Solar System Constraints on a Cosmologically Viable $f(R)$ Theory

Recently, a model $f(R)$ theory is proposed \cite{recent} which is cosmologically viable and distinguishable from $\Lambda$CDM. We use chameleon mechanism to investigate viability of the model in terms of Solar System experiments.


Introduction
There are strong observational evidences that the expansion of the universe is accelerating. These observations are based on type Ia supernova [2], cosmic microwave background radiation [3], large scale structure formation [4], weak lensing [5], etc. The standard explanation invokes an unknown component, usually referred to as dark energy. It contributes to energy density of the universe with Ω d = 0.7 where Ω d is the corresponding density parameter, see e.g., [6] and references therein. The simplest dark energy scenario which seems to be both natural and consistent with observations is the ΛCDM model in which dark energy is identified as a cosmological constant [6] [7] [8]. However, in order to avoid theoretical problems [7], other scenarios have been investigated. Among different scenarios there are modified gravity models [9], in which one modifies the laws of gravity whereby a late time acceleration is produced without recourse to a dark energy component. One family of these modified gravity models is obtained by replacing the Ricci scalar R in the usual Einstein-Hilbert Lagrangian density for some function f (R). These models are cosmologically acceptable if they meet some certain conditions. The most important ones is that they should follow a usual matter-dominated era preceding a late-time accelerated stage. Several models are proposed that admit cosmological solutions with accelerated expansion of the universe at late times [10]. However, among all cosmologically viable f (R) theories there is still an important issue to be pursued, they must be probed at Solar System scale. In fact, changing gravity Lagrangian has consequences not only in cosmological scales but also in galactic and Solar System ones so that it seems to be necessary to investigate the low energy limit of such f (R) theories. Early works on the weak field limit of f (R) theories led to negative results. Using the equivalence of f (R) and scalar-tensor theories [11] [12] [13], it is originally suggested that all f (R) theories should be ruled out [14] since they violate the weak field constraints coming from Solar System experiments. This claim was based on the fact that f (R) theories (in the metric formalism) are equivalent to Brans-Dicke theory with ω = 0 while observations set the constraint ω > 40000 [15]. In this case the post-Newtonian parameter satisfies γ = 1 2 instead of being equal to unity as required by observations. Later, it was noted by many authors that for scalar fields with sufficiently large mass it is possible to drive γ close to unity even for null Brans-Dicke parameter. In this case the scalar field becomes short-ranged and has no effect at Solar System scales. Recently, it is shown that there exists an important possibility that the effective mass of the scalar field be scale dependent [16]. In this chameleon mechanism, the scalar field may acquire a large effective mass in Solar System scale so that it hides local experiments while at cosmological scales it is effectively light and may provide an appropriate cosmological behavior. In the present work we intend to use this criterion to investigate constraints set by local experiments on a cosmologically viable model recently proposed by Miranda et al [1]. It is a two parameter f (R) model which is introduced in the form where α and R 1 are positive parameters. This model is reduced to general relativity for α = 0.
A cosmologically viable f (R) model must start with a radiation-dominated era and have a saddle point matter-dominated phase followed by an accelerated epoch as a final attractor. This is formally stated by introducing the parameters m = R d 2 f /dR 2 dR 2 and r = −R df /dR f [17]. The cosmological dynamics of f (R) models can then be understood by considering m(r) curves in the (r, m) plane. Using this criteria, the authors of [1] showed that the model (1) satisfies all the cosmological requirements for α > 1 and regardless of R 1 † Here we will focus on viability of the model in terms of local gravity experiments. We will show that these local experiments rule out the model as an explanation for the current accelerated expansion of the universe. † In the model proposed in [1], it is stated that the relation (1) is a special case of a general parametrization unifying the models of [18] and [19]. In the latter models there is a parameter with the same dimension of R 1 which is taken to be of the same order of the presently observed cosmological constant. The relation between this parameter and R 1 and also the relevance of R 1 with the time of the beginning of the acceleration phase are important issues that are not explicitly addressed in [1].

