Equivalence principle in chameleon models

Abstract

results of the detailed analyses indicate that those world-lines do not, in general, correspond to geodesics. Quantum aspects further complicate the analysis of the motion of even the simplest things, which one might want to take as paradigmatic point like objects, such as photons [4]. Even though such UFF violating effects are normally very small, they are always there in principle, and therefore these considerations should serve as warnings when we go on to analyze more complex situations.Of particular interest for us here will be any theory in which the local coupling constants are taken to be effectively space-time dependent, while respecting the principles of locality and general covariance, as those entail some kind of fundamental field controlling the spatial dependence, with the field, generically, coupling in different manners to the various types of matter. Such field, usually taken to be a scalar field, generically mediates new forces between macroscopic objects, which would look, at the empirical level, as modifications of gravitation which might, in principle, lead to effective violations of the UFF for test bodies in external gravitational fields. For this reason, most theories that predict variations of fundamental constants also predict effective violations of the WEP [5][6][7][8][9]. For instance, the rest energy of a macroscopic body is made of many contributions related to the energies associated with various kinds of interactions (strong, weak, electromagnetic) and such components would be affected differently by a light scalar field. From the experimental point of view, one needs to confront the very strong limits on possible violations of the WEP that come from Eötvös-Roll-Krotkov-Dicke and Braginsky-Panov experiments and their modern reincarnations [10-15] which explore the differential acceleration of test bodies in gravitational contexts. In fact current bounds reach sensitivities of order ∆a a ≃ 10 −13 (or more), and thus can, in principle, constrain the viability of many models. Recently, there has been a great level of interest in models that claim to be able to avoid the stringent bounds resulting from experimental tests looking for violations of the WEP based on schemes where the effects of the fields are hidden by suitable non-linearities, as in the chameleon models and the Dilaton-Matter-gravity model with strong coupling [16]. Chameleon models were introduced by Khoury and Weltman in 2004 [1] and have been further developed by several authors [18][19][20][21][22][23][24][25]27]. Generically a chameleon consists of a scalar field that is coupled non-minimally to matter and minimally to the curvature (or the other way around via a conformal transformation), and where the field's effective mass depends on the density and pressure of the matter that constitutes the environment. This, in turn, is the result of the nontrivial coupling of the scalar field with the trace of the energy-momentum tensor of the matter sector of the theory 1 . The coupling of the scalar field with matter might...

Thick shell regime in the chameleon two-body problem

“…(9). Under those conditions, and as shown in [1,2], the second term of the effective potential Eq. (3) dominates over the first one and thus we take V in eff ∼ ρ in βϕ M pl .…”

“…In order to determine which of these approximations is the best one, we developed a criterion using a minimization of a suitable energy func- tional. In [1] we found that the energy criterion favors the two body approach over the standard (one body) approach in scenarios where the large object is not in the thick shell regime, the reverse is true for those situations in which the bodies develop a thick shell. In this paper we address this limitation of our previous work and improve our method by implementing the linear approximation V in eff ∼ ρ in βϕ M pl in scenarios where one or both objects are no longer in the thin shell but in the thick shell regime.…”

“…One of the bodies corresponds to a large body, which we take as the main source for the gravitational field, and a small test body whose backreaction on the chameleon field is taken into account, so that the chamaleon mediated force might be more realistically computed. In the analysis carried out in [1] the effective potential was approximated quadratically around each minima (inside and outside the two bodies): V eff (ϕ) ≃ V eff (ϕ min ) + ∂ ϕϕ V eff (ϕ min )[ϕ− ϕ min ] 2 /2, and the problem was studied making use of the axially symmetric of the situation so the solution for Eq. (2) was described as follows:…”

“…The parameters C in1 l , C in2 l , C out1 l and C out2 l are determined from the boundary conditions at the borders of the two bodies R 1 and R 2 using translation coefficients α lm vw and α * lm vw to go from one coordinate system to the other (for details see Ref. [1]). In both cases (when the large body has thin shell and the test body does not, and when both of them are in the thick shell regime) these coefficients depend on the composition of both bodies and on the surrounding environment.…”

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In a previous paper [1] we pointed out some shortcomings of the standard approach to chameleon theories consisting in treating the small bodies used to test the Weak Equivalence Principle (WEP) as test particles, whose presence do not modify the chameleon field configuration. In that paper we developed an alternative method to determine the relevant field configuration which takes into account the influence of both test and source bodies, and computed the chamaleon mediated force.Relying on that analysis we showed that the effective acceleration of test bodies is composition dependent even when the model is based on universal couplings.

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