6.3 Cosmologically-motivated mechanisms

There have been many attempts to explain the anomaly in terms of the expansion of the Universe, motivated by the numerical coincidence aP ≃ cH0, where c the speed of light and H0 is the Hubble constant at the present time (see Section 6.6.2 for details). These attempts were also stimulated by the fact that the initial announcement of the anomaly [24Jump To The Next Citation Point] came almost immediately after reports on the luminosity distances of type Ia supernovae [280310] that were followed by measurements of the angular structure of the cosmic microwave background (CMB) [93], measurements of the cosmological mass densities of large-scale structures [279] that have placed stringent constraints on the cosmological constant Λ and led to a revolutionary conclusion: The expansion of the universe is accelerating. These intriguing numerical and temporal coincidences led to heated discussion (see, e.g., [175273] for contrarian views) of the possible cosmological origin of the Pioneer anomaly.

Below we discuss cosmologically-motivated mechanisms used to explain the Pioneer anomaly.

6.3.1 Cosmological constant as the origin of the Pioneer anomaly

An inverse time dependence for the gravitational constant G produces effects similar to that of an expanding universe. So does a length or momentum scale-dependent cosmological term in the gravitational action functional [231317Jump To The Next Citation Point]. It was claimed that the anomalous acceleration could be explained in the frame of a quasi-metric theory of relativity [276Jump To The Next Citation Point]. The possible influence of the cosmological constant on the motion of inertial systems leading to an additional acceleration has been discussed [317]. Gravitational coupling resulting in an increase of the constant G with scale is analyzed by Bertolami and García-Bellido [52]. A 5-dimensional cosmological model with a variable extra dimensional scale factor in a static external space [4344] was also proposed. It was suggested that the coupling of a cosmological “constant” to matter  [197] may provide a connection with the Pioneer anomaly.

Kagramanova et al. [163] (see also [168Jump To The Next Citation Point332]) have studied the effect of the cosmological constant on the outcome of the various gravitational experiments in the solar system by taking the metric of the Schwarzschild–de Sitter spacetime:

2 2 −1 2 2 2 2 2 ds = α (r)dt − α (r) dr − r (d 𝜃 + sin 𝜃d ϕ ), (6.16 )
α (r) = 1 − 2M--− 1Λr2 (6.17 ) r 3
with Λ being the cosmological constant and M the mass of the source. (Note that Λ < 0 would result in attraction, while Λ > 0 will lead to repulsion.)

The authors of [27Jump To The Next Citation Point] have shown that if the cosmological expansion would be at the origin of the Pioneer anomaly, such a mechanism would produce an opposite sign for the effect. Taken at face value, the anomaly would imply a negative cosmological constant of Λ ∼ − 3 × 10−37 m − 2, which contradicts both the solar system data and the data on cosmological expansion. Indeed, the highest limit on Λ allowable by the solar system tests is set by the data on the perihelion advance, which limit the value of the cosmological constant to −42 −2 Λ ≤ 3 × 10 m [332]. However, the data on the cosmological accelerated expansion yields the value of Λ ≈ 10− 52 m −2, leading Kerr et al. [168] to conclude that the cosmological effects are too small to be measured in the solar system dynamical experiments.

Hackmann and Lämmerzahl [141142] developed the analytic solution of the geodesic equation in Schwarzschild–(anti-)de Sitter spacetimes and show that the influence of the cosmological constant on the orbits of test masses is negligible. They concluded that the cosmological constant cannot be held responsible for the Pioneer anomaly.

Thus, there is now a consensus that the Pioneer anomaly cannot be of a cosmological origin and, specifically, Λ cannot be responsible for the observed anomalous acceleration of the Pioneer 10 and 11 spacecraft. However, the discussion is still ongoing.

