Albareti, F. D.Ruiz Cembranos, José AlbertoCruz Dombriz, Álvaro de laDobado González, Antonio2023-06-192023-06-1920141475-751610.1088/1475-7516/2014/03/012https://hdl.handle.net/20.500.14352/35052© 2014 IOP Publishing Ltd and Sissa Medialab srl. This work has been supported by MICINN (Spain) projects numbers FIS2011-23000, FPA2011-27853-C02-01, FPA2011-27853-C02-01, UCM-BSCH GR58/08 910309 and Consolider-Ingenio MULTIDARK CSD2009-00064. F.D.A. acknowledges financial support from the UAM+CSIC Campus of International Excellence (Spain). A.d.l.C.D. thanks the financial support from ACGC University of Cape Town and IEEC-ICE Barcelona in 2013 as well as the hospitality of Kavli Institute for Theoretical Physics China (KITPC) during the preparation of this manuscript.In this work we address the issue of studying the conditions required to guarantee the Focusing Theorem for both null and time like geodesic congruences by using the Raychaudhuri equation. In particular we study the case of Friedmann-Robertson-Walker as well as more general Bianchi Type I spacetimes. The fulfillment of the Focusing Theorem is mandatory in small scales since it accounts for the attractive character of gravity. However, the Focusing Theorem is not satisfied at cosmological scales due to the measured negative deceleration parameter. The study of the conditions needed for congruences convergence is not only relevant at the fundamental level but also to derive the viability conditions to be imposed on extended theories of gravity describing the different expansion regimes of the universe. We illustrate this idea for f (R) gravity theories.engjournal articlehttp://dx.doi.org/10.1088/1475-7516/2014/03/012http://iopscience.iop.org/http://arxiv.org/abs/1401.7926open access53R-n-gravityDark energyRelativistic theoryMachs principleCpt violationSupergravitySupersymmetrySupernovaeConstantGeometryFísica (Física)22 Física