Field-theoretical formulations of MOND-like gravity

Modified Newtonian dynamics (MOND) is a possible way to explain the flat galaxy rotation curves without invoking the existence of dark matter. It is, however, quite difficult to predict such a phenomenology in a consistent field theory, free of instabilities and admitting a well-posed Cauchy problem. We examine critically various proposals of the literature, and underline their successes and failures both from the experimental and the field-theoretical viewpoints. We exhibit new difficulties in both cases, and point out the hidden fine-tuning of some models. On the other hand, we show that several published no-go theorems are based on hypotheses which may be unnecessary, so that the space of possible models is a priori larger. We examine a new route to reproduce the MOND physics, in which the field equations are particularly simple outside matter. However, the analysis of the field equations within matter (a crucial point which is often forgotten in the literature) exhibits a deadly problem, namely, that they do not remain always hyperbolic. Incidentally, we prove that the same theoretical framework provides a stable and well-posed model able to reproduce the Pioneer anomaly without spoiling any of the precision tests of general relativity. Our conclusion is that all MOND-like models proposed in the literature, including the new ones examined in this paper, present serious difficulties: Not only they are unnaturally fine-tuned, but they also fail to reproduce some experimental facts or are unstable or inconsistent as field theories. However, some frameworks, notably the tensor-vector-scalar one of Bekenstein and Sanders, seem more promising than others, and our discussion underlines in which directions one should try to improve them.

Figure 1

In a theory where different fields have different causal cones, it suffices that their union be embedded in a wider cone for local causality to be satisfied. Initial data for all the fields simultaneously may be specified on a surface exterior to the wider cone, i.e., spacelike with respect to each cone. If the topology of spacetime is such that there does not exist any CTC with respect to the wider cone, then causality is preserved, although some fields may propagate faster than light (i.e., faster than electromagnetic waves, a mere particular case of matter field).

Figure 2

A simple function $f^{\prime }(s)$ reproducing the MOND dynamics for small $s$ (i.e., large distances) and the Newtonian one for large $s$ (i.e., small distances).

Figure 3

Fine-tuned function $f^{\prime }(s)$ such that Newtonian and post-Newtonian predictions are not spoiled in the solar system, although the MOND dynamics is predicted at large distances. The right panel displays the quite unnatural contribution of the scalar field to the acceleration of a test mass, as a function of its distance with respect to the Sun.

Figure 4

Typical shape of a function $f^{\prime }(s)$ such that scalar-field deviations from GR are $\propto r^n$, $n>0$, in the large-$s$ limit.

Figure 5

Feynman diagrams in scalar-tensor theories, where straight, curly, and wavy lines represent, respectively, the scalar field, gravitons, and photons. Matter sources are represented by blobs. (a) Diagrammatic interpretation of the effective gravitational constant $G_{\mathrm{eff}}=G(1+{\alpha }_0^2)$, where each vertex connecting matter to one scalar line involves a factor ${\alpha }_0$. (b) Photons are directly coupled to gravitons but not to the scalar field. (c) Photons feel nevertheless the scalar field indirectly, via its influence on gravitons: The energy-momentum tensor of the scalar field generates a curvature of the Einstein metric $g_{\mu \nu }^{*}$ in which electromagnetic waves propagate.

Figure 6

Diagrammatic illustration of the negative-energy modes spoiling the stability of (a) Lagrangians of the form $R+\gamma \mathrm{GB}+\alpha (R_{\mu \nu \rho \sigma }^2{)}^n$, $n\ge 2$, and (b) the present nonminimal metric coupling (5.1, 5.2). As before, curly lines represent spin-2 degrees of freedom, namely, the usual massless and positive-energy graviton in plain line, and the massive ghost in dotted lines. Contrary to Fig. 5, straight lines represent here matter fields $\psi$ instead of the scalar field $\phi$. In both cases (a) and (b), no ghost degree of freedom can be identified around a flat or empty background. However, a ghost mode does exist in curved backgrounds in case (a), and within matter in case (b). They use the background gravitons or matter fields as “catalyzers” in order to propagate and develop instabilities.