5.2 Chameleon mechanism in f (R) gravity

Let us discuss the chameleon mechanism [344Jump To The Next Citation Point343Jump To The Next Citation Point] in metric f (R) gravity. Unlike the linear expansion approach given in Section 5.1, this corresponds to a non-linear effect arising from a large departure of the Ricci scalar from its background value R0. The mass of an effective scalar field degree of freedom depends on the density of its environment. If the matter density is sufficiently high, the field acquires a heavy mass about the potential minimum. Meanwhile the field has a lighter mass in a low-density cosmological environment relevant to dark energy so that it can propagate freely. As long as the spherically symmetric body has a thin-shell around its surface, the effective coupling between the field and matter becomes much smaller than the bare coupling |Q |. In the following we shall review the chameleon mechanism for general couplings Q and then proceed to constrain f (R) dark energy models from local gravity tests.

5.2.1 Field profile of the chameleon field

The action (2.1View Equation) in f (R) gravity can be transformed to the Einstein frame action (2.32View Equation) with the coupling √ -- Q = − 1∕ 6 between the scalaron field ∘ -------- ϕ = 3∕(2κ2)lnF and non-relativistic matter. Let us consider a spherically symmetric body with radius r&tidle; c in the Einstein frame. We approximate that the background geometry is described by the Minkowski space-time. Varying the action (2.32View Equation) with respect to the field ϕ, we obtain

d2ϕ 2d ϕ dV ----+ ---- − ---eff = 0, (5.15 ) d &tidle;r2 &tidle;r d&tidle;r dϕ
where &tidle;r is a distance from the center of symmetry that is related to the distance r in the Jordan frame via √ -- −Qκϕ &tidle;r = F r = e r. The effective potential Veff is defined by
Qκϕ ∗ Veff(ϕ) = V (ϕ) + e ρ , (5.16 )
where ρ∗ is a conserved quantity in the Einstein frame [343Jump To The Next Citation Point]. Recall that the field potential V (ϕ) is given in Eq. (2.33View Equation). The energy density &tidle;ρ in the Einstein frame is related with the energy density ρ in the Jordan frame via the relation 2 4Q κϕ ρ&tidle;= ρ∕F = e ρ. Since the conformal transformation gives rise to a coupling Q between matter and the field, &tidle;ρ is not a conserved quantity. Instead the quantity ρ∗ = e3Qκϕρ = e−Qκϕ&tidle;ρ corresponds to a conserved quantity, which satisfies &tidle;r3ρ ∗ = r3ρ. Note that Eq. (5.15View Equation) is consistent with Eq. (2.42View Equation).

In the following we assume that a spherically symmetric body has a constant density ρ∗ = ρ A inside the body (&tidle;r < &tidle;rc) and that the energy density outside the body (&tidle;r > &tidle;rc) is ∗ ρ = ρB (≪ ρA). The mass Mc of the body and the gravitational potential Φc at the radius &tidle;rc are given by Mc = (4π ∕3)&tidle;r3cρA and Φc = GMc ∕&tidle;rc, respectively. The effective potential has minima at the field values ϕA and ϕB:

V,ϕ (ϕA ) + κQeQ κϕAρA = 0, (5.17 ) QκϕB V,ϕ(ϕB ) + κQe ρB = 0. (5.18 )
The former corresponds to the region of high density with a heavy mass squared m2 ≡ Veff,ϕϕ(ϕA) A, whereas the latter to a lower density region with a lighter mass squared 2 m B ≡ Veff,ϕϕ(ϕB). In the case of Sun, for example, the field value ϕB is determined by the homogeneous dark matter/baryon density in our galaxy, i.e., 3 ρB ≃ 10−24 g∕cm.
View Image

Figure 3: (Top) The potential 2 2 V (ϕ) = (F R − f)∕(2κ F ) versus the field ∘ -------- ϕ = 3∕(16 π)mpl ln F for the Starobinsky’s dark energy model (4.84View Equation) with n = 1 and μ = 2. (Bottom) The inverted effective potential − Veff for the same model parameters as the top with ρ ∗ = 10Rcm2pl. The field value, at which the inverted effective potential has a maximum, is different depending on the density ∗ ρ, see Eq. (5.22View Equation). In the upper panel “de Sitter” corresponds to the minimum of the potential, whereas “singular” means that the curvature diverges at ϕ = 0.