Chameleon Mechanism
In this section we offer a brief review of the chameleon mechanism. We consider the following where g is the determinant of g µν , f (R) is an unknown function of the scalar curvature R and S m is the matter action depending on the metric g µν and some matter field ψ. We may use a new set of variablesḡ where p ≡ df dR = f ′ (R) and β = 1 6 . This is indeed a conformal transformation which transforms the above action in the Jordan frame to the Einstein frame [11] [12] [13] In the Einstein frame, φ is a scalar field with a self-interacting potential which is given by where r(p) is a solution of the equation f ′ [r(p)]−p = 0 [11]. Note that conformal transformation induces the coupling of the scalar field φ with the matter sector. The strength of this coupling β, is fixed to be 1 6 and is the same for all types of matter fields. In the case of such a strong matter coupling, the role of the potential of the scalar field is important for consistency with local gravity experiments. When the potential satisfies certain conditions it is possible to attribute an effective mass to the scalar field which has a strong dependence on ambient density of matter. A theory in which such a dependence is realized is said to be a chameleon theory [16]. In such a theory the scalar field φ can be heavy enough in the environment of the laboratory tests so that the local gravity constraints suppressed even if β is of the order of unity. Meanwhile, it can be light enough in the low-density cosmological environment to be considered as a candidate for dark energy. Variation of the action (2) with respect toḡ µν and φ, gives the field equations andT =ḡ µνT µν . Covariant differentiation of (7) and the Bianchi identities givē which implies that the matter field is not generally conserved and feels a new force due to gradient of the scalar field. Let us considerT µν as the stress-tensor of dust with energy densitȳ ρ in the Einstein frame. In a static and spherically symmetric spacetime the equation (8) gives wherer is distance from center of the symmetry in the Einstein frame and Here we have used the relationρ = e −4βφ ρ that relates the energy densities in the Jordan and the Einstein frames. We consider a spherically symmetric body with a radiusr c and a constant energy densityρ in (r <r c ). We also assume that the energy density outside the body (r >r c ) is given byρ out . We will denote by ϕ in and ϕ out the field values at two minima of the effective potential V ef f (φ) inside and outside the object, respectively. They must clearly satisfy V  2 which depend on ambient matter density. A region with large mass density corresponds to a heavy mass field while regions with low mass density corresponds to a field with lighter mass. In this way it is possible for the mass field to take sufficiently large values near massive objects in the Solar System scale and to hide the local tests. For a spherically symmetric body there is a thin-shell condition where Φ c = M c /8πr c is the Newtonian potential atr =r c with M c being the mass of the body. In this case, equation (11) with some appropriate boundary conditions gives the field profile outside the object [16] 3 The Model The function f (R) in the Jordan frame is closely related to the potential function of the scalar degree of freedom of the theory in the Einstein frame. Any functional form for the potential function corresponds to a particular class of f (R) theories. To find a viable function f (R) passing Solar System tests one can equivalently work with its corresponding potential function in the Einstein frame and put constraints on the relevant parameters via chameleon mechanism. Taking this as our criterion, we write potential function of the model (1) Assuming that φ << 1, one can find the solution of V ′ ef f (φ) = 0 by substituting (15) into (12) where In the following we shall consider thin-shell condition together with constraints set by equivalence principle and fifth force experiments.

T hin − shell condition
In the chameleon mechanism, the chameleon field is trapped inside large and massive bodies and its influence on the other bodies is only due to a thin-shell near the surface of the body. The criterion for this thin-shell condition is given by (13). If we combine (13) and (16) we obtain where ρ in and ρ out are energy densities inside and outside of the body in the Jordan frame.
Here a = A β − ρ and b = 4αR 2 1 (2α − 1). In weak field approximation, the spherically symmetric metric in the Jordan frame is given by where X(r) and Y (r) are some functions of r. There is a relation between r andr so that r = p 1/2 r. If we consider this relation under the assumption m out r ≪ 1, namely that the Compton wavelength m −1 out is much larger than Solar System scales, then we haver ≈ r. In this case, the chameleon mechanism gives for the post-Newtonian parameter γ [24] We can now apply (20) on the Earth and obtain the condition that the Earth has a thin-shell.

