Improving relativistic MOND with Galileon k-mouflage

Improving relativistic MOND with Galileon k-mouflage


We propose a simple field theory reproducing the MOND phenomenology at galaxy scale, while predicting negligible deviations from general relativity at small scales thanks to an extended Vainshtein (“k-mouflage”) mechanism induced by a covariant Galileon-type Lagrangian. The model passes solar-system tests at the post-Newtonian order, including those of local Lorentz invariance, and its anomalous forces in binary-pulsar systems are orders of magnitude smaller than the tightest experimental constraints. The large-distance behavior is obtained as in Bekenstein’s tensor-vector-scalar (TeVeS) model, but with several simplifications. In particular, no fine-tuned function is needed to interpolate between the MOND and Newtonian regimes, and no dynamics needs to be defined for the vector field because preferred-frame effects are negligible at small distances. The field equations depend on second (and lower) derivatives, and avoid thus the generic instabilities related to higher derivatives. Their perturbative solution around a Schwarzschild background is remarkably simple to derive. We also underline why the proposed model is particularly efficient within the class of covariant Galileons.

04.50.Kd, 95.30.Sf, 95.35.+d

Although the hypothesis of dark matter is consistent with a wide range of observations, it might be an artifact of our Newtonian interpretation of experimental data if the gravitational law happens not to be valid at large distances. In 1983, Milgrom proposed a simple phenomenological rule, called “modified Newtonian dynamics” (MOND) (1), depending on an acceleration scale : In the vicinity of a (baryonic) mass , any test particle undergoes the standard Newtonian acceleration if , but a slower decreasing one if . This law happens to fit remarkably well galaxy rotation curves for a universal constant (2)


Moreover, it automatically recovers the Tully-Fisher law (3) and explains the correlation of dark matter profiles with the baryonic ones (4).

However, reproducing this phenomenological law in a consistent relativistic field theory happens to be quite difficult, as illustrated by almost three decades of abundant literature. In the present paper, we shall refine on one of the best models proposed so far, in our opinion, namely the tensor-vector-scalar (TeVeS) theory constructed by Bekenstein and Sanders (5); (6); (7); (8); (9); (10). We will show that a generalized Galileon action allows us to suppress deviations from general relativity at small distances, thanks to an extension of the Vainshtein mechanism occurring in massive gravity (11); (12). In the model we propose, the magnitude itself of the anomalous force tends towards zero at small distances, and not only its ratio to the gravitational one (as in the standard Vainshtein mechanism).

We consider a scalar-tensor theory of gravity defined by the action


where we use the sign convention of (13) and notably the mostly plus signature. The physical metric will be defined in Eq. (7) below. The scalar field is chosen dimensionless, and its kinetic term is the sum of the following three contributions, where  :


Here denotes the Levi-Civita tensor, related to the fully antisymmetrical symbol (whose values are 0 or ) by . A small mass term , with greater than the largest cluster sizes, might also be added to the above kinetic term.

Denoting as the scale where MOND effects start to manifest around a galaxy of baryonic mass , and as the Vainshtein radius below which scalar effects are suppressed, we will see below that dominates at very large distances , at intermediate ones , and at small scales . As stated above, numerical fits of galaxy rotation curves (2) give the value (1) for , while their flatness up to about (14) imposes that the positive dimensionless constant is smaller than . The constant entering has dimension of a [length], and we will show below that a numerical value


allows the model to pass solar-system tests while predicting MOND effects even for the lightest known dwarf galaxies.

Several ingredients of action (2) are borrowed from the MOND literature. In particular, the scalar kinetic term is well known to generate an extra acceleration on any test mass at distance from a source of baryonic mass (15). Note that we did not write it simply in terms of , but that an absolute value is involved only within the square root, to ensure that the scalar field carries positive energy whatever the sign of  (16). The standard kinetic term has a negligible influence because it is multiplied by the small positive constant , but it ensures that the dynamics of the scalar field is well defined when passes through a vanishing value (16); (17).

