{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T02:03:10Z","timestamp":1776736990908,"version":"3.51.2"},"reference-count":111,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2015,8,27]],"date-time":"2015-08-27T00:00:00Z","timestamp":1440633600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Universe"],"abstract":"<jats:p>Recently, the phenomenology of f(R) gravity has been scrutinized. This scrutiny has been motivated by the possibility to account for the self-accelerated cosmic expansion without invoking dark energy sources. Besides, this kind of modified gravity is capable of addressing the dynamics of several self-gravitating systems alternatively to the presence of dark matter. It has been established that both metric and Palatini versions of these theories have interesting features but also manifest severe and different downsides. A hybrid combination of theories, containing elements from both these two formalisms, turns out to be also very successful accounting for the observed phenomenology and is able to avoid some drawbacks of the original approaches. This article reviews the formulation of this hybrid metric-Palatini approach and its main achievements in passing the local tests and in applications to astrophysical and cosmological scenarios, where it provides a unified approach to the problems of dark energy and dark matter.<\/jats:p>","DOI":"10.3390\/universe1020199","type":"journal-article","created":{"date-parts":[[2015,8,27]],"date-time":"2015-08-27T10:03:37Z","timestamp":1440669817000},"page":"199-238","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":143,"title":["Hybrid Metric-Palatini Gravity"],"prefix":"10.3390","volume":"1","author":[{"given":"Salvatore","family":"Capozziello","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Napoli \u201cFederico I\u201d, Napoli I-80126, Italy"},{"name":"INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, Napoli I-80126, Italy"},{"name":"Gran Sasso Science Institute (INFN), Viale F. Crispi 7, I-67100 L\u2019Aquila, Italy"}]},{"given":"Tiberiu","family":"Harko","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University College London, Gower Street, London WC1E-6BT, UK"}]},{"given":"Tomi","family":"Koivisto","sequence":"additional","affiliation":[{"name":"Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden"}]},{"given":"Francisco","family":"Lobo","sequence":"additional","affiliation":[{"name":"Instituto de Astrof\u00edsica e Ci\u00eancias do Espa\u00e7o, Universidade de Lisboa, Edif\u00edcio C8, Campo Grande, PT1749-016 Lisbon, Portugal"}]},{"given":"Gonzalo","family":"Olmo","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Te\u00f3rica and IFIC, Centro Mixto Universidad de Valencia - CSIC. Universidad de Valencia, Burjassot-46100, Valencia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2015,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"38","DOI":"10.3390\/universe1010038","article-title":"Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein\u2019s Theory of Gravitation in Its Centennial Year","volume":"1","author":"Iorio","year":"2015","journal-title":"Universe"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1086\/307221","article-title":"Measurements of Omega and Lambda from 42 high redshift supernovae","volume":"517","author":"Perlmutter","year":"1999","journal-title":"Astrophys. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1086\/300499","article-title":"Observational evidence from supernovae for an accelerating universe and a cosmological constant","volume":"116","author":"Riess","year":"1998","journal-title":"Astron. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1086\/383612","article-title":"Type Ia supernova discoveries at z > 1 from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution","volume":"607","author":"Riess","year":"2004","journal-title":"Astrophys. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1086\/322348","article-title":"The farthest known supernova: support for an accelerating universe and a glimpse of the epoch of deceleration","volume":"560","author":"Riess","year":"2001","journal-title":"Astrophys. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1103\/PhysRevLett.83.670","article-title":"Constraining dark energy with SNe Ia and large scale structure","volume":"83","author":"Perlmutter","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bennett, C.L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S.S., Page, L., Spergel, D.N., and Tucker, G.S. (2003). First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Preliminary maps and basic results. Astrophys. J. Suppl.","DOI":"10.1086\/377252"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hinshaw, G., Spergel, D.N., Verde, L., Hill, R.S., Meyer, S.S., Barnes, C., Bennett, C.L., Halpern, M., Jarosik, N., and Kogut, A. (2003). First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: The Angular power spectrum. Astrophys. J. Suppl.","DOI":"10.1086\/377225"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1142\/S0218271802002025","article-title":"Curvature quintessence","volume":"11","author":"Capozziello","year":"2002","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1142\/S021827180600942X","article-title":"Dynamics of dark energy","volume":"15","author":"Copeland","year":"2006","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"De Felice, A., and Tsujikawa, S. (2010). f(R) theories. Living Rev. Rel.","DOI":"10.12942\/lrr-2010-3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"043528","DOI":"10.1103\/PhysRevD.70.043528","article-title":"Is cosmic speed-up due to new gravitational physics?","volume":"70","author":"Carroll","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_13","unstructured":"Lobo, F.S.N. (2009). The dark side of gravity: Modified theories of gravity, arXiv:0807.1640."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Capozziello, S., and de Laurentis, M. (2011). Extended Theories of Gravity. Phys. Rept.","DOI":"10.1016\/j.physrep.2011.09.003"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1142\/S0219887807001928","article-title":"Introduction to modified gravity and gravitational alternative for dark energy","volume":"4","author":"Nojiri","year":"2007","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s10509-012-1181-8","article-title":"Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests","volume":"342","author":"Bamba","year":"2012","journal-title":"Astrophys. Space Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.physrep.2011.04.001","article-title":"Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models","volume":"505","author":"Nojiri","year":"2011","journal-title":"Phys. Rept."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1142\/S0218271811018925","article-title":"Palatini Approach to Modified Gravity: f(R) Theories and Beyond","volume":"20","author":"Olmo","year":"2011","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2014.12.002","article-title":"Beyond the Cosmological Standard Model","volume":"568","author":"Joyce","year":"2015","journal-title":"Phys. Rept."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.5506\/APhysPolB.43.2307","article-title":"Screened modified gravity","volume":"43","author":"Brax","year":"2012","journal-title":"Acta Phys. Polon. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"241102","DOI":"10.1103\/PhysRevLett.109.241102","article-title":"Screening Modifications of Gravity through Disformally Coupled Fields","volume":"109","author":"Koivisto","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"083503","DOI":"10.1103\/PhysRevD.82.083503","article-title":"Chameleons with Field Dependent Couplings","volume":"82","author":"Brax","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"044015","DOI":"10.1103\/PhysRevD.86.044015","article-title":"A Unified Description of Screened Modified Gravity","volume":"86","author":"Brax","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2355","DOI":"10.1088\/0264-9381\/23\/7\/009","article-title":"Cosmological perturbations in the Palatini formulation of modified gravity","volume":"23","author":"Koivisto","year":"2006","journal-title":"Class. Quant. Grav."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"083517","DOI":"10.1103\/PhysRevD.73.083517","article-title":"The matter power spectrum in f(r) gravity","volume":"73","author":"Koivisto","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"084021","DOI":"10.1103\/PhysRevD.77.084021","article-title":"Hydrogen atom in Palatini theories of gravity","volume":"77","author":"Olmo","year":"2008","journal-title":"Phys. Rev. D"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"061101","DOI":"10.1103\/PhysRevLett.98.061101","article-title":"Violation of the Equivalence Principle in Modified Theories of Gravity","volume":"98","author":"Olmo","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"084016","DOI":"10.1103\/PhysRevD.85.084016","article-title":"Metric-Palatini gravity unifying local constraints and late-time cosmic acceleration","volume":"85","author":"Harko","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1342006","DOI":"10.1142\/S0218271813420066","article-title":"Hybrid modified gravity unifying local tests, galactic dynamics and late-time cosmic acceleration","volume":"22","author":"Capozziello","year":"2013","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"044016","DOI":"10.1103\/PhysRevD.83.044016","article-title":"Unifying Einstein and Palatini gravities","volume":"83","author":"Amendola","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_31","unstructured":"Koivisto, T.S., Mota, D.F., and Sandstad, M. (2013). Novel aspects of C-theories in Cosmology, arXiv:1305.4754."