A perturbative approach for the study of compatibility between nonminimally coupled gravity and Solar System experiments

We develop a framework for constraining a certain class of theories of nonminimally coupled (NMC) gravity with Solar System observations.


Introduction
We consider the possibility of constraining a class of theories of nonminimally coupled gravity [1] by means of Solar System experiments. NMC gravity is an extension of f (R) gravity where the action integral of General Relativity (GR) is modified in such a way to contain two functions f 1 (R) and f 2 (R) of the space-time curvature R. The function f 1 (R) has a role analogous to f (R) gravity, and the function f 2 (R) yields a nonminimal coupling between curvature and the matter Lagrangian density. For other NMC gravity theories and their potential applications, see, e.g., [2,3,4,5,6].
NMC gravity has been applied to several astrophysical and cosmological problems such as dark matter [7,8], cosmological perturbations [9], post-inflationary reheating [10] or the current accelerated expansion of the Universe [11].
In the present communication, by extending the perturbative study of f (R) gravity in [12], we discuss how a general framework for the study of Solar System constraints to NMC gravity can be developed. The approach is based on a suitable linearization of the field equations of NMC gravity around a cosmological background space-time, where the Sun is considered as a perturbation. Solar System observables are computed, then we apply the perturbative approach to the NMC model by Bertolami, Frazão and Páramos [11], which constitutes a natural extension of 1/R n (n > 0) gravity [13] to the non-minimally coupled case. Such a NMC gravity model is able to predict the observed accelerated expansion of the Universe. We show that, differently from the pure 1/R n gravity case, the NMC model cannot be constrained by this perturbative method so that it remains, in this respect, a viable theory of gravity. Further details about the subject of the present communication can be found in the manuscript [14].

NMC gravity model
We consider a gravity model with an action functional of the type [1], where f i (R) (i = 1, 2) are functions of the Ricci scalar curvature R, L m is the Lagrangian density of matter and g is the metric determinant. By varying the action with respect to the metric we get the field equations We describe matter as a perfect fluid with negligible pressure: the Lagrangian density of matter is L m = −ρ and the trace of the energymomentum tensor is T = −ρ. We write ρ = ρ cos +ρ s , where ρ cos is the cosmological mass density and ρ s is the Sun mass density.
We assume that the metric which describes the spacetime around the Sun is a perturbation of a flat Friedmann-Robertson-Walker (FRW) metric with scale factor a(t): where |Ψ(r, t)| ≪ 1 and |Φ(r, t)| ≪ 1. The Ricci curvature of the perturbed spacetime is expressed as the sum where R 0 denotes the scalar curvature of the background FRW spacetime and R 1 is the perturbation due to the Sun. Following Ref. [12], we linearize the field equations assuming that both around and inside the Sun. This assumption means that the curvature R of the perturbed spacetime remains close to the cosmological value R 0 inside the Sun. In GR such a property of the curvature is not satisfied inside the Sun. However, for f (R) theories which are characterized by a small value of a suitable mass parameter (see next section), condition (2) can be satisfied. For instance, the 1/R n (n > 0) gravity model [13] satisfies condition (2), as shown in [12,15]. Eventually, we assume that functions f 1 (R) and f 2 (R) admit a Taylor expansion around R = R 0 and that terms nonlinear in R 1 can be neglected in the expansion. We use the notation introduced by [12] (for i = 1, 2):

Solution of the linearized field equations
The details of the following computations can be found in the paper [14]. First we linearize the trace of the field equations (1). Using condition (2), we neglect O(R 2 1 ) contributions but we keep the cross-term R 0 R 1 . Introducing the potential where m 2 denotes the mass parameter m 2 (r, t) = 1 3 When f 2 (R) = 0 we recover the mass formula of f (R) gravity theory found in [12]. In the following we assume that |mr| ≪ 1 at Solar System scale. Under this assumption the solution for R 1 outside the Sun is given by where M S is the mass of the Sun. Then we linearize the field equations (1) obtaining Using the divergence theorem and the solution (4) for R 1 , from the first equation we obtain the function Ψ outside of the Sun: where R S is the radius of the Sun. If the following condition is satisfied, then the function Ψ is a Newtonian potential: where G(t) is an effective gravitational constant. Since G depends on slowly varying cosmological quantities we have G(t) ≃ constant, so that Ψ(r, t) ≃ Ψ(r). The solution for the function Φ is computed from the second of the linearized field equations, and we obtain Φ(r) = −γΨ(r), where the PPN parameter γ depends on cosmological quantities and it is given by
When f 2 (R) = 0 we find the known result γ = 1/2 which holds for f (R) gravity theories which satisfy the condition |mr| ≪ 1 and condition |R 1 | ≪ R 0 , as it has been shown in [12]. The 1/R n (n > 0) gravity theory [13], where f (R) is proportional to (R + constant/R n ), is one of such theories that, consequently, have to be ruled out by Cassini measurement.

Application to a NMC cosmological model
We consider the NMC gravity model proposed in [11] to account for the observed accelerated expansion of the Universe: where κ = c 4 /16πG N , G N is Newton's gravitational constant, and R n is a constant. This model yields a cosmological solution with a negative deceleration parameter q < 0, and the scale factor a(t) of the background metric follows the temporal evolution a(t) = a 0 (t/t 0 ) 2(1+n)/3 , where t 0 is the current age of the Universe. Using the properties of the cosmological solution found in [11] the mass parameter (3) can be computed obtaining (we refer to [14] for details of the computation): where µ(n) and ν(n) are rational functions of the exponent n. In [14] it is shown that the condition |mr| ≪ 1 imposes the extremely mild constraint n ≫ (1/6)R 2 S R 0 ∼ 10 −25 . Moreover, from the properties of the cosmological solution [11] we have f 2 R0 ρ cos (t)/κ = −2n/(4n + 1), from which it follows that condition (5) is incompatible with the previous constraint n ≫ 10 −25 : We now check the assumption |R 1 | ≪ R 0 . The previous result shows that we can not rely on the validity of Newtonian approximation. Hence we cannot use the effective gravitational constant G defined in (6) for the estimate of the ratio R 1 /R 0 , so that we resort to Newton's gravitational constant G N = c 4 /16πκ. The value of this ratio outside the Sun can be computed from the exterior solution (4) for R 1 , while the result for the interior solution requires a more involved computation, based on a polynomial model of the mass density ρ s , that can be found in [14]: Though |R 1 | ≪ R 0 for n ≫ 1, the interior solution shows that non-linear terms in the Taylor expansion of f 2 (R) cannot be neglected, contradicting our assumption at the end of Section 2: f 2 (R) = f 2 0 1 − n R 1 R 0 + n(n + 1) 2 R 1 R 0 2 − 1 6 n(n + 1)(n + 2) The lack of validity of the perturbative regime leads us to conclude that the model (7) cannot be constrained by this method, so that it remains, in this respect, a viable theory of gravity.