A Nonminimal Coupling Model and Its Short-Range Solar System Impact

A Nonminimal Coupling Model and its Short-Range Solar System Impact

Nuno Castel-Branco∗ Under the supervision of V´ıtor Cardoso and Jorge Páramos Departamento de F´ısica, Instituto Superior Técnico, Universidade de Lisboa, Portugal (Dated: June 20, 2014) The objective of this work is to present the effects of a nonminimally coupled model of gravity on a Solar System short range regime as studied in Ref. [1]. The action functional of the model involves two functions f 1(R) and f 2(R) of the Ricci scalar curvature R. Using a Taylor expansion around R = 0 for both functions, it was found that the metric around a spherical object is a perturbation of the weak-field Schwarzschild metric. The perturbation of the tt component of the metric, a Newtonian plus Yukawa term, is constrained using the available observational results. Besides, this effect vanishes when the characteristic mass scales of each function are identical, the conclusion is that the Starobinsky model for inflation is not experimentally constrained. Moreover, the geodetic precession effect, obtained also from the radial perturbation of the metric, reveals to be of no relevance for the constraints.

Keywords: f(R) theories; Nonminimal Coupling; Yukawa modiﬁcations on inverse-square law; Solar System

I. INTRODUCTION where a(t) is the cosmological scale factor, dΩ2 is the line element for the 2-sphere and K is related to the General Relativity (GR) was developed by Einstein in curvature of the spatial section of spacetime, where the 1916 and since then it has been able to demonstrate it- spatial coordinates are comoving coordinates. self as a solid theory of physics, at a theoretical and an At very large scales the universe can be described as observational level [2]. The theory is mathematically rep- being a perfect ﬂuid, endowed with an energy-momentum resented by the Einstein-Hilbert action, tensor

Z √ 4 p S = [κ (R − 2Λ) + L ] −g d x, (1) Tµν = ρ + uµuν + pgµν , (4) m c2 where R is the Ricci scalar, Lm is the matter Lagrangian where ρ is the energy density, p is the pressure and uµ density, Λ stands for the Cosmological Constant, g is the is the 4-velocity vector. Inserting this tensor and the determinant of the metric and κ = c4/ (16πG), with G as metric (3) into the EFE (2) gives the cosmology structure Newton’s constant of gravity and c as the velocity of light equations: in a vacuum. The variation of the action with respect to the metric yields the Einstein Field Equations (EFE), a2 a˙ 2 + Kc2 = 8πGρ + Λc2 , (5) 1 3 Gµν + Λgµν = Tµν , (2) a¨ 4πG 3p Λc2 2κ = − ρ + + , (6) a 3 c2 3 1 where Gµν ≡ Rµν − gµν R is the Einstein tensor and 2 wherea ˙ = da/dt. These equations yield a full description T is the energy-momentum tensor of matter, clearly µν of the structure and development of the universe if an showing the relation between matter and geometry. equation of state (EOS) p = p(ρ) is known. Initially, the standard Big-Bang model had some prob- A. Standard Cosmology and Inflation lems to be solved, so inflation theory [5] was developed. 2 Defining critical density as ρc(t) = 3H /8πG, where H =a/a ˙ , and neglecting the cosmological constant term, Following the Cosmological Principle, the universe can the Friedmann equation (5) can be written as be said to have a Friedmann-Lemaˆıtre-Robertson-Walker metric [3] |K|c2 |Ω − 1| = , (7) dr2 a2H2 ds2 = −c2dt2 + a2(t) + r2dΩ2 , (3) 1 − Kr2 where Ω is the density parameter defined as Ω = ρ/ρc. To solve the flatness problem, the r.h.s. should decrease with time, which means that a2H2 =a ˙ 2 should increase, ∗ [email protected] soa ¨ > 0, which is the inflation condition. 2

The origin of inflation is explained by means of a scalar hinting that fR ≡ ϕ acts as an additional degree of free- field named inflaton, φ, with a potential V (φ). The in- dom. Considering the following action for a scalar-field flation condition means that, Z √ 4 Sχ = (2κ [ϕR − U(ϕ)] + Lm) −g d x, (12) a¨ > 0 ⇔ p < −ρc2/3 ⇔ φ˙2 < V (φ), (8) where ϕ = f (χ) = df/dχ and the field potential is This allows to adopt the so-called Slow-Roll Approxima- χ tion (SRA), V (φ) φ˙2. Some very recent experimental observations of inflation are mentioned in Ref. [6]. U(ϕ) = χ(ϕ)ϕ − f(χ(ϕ)), (13)

