Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)

ilustraciones, diagramas

Autores:: Rivera Amezquita, Marlon Zamihir

Tipo de recurso:

Fecha de publicación:: 2022

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: spa

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dc.title.spa.fl_str_mv	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
dc.title.translated.eng.fl_str_mv	Cosmological gravitational radiation in the 1+3 formalism for f(R) gravity theories
title	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
spellingShingle	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R) 530 - Física::539 - Física moderna Gravitational waves Radiación gravitacional Gravitational radiation Ondas Gravitacionales Tensor de Weyl Teorías de gravedad modificada f(R) Formalismo 1+3 Hu-Sawicki Gravitational waves Weyl Tensor Modified gravity theories f(R) 1 + 3 formalism Hu-Sawicki
title_short	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
title_full	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
title_fullStr	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
title_full_unstemmed	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
title_sort	Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)
dc.creator.fl_str_mv	Rivera Amezquita, Marlon Zamihir
dc.contributor.advisor.spa.fl_str_mv	Castañeda Colorado, Leonardo
dc.contributor.author.spa.fl_str_mv	Rivera Amezquita, Marlon Zamihir
dc.contributor.researchgroup.spa.fl_str_mv	Grupo de Astronomía Galáctica, Gravitación y Cosmología
dc.subject.ddc.spa.fl_str_mv	530 - Física::539 - Física moderna
topic	530 - Física::539 - Física moderna Gravitational waves Radiación gravitacional Gravitational radiation Ondas Gravitacionales Tensor de Weyl Teorías de gravedad modificada f(R) Formalismo 1+3 Hu-Sawicki Gravitational waves Weyl Tensor Modified gravity theories f(R) 1 + 3 formalism Hu-Sawicki
dc.subject.lcc.eng.fl_str_mv	Gravitational waves
dc.subject.lemb.spa.fl_str_mv	Radiación gravitacional
dc.subject.lemb.eng.fl_str_mv	Gravitational radiation
dc.subject.proposal.spa.fl_str_mv	Ondas Gravitacionales Tensor de Weyl Teorías de gravedad modificada f(R) Formalismo 1+3 Hu-Sawicki
dc.subject.proposal.eng.fl_str_mv	Gravitational waves Weyl Tensor Modified gravity theories f(R) 1 + 3 formalism Hu-Sawicki
description	ilustraciones, diagramas
publishDate	2022
dc.date.issued.none.fl_str_mv	2022
dc.date.accessioned.none.fl_str_mv	2023-08-01T16:49:44Z
dc.date.available.none.fl_str_mv	2023-08-01T16:49:44Z
dc.type.spa.fl_str_mv	Trabajo de grado - Maestría
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dc.type.content.spa.fl_str_mv	Text
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status_str	acceptedVersion
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dc.identifier.instname.spa.fl_str_mv	Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv	Repositorio Institucional Universidad Nacional de Colombia
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url	https://repositorio.unal.edu.co/handle/unal/84392 https://repositorio.unal.edu.co/
identifier_str_mv	Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv	spa
language	spa
dc.relation.references.spa.fl_str_mv	[1] A. Einstein, “Die Feldgleichungen der Gravitation,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin, pp. 844–847, Jan. 1915. [2] Abbott.;et.al, “Observation of gravitational waves from a binary black hole merger,” Phys. Rv. Lett., vol. 116, p. 061102, Feb 2016. [Online]. Available: https://link.aps.org/doi/10. 1103/PhysRevLett.116.061102 [3] D. Bettoni, J. M. Ezquiaga, K. Hinterbichler, and M. Zumalacárregui, “Speed of gravitational waves and the fate of scalar-tensor gravity,” Phys. Rev. D, vol. 95, p. 084029, Apr 2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.95.084029 [4] G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum, Relativistic Cosmology. Cambridge University Press, 2012. [5] G. F. R. Ellis, “Republication of: Relativistic cosmology,” General Relativity and Gravitation, vol. 41, no. 3, pp. 581–660, Mar. 2009. [6] I. S. W, Hu., “Model of f(r) cosmic acceleration that evade solar system test.phys.rev.d.76,064004.2007.” [7] A. de la Cruz-Dombriz, P. K. S. Dunsby, V. C. Busti, and S. Kandhai, “Tidal forces in f(r) theories of gravity,” Phys. Rev. D, vol. 89, p. 064029, Mar 2014. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.89.064029 [8] S. Basilakos, S. Nesseris, and L. Perivolaropoulos, “Observational constraints on viable f(r) parametrizations with geometrical and dynamical probes,” Phys. Rev. D, vol. 87, p. 123529, Jun 2013. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.87.123529 [9] A. G. Riess and A. V. F. .;et al, “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” The Astronomical Journal, vol. 116, no. 3, pp. 1009–1038, sep 1998. [Online]. Available: https://doi.org/10.1086%2F300499 [10] T. S. A, De Felipe., “f(r) theories.living rev.relativity.2010.” [11] Y. Fujii and K.-i. Maeda, The Scalar-Tensor Theory of Gravitation, ser. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2003. [12] M. Warkentin, “Modification of the laws of gravity in the dgp model by the presence of a second dgp brane,” Journal of High Energy Physics, vol. 2020, no. 3, Mar 2020. [Online]. Available: http://dx.doi.org/10.1007/JHEP03(2020)015 [13] S. Tsujikawa, K. Uddin, and R. Tavakol, “Density perturbations in f(r) gravity theories in metric and palatini formalisms,” Phys. Rev. D, vol. 77, p. 043007, Feb 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.77.043007 [14] T. P. Sotiriou and V. Faraoni, “f(r) theories of gravity,” Rev. Mod. Phys., vol. 82, pp. 451–497, Mar 2010. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.82.451 [15] L. Yang, C.-C. Lee, and C.-Q. Geng, “Gravitational waves in viablef(r) models,” Journal of Cosmology and Astroparticle Physics, vol. 2011, no. 08, pp. 029–029, aug 2011. [Online]. Available: https://doi.org/10.1088%2F1475-7516%2F2011%2F08%2F029 [16] C.-P. Ma and E. Bertschinger, “Cosmological perturbation theory in the synchronous and conformal newtonian gauges,” The Astrophysical Journal, vol. 455, p. 7, dec 1995. [Online]. Available: https://doi.org/10.1086%2F176550 [17] S. Carloni, P. K. S. Dunsby, and A. Troisi, “Evolution of density perturbations in f(r) gravity,” Phys. Rev. D, vol. 77, p. 024024, Jan 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.77.024024 [18] J. Ehlers, “Beiträge zur relativistischen Mechanik kontinuierlicher Medien,” Mainz Akademie Wissenschaften Mathematisch Naturwissenschaftliche Klasse, vol. 11, pp. 792–837, Jan. 1961. [19] W. Kundt and M. Trümper, “Republication of: Contributions to the theory of gravitational radiation fields. Exact solutions of the field equations of the general theory of relativity V,” Gen. Rel. Grav., vol. 48, no. 4, p. 44, 2016. [20] R. Maartens, G. F. R. Ellis, and S. T. C. Siklos, “Local freedom in the gravitational field,” Classical and Quantum Gravity, vol. 14, no. 7, pp. 1927–1936, jul 1997. [Online]. Available: https://doi.org/10.1088/0264-9381/14/7/025 [21] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity. Benjamin Cummings, 2003. [Online]. Available: http://www.amazon.com/Spacetime-GeometryIntroduction-General-Relativity/dp/0805387323 [22] J. D. Barrow, R. Maartens, and C. G. Tsagas, “Cosmology with inhomogeneous magnetic fields,” Physics Reports, vol. 449, no. 6, pp. 131–171, 2007. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0370157307001925 [23] E. Bertschinger, “Cosmological dynamics,” 1995. [Online]. Available: https://arxiv.org/abs/ astro-ph/9503125 [24] K. N. Ananda, S. Carloni, and P. K. S. Dunsby, “Evolution of cosmological gravitational waves in f(r) gravity,” Phys. Rev. D, vol. 77, p. 024033, Jan 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.77.024033 [25] S. W. Hawking, “Gravitational radiation in an expanding universe,” Journal of Mathematical Physics, vol. 9, no. 4, pp. 598–604, 1968. [Online]. Available: https: //doi.org/10.1063/1.1664615 [26] L. Herrera, N. O. Santos, and J. Carot, “Gravitational radiation, vorticity and the electric and magnetic part of weyl tensor,” Journal of Mathematical Physics, vol. 47, no. 5, p. 052502, 2006. [Online]. Available: https://doi.org/10.1063/1.2199027 [27] R. Maartens, “Is the universe homogeneous?” