Higher-Order Theories of Gravity: Types and Examples  

Variational Principles > s.a. actions for gravity [curvature-saturated]; theories of gravity; torsion in physics.
@ General references: Allemandi & Francaviglia IJGMP(07).
@ Action / Lagrangian: Nojiri & Odintsov PRD(00)ht/99, Dyer & Hinterbichler PRD(09)-a0809 [surface terms]; Krasnov PRD(10)-a0911 [with two propagating degrees of freedom, effective Lagrangians]; Caravelli & Modesto PLB(11)-a1001 [from black-hole entropy]; Biswas et al PRL(12)-a1110 [the most general covariant ghost-free theories].
@ Perturbation theory: Havas GRG(77) [linearized]; Multamäki et al PRD(10)-a0910 [stability of solutions]; Bueno & Cano PRD(16)-a1607 [simpler calculations].
@ Boundary terms: Guarnizo et al GRG(10)-a1002; Fatibene et al a1106; Teimouri et al JHEP(16)-a1606; Sáez-Chillón a2011 [Palatini f(R) gravity]; > s.a. Lanczos-Lovelock and scalar-tensor gravity below.
@ Hamiltonian formulation: Szczyrba JMP(87); Ezawa et al CQG(99)gq/98; Querella PhD(98)gq/99; Ezawa et al CQG(06)gq/05; Deruelle et al PRD(09)-a0906, PTP(09)-a0908; Fatibene et al CQG(10)-a1003 [self-dual connection formulation, constraints]; Sanyal et al CQG(12)-a1108; Mandal & Sanyal PRD(17)-a1709 [inequivalent canonical structures]; Sanyal AP(19)-a1807 [ambiguity and the Dirac approach].

f(R) Theories > s.a. energy-momentum pseudotensor; entropic gravity; unimodular gravity.
* Ostrogradski theorem: The only potentially stable, local modification of general relativity is to make the Lagrangian an arbitrary function of the Ricci scalar.
* Field equations: (Extended gravity theories) If \(\cal L\) is a non-linear function of the curvature scalar, then the field equations are of fourth order in the metric; These equations were first discussed by H Weyl in 1918, as an alternative to Einstein's theory; Palatini f(R) theories are mathematically equivalent to Brans-Dicke theories, but not physically.
* R−1 theories: Motivated by the fact that they lead to cosmological acceleration; A 1/R action by itself is ruled out by constraints on scalar-tensor gravity, but could work as an extra contribution to the action.
@ Reviews: Perez Bergliaffa a1107-proc; Capozziello & De Laurentis a1307-MG13; Sporea a1403 [intro].
@ General references: Schmidt IJGMP(07)gq/06-ln [4th-order]; Faraoni PLB(08)-a0806 [f(R) and conformal rescaling]; Jaime et al PRD(11)-a1006 [with R as extra variable]; Yang EPL(11)-a1103 [conformal transformations]; Jorás IJMPA(11)-in [shortcomings]; Paliathanasis et al PRD(11)-a1110 [constraints and solutions]; Tamanini & Koivisto PRD(13)-a1308 [non-minimally coupled, consistency]; Guendelman et al a1312-proc; de Haro EPL(14)-a1403 [holonomy corrections]; Carloni JCAP(15)-a1505 [as dynamical system]; Woodard a1506-en [Ostrogradski theorem]; Capozziello et al a1512-MG14 [frame equivalence]; Nayem & Sanyal IJMPD(18)-a1708 [different canonical forms]; Li et al PRD(18)-a1711 [causality and a-theorem constraints]; Chakraborty et al PRD(19)-a1812 [equivalence of Jordan and Einstein frames].
@ Solutions: Chakraborty & SenGupta EPJC(16)-a1604 [technique for solving field equations]; > s.a. phenomenology.
@ And scalar-tensor gravity: Ezawa & Ohkuwa a1204-ch [and quantization]; Fatibene & Francaviglia IJGMP(14)-a1302; Fatibene & Garruto IJGMP(14), Morais Graça & Bezerra MPLA(15)-a1312 [and Brans-Dicke theories]; Langlois & Noui JCAP(16)-a1512 [Hamiltonian analysis]; Fatibene & Garruto IJGMP(16)-a1601-conf; Nayem & Sanyal IJMPD(17)-a1609; Ruf & Steinwachs PRD(18)-a1711 [quantum equivalence]; Castañeda & Velásquez a1808 [including boundary terms].
@ And other theories: Granda a0812-in [from holographic principle]; Saltas & Hindmarsh CQG(11)-a1002 [equivalence of f(R) and Gauss-Bonnet theories]; Fabbri & Vignolo AdP(12)-a1012 [with torsion and ELKO matter]; Benachour a1202 [theories equivalent to general relativity]; Santos & de Souza Santos AIP(12)-a1212 [and its bimetric structure].
@ Hamiltonian formulation: Olmo & Sanchis-Alepuz PRD(11) [à la Brans-Dicke theory]; Ohkuwa & Ezawa EPJP(15)-a1412 [using Lie derivatives]; Bombacigno et al a1911 [connection variables].
@ R2 theories: Folomeshkin CMP(71); Simon PRD(90) [non-locality]; Meng & Wang CQG(05) [R2 + R−1]; Zhogin a0812-GRF [problems]; Edery & Nakayama PRD(19)-a1902 [Palatini formulation = Einstein gravity, no massless scalar].
@ R2 theories, Hamiltonian: Sanyal GRG(05)ht/04; Klusoň et al PRD(14)-a1311; Debnath et al PRD(14)-a1408 [in higher dimensions].
@ R−1 theories: Chiba PLB(03)ap [and scalar-tensor]; Cline ap/03-wd [as extra term].
@ Other forms: Meng & Wang PLB(04)ht/03 [ln R, Palatini]; Baghram et al PRD(07)ap [\(f(R) = (R^2 − R_0^2)^{1/2}\)]; Kruglov IJTP(13)-a1202 [\(f(R) = \{1 - (1-2\lambda R)^{1/2}\}/\lambda\), Born-Infeld-like]; Kruglov IJMPA(13)-a1204 [f(R) = R eαR]; Kruglov ASS(15)-a1502 [\(f(R) = (1/\beta) \arcsin(\beta R)\), and cosmology]; Kumar a1611 [cosmologically viable].
> Related topics: see Birkhoff's Theorem; geodesics [geodesic deviation]; metric matching; singularities [quantum fields as probes].

