5.1 Linear expansions of perturbations in the spherically symmetric background

First we decompose the quantities R, F (R), and Tμν into the background part and the perturbed part: R = R0 + δR, F = F0(1 + δF), and T μν = (0)Tμν + δTμν about the approximate Minkowski background ((0) gμν ≈ ημν). In other words, although we consider R close to a mean-field value R0, the metric is still very close to the Minkowski case. The linear expansion of Eq. (2.7View Equation) in a time-independent background gives [470Jump To The Next Citation Point250Jump To The Next Citation Point154Jump To The Next Citation Point448Jump To The Next Citation Point]
2 2 κ2-- ∇ δF − M δF = 3F0δT , (5.1 )
where μν δT ≡ η δTμν and
[ ] [ ] 2 1 f,R(R0 ) R0 1 M ≡ -- ---------− R0 = --- -------− 1 . (5.2 ) 3 f,RR(R0 ) 3 m (R0 )
The variable m is defined in Eq. (4.67View Equation). Since 0 < m (R0 ) < 1 for viable f (R) models, it follows that M 2 > 0 (recall that R0 > 0).

We consider a spherically symmetric body with mass Mc, constant density ρ (= − δT ), radius rc, and vanishing density outside the body. Since δF is a function of the distance r from the center of the body, Eq. (5.1View Equation) reduces to the following form inside the body (r < rc):

d2 2 d 2 κ2 --2δF + ----δF − M δF = − ---ρ, (5.3 ) dr rdr 3F0
whereas the r.h.s. vanishes outside the body (r > rc). The solution of the perturbation δF for positive M 2 is given by
−Mr Mr (δF)r<rc = c1 e----+ c2 e---+ -8πG-ρ-, (5.4 ) r r 3F0M 2 e−Mr eMr (δF)r>rc = c3 -----+ c4 ---, (5.5 ) r r
where ci (i = 1,2,3,4) are integration constants. The requirement that (δF )r>rc → 0 as r → ∞ gives c4 = 0. The regularity condition at r = 0 requires that c2 = − c1. We match two solutions (5.4View Equation) and (5.5View Equation) at r = rc by demanding the regular behavior of δF(r) and δ ′(r) F. Since δF ∝ δR, this implies that R is also continuous. If the mass M satisfies the condition M rc ≪ 1, we obtain the following solutions
( ) 4πG ρ 2 r2 (δF)r<rc ≃ -3F--- rc − 3- , (5.6 ) 0 (δ ) ≃ 2GMc--e− Mr. (5.7 ) F r>rc 3F0r

As we have seen in Section 2.3, the action (2.1View Equation) in f (R) gravity can be transformed to the Einstein frame action by a transformation of the metric. The Einstein frame action is given by a linear action in &tidle;R, where &tidle; R is a Ricci scalar in the new frame. The first-order solution for the perturbation hμν of the metric &tidle;gμν = F0 (ημν + h μν) follows from the first-order linearized Einstein equations in the Einstein frame. This leads to the solutions h00 = 2GMc ∕(F0r) and hij = 2GMc ∕(F0r)δij. Including the perturbation δF to the quantity F, the actual metric gμν is given by [448Jump To The Next Citation Point]

&tidle;gμν gμν = ----≃ ημν + hμν − δF ημν. (5.8 ) F
Using the solution (5.7View Equation) outside the body, the (00) and (ii) components of the metric gμν are
(N) (N) 2G-eff-Mc- 2G-eff-Mc- g00 ≃ − 1 + r , gii ≃ 1 + r γ, (5.9 )
where (N ) Geff and γ are the effective gravitational coupling and the post-Newtonian parameter, respectively, defined by
( ) (N ) G 1 −Mr 3 − e−Mr Geff ≡ --- 1 + --e , γ ≡ -----−Mr-. (5.10 ) F0 3 3 + e

For the f (R) models whose deviation from the ΛCDM model is small (m ≪ 1), we have 2 M ≃ R0∕[3m (R0 )] and R ≃ 8πG ρ. This gives the following estimate

(M r )2 ≃ 2--Φc--, (5.11 ) c m (R0)
where 2 Φc = GMc ∕(F0rc) = 4πG ρrc∕(3F0 ) is the gravitational potential at the surface of the body. The approximation M rc ≪ 1 used to derive Eqs. (5.6View Equation) and (5.7View Equation) corresponds to the condition
m (R0 ) ≫ Φc. (5.12 )

Since F0 δF = f,RR(R0 )δR, it follows that

f,R (R0) δR = ---------δF. (5.13 ) f,RR(R0 )
The validity of the linear expansion requires that δR ≪ R0, which translates into δF ≪ m (R0 ). Since δ ≃ 2GM ∕ (3F r ) = 2Φ ∕3 F c 0 c c at r = r c, one has δ ≪ m (R ) ≪ 1 F 0 under the condition (5.12View Equation). Hence the linear analysis given above is valid for m (R0) ≫ Φc.

For the distance r close to rc the post Newtonian parameter in Eq. (5.10View Equation) is given by γ ≃ 1∕2 (i.e., because M r ≪ 1). The tightest experimental bound on γ is given by [616Jump To The Next Citation Point83Jump To The Next Citation Point617Jump To The Next Citation Point]:

|γ − 1 | < 2.3 × 10−5, (5.14 )
which comes from the time-delay effect of the Cassini tracking for Sun. This means that f (R) gravity models with the light scalaron mass (M rc ≪ 1) do not satisfy local gravity constraints [469470Jump To The Next Citation Point245233154448330Jump To The Next Citation Point332Jump To The Next Citation Point]. The mean density of Earth or Sun is of the order of 3 ρ ≃ 1 –10 g ∕cm, which is much larger than the present cosmological density (0) −29 3 ρc ≃ 10 g∕cm. In such an environment the condition δR ≪ R0 is violated and the field mass M becomes large such that M rc ≫ 1. The effect of the chameleon mechanism [344Jump To The Next Citation Point343Jump To The Next Citation Point] becomes important in this non-linear regime (δR ≫ R0[251Jump To The Next Citation Point306Jump To The Next Citation Point134Jump To The Next Citation Point101]. In Section 5.2 we will show that the f (R) models can be consistent with local gravity constraints provided that the chameleon mechanism is at work.
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