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 369, no. 1957, pp. 5115–5137, 2011. [Online]. Available: https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2011.0289 [28] C. Clarkson and R. Maartens, “Inhomogeneity and the foundations of concordance cosmology,” Classical and Quantum Gravity, vol. 27, no. 12, p. 124008, may 2010. [Online]. Available: https://doi.org/10.1088/0264-9381/27/12/124008 [29] A. Abebe, M. Abdelwahab, A. Cruz-Dombriz, and P. Dunsby, “Covariant gauge-invariant per turbations in multifluid f(r) gravity,” Classical and Quantum Gravity - CLASS QUANTUM GRAVITY, vol. 29, 07 2012. [30] Planck Collaboration, Ade, P. A. R., Aghanim, N., Arnaud, M., and Ashdown, M. et.al, “Planck 2015 results - xiii. cosmological parameters,” A&A, vol. 594, p. A13, 2016. [Online]. Available: https://doi.org/10.1051/0004-6361/201525830 [31] A. A. Coley, “Dynamical systems in cosmology,” 1999. [Online]. Available: https: //arxiv.org/abs/gr-qc/9910074 [32] S. Dodelson, Modern Cosmology. Academic Press, Elsevier Science, 2003. [33] E. Lifshitz, “Republication of: On the gravitational stability of the expanding universe,” J. Phys. (USSR), vol. 10, no. 2, p. 116, 1946. [34] K. Nakamura, “Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations,” Progress of Theoretical Physics, vol. 110, no. 4, pp. 723–755, 10 2003. [Online]. Available: https://doi.org/10.1143/PTP.110.723 [35] J. M. Bardeen, “Gauge-invariant cosmological perturbations,” Phys. Rev. D, vol. 22, pp. 1882– 1905, Oct 1980. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.22.1882 [36] H. Kodama and M. Sasaki, “Cosmological Perturbation Theory,” Progress of Theoretical Physics Supplement, vol. 78, pp. 1–166, 01 1984. [Online]. Available: https: //doi.org/10.1143/PTPS.78.1 [37] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological pertur bations,” , vol. 215, no. 5-6, pp. 203–333, Jun. 1992 [38] G. F. R. Ellis and M. Bruni, “Covariant and gauge-invariant approach to cosmological density fluctuations,” Phys. Rev. D, vol. 40, pp. 1804–1818, Sep 1989. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.40.1804 [39] K. A. Malik and D. R. Matravers, “A concise introduction to perturbation theory in cosmology,” Classical and Quantum Gravity, vol. 25, no. 19, p. 193001, sep 2008. [Online]. Available: https://doi.org/10.1088/0264-9381/25/19/193001 [40] K. A. Malik and D. Wands, “Cosmological perturbations,” Physics Reports, vol. 475, no. 1-4, pp. 1–51, may 2009. [Online]. Available: https://doi.org/10.1016%2Fj.physrep.2009.03.001 [41] T. Gebbie and G. Ellis, “1+3 covariant cosmic microwave background anisotropies i: Algebraic relations for mode and multipole expansions,” Annals of Physics, vol. 282, no. 2, pp. 285–320, 2000. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0003491600960330 [42] M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments. Oxford University Press, 10 2007. [Online]. Available: https://doi.org/10.1093/acprof:oso/9780198570745.001. 0001 [43] P. K. S. Dunsby, B. A. C. C. Bassett, and G. F. R. Ellis, “Covariant analysis of gravitational waves in a cosmological context,” Classical and Quantum Gravity, vol. 14, no. 5, pp. 1215–1222, may 1997. [Online]. Available: https://doi.org/10.1088/0264-9381/14/5/023 [44] A. Challinor, “Microwave background anisotropies from gravitational waves: the 1 3 covariant approach,” Classical and Quantum Gravity, vol. 17, no. 4, pp. 871–889, jan 2000. [Online]. Available: https://doi.org/10.1088/0264-9381/17/4/309 [45] “744Bibliography,” in Gravitational Waves: Volume 2: Astrophysics and Cosmology. Oxford University Press, 03 2018. [46] C. Brans and R. H. Dicke, “Mach’s principle and a relativistic theory of gravitation,” Phys. Rev., vol. 124, pp. 925–935, Nov 1961. [Online]. Available: https://link.aps.org/doi/10.1103/ PhysRev.124.925 [47] V. Faraoni, Cosmology in scalar tensor gravity, 2004. [48] G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor. Phys., vol. 10, pp. 363–384, 1974. [49] A. Guarnizo, L. Castaneda, and J. M. Tejeiro, “Boundary Term in Metric f(R) Gravity: Field Equations in the Metric Formalism,” Gen. Rel. Grav., vol. 42, pp. 2713–2728, 2010. [50] A. M. Nzioki, R. Goswami, and P. K. S. Dunsby, “Vibrating Black Holes in f(R) gravity,” 8 2014. [51] T. Chiba, “1/r gravity and scalar-tensor gravity,” Physics Letters B, vol. 575, no. 1, pp. 1–3, 2003. