Higher-Order Theories of Gravity  

In General > s.a. gravity theories; phenomenology [and solutions, weak-field limit]; types of higher-order gravity theories [including actions, Hamiltonians].
* Idea: Theories of gravitation, first discussed by H Weyl in 1918 as alternatives to Einstein's theory, that include quadratic or higher-order curvature terms in the action, e.g., R2, Rab Rab and Rabcd Rabcd (only two of these are independent – the integral of a combination of them gives the Euler number χ(M)).
* Motivation: One can alleviate the divergences in the quantum theory, and modify the predicted cosmological expansion history.
* Action / Lagrangian: In 4D, the most general one with up to quadratic terms is

S = − ∫ (α Rab Rab − β R2 + γ R) dv ;

any other term can be absorbed into these using the Gauss-Bonnet theorem [true only in positive-definite case?]; The only one which gives second-order equations in d > 4 is the Lanczos Lagrangian; But S could have a different curvature dependence.
* Field equations: [@ Stelle PRD(77)]

Hab:= (α − 2β) R;ab− α Rab;c;c − (\(1\over2\)α − 2β) gab \(\square\) R + 2α Rmn Rambn +

− 2β R Rab − \(1\over2\)gab (α Rmn Rmn − βR2) + γ Rab − \(1\over2\)γ gab R = −\(1\over2\)Tab ,

where Hab satisfies H ab;b = 0 identically, as a consequence of the field equations.
@ Reviews: in Fradkin & Tseytlin PRP(85); Cotsakis gq/97-MG8 [status]; Nojiri & Odintsov IJGMP(07)ht/06-ln [and dark energy]; Farhoudi GRG(06) [and trace anomaly relation]; Capozziello & Francaviglia GRG(08)-a0706 [and phenomenology]; Schmidt IJGMP(07)gq/06-ln [4th-order, history and applications]; Sotiriou PhD(07)-a0710; Sotiriou & Faraoni RMP(10)-a0805; Fabbri PhD-a0806; Faraoni a0810-conf [successes and challenges]; Capozziello et al OAJ(10)-a0909; De Felice & Tsujikawa LRR(10)-a1002; Bueno et al PRD(17)-a1610 [linearization].
@ With matter: Páramos a1111-proc [on non-minimal couplings to curvature]; Feng & Lü EPJC(16)-a1512 [non-minimally coupled Maxwell field].
@ Approaches, general references: Stelle GRG(78); Brans CQG(88) [and metric redefinitions]; Essén IJTP(90); Cuzinatto et al EPJC(15)gq/06 [gauge formulation]; Jaime PRD(11)-a1006 [f(R) gravity with R as extra degree of freedom]; Hamad IJMPD(14)-a1408 [entropy-functional formalism]; Belenchia et al RPP(18)-a1612 [degrees of freedom]; Deser GRG(17)-a1705 [bootstrapping from linear theories].

Metric vs Palatini Formulation
* Idea: In general relativity, the metric (second-order) and the Palatini (first-order) formulations are dynamically equivalent, at least classically; This equivalence does not hold for theories of gravity with a non-linear dependence on the curvature, and the set of solutions of the Palatini equations is a non-trivial subset of the solutions of the metric equations; One set of exceptions are the Lovelock gravity theories.
* Hybrid theories: Theories in which an f(R) term constructed à la Palatini is added to the metric Einstein-Hilbert Lagrangian.
@ General references: Olmo & Komp gq/04; Deser CQG(06)gq, comment Kiriushcheva & Kuzmin CQG(07)gq/06 [obstacles with odd-derivative terms]; Sotiriou PLB(07) [instability in metric formulation]; Exirifard & Sheikh-Jabbari PLB(08)-a0705 [equivalence of formulations only for Lovelock gravity]; Popławski IJMPA(08)-a0706; Iglesias et al PRD(07)-a0708 [how (not) to Palatini]; Borunda et al JCAP(08)-a0804; Faraoni PLB(08)-a0806 [Palatini is unphysical]; Bastero-Gil et al AIP(09)-a0901; Capozziello et al MPLA(11)-a1006 [equivalence by divergence-free current]; Vitagliano et al PRD(10)-a1007 [f(R) vs other forms of the action]; Capozziello et al MPLA(10)-a1009 [and Jordan frame vs Einstein frame].
@ Hybrid theories: Capozziello et al JCAP(13)-a1212, IJMPD(13)-a1305-GRF; Tamanini & Böhmer PRD(13); Böhmer et al PRD(13)-a1305 [stability of the Einstein static Universe]; Capozziello et al Univ(15)-a1508 [rev].
@ Special types of theories: Shahid-Saless PRD(87) [R + R2 theory]; Ezawa et al gq/03, NCB(04)gq [f(R) theories]; Amendola et al PRD(11)-a1010 [f(R) theories, unified framework with larger class of theories]; Olmo IJMPD(11)-a1101, a1112-proc [f(R) theories, Palatini approach].

