Thomas P. Kling

Study of Errors in Strong Gravitational Lensing

We examine the accuracy of strong gravitational lensing determinations of the mass of galaxy clusters by comparing the conventional approach with the numerical integration of the fully relativistic null geodesic equations in the case of weak gravitational perturbations on Robertson-Walker metrics. In particular, we study spherically symmetric, three-dimensional singular isothermal sphere models and the three-dimensional matter distribution of Navarro and coworkers which are both commonly used in gravitational lensing studies. In both cases we study two different methods for mass-density truncation along the line of sight: hard truncation and conventional (no truncation). We find that the relative error introduced in the total mass by the thin-lens approximation alone is less than 0.3% in the singular isothermal sphere model and less than 2% in the model of Navarro and coworkers. The removal of hard truncation introduces an additional error of the same order of magnitude in the best case and up to an order of magnitude larger in the worst case studied. Our results ensure that the future generation of precision cosmology experiments based on lensing studies will not require the removal of the thin-lens assumption, but they may require a careful handling of truncation.

The Bianchi identity and weak gravitational lensing

We consider the Bianchi identity as a field equation for the distortion of the shapes of images produced by weak gravitational lensing. Using the spin coefficient formalism of Newman and Penrose (1962 J. Math. Phys. 3 566–78), we show that certain complex components of the Weyl and Ricci curvature tensors are directly related to fundamental observables in weak gravitational lensing. In the case of weak gravitational fields, we then show that the Bianchi identity provides a field equation for the Ricci tensor assuming a known Weyl tensor. From the Bianchi identity, we derive the integral equation for weak lensing presented by Miralda-Escude (1996 IAU Symp. vol 173 p 131), thus making the Bianchi identity a first principles equation of weak gravitational lensing. This equation is integrated in the important case of an axially symmetric lens and explicitly demonstrated in the case of a point lens and a singular isothermal sphere (SIS) model.

Creating a Peer-Led Cooperative Learning Program to Improve STEM Retention

Change • November/December 2015 Thomas P. Kling (tkling@ bridgew.edu) is a professor of physics at Bridgewater State University (BSU) and principle investigator of the NSF-STEP Grant STREAMS (0969109). He has signifi cant experience in student-success data analysis and in faculty development for implementing writing-to-learn principles and inquiry across multiple disciplines. Matthew Salomone (matthew. salomone@bridgew.edu)—an associate professor of mathematics at BSU, director of math services, and STREAMS co-PI—is an expert in peer-led cooperative learning and founding coordinator of Bridgewater’s Quantity Across the Curriculum, a faculty-development program inspired by writing across the curriculum. By Thomas P. Kling and Matthew Salomone

Required Peer-cooperative Learning Improves Retention of STEM Majors

BackgroundPeer-cooperative learning has been shown in the literature to improve student success in gateway science and mathematics courses. Such studies typically demonstrate the impact of students’ attending peer-led learning sessions on their learning or grades in an individual course. In this article, we examine the effects of introducing a required, comprehensive peer-cooperative learning system across five departments simultaneously at a master’s public university, looking not only at students’ success in supported classes, but also their retention within STEM fields two years hence. Combining institutional demographic data with students’ course grades and retention rates, we compare outcomes between 456 students who took their major’s introductory course in the two years prior to implementation of the program, and 552 students who did so after implementation.ResultsWhile these two student groups did not significantly differ in either their demographic profile or their SAT scores, the post-implementation group earned significantly higher grades in their introductory courses in each major, due largely to an erasure of the mediating effect of SAT scores on course grades. Further, this increase in introductory course grades was also associated with an increase in the two-year retention rate of students in STEM majors.ConclusionsThis finding is significant as it suggests that implementing comprehensive educational reform using required peer-led cooperative learning may have the proximate effect of mitigating differences in academic preparation (as measured by SAT scores) for students in introductory STEM courses. Furthermore, this increase in success leads to increased retention rates in STEM, expanding the pipeline of students retained in such fields.

Accuracy of the thin-lens approximation in strong lensing by smoothly truncated dark matter haloes

The accuracy of mass estimates by gravitational lensing using the thin-lens approximation applied to Navarro–Frenk–White mass models with a soft truncation mechanism recently proposed by Baltz, Marshall and Oguri is studied. The gravitational lens scenario considered is the case of the inference of lens mass from the observation of Einstein rings (strong lensing). It is found that the mass error incurred by the simplifying assumption of thin lenses is below 0.5 per cent. As a byproduct, the optimal tidal radius of the soft truncation mechanism is found to be at most 10 times the virial radius of the mass model.

Continuous image distortion by astrophysical thick lenses

Image distortion due to weak gravitational lensing is examined using a non-perturbative method of integrating the geodesic deviation and optical scalar equations along the null geodesics connecting the observer to a distant source. The method we develop continuously changes the shape of the pencil of rays from the source to the observer with no reference to lens planes in astrophysically relevant scenarios. We compare the projected area and the ratio of semi-major to semi-minor axes of the observed elliptical image shape for circular sources from the continuous, thick-lens method with the commonly assumed thin-lens approximation. We find that for truncated singular isothermal sphere and NFW models of realistic galaxy clusters, the commonly used thin-lens approximation is accurate to better than 1 part in 104 in predicting the image area and axes ratios. For asymmetric thick lenses consisting of two massive clusters separated along the line of sight in redshift up to Δz = 0.2, we find that modeling the image distortion as two clusters in a single lens plane does not produce relative errors in image area or axes ratio more than 0.5%.

Null Geodesics and Wave Front Singularities in the Gödel Space–time

We explore wave fronts of null geodesics in the Gödel metric emitted from point sources both at, and away from, the origin. For constant time wave fronts emitted by sources away from the origin, we find cusp ridges as well as blue sky metamorphoses where spatially disconnected portions of the wave front appear, connect to the main wave front, and then later break free and vanish. These blue sky metamorphoses in the constant time wave fronts highlight the non-causal features of the Gödel metric. We introduce a concept of physical distance along the null geodesics, and show that for wave fronts of constant physical distance, the reorganization of the points making up the wave front leads to the removal of cusp ridges.

Angular diameter distances reconsidered in the Newman and Penrose formalism

Using the Newman and Penrose spin coefficient (NP) formalism, we provide a derivation of the Dyer–Roeder equation for the angular diameter distance in cosmological space-times. We show that the geodesic deviation equation written in NP formalism is precisely the Dyer–Roeder equation for a general Friedman–Robertson–Walker (FRW) space-time, and then we examine the angular diameter distance to redshift relation in the case that a flat FRW metric is perturbed by a gravitational potential. We examine the perturbation in the case that the gravitational potential exhibits the properties of a thin gravitational lens, demonstrating how the weak lensing shear and convergence act as source terms for the perturbed Dyer–Roeder equation.