Equivalence principle
We now consider constraints coming from possible violation of weak equivalence principle. We assume that the Earth, together with its surrounding atmosphere, is an isolated body and neglect the effect of the other compact objects such as the Sun, the Moon and the other planets. Far away the Earth, matter density is modeled by a homogeneous gas with energy density ρ G ∼ 10 −24 gr/cm 3 . To proceed further, we first consider the condition that the atmosphere of the Earth satisfies the thin-shell condition [16]. If the atmosphere has a thin-shell the thickness of the shell (∆R a ) must be clearly smaller than that of the atmosphere itself, namely ∆R a < R a , where R a is the outer radius of the atmosphere. If we take thickness of the shell equal to that of the atmosphere itself ∆R a ∼ 10 2 km we obtain ∆Ra Ra < 1.5 × 10 −2 . It is then possible to relate ∆Re Re = ϕa−ϕe 6βΦe and ∆Ra Ra = ϕ G −ϕa 6βΦa where ϕ e , ϕ a and ϕ G are the field values at the local minimum of the effective potential in the regions r < R e , R a > r > R e and r > R a respectively. Using the fact that newtonian potential inside a spherically symmetric object with mass density ρ is Φ ∝ ρR 2 , one can write Φ e = 10 4 Φ a where Φ e and Φ a are Newtonian potentials on the surface of the Earth and the atmosphere, respectively. This gives ∆R e /R e ≈ 10 −4 ∆R a /R a . With these results, the condition for the atmosphere to have a thin-shell is ∆R e R e < 1.5 × 10 −6 The tests of equivalence principle measure the difference of free-fall acceleration of the Moon and the Earth towards the Sun. The constraint on the difference of the two acceleration is given by [15] |a m − a e | a N < 10 −13 (26) where a m and a e are acceleration of the Moon and the Earth respectively and a N is the Newtonian acceleration. The Sun and the Moon are all subject to the thin-shell condition [16] and the field profile outside the spheres are given by (14) with replacement of corresponding § It can be easily checked that our main results do not change when ρ in , ρ out >> R 1 α.
quantities. The accelerations a m and a e are then given by [16] where Φ e = 6.95 × 10 −10 , Φ m = 3.14 × 10 −11 and Φ s = 2.12 × 10 −6 are Newtonian potentials on the surfaces of the Earth, the Moon and the Sun, respectively [25]. This gives a difference of free-fall acceleration Combining this with (26) results in ∆R e R e < 6.74 × 10 −6 (30) which is of the same order of the condition (25) that the atmosphere has a thin-shell. Taking this as the constraint coming from violation of equivalence principle and combining with (23), which is not much different from (24).

F if th f orce
The potential energy associated with a fifth force interaction is parameterized by a Yukawa potential where m 1 and m 2 are masses of the two test bodies separating by distance r, ε is strength of the interaction and λ is the range. Thus fifth force experiment constrains regions of (ε, λ) parameter space. These experiments are usually carried out in a vacuum chamber in which the range of the interaction inside it is of the order of the size of the chamber [16], namely λ ∼ R vac . The tightest bound on the strength of the interaction is ε < 10 −3 [26]. Inside the chamber we consider two identical bodies with uniform densities ρ c , radii r c and masses m c . If the two bodies satisfy the thin-shell condition, their field profile outside the bodies are given by Then the corresponding potential energy of the interaction is The bound on the strength of the interaction translates into One can write for each of the test bodies where ρ vac is energy density of the vacuum inside the chamber. In the experiment carried out in [26], one used a typical test body with mass m c ≈ 40gr and radius r c ≈ 1cm. These correspond to ρ c ≈ 9.5gr/cm 3 and Φ c ∼ 10 −27 . Moreover, the pressure in the vacuum chamber was reported to be 3×10 −8 Torr which is equivalent to ρ vac ≈ 4.8×10 −14 gr/cm 3 . Substituting these into (36) and combining the result with (35) gives the bound which is equivalent to R 1 (1 − 2α) > 10 −1 cm −2 .

Discussion
We have discussed viability of the f (R) model proposed in [1] in terms of local gravity constraints. We have used the correspondence between a general f (R) theory with scalar field theories. In general, in the scalar field representation of a f (R) theory there is a strong coupling of the scalar field with the matter sector. We have considered the conditions that this coupling is suppressed by chameleon mechanism. We have found that in order that the model (1) be consistent both with fifth force and equivalence principle experiments, the two parameters α and R 1 together should satisfy the condition R 1 (1 − 2α) > 10 −1 cm −2 . To have a bound on the parameter α, one should attribute a physical meaning to the dimensional quantity R 1 . Following the models proposed by Hu and Sawicki [18] and Starobinski [19], if we take it as the same order of the observed cosmological constant Λ obs ∼ 10 −58 cm −2 we obtain |2α − 1| > 10 57 . It should be pointed out that this result is obtained under the assumption ρ in , ρ out << R 1 α. To understand the relevance of this assumption, let us consider the case that ρ in , ρ out ∼ R 1 α. Taking energy density inside the Earth as a typical energy density in the Solar System, we obtain R 1 α ∼ 10 −28 cm −2 . In the model (1), if the coefficient R 1 α is so small then it would be hardly distinguishable from general relativity. As the point of view of local gravity experiments, a viable f (R) model should simultaneously satisfy Solar System bounds as well as exhibit an appropriate deviation from general relativity. This requires that R 1 α >> 10 −28 cm −2 , which confirms the assumption that our results are based on. The last point we wish to remark is that, as reported by the authors of [1], the condition that the universe pass through a matter-dominated epoch and finally reach a late-time accelerated phase is that α > 1 regardless of R 1 . This seems not to be consistent with our results in the context of the chameleon mechanism. Although our analysis do not place any experimental bound on the parameter R 1 , however for R 1 > 0 the relation (37) implies that α ∈ (−∞ 1 2 ] which is out of the range reported in [1].