one notices that when χ = R, the action Sχ reduces B. Modern problems to the standard f(R) action S from (9). This form of the action is important because of its similarity with the Modern physics uses the concepts of dark matter and Brans-Dicke (BD) theory of gravity [10], with a vanishing dark energy to advance an explanation for the astrophys- BD parameter ωBD = 0. ical problem of missing matter (seen for example in the flattening of galactic rotation curves) and the cosmological problem of the accelerated expansion of the universe, A. Starobinsky Inflation and Reheating respectively. There are many proposals for dark energy, as the so-called quintessence models [7] and the existence An early model for inflation was posited by Starobin- of scalar fields that account for both dark matter and sky [11], with dark energy [8]. More recent approaches start from the idea of the in- R2 completeness of the fundamental laws of GR, involving, f(R) = 2κ R + 2 , (14) for example, corrections to the Einstein-Hilbert action. 6M Such theories involve a nonlinear correction to the ge- where M ∼ 1013 GeV/c2 is a constant with dimensions ometry part of the action, thus being called f(R) theo- of mass. Inflation is best described by the Starobinsky ries. In the last decade, the study of f(R) theories has version of the cosmological structure equations (5), where been very profitable, as thoroughly discussed in Ref. [9]. one of them is These can be extended to also include a nonminimum coupling (NMC) between the scalar curvature and the matter Lagrangian density. R¨ + 3HR˙ + M 2R = 0. (15) The end of inflation is thus followed by a phase of re- II. f(R) THEORIES heating of the universe. It is in this phase that most of the matter that is observed today in the Universe was This standard modification of GR resorts to the mod- formed. This phenomenon is introduced with a scalar ified action functional field χ with mass mχ, whose action includes a nonminimal coupling with strength ξ of the field with the scalar curvature. As χ is a quantum field, it can be decomposed Z 1 √ S = f(R) + L −g d4x, (9) into Fourier modes χk, whose equation of motion is 2 m k2 where the standard Einstein-Hilbert action is recovered χ¨ + 3Hχ˙ + + m2 + ξR χ = 0, (16) if f(R) = 2κ (R − 2Λ). This action yields the modified k k a2 χ k field equations, where k = |k| is a comoving wavenumber. The case where |ξ| > 1 allows to perform a series of approximations 1 2 [12] so that the above equation becomesχ ¨k + ω χk ' 0. fRRµν − fgµν − ∇µ∇ν fR + fRgµν = Tµν , (10) k 2 Particle production is thus obtained with the resonance of this oscillator - a phenomenon of parametric resonance where f ≡ df/dR, R is the Ricci tensor and is the R µν which can be more detailed by defining z with M(t−t ) = D’Alembertian operator. 0 2z ± π/2, where the opposite signs are related to the sign f(R) theories have an equivalence with scalar-tensor of ξ. The new variable leads to what is called the Mathieu theories, which is suggested by considering the trace of equation the field equations (10),

d2χ k + [A − 2q cos(2z)] χ ' 0, (17) fRR − 2f(R) + 3fR = T, (11) dz2 k k 3 with Ak and q determining the strength of parametric Notice that fR = df/dR and f0 ≡ f(R0). resonance and deﬁned by Applying these expansions into the trace equation yields

2 2 4k 4mχ 8|ξ| Ak = 2 2 + 2 , q = . (18) cos s a M M M(t − t0) fRR − 2f + 3fR = T + T , (24) The oscillatory aspect of the field implies that the res- where T s is the trace of the energy-momentum tensor onance growth changes with the expansion of the Uni- of a spherically symmetric mass source. Considering verse, which also causes the parameters Ak and q to de- that R1(r) is time-independent, so that the operator 2 pend on time. This process of particle production due to becomes ≈ ∇ and that the matter at the local parametric resonance of the field is called preheating. If perturbation is considered with no pressure, such that ξ = 0, which is the standard reheating regime, there still T s = −c2ρ(r) (with c = 1), then the linearised equation is a parametric oscillation with q = 0, due to the expan- for the spatial dependent curvature can be written as sion of the Universe alone. Nonetheless, the preheating scenario is much more efficient than standard reheating. 2 2 ρ ∇ R1 − m R1 = − , (25) 3fRR0 B. Solar System Effect with

The validity of f(R) gravity at the Solar System scale should be assessed. The long-range impact of f(R) is 2 1 fR0 fRR0 m ≡ − R0 − 3 . (26) taken into account by considering the ﬂat FLRW metric 3 fRR0 fRR0 with spherically symmetric perturbations This equation can be solved by resorting to Green functions. ds2 = −[1+2Ψ(r)]dt2 (19) + a(t)2 [1 + 2Φ(r)] dr2 + r2dΩ2 , 1. Long Range Regime where Ψ and Φ are perturbing functions such that |Ψ(r)| 1 and |Φ(r)| 1. Considering ﬁrst the long range regime mr 1, the The background curvature is nonzero, R0(t) 6= 0, thus Green function can be approximated by −1/(4πr). This it can be decomposed into a sum of two components: means that the term linear in R1 from (25) may disappear and the solution for the curvature perturbation becomes