When Q > 0 the effective potential has a minimum for the models with V,ϕ < 0, which occurs, e.g., for the inverse power-law potential V (ϕ) = M 4+nϕ −n. The f (R) gravity corresponds to a negative coupling (√ -- Q = − 1∕ 6), in which case the effective potential has a minimum for V,ϕ > 0. As an example, let us consider the shape of the effective potential for the models (4.83View Equation) and (4.84View Equation). In the region R ≫ Rc both models behave as

[ 2n] f(R ) ≃ R − μRc 1 − (Rc ∕R ) . (5.19 )
For this functional form it follows that
√2-κϕ − (2n+1) F = e 6 = 1 −[2n μ(R ∕Rc) , ] (5.20 ) μR √4 ( − κϕ ) 22nn+1 V (ϕ) = ---c2 e− 6κϕ 1 − (2n + 1) √----- . (5.21 ) 2κ 6nμ
The r.h.s. of Eq. (5.20View Equation) is smaller than 1, so that ϕ < 0. The limit R → ∞ corresponds to ϕ → − 0. In the limit ϕ → − 0 one has 2 V → μRc ∕(2κ ) and V,ϕ → ∞. This property can be seen in the upper panel of Figure 3View Image, which shows the potential V (ϕ ) for the model (4.84View Equation) with parameters n = 1 and μ = 2. Because of the existence of the coupling term √ - e− κϕ∕ 6ρ∗, the effective potential Veff(ϕ) has a minimum at
( ) √ -- -Rc-- 2n+1 κϕM = − 6nμ κ2ρ∗ . (5.22 )
Since R ∼ κ2ρ∗ ≫ Rc in the region of high density, the condition |κϕM | ≪ 1 is in fact justified (for n and μ of the order of unity). The field mass m ϕ about the minimum of the effective potential is given by
( 2 ∗)2 (n+1) m2ϕ = -----1-----Rc κ-ρ-- . (5.23 ) 6n (n + 1 )μ Rc
This shows that, in the regime R ∼ κ2ρ∗ ≫ R c, m ϕ is much larger than the present Hubble parameter H0 (√ --- ∼ Rc). Cosmologically the field evolves along the instantaneous minima characterized by Eq. (5.22View Equation) and then it approaches a de Sitter point which appears as a minimum of the potential in the upper panel of Figure 3View Image.

In order to solve the “dynamics” of the field ϕ in Eq. (5.15View Equation), we need to consider the inverted effective potential (− Veff). See the lower panel of Figure 3View Image for illustration [which corresponds to the model (4.84View Equation)]. We impose the following boundary conditions:

dϕ- d&tidle;r (r&tidle;= 0 ) = 0, (5.24 ) ϕ (&tidle;r → ∞ ) = ϕ . (5.25 ) B
The boundary condition (5.25View Equation) can be also understood as lim &tidle;r→ ∞ dϕ∕d &tidle;r = 0. The field ϕ is at rest at &tidle;r = 0 and starts to roll down the potential when the matter-coupling term κQ ρAeQ κϕ in Eq. (5.15View Equation) becomes important at a radius &tidle;r 1. If the field value at &tidle;r = 0 is close to ϕ A, the field stays around ϕA in the region 0 < &tidle;r < &tidle;r1. The body has a thin-shell if &tidle;r1 is close to the radius &tidle;rc of the body.

In the region 0 < &tidle;r < &tidle;r1 one can approximate the r.h.s. of Eq. (5.15View Equation) as dVeff ∕dϕ ≃ m2A (ϕ − ϕA) around ϕ = ϕA, where m2 = Rc (κ2 ρA∕Rc )2(n+1)∕[6n(n + 1)] A. Hence the solution to Eq. (5.15View Equation) is given by −mAr&tidle; mA &tidle;r ϕ (&tidle;r) = ϕA + Ae ∕r&tidle;+ Be ∕&tidle;r, where A and B are constants. In order to avoid the divergence of ϕ at &tidle;r = 0 we demand the condition B = − A, in which case the solution is

A(e−mA &tidle;r − emA&tidle;r) ϕ(&tidle;r) = ϕA + ----------------- (0 < &tidle;r < &tidle;r1). (5.26 ) &tidle;r
In fact, this satisfies the boundary condition (5.24View Equation).

