Gravitational lensing has developed into one of the most powerful tools for the analysis of the dark universe. This review summarises the theory of gravitational lensing, its main current applications and representative results achieved so far. It has two parts. In the first, starting from the equation of geodesic deviation, the equations of thin and extended gravitational lensing are derived. In the second, gravitational lensing by stars and planets, galaxies, galaxy clusters and large-scale structures is discussed and summarised.

Gravitational Lensing

This paper presents the results of a photometric redshift study of galaxies in the Hubble Deep Field (HDF). The method of determining redshifts from broadband colors is described, and the dangers inherent in using it to estimate redshifts, particularly at very high z, are discussed. In particular, the need for accurate high-z spectral energy distributions is illustrated. The validity of our photometric redshift technique is demonstrated both by direct verification with available HDF spectroscopic data and by comparisons of luminosity functions and luminosity densities with those obtained from z 3. A population of z < 0.5 star-forming dwarfs is observed to M_{F450W_{AB}} = -11. Our results are discussed in the context of recent developments in the understanding of galaxy evolution.

Evolution of the Galaxy Population Based on Photometric Redshifts in the Hubble Deep Field

We study the expected properties and statistics of giant arcs produced by galaxy clusters in a ΛCDM (cold dark matter) universe and investigate how the characteristics of CDM clusters determine the properties of the arcs they generate. Because of the triaxiality and substructure of CDM halos, the giant-arc cross section for individual clusters varies by more than an order of magnitude as a function of viewing angle. In addition, the shallow density cusps and triaxiality of CDM clusters cause systematic alignments of giant arcs that should be testable with larger samples from forthcoming lensing surveys. We compute the predicted statistics of giant arcs for the ΛCDM model and compare them with results from previous surveys. The predicted arc statistics are in excellent agreement with the numbers of giant arcs observed around low-redshift (0.2 z 0.6) clusters from the Extended Medium-Sensitivity Survey sample; however, there are hints of a possible excess of arcs observed around high-redshift (z 0.6) clusters. This excess, if real, appears to be due to the presence of highly massive or concentrated clusters at high redshifts.

https://iopscience.iop.org/article/10.1086/420960/pdf

Statistics of giant arcs in galaxy clusters

We use the semianalytical approach to analyze gravitational lensing of remote quasars by foreground dark halos in various cold dark matter (CDM) cosmologies in order to determine the sensitivity of the predictions for probabilities of image separations to the input assumptions regarding the properties of halos and cosmological models The power spectrum of primordial fluctuations is normalized by the cluster abundance constraints The mass function of dark halos is assumed to be given by the Press-Schechter function The mass density profile of dark halos is alternatively taken to be the singular isothermal sphere (SIS), the Navarro-Frenk-White (NFW) profile, or the generalized NFW profile The cosmologies being considered include the Einstein-de Sitter model (SCDM), the open model (OCDM), and the flat Λ-model (LCDM) As expected, we find that the lensing probability is extremely sensitive to the mass density profile of lenses (dark halos) and somewhat less so to the mean mass density in the universe and the amplitude of primordial fluctuations NFW halos are very much less effective in producing multiple images than SIS halos For NFW lenses, the SCDM model produces fewer lensing events than the OCDM/LCDM models by 2 orders of magnitude For SIS lenses, the SCDM model produces more lensing events with small splitting angles and fewer lensing events with large splitting angles than the OCDM/LCDM models, which is due to the fact that for large-mass halos, the Press-Schechter function is very sensitive to the amplitude of primordial fluctuations In all cases the difference between the OCDM model and the LCDM model is not dramatic None of these models are consistent with current observations: the SIS models predict too many large splitting lenses, while the NFW models predict too few small splitting lenses Essentially, the observed high ratio of small splitting to large splitting lenses is not predicted correctly This indicates that there must be at least two populations of halos in the universe: small-mass halos with a steep inner density slope and large-mass halos with a shallow inner density slope A combination of SIS and NFW halos can reasonably reproduce the current observations if we choose the mass for the transition from SIS to NFW to be ~1013 M☉, as might plausibly occur because of baryonic cooling and contraction in lower mass systems Additionally, there is a tendency for CDM models to have too much power on small scales, ie, too much mass concentration From our sensitivity studies it appears that the cures proposed for other apparent difficulties of CDM would help here as well, an example being the warm dark matter variant, which is shown to produce fewer large splitting lenses than the corresponding CDM model by 1 order of magnitude

https://iopscience.iop.org/article/10.1086/338330/pdf

Semianalytical Models for Lensing by Dark Halos. I. Splitting Angles

We study the impact of merger events on the strong lensing properties of galaxy clusters. Previous lensing simulations were not able to resolve dynamical time scales of cluster lenses, which arise on time scales which are of order a Gyr. In this case study, we first describe qualitatively with an analytic model how some of the lensing properties of clusters are expected to change during merging events. We then analyse a numerically simulated lens model for the variation in its efficiency for producing both tangentia l and radial arcs while a massive substructure falls onto the main cluster body. We find that: ( 1) during the merger, the shape of the critical lines and caustics changes substantially; (2) the lensing cross sections for long and thin arcs can grow by one order of magnitude and reach their maxima when the extent of the critical curves is largest; (3) the cross section for radial arcs also grows, but the cluster can efficiently produce this kind of arcs only while the merging substructure crosses the main cluster centre; (4) while the arc cross sections pass through their m axima as the merger proceeds, the cluster’s X-ray emission increases by a factor of ∼ 5. Thus, we conclude that accounting for these dynamical processes is very important for arc statist ics studies. In particular, they may provide a possible explanation for the arc statistics probl em.

/pdf/the-impact-of-cluster-mergers-on-arc-statistics-4oufaepmri.pdf

The impact of cluster mergers on arc statistics

We study with semi-analytical methods the statistics of pronounced arcs caused by lensing of galaxies by foreground galaxy clusters. For the number density and redshift distribution of rich clusters we use Press-Schechter theory, normalized on the basis of empirical data. For the background sources we make use of observational results in the Hubble Deep Field. We present results for three different lens models, in particular for the universal profile suggested by Navarro, Frenk and White. Our primary concern is the dependence of the expected statistics on the cosmological parameters, M, �. The theoretical estimates are compared with the cluster arcs survey EMSS, and the resulting constraints in the -plane are presented. In spite of considerable theoretical an observational uncertainties a low-density universe is favored. Degeneracy curves for the optical depth and likelihood regions for the arc statistics in the -plane depend only weakly on the cosmological constant.

Giant Arc Statistics and Cosmological Parameters

We study with semi-analytical methods the statistics of pronounced arcs caused by lensing of galaxies by foreground galaxy clusters. For the number density and redshift distribution of rich clusters we use Press-Schechter theory, normalized on the basis of empirical data. For the background sources we make use of observational results in the Hubble Deep Field. We present results for three different lens models, in particular for the universal profile suggested by Navarro, Frenk and White. Our primary concern is the dependence of the expected statistics on the cosmological parameters, $\Omega_M$, $\Omega_\Lambda$. The theoretical estimates are compared with the cluster arcs survey EMSS, and the resulting constraints in the $\Omega$-plane are presented. In spite of considerable theoretical an observational uncertainties a low-density universe is favored. Degeneracy curves for the optical depth and likelihood regions for the arc statistics in the $\Omega$-plane depend only weakly on the cosmological constant.

R. Kaufmann

Papers

Giant Arc Statistics and Cosmological Parameters

Giant Arc Statistics and Cosmological Parameters