R(r, t) ≡ R0(t) + R1(r), (20) 1 M R = S , (27) where R1(r) is considered as a time-independent per- 1 12πf r turbation of the background curvature. The validity of RR0 the following treatment assumes that R1 R0 and this where MS is the total mass of the source. analysis follows the work of Ref. [13]. Considering also Adopting the arguments for linearization from (21), that all derivatives of f(R) are well defined at R0(t0), using the curvature equation and neglecting all the terms where t0 is the present time, a Taylor expansion around that are not linear in R1, Φ and Ψ, the solutions for these R0 can be used to write the functions f(R0 + R1) and perturbations are taken from the time (tt) and radial (rr) fR(R0 + R1). As the framework is a weak-field regime, components of the field equations. The linearization of the expansion terminates by neglecting terms non-linear the tt component of (10) has the form in the perturbation R1, provided that the following conditions are taken into account: 1 f ∇2Ψ + f R − f ∇2R = ρ. (28) R0 2 R0 1 RR0 1 1 (n) n f0 + fR0R1 f (R0)R , n! 1 Inserting equation (25), where the linear term in R1 1 was neglected, reduces the above to f + f R f (n+1)(R )Rn, (21) R0 RR0 1 n! 0 1 2 1 for all n > 1, where f ∇2Ψ = ρ − f R , (29) R0 3 2 R0 1

2 2 which can be decomposed according to Ψ = Ψ0 + Ψ1. fRR0 ≡ d f/dR , (22) R=R0 2 Then, the ﬁrst equation fR0∇ Ψ0 = 2ρ/3, when inte- (n) n n f (R0) = d f/dR . (23) grated by Gauss’s Theorem, becomes R=R0 4

where integrals that have RS as an integration limit ap- pear. 1 m(r) 0 This analytical setup allows to deﬁne the function f(R) Ψ0(r) = 2 , (30) 6πfR0 r as a Taylor expansion around R = 0 [15], similar to the where the prime represents diﬀerentiation with respect Starobinsky action functional from (14), to r and m(r) is the mass of a sphere with radius r. 2 Assuming that limr→∞ Ψ0 = 0, the solution for Ψ0 is R f(R) = 2κ R + + O(R3). (35) 6m2

1 MS Ψ0 = − . (31) In this case, the linearization is done according to the 6πfR0 r expansion of R, Ψ and Φ in powers of c−1 and all the −3 On the other hand, using the long-range approximation terms of the order O(c ) or less are neglected. Inserting mr 1 and a cosmological constraint, it can be shown (35) into the curvature equation (11) and linearizing it, yields that |Ψ1| |Ψ0|, then Ψ1 may be neglected, thus Ψ ≈ Ψ0. Following the same procedures, the linearization of the 8πG ∇2R − m2R = − m2ρ, (36) rr component of (10) ends up yielding c2 which will end up with a twofold solution for the curva- 1 M Φ = S . (32) ture, whose outer (r > RS) version is 12πfR0 r 2GM Comparing the solution Φ with Ψ, it is easy to conclude R↑(r) = S m2A(m, R )e−mr, (37) that Ψ = −2Φ, thus implying that the PPN parameter rc2 S γ = −Φ/Ψ takes the value γ = 1/2, which contradicts with A(m, R ) a form factor, discussed in a subsequent the observational results that yield γ ∼ 1 — implying S section, deﬁned as that, if the additional degree of freedom is long-ranged (mr 1), f(R) gravity is incompatible with experiment. 4π Z RS A(m, R ) = sinh(mr)rρ(r)dr. (38) S mM 2. Short Range Regime S 0 Neglecting all terms smaller than O(1/c2) and com- To study the short range regime of f(R) gravity in the bining it with the same linearisation of the Ricci tensor, Solar System, it is more adequate to consider a pertur- the linearisation of the tt and rr components of the ﬁeld bation of a Minkowski metric of the form equations (10), yields the solution for Ψ and Φ,

