6.4 MOND in the solar system

The primary place to test modified gravity theories is, of course, the solar system, where general relativity has, until now, passed all the proposed tests. Detecting a deviation from Einsteinian gravity in our backyard would actually be the holy grail of modified gravity theories, in the same sense as direct detection in the lab is the holy grail of the CDM paradigm. However, MOND anomalies typically manifest themselves only in the weak-gravity regime, several orders of magnitudes below the typical gravitational field exerted by the sun on, e.g., the inner planets. But in the case of modified inertia (Section 6.1.1), the anomalous acceleration at any location depends on properties of the whole orbit (non-locality), so that anomalies may appear in the motion of Solar system bodies that are on highly-eccentric trajectories taking them to large distances (e.g., long period comets or the Pioneer spacecraft), where accelerations are low [314Jump To The Next Citation Point]. Such MOND effects have been proposed as a possible mechanism for generating the Pioneer anomaly [314Jump To The Next Citation Point, 469], without affecting the motions of planets, whose orbits are fully in the high acceleration regime. On the other hand, in classical, non-relativistic modified gravity theories (Sections 6.1.2 and 6.1.3), small effects could still be observable and would primarily probe two aspects of the theory: (i) the shape of the interpolating function (Section 6.2) in the regime x ≫ 1, and (ii) the external Galactic gravitational field (Section 6.3) acting on the solar system, testing the interpolating function in the regime x ≪ 1.

If, as a first approximation, one considers the solar system as isolated, and the Sun as a point mass, the MOND effect in the inner solar system appears as an anomalous acceleration field in addition to the Newtonian one. In units of a0, the amplitude of the anomalous acceleration is given by x [1 − μ (x )], which can be constrained from the motion of the inner planets, typically their perihelion precession and the (non)-variation of Kepler’s constant [293Jump To The Next Citation Point, 391Jump To The Next Citation Point, 417]. These constraints typically exclude the whole α-family of interpolating functions (Eq. 46View Equation) that are natural for multi-field theories such as TeVeS (see Section 6.2 and Section 7) because they yield x[1 − μ(x)] > 1 for x ≫ 1 while it must be smaller than 0.04 at the orbit of Mars [391Jump To The Next Citation Point]35. Of course, this does not mean that the μ-function cannot be represented by the α-family in the intermediate gravity regime characterizing galaxies, but it must be modified in the strong gravity regime36. Another potential effect of MOND is anomalously strong tidal stresses in the vicinity of saddle points of the Newtonian potential, which might be tested with the LISA pathfinder [37Jump To The Next Citation Point, 49, 255Jump To The Next Citation Point, 464]. The MOND bubble can be quite big and clearly detectable, or the effect could be small and undetectable, depending on the interpolating function [255, 161].

The approximation of an isolated Solar system being incorrect, it is also important to add the effect of the external field from the galaxy. Its amplitude is typically on the order of ∼ 1.5 × a0. From there, Milgrom [314Jump To The Next Citation Point] has predicted (both analytically and numerically) a subtle anomaly in the form of a quadrupole field that may be detected in planetary and spacecraft motions (as subsequently confirmed by [62Jump To The Next Citation Point, 185]). This has been used to constrain the form of the interpolating function in the weak acceleration regime characteristic of the external field itself. Constraints have essentially been set on the n-family of μ-functions from the perihelion precession of Saturn [63, 154], namely that one must have n > 8 in order to fit these data37.

However, it should be noted that it is slightly inconsistant to compare the classical predictions of MOND with observational constraints obtained by a global fit of solar system orbits using a fully-relativistic first-post-Newtonian model. Although the above constraints on classical MOND models are useful guides, proper constraints can only truly be set on the various relativistic theories presented in Section 7, the first-order constraints on these theories coming from their own post-Newtonian parameters [65Jump To The Next Citation Point, 99, 173Jump To The Next Citation Point, 372Jump To The Next Citation Point, 391Jump To The Next Citation Point, 450Jump To The Next Citation Point]. What is more, and makes all these tests perhaps unnecessary, it has recently been shown that it was possible to cancel any deviation from general relativity at small distances in most of these relativistic theories, independently of the form of the μ-function [22Jump To The Next Citation Point].

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