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0370269303014126 [52] S. Kandhai and P. K. S. Dunsby, “Cosmological dynamics of viable f(r) theories of gravity,” 2015. [Online]. Available: https://arxiv.org/abs/1511.00101 [53] R. Arjona, W. Cardona, and S. Nesseris, “Unraveling the effective fluid approach for f(r) models in the subhorizon approximation,” Phys. Rev. D, vol. 99, p. 043516, Feb 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.99.043516 [54] H. Bourhrous, “Cmb tensor anisotropies in f(r) gravity,” 2013. [Online]. Available: https://arxiv.org/abs/1302.1887 [55] J. Lesgourgues, “The cosmic linear anisotropy solving system (class) i: Overview,” 2011. [Online]. Available: https://arxiv.org/abs/1104.2932
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dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia
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institution	Universidad Nacional de Colombia
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spelling	Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Castañeda Colorado, Leonardo09a756690df646a0589f42b538a56da2Rivera Amezquita, Marlon Zamihir8ec30de013537c2d6d58f6af1469cf64Grupo de Astronomía Galáctica, Gravitación y Cosmología2023-08-01T16:49:44Z2023-08-01T16:49:44Z2022https://repositorio.unal.edu.co/handle/unal/84392Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, diagramasLas ondas gravitacionales predichas por Albert Einstein [1], han tomado gran importancia desde su detección en el año 2015 [2]. Además de las fuentes astrofísicas, las ondas gravitacionales también pueden producirse en escenarios cosmológicos y pueden servir como pruebas para estudiar la viabilidad de las teorías de gravedad modificada [3]. En el formalismo covariante e invariante de gauge 1 + 3 [4] , la propagación de ondas gravitacionales esta descrita por la parte eléctrica y magnética del tensor de Weyl, o equivalentemente por el tensor de shear [5] . En este trabajo se realiza un calculo detallado de las diferentes ecuaciones que gobiernan las cantidades cinemáticas y dinámicas del formalismo 1 + 3, luego estudiando la teoría de perturbaciones para llegar a las ecuaciones de Onda para RG y para las teorías de gravedad modificada f(R). Se hallan soluciones para las épocas de radiación y materia en RG, y se estudian posibles soluciones de la ecuaciones de campo de onda para el modelo de Hu-Sawicki [6] , siguiendo las propuestas para la solución de las ecuaciones de campo dadas en [7, 8]. (Texto tomado de la fuente)Gravitational waves, predicted by Albert Einstein [1], have gained great importance since their detection in 2015 [2]. Besides astrophysical sources, gravitational waves can also be produced in cosmological scenarios and can serve as tests to study the viability of modified gravity theories [3]. In the covariant and gauge-invariant 1 + 3 formalism [4], the propagation of gravitational waves is described by the electric and magnetic part of the Weyl tensor, or equivalently by the shear tensor [5]. In this work, a detailed calculation of the different equations governing the kinematic and dynamic quantities of the 1+ 3 formalism is carried out, followed by the study of perturbation theory to obtain the Wave Equations for GR and for modified gravity theories f(R). Solutions are found for the radiation and matter epochs in GR, and possible solutions of the wave field equations are studied for the Hu-Sawicki model [6], following the proposals for solving the field equations given in [7, 8].MaestríaMagíster en Ciencias - FísicaGravedad Modificada103 páginasapplication/pdfspaUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - FísicaFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá530 - Física::539 - Física modernaGravitational wavesRadiación gravitacionalGravitational radiationOndas GravitacionalesTensor de WeylTeorías de gravedad modificada f(R)Formalismo 1+3Hu-SawickiGravitational wavesWeyl TensorModified gravity theories f(R)1 + 3 formalismHu-SawickiRadiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)Cosmological gravitational radiation in the 1+3 formalism for f(R) gravity theoriesTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TM[1] A. Einstein, “Die Feldgleichungen der Gravitation,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin, pp. 844–847, Jan. 1915.