2 2 2 ds = −[1 + 2Ψ (r)]c dt GMS 1 −mr (33) Ψ(r) = − 2 1 + A(m, RS)e , (39) + [1 + 2Φ(r)] dr2 + r2dΩ2, c r 3 where Ψ and Φ are again perturbing functions such that GMS 1 −mr |Ψ(r)| 1 and |Φ(r)| 1, since one assumes that the Φ(r) = 2 1 − A(m, RS)e (1 + mr) . (40) background cosmological dynamics play no short-range c r 3 role. In the GR limit, m → ∞, the exponential term in both In the following, it is assumed that matter behaves as Ψ and Φ vanishes and the weak-field approximation of dust, so that its energy-momentum tensor has the same the Schwarzschild metric is obtained, as expected from a form of (4), but with p = 0. The trace of the energy- weak spherical perturbation on a Minkowski metric. momentum tensor is T = −ρc2. The Lagrangian density 2 of matter is considered as Lm = −ρc (see Ref. [14] for a discussion). The function ρ = ρ(r) is that of a spherically III. NONMINIMAL COUPLING symmetric body with a static radial mass density ρ = ρ(r) and the function ρ(r) and its first derivative are The remarkable range of applications of f(R) grav- assumed to be continuous across the surface of the body, ity, and in particular the Starobinsky inflationary model, dρ prompted physicists to generalize these theories even fur- ρ(RS) = 0 and (RS) = 0, (34) ther. This generalization is made by a NMC between ge- dr ometry and matter, through the product of the matter where RS denotes the radius of the spherical body. These Lagrangian density with another function of the curva- conditions play a relevant role in the following sections, ture, as posited by the action 5

by a second order diﬀerential equation. Indeed, by a cou- Z ple of transformations, one ends up reaching a Mathieu 1 1 2 √ 4 S = f (R) + 1 + f (R) Lm −gd x. (41) equation like Eq. (17) from the pure f(R) case, though 2 with parameter q = 4ξ/z. The results are, therefore, the same as the ones men- The ﬁeld equations, obtained through the variation of the tioned in section II A. action with respect to the metric, are

1 B. Solar System Long Range Regime f 1 + 2f 2 L R − f 1g = R R m µν 2 µν (42) 2 1 2 To assess the NMC effect on the Solar System, the 1 + f Tµν + (µν − gµν ) fR + 2fRLm , perturbed FLRW metric from Eq. (19) is used. As the i i 1 where fR ≡ df (R)/dR. If f (R) = 2κ(R − 2Λ) and curvature can also be decomposed in the form (20), then f 2(R) = 0, the action (41) collapses to the Einstein- the approximations detailed in Eq. (21) are also applied, Hilbert action from (1). In opposition to GR and pure with the exception that now both f 1(R) and f 2(R) are f(R) gravity, these field equations do not respect the subject to it. The curvature solution is obtained through µ covariant conservation of energy, ∇ Tµν 6= 0. the linearisation of the trace of the FE, which yields the A NMC model is also equivalent to scalar-tensor the- equation ories, though with two independent scalar fields, introduced through the action 2 2 1 2 s ∇ U − m U = − 1 + f0 ρ 3 (47) Z 2 2 √ 4 2 s s 2 2 2 s S = ψR − V (φ, ψ) + 1 + f (φ) L −gd x, + f ρ R0 + 2ρ f + 2f ∇ ρ , m 3 R0 R0 R0 (43) with with the potential

1 1 " 1 2 V (φ, ψ) = φψ − f (φ), (44) 2 1 fR0 − fR0Lm 2 m = 1 2 − R0 3 fRR0 + 2fRR0Lm and where the fields are defined as (48) 1 2 cos s 2 # 3 fRR0 − 2fRR0ρ − 6ρ fRR0 1 2 − 1 2 , φ = R, ψ = fR(φ)/2 + fR(φ)Lm. (45) fRR0 + 2fRR0Lm where the potential U(r, t) is defined as A. Preheating and Inflation

1 2 In section II A, the process of particle production after U(r, t) = fRR0(t) + 2fRR0(t)Lm(r, t) R1(r), (49) inflation was explained by means of a preheating mech- 1 2 and fRR0 + 2fRR0Lm 6= 0 is assumed. Naturally, this anism, whose main characteristic was on the parametric expression becomes the one obtained for the pure f(R) resonance of a quantum field χ. The action of this field theories, Eq. (25), when f 2(R) = 0. To work out the included a NMC term between the curvature and the 2 solution of the above equation outside the star, it can be field itself, of the type ξRχ , which amounts to a vari- shown that it is possible to neglect the linear term in U. able mass for the scalar field. This prompts for the more Then, that equation (47) becomes general assumption of the action (41) with the functions

