2.3 Conformal transformation

The action (2.1View Equation) in f (R) gravity corresponds to a non-linear function f in terms of R. It is possible to derive an action in the Einstein frame under the conformal transformation [213609Jump To The Next Citation Point408Jump To The Next Citation Point611Jump To The Next Citation Point249268Jump To The Next Citation Point410]:
2 &tidle;gμν = Ω gμν, (2.24 )
where Ω2 is the conformal factor and a tilde represents quantities in the Einstein frame. The Ricci scalars R and &tidle; R in the two frames have the following relation
R = Ω2 (&tidle;R + 6&tidle;□ ω − 6&tidle;gμν∂ μω∂νω ), (2.25 )
where
∂ω-- &tidle; -1--- ∘ --- μν ω ≡ ln Ω, ∂μω ≡ ∂&tidle;xμ , □ ω ≡ √−-&tidle;g ∂μ( − &tidle;g&tidle;g ∂νω ). (2.26 )

We rewrite the action (2.1View Equation) in the form

∫ ( ) ∫ 4 √ --- -1-- 4 S = d x − g 2κ2 FR − U + d xℒM (gμν,ΨM ), (2.27 )
where
F R − f U = -----2--. (2.28 ) 2 κ
Using Eq. (2.25View Equation) and the relation √ --- √--- − g = Ω− 4 − &tidle;g, the action (2.27View Equation) is transformed as
∫ [ ] ∫ 4 ∘ --- 1 −2 μν −4 4 − 2 S = d x − &tidle;g 2κ2-FΩ (&tidle;R + 6 &tidle;□ω − 6&tidle;g ∂μω ∂νω) − Ω U + d x ℒM (Ω &tidle;gμν,ΨM ).(2.29 )
We obtain the Einstein frame action (linear action in R&tidle;) for the choice
Ω2 = F. (2.30 )
This choice is consistent if F > 0. We introduce a new scalar field ϕ defined by
∘ ---- κϕ ≡ 3∕2 ln F . (2.31 )
From the definition of ω in Eq. (2.26View Equation) we have that ω = κ ϕ∕√6--. Using Eq. (2.26View Equation), the integral ∫ 4 √--- &tidle; d x − &tidle;g□ ω vanishes on account of the Gauss’s theorem. Then the action in the Einstein frame is
∫ [ ] ∫ 4 ∘ --- 1 1 μν 4 −1 SE = d x − &tidle;g 2κ2-&tidle;R − 2&tidle;g ∂μϕ ∂νϕ − V (ϕ) + d xℒM (F (ϕ)&tidle;gμν,ΨM ), (2.32 )
where
V(ϕ ) = U--= F-R-−-f-. (2.33 ) F 2 2κ2F 2
Hence the Lagrangian density of the field ϕ is given by ℒ = − 1&tidle;gμν∂ ϕ ∂ ϕ − V (ϕ) ϕ 2 μ ν with the energy-momentum tensor
2 δ(√ −-&tidle;gℒ ) [1 ] &tidle;Tμ(ϕν) = − √-------------ϕ-= ∂μϕ∂νϕ − &tidle;gμν --&tidle;gαβ∂αϕ ∂βϕ + V (ϕ) . (2.34 ) − &tidle;g δ&tidle;gμν 2

The conformal factor 2 ∘ ---- Ω = F = exp ( 2∕3κϕ ) is field-dependent. From the matter action (2.32View Equation) the scalar field ϕ is directly coupled to matter in the Einstein frame. In order to see this more explicitly, we take the variation of the action (2.32View Equation) with respect to the field ϕ:

( ∂(√ −g&tidle;ℒ )) ∂(√ −-&tidle;gℒ ) ∂ℒ − ∂μ --------ϕ-- + ---------ϕ-+ ---M-= 0, (2.35 ) ∂(∂μϕ) ∂ϕ ∂ϕ
that is
1 ∂ ℒM 1 ∘ --- □&tidle;ϕ − V,ϕ + √--------- = 0, where &tidle;□ ϕ ≡ √----∂ μ( − &tidle;g&tidle;gμν∂νϕ). (2.36 ) − &tidle;g ∂ ϕ − &tidle;g
Using Eq. (2.24View Equation) and the relations √ --- √ --- − &tidle;g = F 2 − g and &tidle;gμν = F −1gμν, the energy-momentum tensor of matter is transformed as
(M ) &tidle;T(M )= − √-2-- δℒM--= Tμν--. (2.37 ) μν − &tidle;g δ&tidle;gμν F
The energy-momentum tensor of perfect fluids in the Einstein frame is given by
&tidle;Tμ (M )= diag (−ρ&tidle; ,P&tidle; , &tidle;P , &tidle;P ) = diag(− ρ ∕F 2,P ∕F 2,P ∕F 2,P ∕F 2). (2.38 ) ν M M M M M M M M
The derivative of the Lagrangian density ℒM = ℒM (gμν) = ℒM (F −1(ϕ)&tidle;gμν) with respect to ϕ is
∂ℒM δℒM ∂g μν 1 δℒM ∂(F (ϕ)&tidle;gμν) ∘ --- F,ϕ ----- = ---μν-----= --------μν----------- = − − &tidle;g ---&tidle;T(μMν )&tidle;gμν. (2.39 ) ∂ϕ δg ∂ϕ F (ϕ)δg&tidle; ∂ ϕ 2F