Other Theories > s.a. higher-order gravity [metric vs Palatini formulations]; scalar-tensor theories [Jordan vs Einstein frames]; spin-2 field theories.
* Quadratic theories: Adding terms quadratic in the curvature to the action renders gravity renormalizable; The price to pay is the presence of a massive ghost.
* Lanczos-Lovelock gravity: The most general theories of gravity in D dimensions which satisfy (a) the principle of equivalence, (b) the principle of general covariance, and (c) have field equations involving derivatives of the metric tensor only up to second order; The m-th order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature tensor; The field equations resulting from it become trivial in the critical dimension \(D = 2m\) and the action itself can be written as the integral of an exterior derivative of an expression involving the vierbeins.
@ Matter couplings: Sotiriou in(08)gq/06; Sotiriou PLB(08)-a0805, Sotiriou & Faraoni CQG(08)-a0805 [with matter coupling to R]; Harko PLB(08)-a0810, PRD(10) [f(R) theories]; Fabbri & Vignolo CQG(11)-a1012 [f(R) theories with torsion and Dirac fields]; Haghani et al IJMPD(14)-a1405-GRF [with RabTab term].
@ Lanczos-Lovelock gravity: Lanczos JMP(69); Madore CQG(86) [action]; Yale & Padmanabhan GRG(11)-a1008; Deser & Franklin CQG(12)-a1110 [canonical analysis]; Kunstatter et al CQG(12) [Hamiltonian for spherically symmetric Lovelock gravity]; Padmanabhan & Kothawala PRP(13)-a1302 [rev]; Chakraborty & Padmanabhan PRD(14)-a1408 [evolution as departure from holographic equipartition]; Chakraborty et al GRG(17) [boundary terms in the action]; > s.a. black-hole geometry; lovelock gravity; quasilocal energy; thermodynamics.
@ Quadratic theories: Borowiec et al CQG(98)gq/96 [\(R_{ab}^{~} R^{ab}\) Lagrangians]; Olmo et al PRD(09) [\(f(R, R^{mn}R_{mn}^{~})\)]; Biswas et al CQG(14)-a1308 [quadratic, non-local but ghost-free and UV asymptotically free theories]; Álvarez-Gaumé et al FdP(16)-a1505 [rev]; Álvarez et al a1710-conf [and UV completion], EPJC(18)-a1802 [physical content]; Salvio FrPh(18)-a1807 [rev]; Morales & Santillán JCAP(19)-a1811 [Cauchy problem].
@ Cubic theories: Caprasse et al IJMPD(93); Bueno & Cano PRD(16)-a1607 [Einsteinian cubic gravity]; > s.a. lensing models.
@ With curvature derivatives: Oliva & Ray PRD(10)-a1004 [six-derivative Lagrangians, classification]; Naruko et al CQG(16)-a1512 ["scalar-tensor" theories, with 2 scalar and 2 tensor degrees of freedom].
@ Fourth-order gravity: Carloni et al GRG(09) [dynamical-systems approach]; Avalos et al a2102, a2102 [energy, positive-energy theorem].
@ Other theories: Hindawi et al PRD(96) [vacua and excitations]; Borowiec et al gq/00-conf [general Ricci-type gravitational Lagrangians]; Navarro & Van Acoleyen JCAP(06)gq/05 [MOND-like, and cosmology]; Gruzinov & Kleban CQG(07)ht/06 [causality constraints]; Ishak & Moldenhauer JCAP(09)-a0808 [minimal set of invariants]; Lü & Pope PRL(11)-a1101 ["critical gravity"]; Atazadeh & Darabi GRG(14)-a1302 [f(R,G), energy conditions and viability]; Colléaux & Zerbini Ent(15)-a1508 [with second-order equations of motion]; Hao & Zhao PRD(17)-a1512 [Ricci polynomial]; Karasu et al PRD(16)-a1602 [minimal extension of Einstein's gravity, quartic theory]; Stachowiak PRD(17)-a1703 [f(Rab), and cosmology]; Bueno et al JHEP(19)-a1906 [generalized quasi-topological gravities]; Amendola et al PLB(20)-a2006 [inverse of the Ricci tensor].
@ Lower-dimensional: Bergshoeff et al AP(10) [3D, and higher-spin gauge theories]; Güllü et al PRD(10)-a1002 [canonical structure]; Gürses et al PRD(12)-a1112 [3D f(Rab) theories, constant-scalar-curvature Type-N and Type-D solutions]; Ohta CQG(12) [3D, classification of unitary and stable theories]; > s.a. 2D gravity; 3D gravity.
@ Higher-dimensional: Ezawa et al CQG(99) [semiclassical stability]; Collins & Holdom JHEP(02) [4+1 with cosmological constant]; Huang et al PRD(10) [and cosmological acceleration]; > s.a. kaluza-klein theory.

