6.2 Possibility for new physics? Modified gravity theories

Many authors investigated the possibility that the origin of the anomalous signal is “new physics” [24Jump To The Next Citation Point27Jump To The Next Citation Point]. This is an interesting conjecture, even though the probability is that some standard physics or some as-yet-unknown systematic will be found to explain this acceleration. Being more specific, one may ask the question, “Is it dark matter or a modification of gravity?” Unfortunately, as we discuss below, it is not easy for either of these solutions to provide a satisfactory answer.

6.2.1 Dark matter

Various distributions of dark matter in the solar system have been proposed to explain the anomaly, e.g., dark matter distributed in the form of a disk in the outer solar system with a density of ∼ 4 × 10−16 kg∕m3, yielding the wanted effect. However, it would have to be a special kind of dark matter that was not seen in other nongravitational processes. Dark matter in the form of a spherical halo of a degenerate gas of heavy neutrinos around the Sun [244] and a hypothetical class of dark matter that would restore the parity symmetry, called the mirror matter [130], have also been discussed. However, it would have to be a special smooth distribution of dark matter that is not gravitationally modulated as normal matter so obviously is.

It was suggested that the observed deceleration in the Pioneer probes can be explained by the gravitational pull of a distribution of undetected dark matter in the solar system [94]. Explanations of the Pioneer anomaly involving dark matter depend on the small scale structure of Navarro–Frenk–White (NFW) haloes, which are not known. N-body simulations to investigate solar system size subhalos would require on the order of 1012 particles [246], while the largest current simulations involve around 108 particles [104]. As a consequence of this lack of knowledge about the small scale structure of dark matter, the existence of a dark matter halo around the Sun is still an open question.

It has been proposed that dark matter could become trapped in the Sun’s gravitational potential after experiencing multiple scatterings [300], perhaps combined with perturbations due to planets [86]. Moreover, the birth of the solar system itself may be a consequence of the existence of a local halo. The existence of dark matter streams crossing the solar system, perhaps forming ring-shaped caustics analogous to the dark matter ring postulated in [94], has also been considered by Sikivie [346]. Considering an NFW dark matter distribution [247], de Diego et al. [94] show that there should be several hundreds of earth masses of dark matter available in the solar system.

Gor’kavyi et al. [139] have shown that the solar system dust distributes in two dust systems and four resonant belts associated with the orbits of the giant planets. The density profile of these belts approximately follows an inverse heliocentric distance dependence law [ρ ∝ (R − k)−1, where k is a constant]. As in the case of dark matter, dust is usually modeled as a collisionless fluid as pressure, stresses, and internal friction are considered negligible. Although dust is subjected to radiation pressure, this effect is very small in the outer solar system. Gravitational pull by dark matter has also recently been considered also by Nieto [254], who also mentioned the possibility of searching for the Pioneer anomaly using the New Horizons spacecraft when the probe crosses the orbit of Saturn.

6.2.2 Modified Newtonian Dynamics (MOND)

The anomalous behavior of galaxy rotational curves led to an extensive search for dark matter particles. Some authors considered the possibility that a modification of gravity is needed to address this challenge. Consequently, there were many attempts made at constructing a theory that modifies Newton’s laws of gravity in the regime of weak gravitational fields. Presently, these efforts aim at constructing a consistent and stable theory that would also be able to account for a range of puzzling phenomena – such as flat galaxy rotational curves, gravitational lensing observations, and recent cosmological data – without postulating the existence of nonbaryonic dark matter or dark energy of yet unknown origin. Some of these novel theories were used to provide a cause of the Pioneer anomaly.

One approach to modify gravity, called Modified Newtonian Dynamics (MOND), is particularly well studied in the literature. MOND is a phenomenological modification that was proposed by Milgrom [42212211213323] to explain the “flat” rotation curves of galaxies by inducing a long-range modification of gravity. In this approach, the Newtonian force law for a test particle with mass m and acceleration a is modified as follows:

{1 if |x | ≫ 1, ma = F → μ (|a|∕a )ma = F, with μ (x) ≃ x if |x | ≪ 1, (6.3 ) 0
where μ(x) is an unspecified function (a frequent, particularly simple choice is μ (x) = x∕(x + 1); other forms of μ are also used) and a0 is some constant acceleration.