∇2U = η(t)ρs(r) + 2f 2 ∇2ρs, (50) R2 R R0 f 1(R) = 2κ R + , f 2(R) = 2ξ . (46) 6M 2 M 2 1 2 η(t) = − 1 + f 2 + f 2 R + 2 f 2 . (51) Logically, this specific action has to be able to explain in- 3 0 3 R0 0 R0 flation at the standard SRA, which means that the NMC Integrating and manipulating this equation, the curva- model only works as part of a perturbative regime with ture R is given by f 2(R) ∼ 0. 1 The NMC parameter ξ is dimensionless and subject to η(t) MS 4 R (r, t) = , (52) the range 1 < ξ < 10 , where the lower bound comes 1 4π (2f 2 ρcos − f 1 ) r from the SRA and the upper bound is obtained by re- RR0 RR0 sorting to the initial inflationary temperature [12]. As in which obviously reduces to Eq. (27) when f 2(R) = 0. Eq. (16), the decomposition of the scalar field from the The R1(r) solution for r < RS is not relevant for the fol- NMC in Fourier modes yields that each mode is governed lowing analysis, so it is not presented here. Nonetheless, 6 one finds it in Ref. [16], as well as all details of the above Yukawa contribution to the Inverse Square Law (ISL), computation. whose potential is then The same linearisation procedure that was used for the curvature may be used for the tt component of the field m U(r) = −G i 1 + αe−r/λ , (57) equations (42). The computation of Ψ once more relies r on the decomposition of the type Ψ = Ψ +Ψ . The final 0 1 where α is the Yukawa strength parameter and λ is a solution for Ψ outside the spherical body is, characteristic length scale. 1 + f 2 + f 2 R + 3 f 2 M These parameters α and λ are constrained by limits Ψ(r, t) = − 0 R0 0 R0 S . (53) 6π (f 1 − 2f 2 ρcos) r imposed from experimental results. Figure 2 shows how R0 R0 large α can be in the different length scales dictated by The Newtonian limit requires that Ψ(r) should be pro- λ. The overall data in this graphic is a composition from portional to MS/r, leading to the following constraint: different experiments, ranging from torsion balance experiments [20] to laser ranging tests between the Earth, the Moon (LLR) and the LAGEOS satellites (Keplerian 2 s 1 2 cos 2fR0 ρ (r) fR0 − 2fR0ρ (t) , r ≤ RS. (54) tests) [21]. The values of α above the curve, in the yellow The linearization of the rr component of the field equa- shaded area, are excluded as physically possible values tions and a subsequent integration gives the solution with a 95%-confidence level. Another constraint that can be made to modifications of GR is related to the post-Newtonian limit of GR. This 2 2 2 1 + f0 + 4fR0R0 + 12fR0 MS name is very appropriate because the post-Newtonian Φ(r, t) = 1 2 cos . (55) 12π (fR0 − 2fR0ρ ) r limit is simply a way to write a general theory of gravitation in the form of lowest-order deviations from the The development of these computations may be found in Newton law of gravitation. Mathematically, this limit Ref. [16]. can be written as an expansion of the Minkowski met- Having the solutions of both potentials Ψ and Φ, the ric in terms of small gravitational potentials, that are PPN parameter γ is then defined in terms of matter variables like the matter den- 2 2 2 sity ρ(x), where x is the position vector. Throughout 1 1 + f0 + 4fR0R0 + 12fR0 γ = , (56) the years, this was generalized by Will in the standard 2 1 + f 2 + f 2 R + 3 f 2 0 R0 0 R0 form known today as the Parameterized Post-Newtonian showing that it is completely defined by the background (PPN) formalism [22]. metric R0, whose value is obtained from the cosmological Accordingly, the current version of the PPN formal- solution of NMC gravity, developed in Ref. [17]. The case ism is written with ten parameters which are the coef- f 2(R) = 0 obviously yields the value γ = 1/2 from the ficients of the metric potentials. These parameters may computations of the previous sections. take different values according to the theory under study. Concluding, all of what was said above means that for Each of the parameters measures or indicates general a NMC model in a long range regime to be compatible properties of the specific theories of gravity. Interest- with Solar System, (i) either the present analysis is not ingly enough, the GR theory only includes the param- adequate (the conditions |mr| 1 and |R1| R0 are not eters γ = β = 1, all the others are null; therefore, the satisfied) or (ii) the validity condition of the Newtonian metric in a resting frame takes the form limit has to satisfy the Cassini constraint γ ∼ 1 [18]. The next step is to explore condition (ii), computing g = −1 + 2U − 2βU 2, g = (1 + 2γU) δ , (58) the parameter of mass m2 in a specific cosmological so- 00 ij ij lution, i.e. in the case where the functions f i(R) assume where i, j = {1, 2, 3} and U is the gravitational potential power-law forms compatible with the description of the defined as U (r, t) ≡ R ρ (r0, t0) / |r − r0| d3x0. current phase of accelerated expansion of the Universe. The experimental data of the Solar System validates Indeed, applying the conditions listed in (i), it becomes the GR theory, so a different theory needs to have very clear that the procedure used does not exclude or even similar PPN parameters as GR. The best way to start constrain the NMC model explored in Ref. [17]. It is would be to compute the value of γ, determined by the a null-result that nevertheless prompts to inspect more Cassini experiment as γ = 1 + (2.1 ± 2.3) × 10−5 [18]. closely the inverse situation of a relevant mass parameter Another way to check if a modified theory of GR is and a negligible cosmological background. viable is if it also predicts the geodetic precession. This effect appears when computing the conservation of the angular momentum of a gyroscope with an intrinsic an- IV. EXPERIMENTAL CONSTRAINTS ON gular momentum vector Sµ = S0, S orbiting a spherical MODIFICATIONS OF GR body like the Earth. The equation becomes

