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Focusing of geodesic congruences in an accelerated expanding Universe

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2012-12-14
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We study the accelerated expansion of the Universe through its consequences on a congruence of geodesics. We make use of the Raychaudhuri equation which describes the evolution of the expansion rate for a congruence of timelike or null geodesics. In particular, we focus on the space-time geometry contribution to this equation. By straightforward calculation from the metric of a Robertson-Walker cosmological model, it follows that in an accelerated expanding Universe the space-time contribution to the Raychaudhuri equation is positive for the fundamental congruence, favoring a non-focusing of the congruence of geodesics. However, the accelerated expansion of the present Universe does not imply a tendency of the fundamental congruence to diverge. It is shown that this is in fact the case for certain congruences of timelike geodesics without vorticity. Therefore, the focusing of geodesics remains feasible in an accelerated expanding Universe. Furthermore, a negative contribution to the Raychaudhuri equation from space-time geometry which is usually interpreted as the manifestation of the attractive character of gravity is restored in an accelerated expanding Robertson-Walker space-time at high speeds.
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© 2012 IOP Publishing Ltd and Sissa Medialab srl. We would like to thank prof. G. F. R. Ellis for drawing our attention to reference [10]. This work has been supported by MICINN (Spain) projects numbers FIS2011-23000, FPA2011-27853-C02-01 and Consolider-Ingenio MULTIDARK CSD2009-00064. FDA would like to thank Theoretical Physics I department, Complutense University of Madrid for providing all the required academic resources. AdlCD acknowledges financial support from NRF and URC research fellowships (South Africa). JARC acknowledges the kind hospitality of ACGC/University of Cape Town while elaborating the manuscript.
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[1] L. Covi, J.E. Kim and L. Roszkowski, Axinos as cold dark matter, Phys. Rev. Lett. 82 (1999) 4180 [hep-ph/9905212] [INSPIRE]; J.L. Feng, A. Rajaraman and F. Takayama, Graviton cosmology in universal extra dimensions, Phys. Rev. D 68 (2003) 085018 [hep-ph/0307375] [INSPIRE]; Probing gravitational interactions of elementary particles, Int. J. Mod. Phys. D 13 (2004) 2355 [Gen. Rel. Grav. 36 (2004) 2575] [hep-th/0405248] [INSPIRE]; J.A. Cembranos, J.L. Feng, A. Rajaraman and F. Takayama, SuperWIMP solutions to small scale structure problems, Phys. Rev. Lett. 95 (2005) 181301 [hep-ph/0507150] [INSPIRE]; J.A. Cembranos, J.L. Feng and L.E. Strigari, Exotic collider signals from the complete phase diagram of minimal universal extra dimensions, Phys. Rev. D 75 (2007) 036004 [hep-ph/0612157] [INSPIRE]; J.A. Cembranos, J.L. Díaz-Cruz and L. Prado, Impact of DM direct searches and the LHC analyses on branon phenomenology, Phys. Rev. D 84 (2011) 083522 [arXiv:1110.0542][INSPIRE]. [2] H. Goldberg, Constraint on the photino mass from cosmology, Phys. Rev. Lett. 50 (1983) 1419 [Erratum ibid. 103 (2009) 099905] [INSPIRE]; J.R. Ellis, J. Hagelin, D.V. Nanopoulos, K.A. Olive and M. Srednicki, Supersymmetric relics from the big bang, Nucl. Phys. B 238 (1984) 453 [INSPIRE]; K. Griest and M. Kamionkowski, Supersymmetric dark matter, Phys. Rept. 333 (2000) 167 [INSPIRE]; J. Cembranos, A. Dobado and A.L. Maroto, Brane world dark matter, Phys. Rev. Lett. 90 (2003) 241301 [hep-ph/0302041] [INSPIRE]; Cosmological and astrophysical limits on brane fluctuations, Phys. Rev. D 68 (2003) 103505 [hep-ph/0307062] [INSPIRE]; Branon radiative corrections to collider physics and precision observables, Phys. Rev. D 73 (2006) 035008 [hep-ph/0510399] [INSPIRE]; Dark matter clues in the muon anomalous magnetic moment, Phys. Rev. D 73 (2006) 057303 [hep-ph/0507066] [INSPIRE]; A.L. Maroto, The nature of branon dark matter, Phys. Rev. D 69 (2004) 043509 [hep-ph/0310272] [INSPIRE]; Brane oscillations and the cosmic coincidence problem, Phys. Rev. D 69 (2004) 101304 [hep-ph/0402278] [INSPIRE]; A. Dobado and A.L. Maroto, The dynamics of the Goldstone bosons on the brane, Nucl. Phys. B 592 (2001) 203 [hep-ph/0007100] [INSPIRE]; J. Cembranos, A. Dobado and A.L. Maroto, Dark geometry, Int. J. Mod. Phys. D 13 (2004) 2275 [hep-ph/0405165] [INSPIRE]; J. Cembranos, A. de la Cruz-Dombriz, A. Dobado and A.L. Maroto, Is the CMB cold spot a gate to extra dimensions?, JCAP 10 (2008) 039 [arXiv:0803.0694] [INSPIRE]. [3] J.A. Cembranos and L.E. Strigari, Diffuse MeV gamma-rays and galactic 511 keV line from decaying WIMP dark matter, Phys. Rev. D 77 (2008) 123519 [arXiv:0801.0630] [INSPIRE]; J.A. Cembranos, J.L. Feng and L.E. Strigari, Resolving cosmic gamma ray anomalies with dark matter decaying now, Phys. Rev. Lett. 99 (2007) 191301 [arXiv:0704.1658] [INSPIRE]; J. Cembranos, A. de la Cruz-Dombriz, A. Dobado, R. Lineros and A. Maroto, Photon spectra from WIMP annihilation, Phys. Rev. D 83 (2011) 083507 [arXiv:1009.4936] [INSPIRE]; J. Cembranos, A. de la Cruz-Dombriz, V. Gammaldi and A. Maroto, Detection of branon dark matter with gamma ray telescopes, Phys. Rev. D 85 (2012) 043505 [arXiv:1111.4448] [INSPIRE]; J. Cembranos, V. Gammaldi and A. Maroto, Possible dark matter origin of the gamma ray emission from the galactic center observed by HESS, Phys. Rev. D 86 (2012) 103506 [arXiv:1204.0655] [INSPIRE]. [4] J. Alcaraz, J. Cembranos, A. Dobado and A.L. Maroto, Limits on the brane fluctuations mass and on the brane tension scale from electron positron colliders, Phys. Rev. D 67 (2003) 075010 [hep-ph/0212269] [INSPIRE]; L3 collaboration, P. Achard et al., Search for branons at LEP, Phys. Lett. B 597 (2004) 145 [hep-ex/0407017] [INSPIRE]; J. Cembranos, A. Rajaraman and F. Takayama, Searching for CPT violation in t¯t production, Europhys. Lett. 82 (2008) 21001 [hep-ph/0609244] [INSPIRE]; J. Cembranos, A. Dobado and A.L. Maroto, Brane skyrmions and wrapped states, Phys. Rev. D 65 (2002) 026005 [hep-ph/0106322] [INSPIRE]; Some model-independent phenomenological consequences of flexible brane worlds, J. Phys. A 40 (2007) 6631 [hep-ph/0611024] [INSPIRE]; Branon search in hadronic colliders, Phys. Rev. D 70 (2004) 096001 [hep-ph/0405286] [INSPIRE]; J. Cembranos, J.L. Feng, A. Rajaraman and F. Takayama, Gravitino and axino superWIMPs, AIP Conf. Proc. 903 (2007) 591 [hep-ph/0701011] [INSPIRE]. [5] Supernova Search Team collaboration, A.G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009 [astro-ph/9805201] [INSPIRE]; Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of and _ from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565 [astro-ph/9812133] [INSPIRE]; Supernova Search Team collaboration, J.L. Tonry et al., Cosmological results from high-z supernovae, Astrophys. J. 594 (2003) 1 [astro-ph/0305008] [INSPIRE]. [6] T. Biswas, J.A. Cembranos and J.I. Kapusta, Thermal duality and Hagedorn transition from p-adic strings, Phys. Rev. Lett. 104 (2010) 021601 [arXiv:0910.2274] [INSPIRE]; Thermodynamics and cosmological constant of non-local field theories from p-adic strings, JHEP 10 (2010) 048 [arXiv:1005.0430] [INSPIRE]; Finite temperature solitons in non-local field theories from p-adic strings, Phys. Rev. D 82 (2010) 085028 [arXiv:1006.4098][INSPIRE]. [7] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (1989) 1 [INSPIRE]. [8] S. Nojiri and S.D. Odintsov, Modified gravity with negative and positive powers of the curvature: unification of the inflation and of the cosmic acceleration, Phys. Rev. D 68 (2003) 123512 [hep-th/0307288] [INSPIRE]; A. Dobado and A.L. Maroto, Inflatonless inflation, Phys. Rev. D 52 (1995) 1895 [Erratum ibid. D 53 (1996) 2262] [hep-ph/9406409][INSPIRE]; G. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE]; J.A. Cembranos, Dark matter from R2-gravity, Phys. Rev. Lett. 102 (2009) 141301 [arXiv:0809.1653] [INSPIRE]; The Newtonian limit at intermediate energies, Phys. Rev. D 73 (2006) 064029 [gr-qc/0507039] [INSPIRE]; J.A. Cembranos, K.A. Olive, M. Peloso and J.-P. Uzan, Quantum corrections to the cosmological evolution of conformally coupled fields, JCAP 07 (2009) 025 [arXiv:0905.1989] [INSPIRE]; S. Nojiri and S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, eConf C 0602061 (2006) 06 [Int. J. Geom. Meth. Mod. Phys. 4 (2007) 115] [hep-th/0601213][INSPIRE]; D. Sáez-Gómez, Modified f(R) gravity from scalar-tensor theory and inhomogeneous EoS dark energy, Gen. Rel. Grav. 41 (2009) 1527 [arXiv:0809.1311] [INSPIRE]; S. Nojiri, S.D. Odintsov and D. Sáez-Gómez, Cosmological reconstruction of realistic modified F(R) gravities, Phys. Lett. B 681 (2009) 74 [arXiv:0908.1269] [INSPIRE]; J. Beltrán Jimenez and A.L. Maroto, A cosmic vector for dark energy, Phys. Rev. D 78 (2008) 063005 [arXiv:0801.1486] [INSPIRE]; Cosmological electromagnetic fields and dark energy, JCAP 03 (2009) 016 [arXiv:0811.0566] [INSPIRE]; Cosmological evolution in vector-tensor theories of gravity, Phys. Rev. D 80 (2009) 063512 [arXiv:0905.1245] [INSPIRE]; Dark energy: the absolute electric potential of the universe, Int. J. Mod. Phys. D 18 (2009) 2243 [arXiv:0905.2589] [INSPIRE]; J. Cembranos, C. Hallabrin, A. Maroto and S.N. Jareno, Isotropy theorem for cosmological vector fields, Phys. Rev. D 86 (2012) 021301 [arXiv:1203.6221] [INSPIRE]; T. Harko, T.S. Koivisto, F.S. Lobo and G.J. Olmo, Metric-Palatini gravity unifying local constraints and late-time cosmic acceleration, Phys. Rev. D 85 (2012) 084016 [arXiv:1110.1049] [INSPIRE]. [9] T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys. Rept. 513 (2012) 1 [arXiv:1106.2476] [INSPIRE]; S.M. Carroll, V. Duvvuri, M. Trodden and M.S. Turner, Is cosmic speed-up due to new gravitational physics?, Phys. Rev. D 70 (2004) 043528 [astro-ph/0306438] [INSPIRE]; A. de la Cruz-Dombriz and A. Dobado, A f(R) gravity without cosmological constant, Phys. Rev. D 74 (2006) 087501 [gr-qc/0607118] [INSPIRE]; A. de la Cruz-Dombriz, A. Dobado and A. Maroto, Black holes in f(R) theories, Phys. Rev. D 80 (2009) 124011 [Erratum ibid. D 83 (2011) 029903] [arXiv:0907.3872] [INSPIRE]; S. Nojiri and S.D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept. 505 (2011) 59 [arXiv:1011.0544] [INSPIRE]; E. Elizalde and D. Sáez-Gómez, F(R) cosmology in presence of a phantom fluid and its scalar-tensor counterpart: towards a unified precision model of the universe evolution, Phys. Rev. D 80 (2009) 044030 [arXiv:0903.2732] [INSPIRE]; O. Bertolami, C.G. Boehmer, T. Harko and F.S. Lobo, Extra force in f(R) modified theories of gravity, Phys. Rev. D 75 (2007) 104016 [arXiv:0704.1733] [INSPIRE]; J. Cembranos, A. de la Cruz-Dombriz and B. Montes Núñez, Gravitational collapse in f(R) theories, JCAP 04 (2012) 021 [arXiv:1201.1289] [INSPIRE]; P.K. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. Sáez-Gómez, On the _CDM universe in f(R) gravity, Phys. Rev. D 82 (2010) 023519 [arXiv:1005.2205] [INSPIRE]; N. Goheer, J. Larena and P.K. Dunsby, Power-law cosmic expansion in f(R) gravity models, Phys. Rev. D 80 (2009) 061301 [arXiv:0906.3860] [INSPIRE]; A.M. Nzioki, P.K. Dunsby, R. Goswami and S. Carloni, A geometrical approach to strong gravitational lensing in f(R) gravity, Phys. Rev. D 83 (2011) 024030 [arXiv:1002.2056] [INSPIRE]; M. Abdelwahab, R. Goswami and P.K. Dunsby, Cosmological dynamics of fourth order gravity: a compact view, Phys. Rev. D 85 (2012) 083511 [arXiv:1111.0171] [INSPIRE]; S. Carloni, R. Goswami and P.K. Dunsby, A new approach to reconstruction methods in f(R) gravity, Class. Quant. Grav. 29 (2012) 135012 [arXiv:1005.1840] [INSPIRE]; A. de la Cruz-Dombriz and D. Sáez-Gómez, On the stability of the cosmological solutions in f(R,G) gravity, Class. Quant. Grav. 29 (2012) 245014 [arXiv:1112.4481] [INSPIRE]; S. Capozziello and D. S´aez-G´omez, Scalar-tensor representation of f(R) gravity and Birkhoff’s theorem, Annalen Phys. 524 (2012) 279 [arXiv:1107.0948] [INSPIRE]; A. de la Cruz-Dombriz, A. Dobado and A.L. Maroto, On the evolution of density perturbations in f(R) theories of gravity, Phys. Rev. D 77 (2008) 123515 [arXiv:0802.2999] [INSPIRE]; Comment on ‘viable singularity-free f(R) gravity without a cosmological constant’, Phys. Rev. Lett. 103 (2009) 179001 [arXiv:0910.1441] [INSPIRE]; A. Abebe, M. Abdelwahab, A. de la Cruz-Dombriz and P.K. Dunsby, Covariant gauge-invariant perturbations in multifluid f(R) gravity, Class. Quant. Grav. 29 (2012) 135011 [arXiv:1110.1191] [INSPIRE]; J. Cembranos, A. de la Cruz-Dombriz and P.J. Romero, Kerr-Newman black holes in f(R) theories, arXiv:1109.4519 [INSPIRE]. [10] G.F. Ellis and H. van Elst, Deviation of geodesics in FLRW space-time geometries, gr-qc/9709060 [INSPIRE]. [11] A. Raychaudhuri, Relativistic cosmology. 1, Phys. Rev. 98 (1955) 1123 [INSPIRE]. [12] R. Sachs, Gravitational waves in general relativity. 6. The outgoing radiation condition, Proc. Roy. Soc. Lond. A 264 (1961) 309 [INSPIRE]. [13] J. Ehlers, Contributions to the relativistic mechanics of continuous media, Gen. Rel. Grav. 25 (1993) 1225 [Abh. Akad. Wiss. Lit. Mainz. Nat. Kl. 11 (1961) 793][INSPIRE]. [14] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973). [15] R.M. Wald, General relativity, University of Chicago Press, Chicago U.S.A. (1984). [16] N. Dadhich, J. Krishna Rao, J.V. Narlikar and C.V. Vishveshwara eds., A random walk in relativity and cosmology. Essays in honor of P.C. Vaidya and A.K. Raychaudhuri, Wiley, New York U.S.A. (1985) [INSPIRE]; F. Paiva, M. Reboucas and A. Teixeira, Time travel in the homogeneous Som-Raychaudhuri universe, Phys. Lett. A 126 (1987) 168 [INSPIRE]; R. Capovilla and J. Guven, Large deformations of relativistic membranes: a generalization of the Raychaudhuri equations, Phys. Rev. D 52 (1995) 1072 [gr-qc/9411061] [INSPIRE]; S. Kar, The generalized Raychaudhuri equations: examples, Phys. Rev. D 53 (1996) 2071 [gr-qc/9505001] [INSPIRE]; Generalized Raychaudhuri equations for strings in the presence of an antisymmetric tensor field, Phys. Rev. D 54 (1996) 6408 [hep-th/9604048] [INSPIRE]; B. Carter, Amalgamated Codazzi-Raychaudhuri identity for foliation, Contemp. Math. 203 (1997) 207 [hep-th/9705083] [INSPIRE]; E. Zafiris, Covariant generalization of Codazzi-Raychaudhuri and area change equations for relativistic branes, J. Geom. Phys. 28 (1998) 271 [hep-th/9710169] [INSPIRE]; Generalized Raychaudhuri and area change equations for classical brane models, Phys. Rev. D 58 (1998) 043509 [INSPIRE]; J. Borgman