It follows from Equation (6.3View Equation) that a test particle separated by r = nr from a large mass M, instead of the standard Newtonian expression a = − GM n∕r2 (which still holds when |a | ≫ a0), is subject to an acceleration that is given phenomenologically by the rule

{ 2 2 √ -GM--∕r ∝ 1∕r if a ≫ a0 (or large forces), |a| → μ(|a|∕a0)a ≃ a0GM ∕r ∝ 1∕r if a ≪ a0 (or small forces). (6.4 )
Such a modification of Newtonian law produces a very distinct modification of galactic rotational curves. The velocities of circular orbits are modified by Equation (6.4View Equation) as
2 { GM ∕r2 { GM ∕r if a ≫ a0 (or small distances), a = v--≃ √a--GM---∕r ⇒ v2 ≃ √a--GM--- if a ≪ a (or large distances). (6.5 ) centrifugal r 0 0 0
With a value of a0 ≃ 1.2 × 10− 10 m ∕s2. MOND reproduces many galactic rotation curves.

Clearly, the original MOND formulation is purely phenomenological, which drew some criticism toward the approach. However, recently a relativistic theory of gravitation that reduces to MOND in the weak-field approximation was proposed by Bekenstein in the form of the tensor-vector-scalar (TeVeS) gravity theory [41]. As the exact form of μ(x) remains unspecified in both MOND and TeVeS, it is conceivable that an appropriately chosen μ(x) might reproduce the Pioneer anomaly even as the theory’s main result, its ability to account for galaxy rotation curves, is not affected.

As far as the Pioneer anomaly is concerned, considering the strong Newtonian regime (i.e., 2 a0 ≪ GM ∕r) and choosing −1 μ (x ) = 1 + ξx, one obtains a modification of Newtonian acceleration in the form a = − GM ∕r2 − ξa0, which reproduces the qualitative behavior implied by the observed anomalous acceleration of the Pioneers. However, Sanders [322] concludes that if the effects of a MONDian modification of gravity are not observed in the motion of the outer planets in the solar system (see Section 6.7.1 for discussion), the acceleration cannot be due to MOND. On the other hand, Bruneton and Esposito-Farèse [65] demonstrate that while it may require model choices that are not justified by underlying symmetry principles, it is possible to simultaneously account for the Pioneer anomalous acceleration and for the tests of general relativity in the solar system within a consistent field theory.

Laboratory experiments have recently reached new levels of precision in testing the proportionality of force and acceleration in Newton’s second law, F = ma, in the limit of small forces and accelerations [2140Jump To The Next Citation Point]. The tests were motivated to explore the acceleration scales implied by several astrophysical puzzles, such as the observed flatness of galactic rotation curves (with MOND-implied acceleration of a = 1.2 × 10−10 m ∕s2 0), the Pioneer anomaly (with a ∼ 9 × 10 −10 m ∕s2 P) and the natural scale set by the Hubble acceleration (−10 2 aH = cH ≃ 7 × 10 m ∕s). Gundlach et al. [140] reported no violation of Newton’s second law at accelerations as small as −14 2 5 × 10 m ∕s. The obtained result does not invalidate MOND directly as the formalism requires that the measurement must be carried out in the absence of any other large accelerations (i.e., those due to the Earth and our solar system). However, the test constrains theoretical formalisms that seek to derive MOND from fundamental principles by requiring that formalism to reproduce F = ma under laboratory conditions similar to those used in the experiment.

Finally, there were suggestions that rather then modifying laws of gravity in order to explain the Pioneer effect, perhaps we needs to modify laws of inertia instead [214]. To that extent, modified-inertia as a reaction to Unruh radiation has been considered in [201].

6.2.3 Large distance modifications of Newton’s potential

Motivated by the puzzle of the anomalous galactic rotation curves, many phenomenological models of modified Newtonian potential (leading to the changes in the gravitational inverse-square law) were considered, a Yukawa-like modification being one of the most popular scenario. Following Sanders [321Jump To The Next Citation Point], consider the ansatz:

( − r∕λ) U (r) = UNewton(r) 1 + αe , (6.6 )
which, as was shown in [321], is able to successfully explain many galactic rotation curves. The same expression may also be used to study the physics of the Pioneer anomaly [229]. Indeed, Equation (6.6View Equation) implies that a body moving in the gravitational field of the Sun is subject to the acceleration law:
M ⊙ α M ⊙ α M ⊙ r a (r ) = − G0--2-+ -----G0 ---2 − ------G0---2-- + ..., (6.7 ) r 1 + α 2λ 1 + α 3λ λ
where G0 = (1 + α )G is the observed gravitational constant in the limit r → 0. Identifying the second term in Equation (6.7View Equation) with the Pioneer acceleration one can solve, for instance, for the parameter α = α(a ,λ) P, and obtain:
2 a = --α---G M-⊙- ⇒ α = ----2λ--aP-----. (6.8 ) P 1 + α 02λ2 G0M ⊙ − 2λ2aP
The denominator in Equation (6.8View Equation) implies that ∘ ------------ λ > GM ⊙∕(2aP ), or 14 λ ≥ 2.8 × 10 m.

