Scalar-Tensor Theories of Gravity |

**In General**
> s.a. gravitation; higher-order theories
and types [equivalence]; mass;
phenomenology; scalar
fields [scalar components of gravity].

* __Idea__: Theories of
gravitation which include, besides the metric, one or more scalar
fields *φ*^{i}, often
called inflatons, and possibly other fields, with an action of the form

*S*[*g*_{ab}, *φ*]
= ∫ d*v* [\(1\over2\)*f*(*φ*) *R*
− \(1\over2\)*ω*(*φ*)
*g*^{ac}
∂_{a} *φ*
∂_{c} *φ*
− *V*(*φ*) + (possible higher-order terms in derivatives
of the *φ*^{i})] ,

where *f*(*φ*) > 0 so that *G*_{eff}
= (8π *f*)^{−1} > 0,
gravity is attractive and the graviton carries positive energy.

* __Motivation__: Slows down
the expansion rate in extended and hyperextended inflation, and allows
bubble nucleation; The scalar field, and others, appears naturally
in low-energy effective actions and dimensional reductions of most
hep-inspired unified theories, including string theory (as dilaton),
supergravity (as partner of a spin-\(1\over2\) particle), Kaluza-Klein
theory, higher-derivative theories.

@ __Reviews, history__: Fujii & Maeda 03;
Brans gq/05 [overview];
Goenner GRG(12) [history 1941–1962, Scherrer, Jordan, Thiry];
Quirós IJMPD(19)-a1901.

@ __General references__:
Bergmann IJTP(68);
Harrison PRD(72),
Serna et al CQG(02)gq [and general relativity];
Charmousis et al PRL(11)-a1106 [with consistent self-tuning mechanism];
Padilla & Sivanesan JHEP(12)-a1206 [boundary terms and junction conditions];
Zhou et al PRD(13)-a1211 [first-order action];
Bloomfield JCAP(13)-a1304 [simplified approach based on Horndeski's theory];
Gao PRD(14)-a1406,
Ezquiaga et al PRD(16)-a1603 [unifying frameworks];
Kozak a1710-MS [Palatini approach].

@ __Cauchy problem, evolution__:
Teyssandier & Tourrenc JMP(83);
Damour & Esposito-Farèse CQG(92);
Damour & Nordtvedt PRL(93),
PRD(93) [general relativity as attractor];
Salgado CQG(06)gq/05;
Salgado & Martínez-del Río JPC(07)-a0712;
Salgado et al PRD(08)-a0801 [hyperbolicity];
Faraoni & Lanahan-Tremblay PRD(08)-a0808.

@ __Hamiltonian approach__: Lin a1702 [Ostrogradski ghost issue].

@ __Related topics__: Wiaux CQG(99) [gauge freedom];
Salgado gq/02/PRD [weak field];
Agarwal & Bean CQG(08)-a0708 [dynamical stability];
Carloni & Dunsby GRG(16)-a1306 [1+1+2 formalism];
Vilson AACA(15)-a1509 [invariant quantities];
Kovacs & Reall a2003
[effective field theory, well-posed initial value formulation];
> s.a. affine connections [non-metricity formulation].

**Conformal Frames**
> s.a. brans-dicke theory; conformal structures.

* __Jordan / Pauli frame__:
Defined by *f*(*φ*) = *φ* in the general
form of the action; The weak equivalence principle is satisfied, but
*h*_{ab}
= *g*_{ab} −
*η*_{ab} is
not the spin-2 massless graviton, and the *φR* term can
become negative so the energy density is not bounded from below, which
is not acceptable classically.

* __Einstein frame__:
Defined by *f*(*φ*) = 1 in the general form of the action;
It is used in inflationary models because equations are easier to work with;
The perturbation *h*_{ab}
= *g*_{ab} −
*η*_{ab}
represents the spin-2 massless graviton and is used for quantization,
but the WEP is not satisfied (weakly, ok with tests).

* __Relationships__:
The metrics in the two frames are conformally related,
*g*_{ab}^{E}
= Ω^{2}
*g*_{ab}^{J},
and the dilatons are related by *φ*^{J}
= 1/*GA*^{2}(*φ*^{E});
Since the transformation is local Chisholm's theorem implies that the
S-matrices are equivalent; The scalar-field transformation shows that the
Einstein-frame theory can be considered as a sector of the Jordan-frame
one, and in this sector there is no instability.

@ __General references__: Carloni et al GRG(10)-a0907 [conformal transformations in cosmology];
Järv et al JPCS(14)-a1501 [parametrizations and general relativity limit].

@ __Jordan vs Einstein frame__:
Cho PRL(92),
CQG(97);
Magnano & Sokołowski PRD(94)gq/93;
Capozziello et al CQG(97),
CQG(97);
in Brans gq/97-fs;
in Faraoni et al FCP(99)gq/98;
Faraoni & Gunzig IJTP(99)ap;
Quirós gq/99,
PRD(00)gq/99,
et al PRD(00)gq/99 [and singularities];
Gong gq/00;
Macías & García GRG(01) [inequivalent];
Casadio & Gruppuso IJMPD(02)gq/01 [and boundary terms];
Álvarez & Conde MPLA(02)gq/01;
Flanagan CQG(04)gq [including higher-order theories];
Bhadra et al MPLA(07)gq/06 [Brans-Dicke theory, light deflection];
Faraoni & Nadeau PRD(07)gq/06;
Järv et al PRD(07)-a0705 [and general relativity limit];
Roberts a0706;
Capozziello et al PLB(10)-a1003 [higher-order gravity, inequivalent];
Corda APP(11)-a1010 [and gravitational-wave astronomy];
Xu & Zhao a1106-wd [inequivalent];
Kozyrev & Daishev a1207;
Postma & Volponi PRD(14)-a1407
[equivalent; action written in terms of conformally invariant quantities].

@ __Quantum aspects__:
Kamenshchik & Steinwachs PRD(15)-a1408;
Banerjee & Majumder PLB(16)-a1601;
Pandey & Banerjee a1610 [equivalence];
> s.a modified approaches to quantum gravity.

**Specific Theories** > s.a. bianchi models;
bimetric theories; higher-dimensional gravity;
non-local theories; unified theories [Weyl-Dirac].

* __Jordan theory__: A generalization of
Brans-Dicke theory; > s.a. kaluza-klein theory.

@ __From large extra dimensions__: Giudice et al NPB(01) [curvature-Higgs mixing].

@ __Other theories__: Graf PRD(03)gq/02,
PMCPA(07)gq/06 [metric + volume element, Ricci flow gravity];
Scholz et al FP(09) [scale-covariant field];
Babichev et al IJMPD(09) [with infrared screening];
Minotti a1106 [with enhanced gravitational effects];
Zumalcárregui & García-Bellido PRD(14)-a1308 [derivative couplings];
Chavineau et al GRG(16)-a1503 [with an external scalar];
Crisostomi et al JCAP(16)-a1602 [extended];
Ezquiaga et al PRD(16)-a1603 [most general];
Quiros et al a1905 [issue with classification];
Borowiec & Kozak a2003 [hybrid metric-Palatini theories].

> __Other theories__:
see Bergmann-Wagoner Theory;
brans-dicke theory; dilaton;
Horndeski Action [including generalizations];
quintessence.

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