Quantum-Gravity Motivated > s.a. higher-order theories of quantum gravity.
@ General references: Bonanno et al CQG(11)-a1006 [renormalization-group improved, inflationary solutions]; Pereira-Dias et al PRD(11)-a1009 [effect of Lorentz-symmetry violating Chern-Simons and Ricci-Cotton terms in the action]; Dzhunushaliev IJMPD(12)-a1201 [f(R) modified gravities from non-perturbative quantum effects].
@ Infinite-derivative, string-inspired: Biswas & Talaganis MPLA(15)-a1412 [rev]; Talaganis & Teimouri a1701 [Hamiltonian]; Talaganis a1704 [towards UV finiteness]; Gording & Schmidt-May JHEP(18)-a1807 [ghost-free]; Edholm a1904-PhD [phenomenological predictions]; Dimitrijevic et al proc(18)-a1902 [derivation of the equations of motion]; > s.a. gravitating bodies; singularities [avoidance].
@ From loop-quantum-gravity effective theory: Bojowald & Skirzewski IJGMP(07)ht/06-ln; Olmo & Singh JCAP(09)-a0806 [lqc].
> Other: see brans-dicke theory; BRST symmetries; Conformal Gravity; Gauss-Bonnet Gravity [including f(G) theories]; hořava-lifshitz gravity; lovelock gravity; Metric-Affine Theories; supergravity; unified theories [fourth-order Weyl].

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