The strength of the coupling between the field and matter can be quantified by the following quantity

Q ≡ − F,ϕ--= − √1-, (2.40 ) 2κF 6
which is constant in f (R) gravity [28Jump To The Next Citation Point]. It then follows that
∂ℒM-- ∘ --- &tidle; ∂ϕ = − &tidle;gκQ T , (2.41 )
where &tidle;T = &tidle;gμνT&tidle;μν(M) = − &tidle;ρM + 3 &tidle;PM. Substituting Eq. (2.41View Equation) into Eq. (2.36View Equation), we obtain the field equation in the Einstein frame:
&tidle;□ϕ − V + κQ T&tidle;= 0. (2.42 ) ,ϕ
This shows that the field ϕ is directly coupled to matter apart from radiation (&tidle;T = 0).

Let us consider the flat FLRW spacetime with the metric (2.12View Equation) in the Jordan frame. The metric in the Einstein frame is given by

2 2 2 2 2 2 d&tidle;s = Ω ds = F (− dt + a (t)dx ), = − d&tidle;t2 + &tidle;a2 (&tidle;t)dx2, (2.43 )
which leads to the following relations (for F > 0)
d&tidle;t = √F--dt, &tidle;a = √F--a, (2.44 )
where
−2Qκϕ F = e . (2.45 )
Note that Eq. (2.45View Equation) comes from the integration of Eq. (2.40View Equation) for constant Q. The field equation (2.42View Equation) can be expressed as
2 d-ϕ-+ 3H&tidle; dϕ-+ V,ϕ = − κQ (&tidle;ρM − 3P&tidle;M ), (2.46 ) d&tidle;t2 d&tidle;t
where
( ) 1 d&tidle;a 1 ˙F H&tidle; ≡ -----= √--- H + --- . (2.47 ) &tidle;a d&tidle;t F 2F
Defining the energy density 1 &tidle; 2 &tidle;ρϕ = 2(d ϕ∕dt) + V(ϕ ) and the pressure &tidle; 1 &tidle; 2 P ϕ = 2(dϕ ∕dt) − V (ϕ), Eq. (2.46View Equation) can be written as
d&tidle;ρϕ- &tidle; &tidle; &tidle; dϕ- d&tidle;t + 3 H (&tidle;ρϕ + P ϕ) = − κQ (&tidle;ρM − 3PM )d&tidle;t . (2.48 )
Under the transformation (2.44View Equation) together with 2 ρM = F &tidle;ρM, 2 &tidle; PM = F PM, and 1∕2 &tidle; &tidle; H = F [H − (dF ∕dt)∕2F ], the continuity equation (2.17View Equation) is transformed as
d&tidle;ρM-- &tidle; &tidle; &tidle; dϕ- d&tidle;t + 3H (&tidle;ρM + PM ) = κQ (&tidle;ρM − 3PM )d &tidle;t. (2.49 )

Equations (2.48View Equation) and (2.49View Equation) show that the field and matter interacts with each other, while the total energy density &tidle;ρT = &tidle;ρϕ + &tidle;ρM and the pressure &tidle;PT = P&tidle;ϕ + P&tidle;M satisfy the continuity equation dρ&tidle;T ∕d&tidle;t + 3H&tidle;(&tidle;ρT + P&tidle;T ) = 0. More generally, Eqs. (2.48View Equation) and (2.49View Equation) can be expressed in terms of the energy-momentum tensors defined in Eqs. (2.34View Equation) and (2.37View Equation):

&tidle; &tidle; μ(ϕ) &tidle;&tidle; &tidle; &tidle;μ(M ) &tidle;&tidle; ∇ μTν = − QT ∇ νϕ, ∇ μTν = Q T ∇ νϕ, (2.50 )
which correspond to the same equations in coupled quintessence studied in [23Jump To The Next Citation Point] (see also [22Jump To The Next Citation Point]).

In the absence of a field potential V (ϕ) (i.e., massless field) the field mediates a long-range fifth force with a large coupling (|Q | ≃ 0.4), which contradicts with experimental tests in the solar system. In f (R) gravity a field potential with gravitational origin is present, which allows the possibility of compatibility with local gravity tests through the chameleon mechanism [344Jump To The Next Citation Point343Jump To The Next Citation Point].

In f (R) gravity the field ϕ is coupled to non-relativistic matter (dark matter, baryons) with a universal coupling Q = − 1∕ √6. We consider the frame in which the baryons obey the standard continuity equation − 3 ρm ∝ a, i.e., the Jordan frame, as the “physical” frame in which physical quantities are compared with observations and experiments. It is sometimes convenient to refer the Einstein frame in which a canonical scalar field is coupled to non-relativistic matter. In both frames we are treating the same physics, but using the different time and length scales gives rise to the apparent difference between the observables in two frames. Our attitude throughout the review is to discuss observables in the Jordan frame. When we transform to the Einstein frame for some convenience, we go back to the Jordan frame to discuss physical quantities.


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