It is relevant to introduce some ways to constrain pos- dS dxν ∇ S = 0 ⇔ µ = Γλ S , (59) sible modifications of GR. The first modification is a ν µ dτ µν λ dτ 7 where τ is the coordinate of proper time and the angular The metric used is the one from (33). For the pur- momentum vector Sµ is defined so as to be orthogonal to pose of the present work the functions Ψ and Φ will be the velocity vector. Defining a new spin vector S as computed at order O(1/c2). The functions f i(R) are assumed to admit Taylor expansions around R = 0, 1 S = (1 + φ) S − v (v · S) , (60) 2 2 1 R 2 R f (R) = 2κ R + 2 , f (R) = 2ξ 2 , (64) and considering the weak field approximation with the 6m m metric according to the PPN formalism1, It should be noted that the Cosmological Constant is dropped, consistent with the assumption that the metric 00 i0 ij is asymptotically flat. g = −1 − 2φ, g = ζi, g = (1 − 2φ) δij, (61) yields, after many manipulations, A. Solutions for R, Ψ and Φ

dS 1 3 = Ω × S, Ω = ∇ × ζ − v × ∇φ, (62) As in the previous sections, the trace of the FE yields dt 2 2 the curvature equation showing that S precesses with angular velocity Ω. Fol- lowing Ref. [4], if the central spherical body is the rotat- 2 2 8πG 2 2ξ 2 ing Earth at rest, then ∇ R − m R = − m ρ − 6 ∇ ρ , (65) c2 m2

which by applying a Green function method, has a r (r · J⊕) J⊕ r × v Ω = 3G − G + 3GM , (63) twofold solution, whose expression outside the star is r5 r3 ⊕ 2r3 where M and J are the mass and angular momentum ⊕ ⊕ 2GM of the Earth. The last term depends only on the mass ↑ S 2 −mr R (r) = 2 m (1 − 12ξ) A(m, RS)e , (66) of the earth and not on its spin and it is called geodetic c r precession. with MS the mass of the spherical body and A(m, RS) a The geodetic precession effect was observationally form factor defined as checked by the Gravity Probe B (GPB) experiment. This experiment consisted in a satellite orbiting the Earth 4π Z RS with extremely precise gyroscopes. The spin axis of the A(m, RS) = sinh(mr)rρ(r)dr. (67) mMS gyroscopes was set to be aligned with a guide star during 0 a certain period of time. Then, the change in the pre- The expression (66) vanishes as r → ∞ and it is consid- cession of the spin axis alignment reveals the predicted ered to be valid only at Solar System scales, since space- geodetic effect. The GPB confirmed the geodetic effect time should assume a De Sitter metric with curvature with 0.28% of accuracy [23]. R0 6= 0 at cosmological scales. It must also be noted ↑ that in the limit m → 0 then R (r) → 0 for any r > RS. As for the solutions of Ψ and Φ, they are obtained V. SHORT RANGE SOLAR SYSTEM IMPACT through the linearization of the tt and rr components of the FE (42), neglecting all terms smaller than O(1/c2). The contents of the following chapters include original The computation of Ψ will once more pass through Ψ0 + work and are completely based in Ref. [1]. Ψ1, and several integrations end up with In this section, the procedure applied in the short range regime of the Solar System impact of pure f(R) theories will be followed, though considering a NMC model. The GMS 1 −mr Ψ(r) = − 1 + − 4ξ A(m, RS)e , (68) action functional used is again the one from Eq. (41). c2r 3 It is once more assumed that matter behaves as dust, ie Eq. (4) with p = 0. A spherically symmetric body with mass density ρ = ρ(r) that respects conditions (34) is GM 1 Φ(r) = S 1 − − 4ξ A(m, R )e−mr(1 + mr) . also considered. c2r 3 S (69) In the GR limit, ξ = 0 and m → ∞, the exponential term in both Ψ and Φ vanishes and the weak-field ap- 1 In these computations for the geodetic precession, all the com- proximation of the Schwarzschild metric is recovered, as putations are made considering c = 1. expected. 8