Relationship with General Relativity and Scalar-Tensor Theories
* Relationship with scalar-tensor gravity: If we have a theory with Lagrangian

\(\cal L\) = |g|1/2 F(g, Ric(g)) + \(\cal L\)m(g, ∂g, φ, ∇gφ) ,

a redefinition of the metric (which need no longer be Lorentzian) gives a scalar-tensor theory with action

\(\cal L\) = |h|1/2 R(h) + \(\cal L\)m + \(\cal L\)+(h, ∂h, φ,∇hφ) ,  where  hab:= |det(δ\(\cal L\)/δRab)|−1/2 (δ\(\cal L\)/δRab) ;

A theory with an independent connection with torsion and/or non-metricity but not coupled to matter is equivalent to ω0 = −3/2 Brans-Dicke theory.
@ And general relativity, scalar-tensor: Goenner in(87); Magnano et al GRG(87); Jakubiec & Kijowski PRD(88), JMP(89); Magnano & Sokołowski PRD(94)gq/93; Magnano gq/95-conf; Sokołowski gq/95-GR14; Núñez & Solganik ht/04; Flanagan CQG(04)gq [equivalence]; Sotiriou CQG(06)gq; Bertolami & Páramos CQG(08)-a0805 [and non-trivial matter coupling]; Bisabr PS(09)-a0808; Sotiriou CQG(09)-a0904 [with non-metricity and torsion, equivalence to Brans-Dicke theory]; Rodrigues et al PRD(11)-a1103 [auxiliary-field representation for modified gravity models]; Castañeda & Velásquez JPComm(20)-a1808 [and cosmological perturbations]; > s.a. f(R) theories.

Special Topics and Results > s.a. higher-order lagrangian theories; massive gravity.
* Bicknell theorem: A conformal relationship between 4th-order gravity and scalar-tensor theory; > s.a. scalar-tensor gravity.
@ Energy: Boulware et al PRL(83); Deser & Yang CQG(89); Borowiec et al GRG(94); Deser & Tekin PRL(02)ht, PRD(03)ht/02, PRD(07)gq; Fatibene et al IJGMP(06) [same solution, different theories]; Multamäki et al CQG(08)-a0712 [energy-momentum complexes]; Baykal PRD(12)-a1212 [quadratic theories]; > s.a. gravitational energy.
@ Noether symmetries: Sanyal et al GRG(05)ap/03; Modak et al GRG(05); Capozziello et al EPJC(12)-a1206 [in Hamiltonian dynamics, rev].
@ Cauchy / initial-value problem, f(R) gravity: Teyssandier & Tourrenc JMP(83) [R + R2]; Capozziello & Vignolo IJGMP(12)-a1103; Tsokaros CQG(14) [for f(R) gravity]; Sáez-Chillón a2103 [3+1 decomposition]; > s.a. initial-value formulation [characteristic formulation].
@ Cauchy / initial-value problem, other: Jakubiec & Kijowski JMP(89) [well-posedness]; Lanahan-Tremblay & Faraoni CQG(07)-a0709, comment Capozziello & Vignolo CQG(09)-a0903, reply CQG(09)-a0906 [well-posed for metric version]; Capozziello & Vignolo IJGMP(09)-a0901, CQG(09) [with perfect fluid]; Paschalidis et al CQG(11)-a1103 [preservation of constraints]; Cayuso et al PRD(17)-a1706; Morales & Santillán JCAP(19)-a1811 [Stelle quadratic gravity].
@ Other related topics: Eliezer NPB(89) [Regge calculus]; Ghoroku pr(90) [lqg-type variables]; Bartoli et al CQG(99)gq/98 [gauge fixing]; Woodard LNP(07)ap/06 [choice of theory]; Carloni & Dunsby JPA(07)gq/06-conf [dynamical systems approach]; Padmanabhan PRD(11)-a1109; Kim et al PRL(13)-a1306 [quasilocal conserved charges]; Mistry et al EPJC(20)-a2001 [spectral action approach]; > s.a. hamilton-jacobi theory; higher-order theories of quantum gravity; metric matching.
> Related results: see Israel's Theorem; Lichnerowicz Theorem; types of higher-order theories [Ostrogradski theorem].

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