The bending of light and lensing in modified gravity

Our modified gravity theory (MOG) was used successfully in the past to explain a range of astronomical and cosmological observations, including galaxy rotation curves, the cosmic microwave background acoustic peaks, and the galaxy mass power spectrum. MOG was also used successfully to explain the unusual features of the Bullet Cluster  1E0657−558  without exotic dark matter. In the present work, we derive the relativistic equations of motion in the spherically symmetric field of a point source in MOG and, in particular, we derive equations for light bending and lensing. Our results also have broader applications in the case of extended distributions of matter, and they can be used to validate the Bullet Cluster results and provide a possible explanation for the merging clusters in Abell 520.     

The Δθ–zs relation for gravitational lenses as a cosmological test

Recently, Park &38; Gott claimed that there is a statistically significant, strong, negative correlation between the image separation Δθ and source redshift z _s for gravitational lenses. This is somewhat puzzling if one believes in a flat ( k  = 0) universe, since in this case the typical image separation is expected to be independent of the source redshift, while one expects a negative correlation in a k  = −1 universe and a positive one in a k  = +1 universe. Park &38; Gott explored several effects that could cause the observed correlation, but no combination of these can explain the observations with a realistic scenario. Here, I explore this test further in three ways. First, I show that in an inhomogeneous universe a negative correlation is expected regardless of the value of k . Secondly, I test whether the Δθ– z _s relation can be used as a test to determine λ₀ and Ω₀, rather than just the sign of k . Thirdly, I compare the results of the test from the Park &38; Gott sample with those using other samples of gravitational lenses, which can illuminate (unknown) selection effects and probe the usefulness of the Δθ– z _s relation as a cosmological test.     

The continuous limit of the multiple lens effect and the optical scalar equation

We study the continuous limit of the multiple gravitational lensing theory based on the thin lens approximation. We define a new, light-path dependent angular diameter distance     and show that it satisfies the optical scalar equation. The distance provides relations between quantities used in gravitational lensing theory (the convergence, the shear and the twist terms) and those used in scalar optics theory (the rates of expansion, shear and rotation).     

The three-dimensional power spectrum of dark and luminous matter from the VIRMOS-DESCART cosmic shear survey

We present the first optimal power spectrum estimation and three-dimensional deprojections for the dark and luminous matter and their cross-correlations. The results are obtained using a new optimal fast estimator, deprojected using minimum variance and Singular Value Decomposition (SVD) techniques. We show the resulting 3D power spectra for dark matter and galaxies, and their covariance for the VIRMOS-DESCART weak lensing shear and galaxy data. The survey is most sensitive to non-linear scales   k _NL∼ 1 h Mpc⁻¹  . On these scales, our 3D power spectrum of dark matter is in good agreement with the RCS 3D power spectrum found by Tegmark & Zaldarriaga. Our galaxy power is similar to that found by the 2MASS survey, and larger than that of SDSS, APM and RCS, consistent with the expected difference in galaxy population.
We find an average bias   b = 1.24 ± 0.18  for the I -selected galaxies, and a cross-correlation coefficient   r = 0.75 ± 0.23  . Together with the power spectra, these results optimally encode the entire two point information about dark matter and galaxies, including galaxy–galaxy lensing. We address some of the implications regarding galaxy haloes and mass-to-light ratios. The best-fitting 'halo' parameter   h ≡ r / b = 0.57 ± 0.16  , suggesting that dynamical masses estimated using galaxies systematically underestimate total mass.
Ongoing surveys, such as the Canada–France–Hawaii Telescope Legacy Survey, will significantly improve on the dynamic range, and future photometric redshift catalogues will allow tomography along the same principles.     

Giant cluster arcs as a constraint on the scattering cross-section of dark matter

We carry out ray tracing through five high-resolution simulations of a galaxy cluster, to study how its ability to produce giant gravitationally lensed arcs is influenced by the collision cross-section of its dark matter. In three cases typical dark matter particles in the cluster core undergo between 1 and 100 collisions per Hubble time; two more explore the long ('collisionless') and short ('fluid') mean free path limits. We study the size and shape distributions of arcs and compute the cross-section for producing 'extreme' arcs of various sizes. Even a few collisions per particle modifies the core structure enough to destroy the ability of the cluster to produce long, thin arcs. For larger collision frequencies the cluster must be scaled up to unrealistically large masses before it regains the ability to produce giant arcs. None of our models with self-interacting dark matter (except the 'fluid' limit) is able to produce radial arcs; even the case with the smallest scattering cross-section must be scaled to the upper limit of observed cluster masses before it produces radial arcs. Apparently the elastic collision cross-section of dark matter in clusters must be very small, below 0.1 cm² g⁻¹, to be compatible with the observed ability of clusters to produce both radial arcs and giant arcs.