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"101501","DOI":"10.1103\/PhysRevD.83.101501","article-title":"On new variational principles as alternatives to the Palatini method","volume":"83","author":"Koivisto","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"015014","DOI":"10.1088\/0264-9381\/28\/1\/015014","article-title":"A Unified Approach to Variational Derivatives of Modified Gravitational Actions","volume":"28","author":"Baykal","year":"2011","journal-title":"Class. Quant. Grav."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"084024","DOI":"10.1103\/PhysRevD.86.084024","article-title":"The Bimetric variational principle for General Relativity","volume":"86","author":"Jimenez","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"084041","DOI":"10.1103\/PhysRevD.88.084041","article-title":"Multi-Scalar-Tensor Equivalents for Modified Gravitational Actions","volume":"88","author":"Baykal","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"155005","DOI":"10.1088\/0264-9381\/30\/15\/155005","article-title":"Non-locality of the C- and D-theories","volume":"30","author":"Sandstad","year":"2013","journal-title":"Class. Quant. Grav."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"135002","DOI":"10.1088\/0264-9381\/31\/13\/135002","article-title":"Extended Gauss-Bonnet gravities in Weyl geometry","volume":"31","author":"Jimenez","year":"2014","journal-title":"Class. Quant. Grav."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"021","DOI":"10.1088\/1475-7516\/2015\/05\/021","article-title":"ADM Analysis of Gravity Models within the Framework of Bimetric Variational Formalism","volume":"2015","author":"Golovnev","year":"2015","journal-title":"J. Cosmo. Astropart. Phys"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4289","DOI":"10.1088\/0264-9381\/23\/12\/N01","article-title":"Covariant conservation of energy momentum in modified gravities","volume":"23","author":"Koivisto","year":"2006","journal-title":"Class. Quant. Grav."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"063505","DOI":"10.1103\/PhysRevD.72.063505","article-title":"Dark energy dominance and cosmic acceleration in first order formalism","volume":"72","author":"Allemandi","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.physletb.2014.11.034","article-title":"Brane-world and loop cosmology from a gravity-matter coupling perspective","volume":"740","author":"Olmo","year":"2015","journal-title":"Phys. Lett. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1140\/epjc\/s10052-010-1467-3","article-title":"f(R,Lm) gravity","volume":"70","author":"Harko","year":"2010","journal-title":"Eur. Phys. J. C"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"104016","DOI":"10.1103\/PhysRevD.75.104016","article-title":"Extra force in f(R) modified theories of gravity","volume":"75","author":"Bertolami","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"084018","DOI":"10.1103\/PhysRevD.77.084018","article-title":"Do f(R) theories matter?","volume":"77","author":"Bertolami","year":"2008","journal-title":"Phys. Rev. D"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"064036","DOI":"10.1103\/PhysRevD.78.064036","article-title":"Non-minimum coupling of perfect fluids to curvature","volume":"78","author":"Bertolami","year":"2008","journal-title":"Phys. Rev. D"},{"key":"ref_46","unstructured":"Bertolami, O., Paramos, J., Harko, T., and Lobo, F.S.N. (2008). Non-minimal curvature-matter couplings in modified gravity, arXiv:0811.2876 [gr-qc]."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1467","DOI":"10.1142\/S0217732311035869","article-title":"Palatini formulation of modified gravity with a nonminimal curvature-matter coupling","volume":"26","author":"Harko","year":"2011","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"024020","DOI":"10.1103\/PhysRevD.84.024020","article-title":"f(R, T) gravity","volume":"84","author":"Harko","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"044023","DOI":"10.1103\/PhysRevD.88.044023","article-title":"Further matters in space-time geometry: f(R, T, R\u03bc\u03bdT\u03bc\u03bd) gravity","volume":"88","author":"Haghani","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/j.physletb.2013.07.026","article-title":"f(R, T, R\u03bc\u03bdT\u03bc\u03bd) gravity phenomenology and \u039bCDM universe","volume":"725","author":"Odintsov","year":"2013","journal-title":"Phys. Lett. