FLRW cosmology in Weyl-Integrable Space-Time
  We investigate the Weyl space-time extension of general relativity (GR) for studying the FLRW cosmology through focusing and defocusing of the geodesic congruences. We have derived the equations of evolution for expansion, shear and rotation in the Weyl space-time. In particular we consider the Starobinsky modification, $f(R)=R+\beta R^2-2\Lambda$, of gravity in the Einstein-Palatini formalism, which turns out to reduce to the Weyl integrable space-time (WIST) with the Weyl vector being a gradient. The modified Raychaudhuri equation takes the form of the Hill-type equation which is then analysed for studying formation of the caustics. In this model, it is possible to have a Big Bang singularity free cyclic Universe but unfortunately the periodicity turns out to be extremely short. 
Radouane Gannouji, Hemwati Nandan, Naresh Dadhich
gr-qc astro-ph.CO