A combination of log λ > 16 with log |α | ≈ 0 is compatible with the existing solar system data and the Yukawa modification in the form of Equation (6.6View Equation) may provide a viable model for the Pioneer anomaly [176Jump To The Next Citation Point]. Furthermore, after rearranging the terms in Equation (6.7View Equation) as

M ⊙ 2 r a(r) = − G0--2-+ aP − --aP--+ ..., (6.9 ) r 3 λ
one can note that the third term in Equation (6.9View Equation) is smaller than aP by a factor of 2 r 3 λ ≤ 0.06 and may account for the small decrease in the observed acceleration. The values for the parameters α and λ obtained for the Pioneer anomaly are also compatible with the analysis of the galactic rotation curves. Indeed, for the case of log λ > 16 the Pioneer anomalous acceleration implies α + 1 ≤ 10−5 while the galactic curves data yields a weaker limit of − 1 α + 1 ≤ 10.

A modification of the gravitational field equations for a metric theory of gravity, by introducing a momentum-dependent linear relation between the Einstein tensor and the energy-momentum tensor, has been developed by Jaekel and Reynaud [156157158159160] and was shown to be able to account for a P. The authors identify two sectors, characterized by the two potentials

g00 ≃ 1 + 2ΦN , g00grr ≃ 1 + 2ΦP , (6.10 )
where g00 and grr are components of the metric written in Eddington isotropic coordinates. The Pioneer anomaly could be accounted for by an anomaly in the Newtonian potential, δΦ ≃ (r − r )∕ℓ N 1 P with a characteristic length scale given by −1 2 ℓP ≡ aP ∕c. However, this model is likely excluded by measurements such as Viking Mars radio ranging. On the other hand, an anomaly due to the potential in the second sector in the form
(r − r )2 + μ (r − r ) δΦP ≃ − ------1------P------1-, (6.11 ) 3κℓP
with κ given by 1 ∕3κℓP ≃ 4 × 108 AU − 2 and μP being a further characteristic length representing the radial derivative of the metric anomaly at the Earth’s orbit, could account for the Pioneer anomaly, and the conflict with Viking ranging data can be resolved [159].

Other related proposals include Yukawa-like or higher-order corrections to the Newtonian potential [27Jump To The Next Citation Point] and Newtonian gravity as a long wavelength excitation of a scalar condensate inducing electroweak symmetry breaking [81].

6.2.4 Scalar-tensor extensions of general relativity

There are many proposals that attempt to explain the Pioneer anomaly by invoking scalar fields. In scalar-tensor theories of gravity, the gravitational coupling strength exhibits a dependence on a scalar field φ. A general action for these theories can be written as

3 ∫ [ ] ∑ S = -c--- d4x √ − g 1-f(φ)R − 1g (φ )∂μφ ∂μφ + V (φ) + qi(φ )ℒi, (6.12 ) 4πG 4 2 i
where f(φ), g(φ), and V (φ) are generic functions, qi(φ ) are coupling functions, and ℒi is the Lagrangian density of matter fields, as prescribed by the Standard Model of particles and fields.

Effective scalar fields are prevalent in supersymmetric field theories and string/M-theory. For example, string theory predicts that the vacuum expectation value of a scalar field, the dilaton, determines the relationship between the gauge and gravitational couplings. A general, low energy effective action for the massless modes of the dilaton can be cast as a scalar-tensor theory (as in Equation (6.12View Equation)) with a vanishing potential, where f(φ), g(φ), and qi(φ) are the dilatonic couplings to gravity, the scalar kinetic term, and the gauge and matter fields, respectively, which encode the effects of loop effects and potentially nonperturbative corrections.

Brans–Dicke theory [60] is the best known alternative scalar theory of gravity. It corresponds to the choice

ω f (φ) = φ, g(φ) = -, V (φ ) = 0. (6.13 ) φ
In Brans–Dicke theory, the kinetic energy term of the field φ is noncanonical and the latter has a dimension of energy squared. In this theory, the constant ω marks observational deviations from general relativity, which is recovered in the limit ω → ∞. In the context of Brans–Dicke theory, one can operationally introduce Mach’s principle, which states that the inertia of bodies is due to their interaction with the matter distribution in the Universe. Indeed, in this theory the gravitational coupling is proportional to φ −1, which depends on the energy-momentum tensor of matter through the field equation □2φ = 8π∕(3 + 2ω )T where T is the trace of the matter stress-energy tensor defined as the variation of ℒi with respect to the metric tensor.