[2] Abbott.;et.al, “Observation of gravitational waves from a binary black hole merger,” Phys. Rv. Lett., vol. 116, p. 061102, Feb 2016. [Online]. Available: https://link.aps.org/doi/10. 1103/PhysRevLett.116.061102[3] D. Bettoni, J. M. Ezquiaga, K. Hinterbichler, and M. Zumalacárregui, “Speed of gravitational waves and the fate of scalar-tensor gravity,” Phys. Rev. D, vol. 95, p. 084029, Apr 2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.95.084029[4] G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum, Relativistic Cosmology. Cambridge University Press, 2012.[5] G. F. R. Ellis, “Republication of: Relativistic cosmology,” General Relativity and Gravitation, vol. 41, no. 3, pp. 581–660, Mar. 2009.[6] I. S. W, Hu., “Model of f(r) cosmic acceleration that evade solar system test.phys.rev.d.76,064004.2007.”[7] A. de la Cruz-Dombriz, P. K. S. Dunsby, V. C. Busti, and S. Kandhai, “Tidal forces in f(r) theories of gravity,” Phys. Rev. D, vol. 89, p. 064029, Mar 2014. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.89.064029[8] S. Basilakos, S. Nesseris, and L. Perivolaropoulos, “Observational constraints on viable f(r) parametrizations with geometrical and dynamical probes,” Phys. Rev. D, vol. 87, p. 123529, Jun 2013. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.87.123529[9] A. G. Riess and A. V. F. .;et al, “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” The Astronomical Journal, vol. 116, no. 3, pp. 1009–1038, sep 1998. [Online]. Available: https://doi.org/10.1086%2F300499[10] T. S. A, De Felipe., “f(r) theories.living rev.relativity.2010.”[11] Y. Fujii and K.-i. Maeda, The Scalar-Tensor Theory of Gravitation, ser. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2003.[12] M. Warkentin, “Modification of the laws of gravity in the dgp model by the presence of a second dgp brane,” Journal of High Energy Physics, vol. 2020, no. 3, Mar 2020. [Online]. Available: http://dx.doi.org/10.1007/JHEP03(2020)015[13] S. Tsujikawa, K. Uddin, and R. Tavakol, “Density perturbations in f(r) gravity theories in metric and palatini formalisms,” Phys. Rev. D, vol. 77, p. 043007, Feb 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.77.043007[14] T. P. Sotiriou and V. Faraoni, “f(r) theories of gravity,” Rev. Mod. Phys., vol. 82, pp. 451–497, Mar 2010. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.82.451[15] L. Yang, C.-C. Lee, and C.-Q. Geng, “Gravitational waves in viablef(r) models,” Journal of Cosmology and Astroparticle Physics, vol. 2011, no. 08, pp. 029–029, aug 2011. [Online]. Available: https://doi.org/10.1088%2F1475-7516%2F2011%2F08%2F029[16] C.-P. Ma and E. Bertschinger, “Cosmological perturbation theory in the synchronous and conformal newtonian gauges,” The Astrophysical Journal, vol. 455, p. 7, dec 1995. [Online]. Available: https://doi.org/10.1086%2F176550[17] S. Carloni, P. K. S. Dunsby, and A. Troisi, “Evolution of density perturbations in f(r) gravity,” Phys. Rev. D, vol. 77, p. 024024, Jan 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.77.024024[18] J. Ehlers, “Beiträge zur relativistischen Mechanik kontinuierlicher Medien,” Mainz Akademie Wissenschaften Mathematisch Naturwissenschaftliche Klasse, vol. 11, pp. 792–837, Jan. 1961.[19] W. Kundt and M. Trümper, “Republication of: Contributions to the theory of gravitational radiation fields. Exact solutions of the field equations of the general theory of relativity V,” Gen. Rel. Grav., vol. 48, no. 4, p. 44, 2016.[20] R. Maartens, G. F. R. Ellis, and S. T. C. 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Radiación gravitacional cosmológica en el formalismo 1+3 para las teorías de gravedad f(R)

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