In the region &tidle;r1 < &tidle;r < &tidle;rc the field |ϕ(&tidle;r)| evolves toward larger values with the increase of &tidle;r. In the lower panel of Figure 3View Image the field stays around the potential maximum for 0 < &tidle;r < &tidle;r1, but in the regime &tidle;r1 < &tidle;r < &tidle;rc it moves toward the left (largely negative ϕ region). Since Qκϕ |V,ϕ| ≪ |κQ ρAe | in this regime we have that dVeff∕dϕ ≃ κQ ρA in Eq. (5.15View Equation), where we used the condition Q κϕ ≪ 1. Hence we obtain the following solution

1- 2 C- ϕ(&tidle;r) = 6 κQ ρA&tidle;r − &tidle;r + D (&tidle;r1 < &tidle;r < &tidle;rc), (5.27 )
where C and D are constants.

Since the field acquires a sufficient kinetic energy in the region &tidle;r1 < &tidle;r < &tidle;rc, the field climbs up the potential hill toward the largely negative ϕ region outside the body (&tidle;r > &tidle;rc). The shape of the effective potential changes relative to that inside the body because the density drops from ρA to ρB. The kinetic energy of the field dominates over the potential energy, which means that the term dVeff∕d ϕ in Eq. (5.15View Equation) can be neglected. Recall that one has |ϕB | ≫ |ϕA | under the condition ρA ≫ ρB [see Eq. (5.22View Equation)]. Taking into account the mass term m2 = Rc(κ2ρB ∕Rc)2(n+1)∕ [6n (n + 1)] B, we have dVe ff∕dϕ ≃ m2 (ϕ − ϕB ) B on the r.h.s. of Eq. (5.15View Equation). Hence we obtain the solution ϕ (&tidle;r) = ϕ + Ee −mB (&tidle;r− &tidle;rc)∕&tidle;r + FemB (&tidle;r− &tidle;rc)∕&tidle;r B with constants E and F. Using the boundary condition (5.25View Equation), it follows that F = 0 and hence

−mB (&tidle;r− &tidle;rc) ϕ (&tidle;r) = ϕB + E e--------- (&tidle;r > r&tidle;c). (5.28 ) &tidle;r

Three solutions (5.26View Equation), (5.27View Equation) and (5.28View Equation) should be matched at &tidle;r = &tidle;r1 and &tidle;r = &tidle;rc by imposing continuous conditions for ϕ and dϕ ∕d&tidle;r. The coefficients A, C, D and E are determined accordingly [575Jump To The Next Citation Point]:

s1s2[(ϕB − ϕA ) + (&tidle;r21 − &tidle;r2c)κQ ρA∕6 ] + [s2&tidle;r21(e− mA&tidle;r1 − emA&tidle;r1) − s1&tidle;r2c]κQ ρA ∕3 C = ---------------------------−mA-&tidle;r1---mA-&tidle;r1--------------------------------, (5.29 ) mA (e + e )s2 − mBs1 A = − 1-(C + κQ ρ &tidle;r3∕3), (5.30 ) s1 A 1 1 E = − --(C + κQ ρA&tidle;r3c∕3), (5.31 ) s2 D = ϕ − 1κQ ρ &tidle;r2+ 1-(C + E ), (5.32 ) B 6 A c &tidle;rc
s1 ≡ mA &tidle;r1(e− mA&tidle;r1 + emA&tidle;r1) + e− mA&tidle;r1 − emA&tidle;r1, (5.33 ) s2 ≡ 1 + mB &tidle;rc. (5.34 )
If the mass mB outside the body is small to satisfy the condition mB &tidle;rc ≪ 1 and mA ≫ mB, we can neglect the contribution of the mB-dependent terms in Eqs. (5.29View Equation) – (5.32View Equation). Then the field profile is given by [575Jump To The Next Citation Point]
[ ] 1 1 2 2 e−mA &tidle;r − emA &tidle;r ϕ (&tidle;r) = ϕA − -----−-mA&tidle;r1----mA&tidle;r1- ϕB − ϕA + -κQ ρA(&tidle;r1 − &tidle;rc) -------------, mA (e + e ) 2 &tidle;r (0 < r&tidle;< &tidle;r1), (5.35 ) 1 κQ ρA&tidle;r3 ϕ (&tidle;r) = ϕB + -κQ ρA(&tidle;r2 − 3&tidle;r2c) + ------1- [ 6 − m &tidle;r1 m &tidle;r1 3&tidle;r][ ] − 1 + ---e---A--−--e-A------ ϕ − ϕ + 1-κQ ρ (&tidle;r2 − &tidle;r2) &tidle;r1, mA &tidle;r1(e−mA &tidle;r1 + emA &tidle;r1) B A 2 A 1 c &tidle;r (&tidle;r < &tidle;r < &tidle;r ), (5.36 ) 1 [ c ( ) ( ) 1 3 &tidle;r1 &tidle;r1 2 ϕ (&tidle;r) = ϕB − &tidle;r1(ϕB − ϕA) + 6κQ ρA &tidle;rc 2 + &tidle;r- 1 − &tidle;r- c c ] e− mA&tidle;r1 − emA&tidle;r1 { 1 } e −mB(&tidle;r− &tidle;rc) + -------------------- ϕB − ϕA + -κQ ρA (&tidle;r21 − &tidle;r2c) ----------, mA (e−mA &tidle;r1 + emA &tidle;r1) 2 &tidle;r (&tidle;r > &tidle;r ). (5.37 ) c
Originally a similar field profile was derived in [344Jump To The Next Citation Point343Jump To The Next Citation Point] by assuming that the field is frozen at ϕ = ϕA in the region 0 < &tidle;r < &tidle;r1.