6.3.2 The effect of cosmological expansion on local systems

The effect of cosmological expansion on local systems had been studied by a number of authors [2182Jump To The Next Citation Point118137202], (for reviews, see [73Jump To The Next Citation Point124Jump To The Next Citation Point177Jump To The Next Citation Point422Jump To The Next Citation Point]). To study the behavior of small isolated mass in expanding universe, one starts with the weak field ansatz [73Jump To The Next Citation Point177Jump To The Next Citation Point]:

g μν = bμν + hμν, hμν ≪ bμν, (6.18 )
and derives the linearized Einstein equations for hμν:
bρσD ρD σ¯hμν + bρσRκμρν¯hκσ = 16G πTμν. (6.19 )
The relevant solution with modified Newtonian potential is given below
√ -- ( ) 2GM---cos(--6|R ˙|r) 2GM--- 2 2 h00 = R r = Rr 1 − 3H (Rr ) + .... (6.20 )
The first part in Equation (6.20View Equation) is the standard Newtonian potential with the measured distance R (t)r in the denominator. Lëmmerzahl et al. [177Jump To The Next Citation Point] observed that the additional acceleration towards the gravitating body is of the second order in the Hubble constant H. As such, this potential practically does not participate in the cosmic expansion; thus, there is no support for the cosmological origin of aP.

Oliveira [273] conjectured that the solar system has escaped the gravity of the Galaxy, as evidenced by its orbital speed and radial distance and by the visible mass within the solar system radius. Spacecraft unbound to the solar system would also be unbound to the galaxy and subject to the Hubble law. However, this hypothesis produces practically unnoticeable effects.

6.3.3 The cosmological effects on planetary orbits

The cosmological effects on the planetary orbits has been addressed in many papers recently (for instance, [124]). To study the effect of cosmological modification of planetary orbit, one considers the action

∫ ∫ 2 ( U (x ) 2 ˙x2) 12 ( 2 m 2 2 2 2 ) S = − mc 1 − 2-c2--− R (t)c2- dt ≈ − mc + mU (r) + 2-R (t)(r˙ + r ˙φ ) dt.(6.21 )
In the case of weak time dependence of R(t) the action above has two adiabatic invariants:
∘ ----- GmM m Iφ = L, Ir = − L + --R---- 2-|E-|, (6.22 )
which determine the energy, E, momentum, p and the eccentricity, e, of an orbit:
3 2 2 I2 [ ]2 E = − m--G--M---, p = ----φ-----, e2 = 1 − --Iφ--- . (6.23 ) 2(Ir + Iφ) m2M R (t) Ir + Iφ
Lämmerzahl [176Jump To The Next Citation Point] noted that in this scenario E, e and R (t)p stay nearly constant. In addition, R (t)p and e of the planetary orbits practically do not participate in cosmic expansion. Remember that the stability of adiabatic invariants is governed by the factor of e −τ∕T, where τ is the characteristics time of change of external parameter (here: τ = 1∕H) and T is the characteristic time of the periodic motion (here: T = periods of planets). In the the case of periodic bound orbits 10 τ∕T ∼ 10, thus, any change of the adiabatic invariants is truly negligible.

6.3.4 Gravitationally bound systems in an expanding universe

The question of whether or not the cosmic expansion has an influence on the size of the Solar system was addressed in conjunction to the study of the Pioneer anomaly. In particular, is there a difference between locally bound and escape orbits? If the former are proven to be practically immune to cosmic expansion, what about the latter? In fact, the properties of bound (either electrically or gravitationally) systems in an expanding universe have been discussed controversially in many papers, notably by [21203].

Effects of cosmological expansion on local systems were addressed by a number of authors (for reviews see [73Jump To The Next Citation Point176Jump To The Next Citation Point]). Gautreu [137] studied the behavior of a spherical mass with the energy-stress tensor taken in the form of an ideal fluid. The obtained results show that the outer planets would tend to out-spiral away from the solar system. Anderson [21] has studied the behavior of local systems in the cosmologically-curved background. He obtained cosmological modifications of local gravitational fields with an additional drag term for escape orbits and demonstrated that Rr = const for bound orbits. Cooperstock et al. [82] used the Einstein–de Sitter universe 2 2 2 2 2 2 ds = c dt − R (t)(dr + dΩ ) and derived a geodesic deviation equation in Fermi normal coordinates ¨x − (R¨âˆ•R )x = 0. Clearly, the additional terms are too small be observed in the solar system.