As in Refs. (8); (9); (7); (10), the matter action assumes that all matter fields are minimally coupled to a physical metric


where is a timelike unit vector field, i.e., . Light deflection by galaxies and clusters would indeed be inconsistent with experiment if matter was merely coupled to the scalar field via a conformal metric (5); (6); (16). In a locally inertial frame where , and choosing the observer’s velocity such that lies along his proper time direction, Eq. (7) merely means that but (mimicking thus the behavior of pure general relativity in presence of dark matter). Previous field theories attempting at reproducing the MOND dynamics predicted large preferred-frame effects in the solar system, inconsistent with experiment, if was assumed to be a constant vector field (7); (16). This is why it was assumed to be dynamical in the TeVeS model (8); (9), with the idea that it could align with the matter’s local proper time direction. However, making the vector field dynamical by adding a kinetic term , or any function of it, causes this vector to be unstable (18); (16); (19). In the present paper, we will see that preferred-frame effects remain negligible in the solar system even if is assumed to be constant, thanks to the Vainshtein mechanism at small distances. This mechanism also allows us to choose other forms than (7), in contrast to Refs. (8); (9); (7); (10). We could also choose the physical metric as , but would need to be a fine-tuned function of both and its standard kinetic term, , in order to be consistent with the observed light deflection by galaxies and clusters. Moreover, the conditions for consistency of the theory within matter would be quite involved (16). The choice (7), borrowed from (8); (9); (7); (10), is thus the most natural one in the present framework.

The reason why a mere action , Eqs. (3) and (4) above, does not suffice to define a consistent relativistic field theory of MOND is that it would also predict an extra force within the solar system (where denotes the mass of the Sun), in addition to the Newtonian one and its post-Newtonian corrections. [We often use the word “force” instead of “acceleration” in the present paper, i.e., do not write the mass of the test particle to simplify.] This would be ruled out by tests of Kepler’s third law and those of post-Newtonian dynamics. The literature considered thus “Relativistic AQUAdraric Lagrangians” (RAQUAL), also known as “k-essence” theories in the cosmological framework, i.e., a scalar kinetic term depending on a function of . In order to reproduce the MOND dynamics, this kinetic term was assumed to take the form (4) for small accelerations (i.e., small values of ), while the scalar field behavior within the solar system depended on the shape of for large . The clever choice of the literature was to impose that tends towards a constant value for large , say , in order to recover a Brans-Dicke behavior at small distances, so that the physical metric (7) reproduce the standard Schwarzschild solution up to a rescaling of the gravitational constant . The parametrized post-Newtonian (PPN) parameters and (20) take then strictly their general relativistic values , and classical solar-system tests are passed. However, binary-pulsar tests are directly sensitive to the matter-scalar coupling strength , independently of the fact that the physical metric (7) contains a “disformal” contribution proportional to , and they impose (21). As discussed in (16), such a large value is difficult to reconcile with the expression (4) needed for small . One needs a fine-tuned interpolating function , of a shape similar to Fig. 3 of (16), in order to predict MOND effects while passing binary-pulsar tests. Another idea would thus be to choose a function such that the scalar force at small distances is negligible with respect to the Newtonian one , instead of keeping the same radial dependence . As underlined in (22), nonlinear kinetic terms (i.e., those of k-essence/RAQUAL models) can reduce scalar effects at small distances, acting as a camouflage for the scalar (hence the name “k-mouflage”). However, RAQUAL models must satisfy two conditions to have a Hamiltonian bounded by below and a well-posed Cauchy problem (16): and . These conditions suffice to prove that , i.e., that is a decreasing function of , and the best we can obtain is thus an almost constant force within the solar system, the value being imposed by the MOND regime for . But solar-system tests are precise enough to rule out a constant anomalous acceleration even numerically as tiny as (1). We must therefore look for other possible scalar kinetic terms, and this is where generalized Galileons enter our discussion.

Galileons were first introduced in the cosmological context in (23) (although they had actually already been studied in (24) in a quite different framework). In flat spacetime, they are theories whose field equations depend only on second derivatives of a scalar field, but not on their lower (0th and 1st) nor on higher derivatives. One of their initial motivations was to generalize the key features of the decoupling limit of the DGP brane model (25), which yields an equation of motion for a scalar degree of freedom, playing a crucial role in cosmology (26); (27). References (28); (29) showed how to extend Galileon models to curved spacetime without introducing higher derivatives (while making now first derivatives also enter them). The same Lagrangians can be obtained in a suitable limit of brane models including Gauss-Bonnet-Lovelock densities (30), or from dimensional reduction of such densities (31). They can also be extended, in any dimension, to arbitrary -forms possibly coupled to each other (32), or to general nonlinear models whose field equations depend on at most second derivatives (33). Throughout this paper, we call Galileons this full class of models, although they go beyond the initial ones of (23).