The radius r 1 is determined by the following condition

m2 [ϕ(&tidle;r ) − ϕ ] = κQ ρ . (5.38 ) A 1 A A
This translates into
1 2 2 6Q Φc mA &tidle;r1(emA &tidle;r1 + e− mA&tidle;r1) ϕB − ϕA + --κQ ρA(&tidle;r1 − &tidle;rc) = --------2-----mA&tidle;r1----−mA-&tidle;r1----, (5.39 ) 2 κ(mA &tidle;rc) e − e
where Φc = κ2Mc ∕(8π &tidle;rc) = κ2 ρA&tidle;r2c∕6 is the gravitational potential at the surface of the body. Using this relation, the field profile (5.37View Equation) outside the body reduces to
[ { } ] 2Q Φc &tidle;r31 &tidle;r1 1 mA r&tidle;1(emA &tidle;r1 + e− mA&tidle;r1) &tidle;rce−mB (&tidle;r− &tidle;rc) ϕ(&tidle;r) = ϕB − --κ--- 1 − &tidle;r3 + 3 &tidle;r-(m--&tidle;r-)2 ----emA&tidle;r1 −-e−mA-&tidle;r1----− 1 ------&tidle;r-----, c c A c (&tidle;r > &tidle;rc). (5.40 )

If the field value at &tidle;r = 0 is away from ϕA, the field rolls down the potential for &tidle;r > 0. This corresponds to taking the limit &tidle;r1 → 0 in Eq. (5.40View Equation), in which case the field profile outside the body is given by

ϕ (r&tidle;) = ϕB − 2Q-GMc--e− mB(&tidle;r− &tidle;rc). (5.41 ) κ &tidle;r
This shows that the effective coupling is of the order of Q and hence for |Q | = 𝒪 (1) local gravity constraints are not satisfied.

5.2.2 Thin-shell solutions

Let us consider the case in which r&tidle;1 is close to &tidle;rc, i.e.

Δ &tidle;r ≡ &tidle;r − &tidle;r ≪ &tidle;r . (5.42 ) c c 1 c
This corresponds to the thin-shell regime in which the field is stuck inside the star except around its surface. If the field is sufficiently massive inside the star to satisfy the condition mA &tidle;rc ≫ 1, Eq. (5.39View Equation) gives the following relation
𝜖th ≡ κ(ϕB-−--ϕA)-≃ Δ-&tidle;rc+ --1--, (5.43 ) 6Q Φc &tidle;rc mA &tidle;rc
where 𝜖 th is called the thin-shell parameter [344Jump To The Next Citation Point343Jump To The Next Citation Point]. Neglecting second-order terms with respect to Δ &tidle;rc∕&tidle;rc and 1∕(mA &tidle;rc) in Eq. (5.40View Equation), it follows that
2Qe-ffGMc-- −mB (&tidle;r− &tidle;rc) ϕ(&tidle;r) ≃ ϕB − κ &tidle;r e , (5.44 )
where Qeff is the effective coupling given by
Qeff = 3Q 𝜖th. (5.45 )

Since 𝜖th ≪ 1 under the conditions Δ &tidle;rc∕&tidle;rc ≪ 1 and 1∕(mA &tidle;rc) ≪ 1, the amplitude of the effective coupling Qe ff becomes much smaller than 1. In the original papers of Khoury and Weltman [344Jump To The Next Citation Point343Jump To The Next Citation Point] the thin-shell solution was derived by assuming that the field is frozen with the value ϕ = ϕA in the region 0 < &tidle;r < r&tidle;1. In this case the thin-shell parameter is given by 𝜖th ≃ Δ &tidle;rc∕&tidle;rc, which is different from Eq. (5.43View Equation). However, this difference is not important because the condition Δr&tidle;∕&tidle;r ≫ 1∕(m &tidle;r ) c c A c is satisfied for most of viable models [575].

