is designed to reduce the uncertainties in the Cosmic Lens All-Sky Survey (CLASS) gravitational lensing statistics arising from the lack of knowledge about the parent population luminosity function. From observations at 4.86 GHz with the Very Large Array, we have selected a sample of 117 flat-spectrum radio sources with flux densities greater than 5 mJy. These sources were selected in a similar manner to the CLASS complete sample and are therefore representative of the parent population at low flux densities. The vast majority (~90 per cent) of the JBF sample are found to be compact on the arcsecond scales probed here and show little evidence of any extended radio jet emission. Using the JBF and CLASS complete samples we find the differential number counts slope of the parent population above and below the CLASS 30 mJy flux density limit to be -2.07+/-0.02 and -1.96+/-0.12, respectively.

Constraints on the velocity profiles of galaxies from strong lensing statistics and semi-analytical modelling of galaxy formation

Semi-analytical models of galaxy formation can be used to predict the evolution of the number density of early-type galaxies as a function of the circular velocity at the virial radius, v_{c,vir}. Gravitational lensing probability and separation distribution on the other hand are sensitive to the velocity dispersion (or circular velocity) at about the effective radius. We adopt the singular isothermal ellipsoid (SIE) lens model to estimate the velocity dispersion at the effective radius. We use radio lenses from the Cosmic Lens All-Sky Survey and the PMN-NVSS Extragalactic Lens Survey to study how the velocity dispersions, \sigma_SIE, are related to v_{c,vir}. When we include both the lensing probability and separation distribution as our lensing constraints, we find \sigma_SIE /(200\kms) = [(1.17_{-0.26}^{+0.40}) v_{c,vir}/ (200\kms)]^{0.22^{+0.05}_{-0.04}} for 200\kms \la \sigma_SIE \la 260\kms; at \sigma_SIE = 200\kms, the ratio \sqrt{2} \sigma_SIE / v_{c,vir} is about 1.65^{+0.57}_{-0.37} (68% CL) but decreases to 0.65_{-0.12}^{+0.15} (68% CL) for \sigma_SIE = 260\kms. These results are consistent with those of Seljak (2002) obtained from galaxy-galaxy weak lensing for galaxies of around L_*. However, our results clearly suggest that the ratio must vary significantly as \sigma_SIE is varied and are marginally discrepant with the Seljak results at \sigma_SIE = 260\kms. The scaling \sigma_SIE \propto v_{c,vir}^{0.22\pm 0.05} is broadly consistent with those from galaxy occupation statistics studies and the most recent galaxy-galaxy weak lensing study. (Abridged)

Arc sensitivity to cluster ellipticity, asymmetries, and substructures

Aims.We investigate how ellipticity, asymmetries and substructures separately affect the ability of galaxy clusters to produce strong lensing events, i.e. gravitational arcs, and how they influence the arc morphologies and fluxes. This is important for studies aiming, for example, at constraining cosmological parameters from statistical lensing, or at determining the inner structure of galaxy clusters through gravitational arcs.

Lensing Reconstruction Using Redshifted 21 Centimeter Fluctuations

We investigate the potential of second generation measurements of redshifted 21 cm radiation from before and during the epoch of reionization (EOR) to reconstruct the matter density fluctuations along the line of sight. To do so we generalize the quadratic methods developed for the cosmic microwave background (CMB) to 21 cm fluctuations. We show that the three-dimensional signal can be decomposed into a finite number of line-of-sight Fourier modes that contribute to the lensing reconstruction. Our formalism properly takes into account correlations along the line of sight and uses all the information contained in quadratic combinations of the signal. In comparison with the CMB, 21 cm fluctuations have the disadvantage of a relatively scale invariant unlensed power spectrum, which suppresses the lensing effect. The smallness of the lensing effect is compensated by using information from a range of observed redshifts. We estimate the size of experiments that are needed to measure this effect. With a square kilometer of collecting area and a maximal baseline of 3 km, one can achieve lensing reconstruction noise levels an order of magnitude below CMB quadratic estimator constraints at L = 1000, and map the deflection field out to less then a tenth of a degree (L > 2000) within a season of observations on one field. Statistical lensing power spectrum detections will be possible to subarcminute scales, even with the limited sky coverage that currently conceived experiments have. One should be able to improve constraints on cosmological parameters by using this method. With larger collecting areas or longer observing times, one could probe arcminute scales of the lensing potential and thus individual clusters. We address the effect that foregrounds might have on lensing reconstruction with 21 cm fluctuations.

Designing weak lensing surveys: a generalized eigenmode analysis

We study the estimators of various second-order weak lensing statistics such as the shear correlation functions ξ± and the aperture mass dispersion (M 2 ap ) which can directly be constructed from weak lensing shear maps. We compare the efficiency with which these estimators can be used to constrain cosmological parameters. To this end we introduce the Karhunen-Loeve (KL) eigenmode analysis techniques for weak lensing surveys. These tools are shown to be very effective as a diagnostics for optimizing survey strategies. The usefulness of these tools to study the effect of angular binning, the depth and width of the survey and noise contributions due to intrinsic ellipticities and number density of source galaxies on the estimation of cosmological parameters is demonstrated. Results from independent analysis of various parameters and joint estimations are compared. We also study how degeneracies among various cosmological and survey parameters affect the eigenmodes associated with these parameters.

Radial distribution and strong lensing statistics of satellite galaxies and substructure using high-resolution CDM hydrodynamical simulations

We analyse the number density and radial distribution of substructures and satellite galaxies using cosmological simulations that follow the gas dynamics of a baryonic component, including shock heating, radiative cooling and star formation within the hierarchical concordance Lambda cold dark matter model. We find that the dissipation of the baryons greatly enhances the survival of subhaloes, especially in the galaxy core, resulting in a radial distribution of satellite galaxies that closely follows the overall mass distribution. Hydrodynamical simulations are necessary to resolve the adiabatic contraction and dense cores of galaxies, resulting in a total number of satellites a factor of 2 larger than that found in pure dark matter simulation, in good agreement with the observed spatial distribution of satellite galaxies within galaxies and clusters. Convergence tests show that the cored distribution found by previous authors in pure N-body simulations was due to the physical overmerging of dark matter only structures. We proceed to use a ray-shooting technique in order to study the impact of these additional substructures on the number of violations of the cusp caustic magnification relation. We develop a new approach to try to disentangle the effect of substructures from the intrinsic discreteness of N-body simulations. Even with the increased number of substructures in the centres of galaxies, we are not able to reproduce the observed high numbers of discrepancies observed in the flux ratios of multiply lensed quasars.

Characterizing a cosmic string with the statistics of string lensing

The deep imaging of the field of an observed lensing event by a cosmic string reveals many additional lensing events. We study the statistics of such string lensing. We derive explicit expressions for the distributions of image separations of lensing by a cosmic string and point out that they are quite sensitive to parameters which characterize the cosmic string, such as the redshift and tension of the cosmic string. Thus the statistics of string lensing events add new important information on the cosmic string which cannot be obtained from the detailed investigation of one lensing event.