The new ingredient of the present paper is the third scalar kinetic term of action (2), , which strongly suppresses all scalar-field effects at small distances, as we will show below. This action (up to the factor ) was obtained in Ref. (31) for the first time by dimensional reduction of the Gauss-Bonnet-Lovelock density. It is written here in the compact form of (29), and is also in the general classes considered in (32); (33). It is easy to check that all field equations involve at most second derivatives, because of the antisymmetry of the Levi-Civita tensors entering (5).

Although it is straightforward to write the full field equations deriving from action (2), for both and the metric, it will suffice in the present paper to obtain the perturbative solution for a static and spherically symmetric in a Schwarzschild background , where denotes the Schwarzschild radius. It can be checked a posteriori that the backreaction of the scalar field on the metric has negligible effect as compared to present experimental bounds. We will even assume that to simplify the expressions. More detail will be provided in a forthcoming publication (34). Denoting as before , and integrating once the field equation for , we find


where the origin of the different terms is obvious from (3)–(5). The large number of field derivatives involved in , Eq. (5), is responsible for the negative power of in the first term of (8). This is the central idea of the Vainshtein mechanism, since it makes this first term dominate at small distances. Imposing now that for (otherwise would diverge at infinity), we can immediately write the unique solution of (8), which is a mere second-order polynomial equation for :


We thus easily recover the asymptotic Brans-Dicke behavior for very large distances , the MOND regime at intermediate scales, but a small derivative at small distances. Note that this small-distance behavior of does not depend at all on the mass , which not only generates the right-hand side (source) of Eq. (8), but also the background Schwarzschild geometry entering the first term of (8) via the Riemann tensor of (5). This universal small-distance behavior means that, paradoxically, any body generates strictly the same scalar force in its vicinity. Figure 1 illustrates the three regimes of solution (9).

Figure 1: Behavior of the scalar force in the three regimes of the theory: at small distances (k-mouflage mechanism), at intermediate range (MOND regime), and at large distances (asymptotic Brans-Dicke theory).

Neglecting its contribution, the maximum value is reached at , which defines thus a transition radius.

In order to predict the MOND phenomenology in a galaxy of baryonic mass , we need , i.e., . The lightest dwarf galaxies for which we have evidence for dark matter or MOND effects give thus an upper bound for the constant . Draco-like dwarfs correspond to a baryonic mass between and solar masses, but even tiny clusters of only seem to be dominated by dark matter (35). Let us thus be conservative and impose . This is the numerical value given in Eq. (6) above.

We can now estimate the order of magnitude of scalar effects within the solar system. Since we chose a physical metric of the disformal form (7), like in TeVeS, one could naively conclude that the PPN parameters and keep their general relativistic values . However, there is now no meaning to define such parameters, because the PPN formalism assumes that no length scale enters the theory, and it needs the gravitational potential to be at the Newtonian order. Therefore, the consequences of our anomalous potential at small distances cannot be analyzed in the standard way. Let us thus merely compare the anomalous force it generates on a test particle, with the post-Newtonian forces which have been precisely tested in the solar system. Since their ratio is for the chosen value (6) of the constant , the scalar effects are thus smaller than times post-Newtonian ones at the Earth’s orbit, i.e., negligible with respect to the best experimental constraints. This ratio becomes even smaller for inner planets, and is for Mercury’s orbit. On the other hand, scalar effects grow for outer planets, but they remain smaller than post-Newtonian forces at the orbit of Mars, and at Jupiter’s. This is consistent with the most precise planetary data. When considering the Moon’s orbit, the contribution of the Earth to the Riemann tensor entering (5) dominates over that of the Sun, and is almost spherically symmetric with respect to the Earth’s center. Denoting now as the distance to this center, we get again the universal behavior , which must be compared to the Newtonian accelerations caused by both the Earth and the Sun and their post-Newtonian corrections. We find that scalar effects on the Moon’s motion are smaller than post-Newtonian ones, i.e., four orders of magnitude smaller than the tightest experimental constraints derived from Lunar Laser Ranging (20).

