Measuring angular diameter distances of strong gravitational lenses

The distance-redshift relation plays a fundamental role in constraining cosmological models. In this paper, we show that measurements of positions and time delays of strongly lensed images of a background galaxy, as well as those of the velocity dispersion and mass profile of a lens galaxy, can be combined to extract the angular diameter distance of the lens galaxy. Physically, as the velocity dispersion and the time delay give a gravitational potential ($GM/r$) and a mass ($GM$) of the lens, respectively, dividing them gives a physical size ($r$) of the lens. Comparing the physical size with the image positions of a lensed galaxy gives the angular diameter distance to the lens. A mismatch between the exact locations at which these measurements are made can be corrected by measuring a local slope of the mass profile. We expand on the original idea put forward by Paraficz and Hjorth, who analyzed singular isothermal lenses, by allowing for an arbitrary slope of a power-law spherical mass density profile, an external convergence, and an anisotropic velocity dispersion. We find that the effect of external convergence cancels out when dividing the time delays and velocity dispersion measurements. We derive a formula for the uncertainty in the angular diameter distance in terms of the uncertainties in the observables. As an application, we use two existing strong lens systems, B1608+656 ($z_{\rm L}=0.6304$) and RXJ1131$-$1231 ($z_{\rm L}=0.295$), to show that the uncertainty in the inferred angular diameter distances is dominated by that in the velocity dispersion, $\sigma^2$, and its anisotropy. We find that the current data on these systems should yield about 16% uncertainty in $D_A$ per object. This improves to 13% when we measure $\sigma^2$ at the so-called sweet-spot radius. Achieving 7% is possible if we can determine $\sigma^2$ with 5% precision.Comment: Accepted to JCA

Cosmography from two-image lens systems: overcoming the lens profile slope degeneracy

The time delays between the multiple images of a strong lens system, together with a model of the lens mass distribution, allow a one-step measurement of a cosmological distance, namely, the "time-delay distance" of the lens (D_dt) that encodes cosmological information. The time-delay distance depends sensitively on the radial profile slope of the lens mass distribution; consequently, the lens slope must be accurately constrained for cosmological studies. We show that the slope cannot be constrained in two-image systems with single-component compact sources, whereas it can be constrained in systems with two-component sources provided the separation between the image components can be measured with milliarcsecond precisions, which is not feasible in most systems. In contrast, we demonstrate that spatially extended images of the source galaxy in two-image systems break the radial slope degeneracy and allow D_dt to be measured with uncertainties of a few percent. Deep and high-resolution imaging of the lens systems are needed to reveal the extended arcs, and stable point spread functions are required for our lens modelling technique. Two-image systems, no longer plagued by the radial profile slope degeneracy, would augment the sample of useful time-delay lenses by a factor of ~6, providing substantial advances for cosmological studies.Comment: 14 pages, 9 figures, revisions based on referee's comments, accepted for publication in MNRA

The anatomy of a quadruply imaged gravitational lens system

The key to using a strong gravitational lens system to measure the Hubble constant is to obtain an accurate model of the lens potential. In this paper, we investigate the properties of gravitational lens B1608+656, a quadruply-imaged lens system with an extended source intensity distribution. Our analysis is valid for generic quadruply-lensed systems. Limit curves and isophotal separatrices are defined for such systems, and we show that the isophotal separatrices must intersect at the critical curves and the satellite isophotes must be tangent to the limit curves. The current model of B1608+656 (Koopmans et al. 2003) satisfies these criteria for some, but not all, of the isophotal separatrices within the observational uncertainty. We study a non-parametric method of potential reconstruction proposed by Blandford, Surpi & Kundic (2001) and demonstrate that although the method works in principle and elucidates image formation, the initial potential only converges to the true model when it is within ~ 1 percent of the true model.Comment: 12 pages, 12 figures. Minor revisions based on referee's comments after initial submission to MNRA

Interlinkages between Indicators of Sustainable Development Goals: Evidence from Seven Low Income and Lower Middle-Income Countries

In 2015, 17 Sustainable Development Goals (SDGs) were adopted by member states of United Nations as a blueprint to achieve a better and more sustainable future for all. Up to date, the global indicator of framework of SDGs has 247 indicators (including repetitive indicators), which monitor progress of achieving the SDGs. The interlinkages and integration of SDG indicators have attracted attention from scholars and practitioners, however, there is limited existing knowledge of the interlinkages and interactions between SDG indicators in low income and lower middle-income countries. This article therefore uses data from seven low income and lower middle-income countries to analyze the interlinkages between SDG indicators 9.2.1 and 9.4.1, which monitors the level of industrialization and energy efficiency in manufacturing sector respectively. It is widely believed that higher level industrialization will reduce energy emission per manufacturing value added. However, the results suggest that beyond traditional knowledge, in the seven selected countries, SDG indicators 9.2.1 and 9.4.1 are positively associated with each other. A possible reason of this unconventional finding is the country context, as in low income and lower middle-income countries the manufacturing industries are usually not energy-extensive, although not technology-intensive either. This article also calls for holistic and integrated approaches to use and interpret SDG indicators in line with the national and regional context

Higher education and Sustainable Development Goals during COVID-19: coping strategies of a university in Wuhan, China

Background: It is widely perceived that COVID-19 has significant influence on higher education and also contribution to development including Sustainable Development Goals (SDGs). However there is insufficient evidence about investigations on such influences, especially at micro level. Design and method: A university located in Wuhan, China, is selected for the case study to explore how COVID-19 affects higher education and how universities’ coping strategies of COVID-19 can contribute to SDGs. The method is an analysis of 32 institutional documents published by the university. Results: The university in the case study has taken a number of coping strategies of COVID-19, largely in four aspects including medical services, online education, logistic support, and graduate employment promotion. These coping strategies contribute to achieving SDGs, especially SDGs 1, 3, 4, 5, 8, and 10. Conclusions: The case study provides micro-level empirical evidence, which supports that appropriate university coping strategies of COVID-19 can contribute to SDGs, even it is widely perceived that the pandemic has brought strong negative impact on higher education and sustainable development. The selection of a university in Wuhan, China, can generate more practical implications, as Wuhan is the first city that experienced the unprecedented lockdown, and China is the first country that reopened university campuses after the lockdown