10.3 Local gravity constraints

We study local gravity constraints (LGC) for BD theory given by the action (10.10View Equation). In the absence of the potential U(ϕ) the BD parameter ωBD is constrained to be 4 ωBD > 4 × 10 from solar-system experiments [61683617]. This bound also applies to the case of a nearly massless field with the potential U (ϕ) in which the Yukawa correction e−Mr is close to unity (where M is a scalar-field mass and r is an interaction length). Using the bound ω > 4 × 104 BD in Eq. (10.11View Equation), we find that
|Q | < 2.5 × 10 −3. (10.29 )
This is a strong constraint under which the cosmological evolution for such theories is difficult to be distinguished from the ΛCDM model (Q = 0).

If the field potential is present, the models with large couplings (|Q| = 𝒪 (1)) can be consistent with local gravity constraints as long as the mass M of the field ϕ is sufficiently large in the region of high density. For example, the potential (10.23View Equation) is designed to have a large mass in the high-density region so that it can be compatible with experimental tests for the violation of equivalence principle through the chameleon mechanism [596Jump To The Next Citation Point]. In the following we study conditions under which local gravity constraints can be satisfied for the model (10.23View Equation).

As in the case of metric f (R) gravity, let us consider a configuration in which a spherically symmetric body has a constant density ρA inside the body with a constant density ρ = ρB (≪ ρA) outside the body. For the potential 2 V = U ∕F in the Einstein frame one has p−1 V,ϕ ≃ − 2U0QpC (2Q ϕ) under the condition |Qϕ | ≪ 1. Then the field values at the potential minima inside and outside the body are

( )1∕(1− p) ϕ ≃ -1- 2U0pC-- , i = A,B. (10.30 ) i 2Q ρi
The field mass squared 2 m i ≡ V,ϕϕ at ϕ = ϕi (i = A,B) is approximately given by
( ) (2− p)∕(1−p) 2 ---1-−-p----- 2 -ρi m i ≃ (2ppC )1∕(1− p)Q U0 U0. (10.31 )

Recall that U0 is roughly the same order as the present cosmological density ρ0 ≃ 10−29 g∕cm3. The baryonic/dark matter density in our galaxy corresponds to ρB ≃ 10−24 g∕cm3. The mean density of Sun or Earth is about 3 ρA = 𝒪 (1) g∕cm. Hence mA and mB are in general much larger than H0 for local gravity experiments in our environment. For mA &tidle;rc ≫ 1 the chameleon mechanism we discussed in Section 5.2 can be directly applied to BD theory whose Einstein frame action is given by Eq. (10.6View Equation) with F = e−2Qϕ.

The bound (5.56View Equation) coming from the possible violation of equivalence principle in the solar system translates into

(2U0pC ∕ρB)1∕(1− p) < 7.4 × 10−15|Q|. (10.32 )
Let us consider the case in which the solutions finally approach the de Sitter point (e) in Table 1. At this de Sitter point we have 3F1H2 = U0 [1 − C (1 − F1)p] 1 with C given in Eq. (10.26View Equation). Then the following relation holds
2 U0 = 3H 1 [2 + (p − 2 )F1 ]∕p. (10.33 )
Substituting this into Eq. (10.32View Equation) we obtain
1∕(1−p) −15 (R1∕ ρB) (1 − F1) < 7.4 × 10 |Q|, (10.34 )
where R1 = 12H2 1 is the Ricci scalar at the de Sitter point. Since (1 − F1) is smaller than 1/2 from Eq. (10.28View Equation), it follows that 1∕(1− p) −14 (R1 ∕ρB) < 1.5 × 10 |Q|. Using the values − 29 3 R1 = 10 g∕cm and −24 3 ρB = 10 g∕cm, we get the bound for p [596Jump To The Next Citation Point]:
p > 1 − -------5-------. (10.35 ) 13.8 − log10|Q|
When −1 |Q | = 10 and |Q | = 1 we have p > 0.66 and p > 0.64, respectively. Hence the model can be compatible with local gravity experiments even for |Q | = 𝒪 (1 ).
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