Gravitational Lensing of Dark Energy Models and $Λ$CDM Using Observational data in Loop Quantum Cosmology

Abstract: This paper investigates the accelerated cosmic expansion in the late Universe by examining two dark energy models, viscous modified Chaplygin gas (VsMCG) and variable modified Chaplygin gas (VMCG), within loop quantum cosmology alongside the $\Lambda$CDM model. The objective is to constrain cosmic parameters using the $\Lambda$CDM model and 30 of the latest $H(z)$ measurements from cosmic chronometers (CC), including Type Ia Supernovae, Gamma-Ray Bursts (GRB), Quasars, and 24 uncorrelated baryon acoustic oscillations (BAO) measurements across a redshift range from 0.106 to 2.33. The latest Hubble constant measurement from Riess in 2022 is included to enhance constraints. In the $\Lambda$CDM, VsMCG, and VMCG frameworks, best-fit parameters for the Hubble parameter ($H_0$) and sound horizon ($r_d$) are obtained. The results highlight significant disparities between $H_0$ and $r_d$ values from late-time observational measurements, reflecting the known $H_0$ and $r_d$ tensions. The gravitational lensing optical depth of the two dark energy models is studied by plotting $\log(\tau(z_l)/H_0^{-3} \tau_N)$ vs $z_l$. The probability of finding gravitational lenses (optical depth) in both models increases with lens redshift $z_l$. The change in optical depth behavior for different parameter constraints is graphically analyzed. A joint analysis of VsMCG and VMCG with $\Lambda$CDM is conducted. While the models diverge in the early Universe, they are indistinguishable at low redshift. Using the Akaike information criteria, the analysis indicates that neither dark energy model can be