The ω parameter can be directly related to the Eddington–Robertson (PPN) parameter γ by the relation [418Jump To The Next Citation Point]: γ = (ω + 1)∕(ω + 2 ). The stringent observational bound resulting from the 2003 experiment with the Cassini spacecraft require that |ω | ≳ 40000 [57418]. On the other hand, ω = − 3∕2 may be favored by cosmological observations and also offer a resolution of the Pioneer anomaly [85]. A possible resolution can be obtained by incorporating a Gauss–Bonnet term in the form of ℒGB = R μνρσR μνρσ − 4R μνR μν + R2 into the Brans–Dicke version of the Lagrangian Equation (6.12View Equation) with the choice of Equation (6.13View Equation), which may allow the Eddington parameter γ to be arbitrarily close to 1, while choosing an arbitrary value for ω [10]. Another scalar-tensor model, proposed by Novati et al. [68], was also motivated in part by the observed anomalous acceleration of the two Pioneer spacecraft.

Other scalar-tensor approaches using different forms of the Lagrangian Equation (6.12View Equation) were used to investigate the anomaly. Capozziello et al.  [70] developed a proposal based on flavor oscillations of neutrinos in Brans–Dicke theory; Wood [426] proposed a theory of conformal gravity with dynamical mass generation, including the Higgs scalar. Cadoni [67] studied the coupling of gravity with a scalar field with an exponential potential, while Bertolami and Páramos [53] also applied a scalar field in the context of the braneworld scenarios. In particular, Bertolami and Parámos [53] have shown that a generic scalar field cannot explain aP; on the other hand, a non-uniformly-coupled scalar could produce the wanted effect. In addition, although braneworld models with large extra dimensions may offer a richer phenomenology than standard scalar-tensor theories, it seems difficult to find a convincing explanation for the Pioneer anomaly [54].

6.2.5 Scalar-tensor-vector modified gravity theory (MOG)

Moffat [233] attempted to explain the anomaly in the framework of Scalar-Tensor-Vector Gravity (STVG) theory. The theory originates from investigations of a nonsymmetric generalization of the metric tensor, which gives rise to a skew-symmetric field. Endowing this field with a mass led to the Metric-Skew-Tensor Gravity (MSTG) theory, while the further step of replacing the skew-symmetric field with the curl of a vector field yields STVG. The theory successfully accounts for observed galactic rotation curves, galaxy cluster mass profiles, gravitational lensing in the Bullet Cluster (1E0657-558), and cosmological observations.

The STVG Lagrangian takes the form,

√ ---{ 1 1 [ 1 1 ] ℒ = − g − ------(R + 2Λ ) − ---ω -B μνBμν − --μ2ϕμϕ μ + V ϕ(ϕ) [ 1(6πG 4π 4 2) ] } -1 1-μν ∇-μG-∇-νG- ∇-μμ∇-νμ- VG-(G)- V-μ(μ) − G 2g G2 + μ2 − ∇ μω ∇ νω + G2 + μ2 + Vω(ω ) , (6.14 )
where g is the determinant of the metric tensor, R is the Ricci-scalar, Λ is the cosmological constant, ϕ ν is a massive vector field with (running) mass μ, B μν = ∂μϕν − ∂νϕ μ, G is the (running) gravitational constant, ω is the (running) vector field coupling constant, and Vϕ, VG, V μ and Vω are the potentials associated with the vector field and the three running scalar fields.

The spherically symmetric, static vacuum solution of Equation (6.14View Equation) yields, in the weak field limit, an effective gravitational potential that is a combination of a Newtonian and a Yukawa-like term, and can be written as

( ) − μr Geff = GN 1 + α − α(1 + μr )e . (6.15 )
In earlier papers, the values of α and μ were treated as fitted parameters. This allowed Brownstein and Moffat [62232] to reproduce an anomalous acceleration of the correct magnitude and also account for the anomaly’s apparent “onset” at a distance of ∼ 10 AU from the Sun. More recently, the values of α and μ were derived successfully as functions of the gravitational source mass [234]. This later approach results in the prediction of Newtonian behavior within the solar system, indeed within all self-gravitating systems with a mass below several times 6 10 M ⊙.
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