Exploring the Possible Evolution of the Mass Density Power-law Index of Strong Gravitational Lenses with a Model-independent Method

Abstract

Abstract In this work, we adopt a cosmological model-independent approach for the first time to test the question of whether the mass density power-law index (γ) of strong gravitational lensing systems (SGLSs) evolves with redshift, and the joint light-curve Type Ia supernova (SNe Ia) sample and the quasar sample from Risaliti & Lusso are used to provide the luminosity distances to be calibrated. Our work is based on the flat universe assumption and the cosmic distance duality relation. A reliable data-matching method is used to pair SGLS–SNe and SGLS–quasars. By using the maximum likelihood method to constrain the luminosity distance and γ index, we obtain the likelihood function values for the evolved and nonevolved cases, and then use the Akaike weights and the Bayesian Information Criterion (BIC) selection weights to compare the advantages and disadvantages of these two cases. We find that the γ index is slightly more likely to be a nonevolutionary model for γ = 2 in the case of the currently used samples with low redshift (z l < ∼ 0.66). With Akaike weights, the relative probabilities are 66.3% versus 33.7% and 69.9% versus 30.1% for the SGLS + SNe Ia sample and SGLS + quasar sample, respectively, and with the BIC selection weights, the relative probabilities are 87.4% versus 12.6% and 52.0% versus 48.0% for the two samples. In the evolving case for the relatively low-redshift lenses (SGLS + SNe Ia), with redshift 0.0625–0.659, γ = 2.058 − 0.040 + 0.041 – 0.136 − 0.165 + 0.163 z . At high redshift (SGLS + quasar), with redshift 0.0625–1.004, γ = 2.051 − 0.077 + 0.076 – 0.171 − 0.196 + 0.214 z . Although not the more likely model, this evolved γ case also fits the data well, with a negative and mild evolution for both low- and high-redshift samples.