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"124034","DOI":"10.1103\/PhysRevD.86.124034","article-title":"Geodesic deviation, Raychaudhuri equation, and tidal forces in modified gravity with an arbitrary curvature-matter coupling","volume":"86","author":"Harko","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"064052","DOI":"10.1103\/PhysRevD.88.064052","article-title":"Consistency of nonminimally coupled f(R) gravity","volume":"88","author":"Tamanini","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"104003","DOI":"10.1103\/PhysRevD.91.104003","article-title":"Consistency of universally nonminimally coupled f(R, T, R\u03bc\u03bdT\u03bc\u03bd) theories","volume":"91","author":"Ayuso","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"011","DOI":"10.1088\/1475-7516\/2013\/04\/011","article-title":"Cosmology of hybrid metric-Palatini f(X)-gravity","volume":"2013","author":"Capozziello","year":"2013","journal-title":"J. Cosmo. Astropart. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"261102","DOI":"10.1103\/PhysRevLett.95.261102","article-title":"The Gravity Lagrangian according to Solar System experiments","volume":"95","author":"Olmo","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"083505","DOI":"10.1103\/PhysRevD.72.083505","article-title":"Post-Newtonian constraints on f(R) cosmologies in metric and Palatini formalism","volume":"72","author":"Olmo","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1063\/1.3292516","article-title":"Cosmology of modified (but second order) gravity","volume":"1206","author":"Koivisto","year":"2010","journal-title":"AIP Conf. Proc."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1142\/S0219887809003874","article-title":"On the well formulation of the initial value problem of metric-affine f(R)-gravity","volume":"6","author":"Capozziello","year":"2009","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"175013","DOI":"10.1088\/0264-9381\/26\/17\/175013","article-title":"The Cauchy problem for metric-affine f(R)-gravity in presence of perfect-fluid matter","volume":"26","author":"Capozziello","year":"2009","journal-title":"Class. Quant. Grav."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1142\/S0219887811005063","article-title":"The Cauchy problem for metric-affine f(R)-gravity in presence of a Klein-Gordon scalar field","volume":"8","author":"Capozziello","year":"2011","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1250006","DOI":"10.1142\/S0219887812500065","article-title":"The Cauchy problem for f(R)-gravity: An Overview","volume":"9","author":"Capozziello","year":"2012","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"104036","DOI":"10.1103\/PhysRevD.83.104036","article-title":"Hamiltonian Formulation of Palatini f(R) theories a la Brans-Dicke","volume":"83","author":"Olmo","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Choquet\u2013Bruhat, Y. (2009). General Relativity and the Einstein Equations, Oxford University Press Inc.","DOI":"10.1093\/acprof:oso\/9780199230723.001.0001"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"4719","DOI":"10.1088\/0264-9381\/23\/14\/010","article-title":"The Cauchy problem of scalar tensor theories of gravity","volume":"23","author":"Salgado","year":"2006","journal-title":"Class. Quant. Grav."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1450042","DOI":"10.1142\/S021988781450042X","article-title":"The Cauchy problem in hybrid metric-Palatini f(X)-gravity","volume":"11","author":"Capozziello","year":"2014","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Wald, R.M. (1984). General Relativity, University of Chicago Press.","DOI":"10.7208\/chicago\/9780226870373.001.0001"},{"key":"ref_67","unstructured":"Leray, J. (1953). Hyperbolic Differential Equations, Institute for Advanced Study Pub."},{"key":"ref_68","unstructured":"Witten, L. (1962). Cauchy Problem, in Gravitation: An Introduction to Current Research, Wiley."},{"key":"ref_69","unstructured":"Four\u00e9s\u2013Bruhat, Y. (1958). Bull. de la SMF, Societe Matematique de France."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/978-3-540-71013-4_14","article-title":"Avoiding dark energy with 1\/r modifications of gravity","volume":"720","author":"Woodard","year":"2007","journal-title":"Lect. Notes Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"104030","DOI":"10.1103\/PhysRevD.87.104030","article-title":"Ghosts in pure and hybrid formalisms of gravity theories: A unified analysis","volume":"87","author":"Koivisto","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"031101","DOI":"10.1103\/PhysRevLett.108.031101","article-title":"Towards singularity and ghost free theories of gravity","volume":"108","author":"Biswas","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_73","unstructured":"Biswas, T., Koivisto, T., and Mazumdar, A. (2013). Nonlocal theories of gravity: The flat space propagator, arXiv:1302.0532 [gr-qc]."