5.2.3 Post Newtonian parameter

We derive the bound on the thin-shell parameter from experimental tests of the post Newtonian parameter in the solar system. The spherically symmetric metric in the Einstein frame is described by [251Jump To The Next Citation Point]

d &tidle;s2 = &tidle;gμνd&tidle;xμd &tidle;xν = − [1 − 2𝒜&tidle;(&tidle;r)]dt2 + [1 + 2 &tidle;ℬ(&tidle;r)]d&tidle;r2 + &tidle;r2dΩ2, (5.46 )
where &tidle;𝒜 (&tidle;r) and &tidle;ℬ(&tidle;r) are functions of &tidle;r and dΩ2 = d𝜃2 + (sin2 𝜃)dϕ2. In the weak gravitational background (&tidle; 𝒜 (&tidle;r) ≪ 1 and &tidle; ℬ(&tidle;r) ≪ 1) the metric outside the spherically symmetric body with mass Mc is given by 𝒜&tidle;(&tidle;r) ≃ ℬ&tidle;(&tidle;r) ≃ GMc ∕&tidle;r.

Let us transform the metric (5.46View Equation) back to that in the Jordan frame under the inverse of the conformal transformation, gμν = e2Qκϕ&tidle;gμν. Then the metric in the Jordan frame, ds2 = e2Qκϕd &tidle;s2 = gμνdxμdx ν, is given by

ds2 = − [1 − 2𝒜 (r)]dt2 + [1 + 2ℬ (r)]dr2 + r2d Ω2. (5.47 )
Under the condition |Q κϕ | ≪ 1 we obtain the following relations
Q κϕ &tidle; &tidle; dϕ(&tidle;r)- &tidle;r = e r, 𝒜 (r) ≃ 𝒜 (&tidle;r) − Q κϕ(&tidle;r), ℬ(r) ≃ ℬ (r&tidle;) − Q κ&tidle;r d&tidle;r . (5.48 )
In the following we use the approximation r ≃ &tidle;r, which is valid for |Q κϕ| ≪ 1. Using the thin-shell solution (5.44View Equation), it follows that
GMc--[ 2 ] GMc--( 2 ) 𝒜 (r) = r 1 + 6Q 𝜖th (1 − r∕rc) , ℬ (r) = r 1 − 6Q 𝜖th , (5.49 )
where we have used the approximation |ϕB| ≫ |ϕA | and hence ϕB ≃ 6Q Φc𝜖th∕κ.

The term Q κ ϕB in Eq. (5.48View Equation) is smaller than 𝒜(r) = GMc ∕r under the condition r∕rc < (6Q2 𝜖th)−1. Provided that the field ϕ reaches the value ϕB with the distance rB satisfying the condition rB ∕rc < (6Q2 𝜖th)−1, the metric 𝒜 (r) does not change its sign for r < rB. The post-Newtonian parameter γ is given by

ℬ (r ) 1 − 6Q2 𝜖 γ ≡ -----≃ -------2-----th-----. (5.50 ) 𝒜 (r) 1 + 6Q 𝜖th(1 − r∕rc)
The experimental bound (5.14View Equation) can be satisfied as long as the thin-shell parameter 𝜖th is much smaller than 1. If we take the distance r = rc, the constraint (5.14View Equation) translates into
𝜖th,⊙ < 3.8 × 10−6∕Q2, (5.51 )
where 𝜖th,⊙ is the thin-shell parameter for Sun. In f (R) gravity (√ -- Q = − 1∕ 6) this corresponds to − 5 𝜖th,⊙ < 2.3 × 10.