As far as the Pioneer anomaly is concerned, the papers above are consistent in saying that planetary orbits in the solar system should see the effects of the anomaly and aP may not be of gravitational origin. However, Anderson [21] found an interesting result that suggests that the expansion couples to escape orbits, while it does not couple to bound orbits.

To study this possibility Lämmerzahl [176Jump To The Next Citation Point] used a PPN-inspired spacetime metric:

U U 2 ( U ) 2 g00 = 1 − 2α c2 + 2β-c4 , gij = − 1 + 2γ c2 R (t)δij, (6.24 )
that led to the following equation of motion (written in terms of measured distances and times):
[ ( )2 ] [ ] d2Xi- -∂U- -1 dX-- ( 2 )U- ( ) 1-∂U--dXj-- R˙ dXi- dT 2 = ∂Xi α + γc2 dT + 2 α − γα − 2β c2 − α + γ c2∂Xj dT + R dT .(6.25 )
Using Equation (6.25View Equation) Lämmerzahl [176Jump To The Next Citation Point] concludes that in cosmological context the behavior of bound orbits is different from that of unbound ones. However, cosmological expansion results in a decelerating drag term, which is a factor v∕c too small to account for the Pioneer effect.

6.3.5 Dark-energy-inspired f (R ) gravity models

The idea that the cosmic acceleration of the Universe may be caused by a modification of gravity at very large distances, and not by a dark energy source, has recently received a great deal of attention (see [319351Jump To The Next Citation Point]). Such a modification could be triggered by extra space dimensions, to which gravity spreads over cosmic distances. Recently, models involving inverse powers of the curvature or other invariants have been proposed as an alternative to dark energy. Although such theories can lead to late-time acceleration, they typically result in one of two problems: either they are in conflict with tests of general relativity in the solar system, due to the existence of additional dynamical degrees of freedom [77], or they contain ghost-like degrees of freedom that seem difficult to reconcile with fundamental theories.

Gravity theories that supplement the Einstein–Hilbert Lagrangian with a nonlinear function f(R) of the curvature scalar have attracted interest in the context of inflationary cosmological scenarios, and because these theories may explain the accelerated expansion of the universe without the need to postulate scalar fields or a cosmological constant (for a review, see [351]).

As an upshot of these efforts, Bertolami et al. [50] explored a particular family of f(R ) Lagrangians that gives rise to an extra acceleration in the form

2 2 aE = f-r--+ 2f. (6.26 ) GM
Assuming that f ∼ αGM ∕r where α is a constant leads to aE ∝ α2 = const in the limit of large r. For a galaxy, a E is naturally associated with the MOND acceleration a 0, while in the solar system, it may acquire the Pioneer acceleration value, aP.

A similar result was obtained by Saffari and Rahvar [320] who used a metric formalism in f (R ) modified gravity to study the dynamics of various systems from the solar system to the cosmological scales. Replacing f (R) with F (R ) = df ∕dR, the authors postulated the ansatz F (r) = (1 + r∕d)−α where α ≪ 1 is a dimensionless constant and αr ∕d ≪ 1. They obtained an anomalous acceleration term of − α∕2d, which agrees with the Pioneer anomaly for α ∕d ≃ 10− 26 m −1.

On the other hand, Exirifard [122] demonstrated that a correction to the Einstein–Hilbert Lagrangian in the form of Δ â„’ = R κλμνR κλμν cannot offer a covariant resolution to the Pioneer anomaly.

Capozziello et al., [72] developed a general analytic procedure to investigate the Newtonian limit of f (R ) gravity. The authors discussed the Newtonian and the post-Newtonian limits of these gravity models, including the investigation of their possible relevance to the Pioneer anomaly. Capozziello et al. [69] mention that, by treating the Pioneer anomaly as a correction to the Newtonian potential, it could be studied in the general theoretical scheme of f(R ) gravity theories.

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