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, and Einsteinian cubic gravity].
@ Boundary terms: Guarnizo et al GRG(10)-a1002; Fatibene et al a1106; Teimouri et al JHEP(16)-a1606; > s.a. Lanczos-Lovelock 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 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 a1708 [different canonical forms]; Li et al a1711 [causality and a-theorem constraints].
@ 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 a1711 [quantum equivalence].
@ 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].
@ R2 theories: Folomeshkin CMP(71); Simon PRD(90) [non-locality]; Meng & Wang CQG(05) [R2 + R–1]; Zhogin a0812-GRF [problems].
@ 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) = (R2R02)1/2]; Kruglov IJTP(13)-a1202 [f(R) = {1 – (1 – 2λ R)1/2}/λ, 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 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 [Rab Rab Lagrangians]; Olmo et al PRD(09) [f(R, RmnRmn)]; 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], a1802 [physical content]; Salvio a1807-FrPh [rev].
@ 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].
@ Other theories: Caprasse et al IJMPD(93) [cubic]; 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]; Carloni et al GRG(09) [fourth-order, dynamical-systems approach]; 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 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].
@ 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].
@ String-inspired, infinite-derivative: Biswas & Talaganis MPLA(15)-a1412 [rev]; Talaganis & Teimouri a1701 [Hamiltonian]; Talaganis a1704 [towards UV finiteness]; Gording & Schmidt-May a1807 [ghost-free]; > 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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