and L. Ford, Stochastic gravity and the Langevin-Raychaudhuri equation, Int. J. Mod. Phys. A 20 (2005) 2364 [INSPIRE]; N. Dadhich, Derivation of the Raychaudhuri equation, gr-qc/0511123 [INSPIRE]; J. Ehlers, A.K. Raychaudhuri and his equation, Int. J. Mod. Phys. D 15 (2006) 1573 [INSPIRE]; N. Ahmadi and M. Nouri-Zonoz, Quantum gravitational optics: effective Raychaudhuri equation, Phys. Rev. D 74 (2006) 044034 [gr-qc/0605009] [INSPIRE]; G.F.R. Ellis, On the Raychaudhuri equation, Pramana 69 (2007) 15; S. Kar and S. SenGupta, The Raychaudhuri equations: a brief review, Pramana 69 (2007) 49 [gr-qc/0611123] [INSPIRE]; C. Mukku, S.M. Mahajan and B.A. Bambah, On a Raychaudhuri equation for hot gravitating fluids, Pramana 69 (2007) 137 [gr-qc/0610150] [INSPIRE]; N. Ahmadi and M. Nouri-Zonoz, Raychaudhuri equation in quantum gravitational optics, Pramana 69 (2007) 147 [INSPIRE]; N. Dadhich, Singularity: Raychaudhuri equation once again, Pramana 69 (2007) 23 [gr-qc/0702095] [INSPIRE]; S. Kar, An introduction to the Raychaudhuri equations, Resonance J. Sci. Educ. 13 (2008) 319; M. Wanas and M. Bakry, Effect of spin-torsion interaction on Raychaudhuri equation, Int. J. Mod. Phys. A 24 (2009) 5025 [arXiv:0807.0898] [INSPIRE]; A.P. Kouretsis and C.G. Tsagas, Raychaudhuri’s equation and aspects of relativistic charged collapse, Phys. Rev. D 82 (2010) 124053 [arXiv:1010.4211] [INSPIRE]; G. Abreu and M. Visser, Some generalizations of the Raychaudhuri equation, Phys. Rev. D 83 (2011) 104016 [arXiv:1012.4806] [INSPIRE]; C. Valls, Darbouxian integrals for generalized Raychaudhuri equations, J. Math. Phys. 52 (2011) 032703 [INSPIRE]. [17] L.P. Eisenhart, Riemannian geometry, Princeton University Press, Princeton U.S.A. (1926). [18] S. Capozziello, G. Lambiase and C. Stornaiolo, Geometric classification of the torsion tensor in space-time, Annalen Phys. 10 (2001) 713 [gr-qc/0101038] [INSPIRE]. [19] R. Bousso, The holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE]. [20] A. Friedmann, ¨Uber die Kr¨ummung des Raumes (in German), Z. Phys. A 10 (1922) 377; ¨Uber die M¨oglichkeit einer Welt mit konstanter negativer Kr¨ummung des Raumes (in German), Z. Phys. A 21 (1924) 326; G. Lemaˆıtre, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae, Mon. Not. Roy. Astron. Soc. 91 (1931) 483 [INSPIRE]; The expanding universe, Mon. Not. Roy. Astron. Soc. 91 (1931) 490 [INSPIRE]; Republication of: the beginning of the world from the point of view of quantum theory, Nature 127 (1931) 706 [Gen. Rel. Grav. 43 (2011) 2929] [INSPIRE]; The expanding universe, Gen. Rel. Grav. 29 (1997) 641 [Annales Soc. Sci. Brux. Ser. I Sci. Math. Astron. Phys. A 53 (1933) 51] [INSPIRE]; H. Robertson, Kinematics and world-structure, Astrophys. J. 82 (1935) 284 [INSPIRE]; Kinematics and world-structure. 2, Astrophys. J. 83 (1935) 187 [INSPIRE]; Kinematics and world-structure. 3, Astrophys. J. 83 (1936) 257 [INSPIRE]; H.P. Robertson and T.W. Noonan, Relativity and cosmology, AD-678-054 [INSPIRE]; A.G. Walker, On Milne’s theory of world-structure, Proc. Lond. Math. Soc. 42 (1937) 90. [21] N. Jarosik et al., Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: sky maps, systematic errors and basic results, Astrophys. J. Suppl. 192 (2011) 14 [arXiv:1001.4744] [INSPIRE]; WMAP collaboration, E. Komatsu et al., Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation, Astrophys. J. Suppl. 192 (2011) 18 [arXiv:1001.4538] [INSPIRE]. [22] R. Lazkoz, V. Salzano and I. Sendra, Revisiting a model-independent dark energy reconstruction method, Eur. Phys. J. C 72 (2012) 2130 [arXiv:1202.4689] [INSPIRE].
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