# Testing the Distance-Duality Relation from Strong Gravitational Lensing, Type Ia Supernovae and Gamma-Ray Bursts Data up to redshift $z\sim3.6$

## TL;DR

This study tests the cosmic distance-duality relation (CDDR) up to redshift 3.6 using strong gravitational lensing, Type Ia supernovae, and gamma-ray burst data, finding compatibility within 1 sigma confidence level.

## Contribution

It introduces a model-independent approach combining SGL, SNIa, and GRB data to test CDDR over a wider redshift range, extending previous analyses.

## Key findings

- CDDR is consistent with observations at 1 sigma confidence level.
- Method B allows testing CDDR at higher redshifts.
- Results support the validity of CDDR in a broader redshift domain.

## Abstract

In this paper, we perform a cosmological model-independent test of the cosmic distance-duality relation (CDDR) in terms of the ratio of angular diameter distance (ADD) $D=D_{\rm A}^{\rm sl}/D_{\rm A}^{\,\rm s}$ from strong gravitational lensing (SGL) and the ratio of luminosity distance (LD) $D^\ast=D_{\rm L}^{\,\rm l}/D_{\rm L}^{\,\rm s}$ obtained from the joint of type Ia supernovae (SNIa) Union2.1 compilation and the latest Gamma-Ray Bursts (GRBs) data, where the superscripts s and l correspond to the redshifts $z_{\,\rm s}$ and $z_{\,\rm l}$ at the source and lens from SGL samples. The purpose of combining GRB data with SNIa compilation is to test CDDR in a wider redshift range. The LD associated with the redshits of the observed ADD, is obtained through two cosmological model-independent methods, namely, method A: binning the SNIa+GRBs data, and method B: reconstructing the function of DL by combining the Crossing Statistic with the smoothing method. We find that CDDR is compatible with the observations at $1\sigma$ confidence level for the power law model which is assumed to describe the mass distribution of lensing systems with method B in a wider redshift range.