B. Discussion of Yukawa potential which yields the PPN parameter 1 1 + 6ξ From the tt component of the metric, it is possible to γ = . (75) 2 1 − 3ξ identify a Newtonian potential plus a Yukawa perturbation: In the absence of a NMC, ξ = 0, this yields γ = 1/2, a GM strong departure from GR that is disallowed by current U(r) = − S 1 + αA(m, R )e−r/λ , (70) r S experimental bounds, as seen before. Nonetheless, this result appears to show that a NMC allows f(R) theories defining the characteristic length λ = 1/m and the to remain compatible with observations, as long as ξ = strength of the Yukawa addition 1/12 — which is just a restatement of the previously 1 obtained result. Again, the path towards obtaining the α = − 4ξ, (71) 3 γ parameter depicted above is presented for illustration only. so that, if ξ = 0, the Yukawa strength for pure f(R) theories is obtained, α = 1/3; it should also be noticed that a positive NMC yields α ≤ 1/3. Strikingly, a NMC C. Experimental constraints to NMC gravity with ξ = 1/12 cancels the Yukawa contribution. parameters The dimensionless form factor, defined in (67), was found by integrating the field equations of NMC grav- As it was shown before, the Yukawa potential (70) is ity, but is not specific of it nor of f(R) theories. This not new in physics as an alternative way to account for form factor can then be evaluated in several ways, ac- deviations from Newtonian gravity or other forces of na- cording to the function of mass density ρ(r). Taking the ture. Fig. 2 shows the exclusion plot for the phase space limit of a point source, r → 0 allows to expand around (λ, α), which is used to constrain the phase space of the mr 1, so that sinh (mr) ≈ mr[1 + (mr)2/6] and then model (41) under scrutiny. It is seen that a NMC with A(m, R ) ∼ 1. This can be verified explicitly by making S ξ = 1/12 cancels this contribution, as shown by the val- all computations with a test mass density (such as a uni- ues of |α| → 0 overlaid on the exclusion plot. Also, it form profile) and, in the end, taking the limit R → 0. S should be noticed that large values of ξ lead to a large, Indeed, if ρ = 3M /(4πR3 ), then A(m, R ) admits the 0 S S S negative strength α ∼ −4ξ. limiting cases Further insight is obtained by casting the NMC presented in Eq. (64) as f 2(R) = R/(6M 2), so that it is (mR )2 A(m, R ) ≈ 1 + S ∼ 1 , mR 1, (72) characterized by a distinct mass scale M, instead of the S 10 S relative strength parameter ξ: by making the transfor- 3 emRS mation ξ = (m/M)2/12, the suggestive form A(m, RS) ≈ 2 , mRS 1. 2 (mRS) 1 m 2 α = 1 − (76) If the central body is the Sun (with radius R ), a 3 M more accurate density NASA polynomial profile [24] can instead be considered, obtaining the limiting cases is thus obtained, which, inverting, allows to plot the exclusion plot for the phase space (m, M) in Fig. 3.

−2 2 AHm, R L A(m, R ) ≈ 1 + 6 × 10 (mR ) ∼ 1 , mR 1,

mR e 1035 A(m, R ) ≈ 7.3 3 , mR 1. (73) (mR ) 1028 Both forms for A(m, R ) are plotted in Fig. 1, showing 21 that it grows with mR . 10 It should also be noted that the PPN formalism is in- 14 compatible with the presence of a Yukawa term in the 10 gravitational potential, since the latter cannot be ex- 107 panded in powers of 1/r. Nevertheless, for consistency, mR what happens if the condition mr 1 is valid through- 20 40 60 80 100 out the Solar System region can be considered: in this case, the metric (33) with the above limits for the form Figure 1. Form factor A(m, R ) for a constant density (full) factor of ∼ 1 is well approximated by and fourth-order density proﬁle for the Sun. 2GM 4 ds2 = − 1 − S − 4ξ c2dt2 These two ﬁgures 2 and 3 show that, if m falls within c2r 3 −22 (74) the range 10 eV < m < 1 meV (corresponding to 2GM 2 lengthscales λ ranging from the millimeter to Solar Sys- + 1 + S + 4ξ dr2 + r2dΩ2, c2r 3 tem scales), then the strong constraints available on the 9