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1088\/0264-9381\/21\/2\/006","article-title":"Higher order gravity theories and scalar tensor theories","volume":"21","author":"Flanagan","year":"2003","journal-title":"Class. Quant. Grav."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"084031","DOI":"10.1103\/PhysRevD.87.084031","article-title":"Generalized hybrid metric-Palatini gravity","volume":"87","author":"Tamanini","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Carloni, S., Koivisto, T., and Lobo, F.S.N. (2015). A dynamical system analysis of hybrid metric-Palatini cosmologies, arXiv:1507.04306 [gr-qc].","DOI":"10.1103\/PhysRevD.92.064035"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"104019","DOI":"10.1103\/PhysRevD.88.104019","article-title":"Einstein static Universe in hybrid metric-Palatini gravity","volume":"88","author":"Lobo","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"083527","DOI":"10.1103\/PhysRevD.89.083527","article-title":"Dynamics of Linear Perturbations in the hybrid metric-Palatini gravity","volume":"89","author":"Lima","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_79","doi-asserted-by":"crossref","unstructured":"Lima, N.A., and Smer-Barreto, V. (2015). Constraints on hybrid metric-Palatini models from background evolution, arXiv:1501.05786 [astro-ph.CO].","DOI":"10.3847\/0004-637X\/818\/2\/186"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"023517","DOI":"10.1103\/PhysRevD.91.023517","article-title":"Invariant solutions and Noether symmetries in Hybrid Gravity","volume":"91","author":"Borowiec","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1086\/176550","article-title":"Cosmological perturbation theory in the synchronous and conformal Newtonian gauges","volume":"455","author":"Ma","year":"1995","journal-title":"Astrophys. J."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"2236","DOI":"10.1103\/PhysRevLett.85.2236","article-title":"Reconstruction of a scalar tensor theory of gravity in an accelerating universe","volume":"85","author":"Boisseau","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"Koivisto, T.S., Saridakis, E.N., and Tamanini, N. (2015). Scalar-Fluid theories: Cosmological perturbations and large-scale structure, arXiv:1505.07556 [astro-ph.CO].","DOI":"10.1088\/1475-7516\/2015\/09\/047"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"123515","DOI":"10.1103\/PhysRevD.77.123515","article-title":"On the evolution of density perturbations in f(R) theories of gravity","volume":"77","author":"Dobado","year":"2008","journal-title":"Phys. Rev. D"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"135011","DOI":"10.1088\/0264-9381\/29\/13\/135011","article-title":"Covariant gauge-invariant perturbations in multifluid f(R) gravity","volume":"29","author":"Abebe","year":"2012","journal-title":"Class. Quant. Grav."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"161101","DOI":"10.1103\/PhysRevLett.110.161101","article-title":"Releasing scalar fields: cosmological simulations of scalar-tensor theories for gravity beyond the static approximation","volume":"110","author":"Llinares","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Sawicki, I., and Bellini, E. (2015). Limits of Quasi-Static Approximation in Modified-Gravity Cosmologies, arXiv:1503.06831 [astro-ph.CO].","DOI":"10.1103\/PhysRevD.92.084061"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"141301","DOI":"10.1103\/PhysRevLett.102.141301","article-title":"Dark Matter from R2-gravity","volume":"102","author":"Cembranos","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"049","DOI":"10.1088\/1475-7516\/2012\/02\/049","article-title":"Cosmological evolution in R2 gravity","volume":"2012","author":"Arbuzova","year":"2012","journal-title":"J. Cosmo. Astropart. Phys"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"121502","DOI":"10.1103\/PhysRevD.84.121502","article-title":"The post-Newtonian limit in C-theories of gravitation","volume":"84","author":"Koivisto","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"1530015","DOI":"10.1142\/S0218271815300153","article-title":"Gravitational anomalies in the solar system?","volume":"24","author":"Iorio","year":"2015","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"086901","DOI":"10.1088\/0034-4885\/75\/8\/086901","article-title":"21-cm cosmology","volume":"75","author":"Pritchard","year":"2012","journal-title":"Rept. Prog. Phys."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1086\/158003","article-title":"Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122 kpc)","volume":"238","author":"Rubin","year":"1980","journal-title":"Astrophys. J."