2.2 Equivalence with Brans–Dicke theory

The f (R) theory in the metric formalism can be cast in the form of Brans–Dicke (BD) theory [100Jump To The Next Citation Point] with a potential for the effective scalar-field degree of freedom (scalaron). Let us consider the following action with a new field χ,
1 ∫ 4 √ --- ∫ 4 S = --2- d x − g [f(χ) + f,χ(χ )(R − χ)] + d x ℒM (gμν,ΨM ). (2.18 ) 2κ
Varying this action with respect to χ, we obtain
f,χχ(χ)(R − χ) = 0. (2.19 )
Provided f,χχ(χ ) ⁄= 0 it follows that χ = R. Hence the action (2.18View Equation) recovers the action (2.1View Equation) in f (R) gravity. If we define
φ ≡ f,χ (χ ), (2.20 )
the action (2.18View Equation) can be expressed as
∫ [ ] ∫ S = d4x √ −-g -1-φR − U (φ) + d4xℒ (g ,Ψ ), (2.21 ) 2κ2 M μν M
where U(φ ) is a field potential given by
χ (φ)φ − f (χ(φ)) U (φ) = ---------2-------. (2.22 ) 2κ

Meanwhile the action in BD theory [100Jump To The Next Citation Point] with a potential U (φ) is given by

∫ [ ] ∫ 4 √ --- 1 ωBD 2 4 S = d x − g 2φR − 2φ--(∇φ ) − U(φ ) + d x ℒM (gμν,ΨM ), (2.23 )
where ωBD is the BD parameter and (∇ φ)2 ≡ gμν∂ μφ∂νφ. Comparing Eq. (2.21View Equation) with Eq. (2.23View Equation), it follows that f (R) theory in the metric formalism is equivalent to BD theory with the parameter ω = 0 BD [467Jump To The Next Citation Point579Jump To The Next Citation Point152Jump To The Next Citation Point] (in the unit κ2 = 1). In Palatini f (R) theory where the metric gμν and the connection α Γ βγ are treated as independent variables, the Ricci scalar is different from that in metric f (R) theory. As we will see in Sections 9.1 and 10.1, f (R) theory in the Palatini formalism is equivalent to BD theory with the parameter ωBD = − 3∕2.
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