Preferred-frame effects can be estimated in a similar way. Assuming that the solar system is moving with a velocity with respect to the preferred frame where , we compute the contributions proportional to in and to in , and their radial derivatives give us the magnitude of the anomalous scalar forces. Comparing them to those generated by the -term in the PPN formalism (20) (while the terms corresponding to and vanish in the present model (7)), we find that their ratio is , giving thus scalar effects similar to those of an at the orbit of Mars. As above, when considering the Moon’s orbit around the Earth, the local value of must be used ( denoting now the Earth-Moon distance), and we find that preferred-frame effects caused by the scalar field are similar to those of an . Since the tightest constraint comes from Lunar Laser Ranging, we conclude that the model (2)–(7) does not predict any detectable violation of local Lorentz invariance in the solar system.

Binary-pulsar tests are much more subtle to compute. A precise analysis would need either to study the (time-dependent) dynamics of a binary system at least up to order , or to be able to relate scalar multipoles at infinity to their local matter sources in spite of the nonlinearities of the Vainshtein mechanism. In the present paper, we shall only estimate the rough order of magnitude of scalar radiation by comparing the local scalar forces to those of the precisely studied Brans-Dicke-like theories. Let us first note that the monopolar radiation, naively of order , is actually reduced to order because the local scalar solution generated by any body does not depend on time; this is similar to the case of standard scalar-tensor theories (36). On the other hand, the dipolar radiation starts at order in spite of the fact that the local scalar solution is the same around any body, independently of its mass. Indeed, if the two bodies of a binary system do not have the same mass (say, , they do not move on the same orbit around their common center of mass, and the global scalar field they generate defines a preferred (oriented) direction in space. The dominant scalar radiation is thus a dipole, and we can estimate its order of magnitude by multiplying the one predicted in standard scalar-tensor theories (see, e.g. Eq. (6.52b) of Ref. (36)) by the square of the ratio of the present scalar force between the two bodies () and the standard one. We get that the scalar field contribution to the time derivative of the orbital period is of order , and numerically at least smaller than the tightest experimental uncertainties. Although this estimate might be erroneous by some large numerical coefficient, we can anyway conclude that the present model should easily pass all binary-pulsar tests.

Although the Galileon field equations involve at most second derivatives, and avoid thus the generic instability related to higher derivatives, this does not suffice to prove that these models are stable. One should carefully analyze both the boundedness by below of their Hamiltonian density and the well-posedness of their Cauchy problem. The Hamiltonian of flat-space Galileons is straightforward to derive (37); (34), but the situation is much more complex in curved spacetime, because all field equations for and involve second derivatives of both of them. Around a given background, one should thus diagonalize the kinetic terms in order to test the stability of perturbations and the hyperbolicity of their field equations. For instance, it would have no meaning to freeze and study the perturbations of only in the scalar field equation (similarly to Brans-Dicke theory, which seems to contain a ghost scalar field for if one freezes the Jordan metric, whereas studying simultaneously the dynamics of and shows that the theory is stable even for such a slightly negative ). This problem of the consistency of Galileon field theories goes thus beyond the scope of this paper, and we postpone it to a forthcoming publication (34).

Let us finally comment on our choice of Lagrangian (5) to obtain a k-mouflage mechanism reducing scalar effects at small distances. It happens to be the most efficient one amongst all those that we have analyzed. The highest-order Galileon Lagrangian which is nontrivial in 4-dimensional flat space (23); (28); (29), namely , also generates a small at small distances, but its slower radial dependence makes scalar effects still marginally detectable in the solar system (while giving again fully negligible preferred-frame effects). Imposing as above that the MOND phenomenology should occur in the lightest known dwarf galaxies, we get that scalar effects in the solar system are times post-Newtonian forces, i.e., about the order of magnitude of the most precise bounds. Similarly, scalar effects on the Moon’s motion are smaller than post-Newtonian ones, i.e., of the order of current limits. A full fit of planetary data taking into account the possible presence of such a small scalar force would thus be necessary to test whether it is already excluded or not. If not, this opens the exciting possibility to detect them in future higher-precision solar-system observations. All the other known covariant Galileon actions are either total derivatives in 4 dimensions, or yield a vanishing scalar field equation around a Schwarzschild background (like Eq. (23) of Ref. (31)), or predict a negative and therefore too large scalar forces in the solar system. On the other hand, generalized Galileon actions (33) involving non-differentiated fields and/or negative powers of can provide alternative models (34), but anyway less natural than (5).


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