5.2.4 Experimental bounds from the violation of equivalence principle

Let us next discuss constraints on the thin-shell parameter from the possible violation of equivalence principle (EP). The tightest bound comes from the solar system tests of weak EP using the free-fall acceleration of Earth (a ⊕) and Moon (aMoon) toward Sun [343Jump To The Next Citation Point]. The experimental bound on the difference of two accelerations is given by [616Jump To The Next Citation Point83Jump To The Next Citation Point617Jump To The Next Citation Point]

|a⊕ − aMoon| ---------------< 10−13. (5.52 ) |a⊕ + aMoon |∕2

Provided that Earth, Sun, and Moon have thin-shells, the field profiles outside the bodies are given by Eq. (5.44View Equation) with the replacement of corresponding quantities. The presence of the field ϕ (r ) with an effective coupling Qeff gives rise to an extra acceleration, a fifth = |Qe ff∇ ϕ(r)|. Then the accelerations a⊕ and aMoon toward Sun (mass M ⊙) are [343Jump To The Next Citation Point]

[ ] GM ⊙ 2 2 Φ ⊕ a⊕ ≃ ---2-- 1 + 18Q 𝜖th,⊕--- , (5.53 ) r [ Φ ⊙ ] GM--⊙- 2 2 ---Φ2⊕--- aMoon ≃ r2 1 + 18Q 𝜖th,⊕Φ ⊙ΦMoon , (5.54 )
where 𝜖th,⊕ is the thin-shell parameter of Earth, and Φ⊙ ≃ 2.1 × 10 −6, Φ ⊕ ≃ 7.0 × 10− 10, ΦMoon ≃ 3.1 × 10−11 are the gravitational potentials of Sun, Earth and Moon, respectively. Hence the condition (5.52View Equation) translates into [134Jump To The Next Citation Point596Jump To The Next Citation Point]
−7 𝜖th,⊕ < 8.8 × 10 ∕|Q |, (5.55 )
which corresponds to 𝜖th,⊕ < 2.2 × 10 −6 in f (R) gravity. This bound provides a tighter bound on model parameters compared to (5.51View Equation).

Since the condition |ϕB| ≫ |ϕA | is satisfied for ρA ≫ ρB, one has 𝜖th,⊕ ≃ κ ϕB∕(6Q Φ ⊕) from Eq. (5.43View Equation). Then the bound (5.55View Equation) translates into

|κϕB,⊕ | < 3.7 × 10−15. (5.56 )

5.2.5 Constraints on model parameters in f (R) gravity

We place constraints on the f (R) models given in Eqs. (4.83View Equation) and (4.84View Equation) by using the experimental bounds discussed above. In the region of high density where R is much larger than Rc, one can use the asymptotic form (5.19View Equation) to discuss local gravity constraints. Inside and outside the spherically symmetric body the effective potential Veff for the model (5.19View Equation) has two minima at

√ -- ( Rc )2n+1 √ -- ( Rc )2n+1 κ ϕA ≃ − 6nμ -2--- , κ ϕB ≃ − 6n μ -2--- . (5.57 ) κ ρA κ ρB

The bound (5.56View Equation) translates into

( )2n+1 -nμ--- -R1-- < 1.5 × 10−15, (5.58 ) x2dn+1 κ2ρB
where xd ≡ R1 ∕Rc and R1 is the Ricci scalar at the late-time de Sitter point. In the following we consider the case in which the Lagrangian density is given by (5.19View Equation) for R ≥ R1. If we use the models (4.83View Equation) and (4.84View Equation), then there are some modifications for the estimation of R1. However this change should be insignificant when we place constraints on model parameters.

At the de Sitter point the model (5.19View Equation) satisfies the condition μ = x2n+1∕[2(x2n − n − 1)] d d. Substituting this relation into Eq. (5.58View Equation), we find

( )2n+1 -------n------- -R1-- < 1.5 × 10− 15. (5.59 ) 2(x2nd − n − 1) κ2ρB
For the stability of the de Sitter point we require that m (R1 ) < 1, which translates into the condition 2n 2 x d > 2n + 3n + 1. Hence the term 2n n ∕[2 (x d − n − 1)] in Eq. (5.59View Equation) is smaller than 0.25 for n > 0.

We use the approximation that R1 and ρB are of the orders of the present cosmological density 10− 29 g∕cm3 and the baryonic/dark matter density 10− 24 g∕cm3 in our galaxy, respectively. From Eq. (5.59View Equation) we obtain the bound [134Jump To The Next Citation Point]

n > 0.9. (5.60 )
Under this condition one can see an appreciable deviation from the ΛCDM model cosmologically as R decreases to the order of Rc.

If we consider the model (4.81View Equation), it was shown in [134] that the bound (5.56View Equation) gives the constraint −10 n < 3 × 10. This means that the deviation from the ΛCDM model is very small. Meanwhile, for the models (4.83View Equation) and (4.84View Equation), the deviation from the ΛCDM model can be large even for n = 𝒪 (1), while satisfying local gravity constraints. We note that the model (4.89View Equation) is also consistent with local gravity constraints.

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