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/mnras\/278.1.27","article-title":"The Universal rotation curve of spiral galaxies: 1. The Dark matter connection","volume":"281","author":"Persic","year":"1996","journal-title":"Mon. Not. Roy. Astron. Soc."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"125016","DOI":"10.1103\/PhysRevD.63.125016","article-title":"An Alternative approach to the galactic dark matter problem","volume":"63","author":"Nucamendi","year":"2001","journal-title":"Phys. Rev. D"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"084050","DOI":"10.1103\/PhysRevD.81.084050","article-title":"Galactic rotation curves in modified gravity with non-minimal coupling between matter and geometry","volume":"81","author":"Harko","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"051101","DOI":"10.1103\/PhysRevLett.92.051101","article-title":"Galactic potentials","volume":"92","author":"Lake","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.astropartphys.2013.09.005","article-title":"Galactic rotation curves in hybrid metric-Palatini gravity","volume":"50","author":"Capozziello","year":"2013","journal-title":"Astropart. Phys."},{"key":"ref_99","unstructured":"Melchiorri, F., and Rephaeli, Y. (2004, January 6\u201316). X-ray Observations of Clusters of Galaxies. Background Microwave Radiation and Intracluster Cosmology, Proceedings of the International School of Physics \u201cEnrico Fermi\u201d, Varenna, Italy."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1086\/338753","article-title":"The Mass function of an X-ray flux-limited sample of galaxy clusters","volume":"567","author":"Reiprich","year":"2002","journal-title":"Astrophys. J."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1051\/0004-6361:20010282","article-title":"Cosmic mass functions from Gaussian stochastic diffusion processes","volume":"370","author":"Schuecker","year":"2001","journal-title":"Astron. Astrophys."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"1684","DOI":"10.1111\/j.1365-2966.2009.15987.x","article-title":"Hydrodynamical N-body simulations of coupled dark energy cosmologies","volume":"403","author":"Baldi","year":"2010","journal-title":"Mon. Not. Roy. Astron. Soc."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"024007","DOI":"10.1103\/PhysRevD.83.024007","article-title":"N-Body Simulations for Coupled Scalar Field Cosmology","volume":"83","author":"Li","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"024","DOI":"10.1088\/1475-7516\/2013\/07\/024","article-title":"The virial theorem and the dark matter problem in hybrid metric-Palatini gravity","volume":"2013","author":"Capozziello","year":"2013","journal-title":"J. Cosmo. Astropart. Phys."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"044013","DOI":"10.1103\/PhysRevD.76.044013","article-title":"The Virial theorem and the dynamics of clusters of galaxies in the brane world models","volume":"76","author":"Harko","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_106","first-page":"595","article-title":"Collision free gases in spatially homogeneous space-times","volume":"22","author":"Maartens","year":"1990","journal-title":"J. Math. Phys."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1088\/0264-9381\/6\/8\/017","article-title":"Transport Equations for Freely Propagating Photons in Curved Space-times: A Derivation by Wigner Transformation","volume":"6","author":"Bildhauer","year":"1989","journal-title":"Class. Quant. Grav."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1093\/mnras\/148.3.249","article-title":"The dynamics of clusters of galaxies in universes with non-zero cosmological constant, and the virial theorem mass discrepancy[J]. Monthly Notices of the Royal Astronomical Society","volume":"148","author":"Jackson","year":"1970","journal-title":"Month. Not. R. Astr. Soc."},{"key":"ref_109","doi-asserted-by":"crossref","unstructured":"Borka, B., Capozziello, S., Jovanovic, P., and Jovanovic, V.B. (2015). Probing hybrid modified gravity by stellar motion around Galactic Centre, arXiv:1504.07832 [gr-qc].","DOI":"10.1016\/j.astropartphys.2016.03.002"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"6","DOI":"10.12942\/lrr-2013-6","article-title":"Cosmology and fundamental physics with the Euclid satellite","volume":"16","author":"Amendola","year":"2013","journal-title":"Living Rev. Rel."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"127504","DOI":"10.1103\/PhysRevD.86.127504","article-title":"Wormholes supported by hybrid metric-Palatini gravity","volume":"86","author":"Capozziello","year":"2012","journal-title":"Phys. Rev. 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