We propose an accurate test of the distance-duality （DD） relation, η = DL（z）（1 ＋ z）^-2/DA（z） = 1 （where DL and DA are the luminosity distances and angular diameter distances, respectively）, with a combination of cosmological observational data of Type Ia Supernovae （SNe Ia） from the Union2 set and the galaxy cluster sample under an assumption of the spherical model. In order to avoid bias brought on by redshift non-coincidence between observational data and to consider redshift error bars of both clusters and SNe Ia in the analysis, we carefully choose the SNe Ia points which have the minimum acceptable redshift difference of the galaxy cluster sample （│△Z│min = σz,SN ＋σz,cluster）. By assuming η to be a constant and defined as functions of the redshift parameterized by six different expressions, we find that there exists no observable evidence for variations in the DD relation based on the collected data, since related statistical tests are well satisfied within the 1σ confidence level for most cases. Further, considering different values of △z as constraints, we also find that the choice of △z may play an important role in this model-independent test of the DD relation for the spherical sample of galaxy clusters.
The validity of distance duality relation, η = D L (z)(1 + z) 2 /D A (z) = 1, an exact result required by the Etherington reciprocity theorem, where D A (z) and D L (z) are the angular and luminosity distances, plays an essential part in cosmological observations and model constraints. In this paper, we investigate some consequences of such a relation by assuming η a constant or a function of the redshift. In order to constrain the parameters concerning η, we consider two groups of cluster gas mass fraction data including 52 X-ray luminous galaxy clusters observed by Chandra in the redshift range from 0.3 to 1.273 and temperature range T gas > 4 keV, under the assumptions of two different temperature profiles. We find that the constant temperature profile is in relatively good agreement with no violation of the distance duality relation for both parameterizations of η, while the one with temperature gradient (the Vikhlinin et al. temperature profile) seems to be incompatible even at 99% CL.
We present a multi-wavelength study of the gravitational lens COSMOS J095930＋023427 （Zl = 0.892）, together with the associated galaxy group along the line of sight located at z 0.7, and the lensed background galaxy. The source redshift is currently unknown, but estimated to be at zs ~ 2. This analysis is based on publicly available HST, Subaru and Chandra imaging data, as well as VLT spectroscopy. The lensing system is an early-type galaxy showing a strong [OII] emission line, and pro- duces four bright images of the distant background source. It has an Einstein radius of 0.79＂, about four times larger than the effective radius. We perform a lensing anal- ysis using both a singular isothermal ellipsoid and a peudo-isothermal elliptical mass distribution for the lensing galaxy, and find that the final results on the total mass, the dark matter （DM） fraction within the Einstein radius and the external shear due to a foreground galaxy group are robust with respect to the choice of the parametric model and the source redshift （yet unknown）. We measure the luminous mass from the pho- tometric data, and find the DM fraction within the Einstein radius fDM to be between 0.71 ~ 0.13 and 0.79 ~ 0.15, depending on the unknown source redshift. Meanwhile, the non-null external shear found in our lensing models supports the presence and structure of a galaxy group at z ~ 0.7, and an independent measurement of the 0.5- 2 keV X-ray luminosity within 20＂ around the X-ray centroid provides a group mass of M = （3 - 10） x 1013 Mo, in good agreement with the previous estimate derived through weak lensing analysis. Finally, by inverting the HST/ACS/814 image with the lensing equation, we obtain the reconstructed image of the magnified source galaxy, which has a scale of about 3.3 kpc at z~ = 2 （2.7 kpc at zs = 4） and the typical disturbed disk-like appearance observed in low-mass star-forming galaxies at z ~ 3. However, deep, spatially resolved spectroscopic data for similar lensed sources are still required
