10.1 Brans–Dicke theory and the equivalence with f (R) theories

Let us start with the following 4-dimensional action in BD theory
∫ √--- [1 ω ] S = d4x − g --φR − -BD-(∇ φ)2 − U(φ ) + SM (gμν,ΨM ), (10.1 ) 2 2 φ
where ωBD is the BD parameter, U (φ) is a potential of the scalar field φ, and SM is a matter action that depends on the metric gμν and matter fields ΨM. In this section we use the unit κ2 = 8 πG = 1∕M 2 = 1 pl, but we recover the gravitational constant G and the reduced Planck mass M pl when the discussion becomes transparent. The original BD theory [100] does not possess the field potential U(φ ).

Taking the variation of the action (10.1View Equation) with respect to gμν and φ, we obtain the following field equations

1 1 1 1 Rμν(g) − --gμνR(g ) = -T μν − -gμνU (φ) + --(∇ μ∇ νφ − gμν□ φ) 2 φ [ φ φ ] ωBD- 1- 2 + φ2 ∂μφ ∂νφ − 2 gμν(∇ φ) , (10.2 ) (3 + 2ωBD )□ φ + 4U (φ) − 2φU,φ = T , (10.3 )
where R(g) is the Ricci scalar in metric f (R) gravity, and Tμν is the energy-momentum tensor of matter. In order to find the relation with f (R) theories in the metric and Palatini formalisms, we consider the following correspondence
RF--−--f φ = F(R ), U (φ) = 2 . (10.4 )
Recall that this potential (which is the gravitational origin) already appeared in Eq. (2.28View Equation). We then find that Eqs. (2.4View Equation) and (2.7View Equation) in metric f (R) gravity are equivalent to Eqs. (10.2View Equation) and (10.3View Equation) with the BD parameter ωBD = 0. Hence f (R) theory in the metric formalism corresponds to BD theory with ωBD = 0 [467579152246112]. In fact we already showed this by rewriting the action (2.1View Equation) in the form (2.21View Equation). We also notice that Eqs. (9.4View Equation) and (9.2View Equation) in Palatini f (R) gravity are equivalent to Eqs. (2.4View Equation) and (2.7View Equation) with the BD parameter ωBD = − 3∕2. Then f (R) theory in the Palatini formalism corresponds to BD theory with ωBD = − 3∕2 [262470551]. Recall that we also showed this by rewriting the action (2.1View Equation) in the form (9.8View Equation).

One can consider more general theories called scalar-tensor theories [268] in which the Ricci scalar R is coupled to a scalar field φ. The general 4-dimensional action for scalar-tensor theories can be written as

∫ √ ---[ ] S = d4x − g 1-F (φ )R − 1ω(φ )(∇ φ )2 − U (φ) + SM (gμν,ΨM ), (10.5 ) 2 2
where F(φ ) and U (φ) are functions of φ. Under the conformal transformation &tidle;g = F g μν μν, we obtain the action in the Einstein frame [408611]
∫ [ ] 4 ∘ --- 1- 1- 2 −1 SE = d x − &tidle;g 2 &tidle;R − 2 (&tidle;∇ϕ ) − V (ϕ) + SM (F &tidle;gμν,ΨM ), (10.6 )
where V = U∕F 2. We have introduced a new scalar field ϕ to make the kinetic term canonical:
∘ --------------- ∫ ( )2 ϕ ≡ d φ 3- F,φ- + ω-. (10.7 ) 2 F F

We define a quantity Q that characterizes the coupling between the field ϕ and non-relativistic matter in the Einstein frame:

[ ] −1∕2 F,ϕ F,φ 3 (F,φ )2 ω Q ≡ − --- = − ---- -- ---- + -- . (10.8 ) 2F F 2 F F
Recall that, in metric f (R) gravity, we introduced the same quantity Q in Eq. (2.40View Equation), which is constant (√ -- Q = − 1∕ 6). For theories with Q =constant, we obtain the following relations from Eqs. (10.7View Equation) and (10.8View Equation):
( )2 F = e−2Qϕ, ω = (1 − 6Q2 )F dϕ- . (10.9 ) dφ
In this case the action (10.5View Equation) in the Jordan frame reduces to [596Jump To The Next Citation Point]
∫ √ ---[ 1 1 ] S = d4x − g -F (ϕ)R − -(1 − 6Q2 )F (ϕ)(∇ ϕ)2 − U (ϕ) + SM (gμν,ΨM ), with F (ϕ) = e(−120Q.ϕ1.0) 2 2
In the limit that Q → 0 we have F (ϕ) → 1, so that Eq. (10.10View Equation) recovers the action of a minimally coupled scalar field in GR.

Let us compare the action (10.10View Equation) with the action (10.1View Equation) in BD theory. Setting −2Qϕ φ = F = e, the former is equivalent to the latter if the parameter ωBD is related to Q via the relation [343Jump To The Next Citation Point596Jump To The Next Citation Point]

3 + 2ω = -1--. (10.11 ) BD 2Q2
This shows that the GR limit (ωBD → ∞) corresponds to the vanishing coupling (Q → 0). Since √ -- Q = − 1∕ 6 in metric f (R) gravity one has ωBD = 0, as expected. The Palatini f (R) gravity corresponds to ωBD = − 3∕2, which corresponds to the infinite coupling (Q2 → ∞). In fact, Palatini gravity can be regarded as an isolated “fixed point” of a transformation involving a special conformal rescaling of the metric [247]. In the Einstein frame of the Palatini formalism, the scalar field ϕ does not have a kinetic term and it can be integrated out. In general, this leads to a matter action which is non-linear, depending on the potential U (ϕ). This large coupling poses a number of problems such as the strong amplification of matter density perturbations and the conflict with the Standard Model of particle physics, as we have discussed in Section 9.

Note that BD theory is one of the examples in scalar-tensor theories and there are some theories that give rise to non-constant values of Q. For example, the action of a nonminimally coupled scalar field with a coupling ξ corresponds to F (φ) = 1 − ξφ2 and ω(φ) = 1, which gives the field-dependent coupling Q (φ) = ξφ ∕[1 − ξ φ2(1 − 6ξ)]1∕2. In fact the dynamics of dark energy in such a theory has been studied by a number of authors [22Jump To The Next Citation Point601Jump To The Next Citation Point1516849144505]. In the following we shall focus on the constant coupling models with the action (10.10View Equation). We stress that this is equivalent to the action (10.1View Equation) in BD theory.

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