Constraining cosmic curvature by using age of galaxies and gravitational   lenses
  We use two model-independent methods to constrain the curvature of the universe. In the first method, we study the evolution of the curvature parameter ($\Omega_k^0$) with redshift by using the observations of the Hubble parameter and transverse comoving distances obtained from the age of galaxies. Secondly, we also use an indirect method based on the mean image separation statistics of gravitationally lensed quasars. The basis of this methodology is that the average image separation of lensed images will show a positive, negative or zero correlation with the source redshift in a closed, open or flat universe respectively. In order to smoothen the datasets used in both the methods, we use a non-parametric method namely, Gaussian Process (GP). Finally from first method we obtain $\Omega_k^0= 0.025\pm0.57$ for a presumed flat universe while the cosmic curvature remains constant throughout the redshift region $0<z<1.37$ which indicates that the universe may be homogeneous. Moreover, the combined result from both the methods suggests that the universe is marginally closed. However, a flat universe can be incorporated at $3\sigma$ level. 
Akshay Rana, Deepak Jain, Shobhit Mahajan and Amitabha Mukherjee
astro-ph.CO