a The ﬁnal results of the GPB experiment report an ac- 108 GPB r=const curacy of 0.28% in the measurement of geodetic preces- 106 Excluded » » sion [23], which corresponds to the following constraint 104 x=2500 on NMC gravity parameters: 102 H L x=25 Torsion GPB point source GR 100 Balance ΩG − ΩG x=0 GR < 0.0028, (78) -2 Ω 10 Keplerian G -4 -3 Tests H L 10 x= 1-10 12 Planetary where only the modulus of angular velocity is considered, -6 Tests 10 and ΩGR denotes the value of geodetic precession in GR. x= 1-10-6 12 LLR G 10-8 H Lê Substituting the expression of NMC geodetic precession -10 10 x= 1-10-9 12 in this constraint and knowing that x ≡ αA(m, RS)(1 + −mr H Lê l m mr)e has to respect x 1, then the condition that -6 -4 -2 0 2 4 6 8 10 12 14 10 10 10 10 10 10 10 10 10 10 10 remains to be analysed is H Lê Figure 2. Yukawa exclusion plot for α and λ. Adapted fromH L Refs. [25]. 0.0168 emr |α| < . (79) 1 + mr A(m, RS) M eV 10-1 From the assumption (34) of continuity of mass density 10-3 Torsion and its derivative across the surface of the Earth, it is H L Excluded Balance 10-5 possible to consider the form factor A(m, RS) to converge 10-7 to the value corresponding to the uniform density model, Keplerian -9 Eq. (72), hence the inequality (79) reads 10 Tests -11 10 4 x10 |α| < 0.0168 , mR⊕ 1 , (80) - 10 13 2 mR⊕ m(r−R ) 10-15 |α| < 0.0112 e ⊕ , mR 1 , LLR r ⊕ 10-17 10-19 Planetary where R⊕ ≈ 6371 km is the radius of the Earth. Knowing Tests 10-21 Excluded that the GPB satellite orbits the Earth at a height of r ∼ 650 km, this condition can be seen in the (λ, α) m eV 10-21 10-19 10-17 10-15 10-13 10-11 10-9 10-7 10-5 10-3 exclusion plot, as shown in Fig. 2. It is found that the condition is well-within the already excluded phase space, Figure 3. Exclusion plot for the characteristic mass scales MH L so that the current bounds on geodetic precession do not and m. add any new constraint on the model parameters.

Yukawa strength, |α| 1, require that ξ ∼ 1/12 — or, VI. DISCUSSION AND OUTLOOK equivalently, that both characteristic mass scales are very similar, m ∼ M. In this work it has been computed the effect of a NMC model, specified by (64), in a perturbed weak-field 1. Geodetic precession Schwarzschild metric, as depicted in Eq. (68) and (69). In the weak-field limit, this translates into a Yukawa perturbation to the usual Newtonian potential, with charac- In this section it is assumed that the Earth can be ap- teristic range and coupling strength proximated as a spherically symmetric body. Following the computations from section IV on geodetic precession 1 1 1 m 2 and since (dS/dt) = Ω × S, the angular velocity vec- λ = , α = − 4ξ = 1 − . (81) sec G m 3 3 M tor ΩG of geodetic precession is given by

3/2 This result is quite natural and can be interpreted 3 (GM ) 1/2 Ω = S 1 + αA(m, R )(1 + mr)e−mr straightforwardly: a minimally coupled f(R) theory in- G 2 c2r5/2 S troduces a new massive degree of freedom, leading to αA(m, R ) a Yukawa contribution with characteristic lengthscale × 1 − S (1 + mr)e−mr n, 3 λ = 1/m and coupling strength α = 1/3. (77) The introduction of a NMC has no dynamical effect in the vacuum, as there is no matter to couple the scalar where n is the unit vector perpendicular to the plane of curvature to. As a result, it is not expected any modifi- the orbit. If ξ = 0, the above expression reduces to the cation in the range of this Yukawa addition. Conversely, case for f(R) models, as expected [15]. a NMC has an impact on the description of the interior 10 of the central body leading to a correction to the lat- In particular, the Starobinsky inflationary model, which ter’s coupling strength, which has a negative sign since requires the much heavier mass scale m ≈ 3 × 1013 GeV −6 −29 Lm = −ρ. ∼ 10 MP , manifests itself at a lengthscale λ ∼ 10 m. Using the available experimental constraints, it was This implies that the generalized preheating scenario found that, for 10−22 eV < m < 1 meV, the NMC pa- posited previously, which requires 1 < ξ < 104, is thus rameter must be ξ ∼ 1/12 or, equivalently, both mass completely allowed by experiment and unconstrained by scales m and M of the non-trivial functions f 1(R) and this work. f 2(R) must be extremely close. Finally, by computing the perturbation induced on If this is the case, the latter relation is not interpreted geodetic precession, it was found that no significant new as an undesirable fine-tuning, but instead is suggestive constraint arises, as this is already included in the exist- of a common origin for both non-trivial functions f 1(R) ing Yukawa exclusion plot. and f 2(R). Higher-order computations (on 1/c2) might reveal Conversely, for values of m (or λ) away from the range other effects, mainly when one begins to take into account mentioned above the Yukawa coupling strength, α can be the competing effect of the cosmological background dy- much larger than unity, so that ξ can assume any value namics (modelled for example by a Cosmological Con- and the mass scales m and M can differ considerably. stant).

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