A set of particles of mass moving in a gravitational field generated by the matter density
distribution and described by the Newtonian potential has the following
action^{22}:

The first possibility, modified inertia, has been investigated by Milgrom [300, 321], who constructed modified kinetic
actions^{24}
(the first term in Eq. 13) that are functionals depending on the trajectory of the particle as well as
on the acceleration constant . By construction, the gravitational potential is then still determined from
the Newtonian Poisson equation, but the particle equation of motion becomes, instead of Newton’s second
law,

Milgrom [300] investigated theories of this vein and rigorously showed that
they always had to be time-nonlocal (see also Section 7.10) to be Galilean
invariant^{25}.
Interestingly, he also showed that quantities such as energy and momentum had to be redefined
but were then enjoying conservation laws: this even leads to a generalized virial relation for
bound trajectories, and in turn to an important and robust prediction for circular orbits in
an axisymmetric potential, shared by all such theories. Eq. 14 becomes for such trajectories:

The idea of modified gravity is, on the one hand, to preserve the particle equation of motion by preserving the kinetic action, but, on the other hand, to change the gravitational action, and thus modify the Poisson equation. In that case, all the usual conservation laws will be preserved by construction.

A very general way to do so is to write [38]:

where can be any dimensionless function. The Lagrangian being non-quadratic in , this has been dubbed by Bekenstein & Milgrom [38] Aquadratic Lagrangian theory (AQUAL). Varying the action with respect to then leads to a non-linear generalization of the Newtonian Poisson equationIn vacuum and at very large distances from a body of mass , the isopotentials always tend to become spherical and the curl field tends to zero, while the gravitational acceleration falls well below (a regime known as the “deep-MOND” regime), so that:

An important point, demonstrated by Bekenstein & Milgrom [38], is that a system with a low center-of-mass acceleration, with respect to a larger (more massive) system, sees the motion of its constituents combine to give a MOND motion for the center-of-mass even if it is made up of constituents whose internal accelerations are above (for instance a compact globular cluster moving in an outer galaxy). The center-of-mass acceleration is independent of the internal structure of the system (if the mass of the system is small), namely the Weak Equivalence Principle is satisfied.

In a modified gravity theory, any time-independent system must still satisfy the virial theorem:

where is the total kinetic energy of the system, being the total mass of the system, the second moment of the velocity distribution, and is the “virial”, proportional to the total potential energy. Milgrom [301, 302] showed that, in Bekenstein–Milgrom MOND, the virial is given by: For a system entirely in the extremely weak field limit (the “deep-MOND” limit ) where and , the second term vanishes and we get (see [301] for the specific conditions for this to be valid). In this case, we can get an analytic expression for the two-body force under the approximation that the two bodies are very far apart compared to their internal sizes [301, 509, 511]. Since the kinetic energy can be separated into the orbital energy and the internal energy of the bodies , we get from the scalar virial theorem of a stationary system: We can then assume an approximately circular velocity such that the two-body force (satisfying the action and reaction principle) can be written analytically in the deep-MOND limit as :The latter equation is not valid for N-body configurations, for which the Bekenstein–Milgrom (BM) modified Poisson equation (Eq. 17) must be solved numerically (apart from highly-symmetric N-body configurations). This equation is a non-linear elliptic partial differential equation. It can be solved numerically using various methods [50, 77, 96, 147, 250, 457]. One of them [77, 457] is to use a multigrid algorithm to solve the discrete form of Eq. 17 (see also Figure 17):

where- is the density discretized on a grid of step ,
- is the MOND potential discretized on the same grid of step ,
- , and , are the values of at points and corresponding to and respectively (Figure 17).

The gradient component , in , is approximated in the case of by (see Figure 17).

In [457] the Gauss–Seidel relaxation with red and black ordering is used to solve this discretized equation, with the boundary condition for the Dirichlet problem given by Eq. 20 at large radii. It is obvious that subsequently devising an evolving N-body code for this theory can only be done using particle-mesh techniques rather than the gridless multipole expansion treecode schemes widely used in standard gravity.

Finally, let us note that it could be imagined that MOND, given some of its observational problems (developed in Section 6.6), is incomplete and needs a new scale in addition to . There are several ways to implement such an idea, but for instance, Bekenstein [36] proposed in this vein a generalization of the AQUAL formalism by adding a velocity scale , in order to allow for effective variations of the acceleration constant as a function of the deepness of the potential, namely:

leading to where . Interestingly, with this “modified MOND”, Gauss’ theorem (or Newton’s second theorem) would no longer be valid in spherical symmetry. A suitable choice of (e.g., on the order of ; see [36]) could affect the dynamics of galaxy clusters (by boosting the modification with an effectively higher value of ) compared to the previous MOND equation, while keeping the less massive systems such as galaxies typically unaffected compared to usual MOND, while other (lower) values of could allow (modulo a renormalization of ) for a stronger modification in galaxy clusters as well as milder modification in subgalactic systems such as globular clusters, which, as we shall soon see could be interesting from a phenomenological point of view (see Section 6.6). However, the possibility of too strong a modification should be carefully investigated, as well as, in a relativistic (see Section 7) version of the theory, the consequences on the dynamics of a scalar-field with a similar action.Another way [319] of modifying gravity in order to reproduce Milgrom’s law is to still keep the “matter
action” unchanged , thus ensuring that varying the action of a test
particle with respect to the particle degrees of freedom leads to , but to invoke an
auxiliary acceleration field in the gravitational action instead of invoking an aquadratic
Lagrangian in . The addition of such an auxiliary field can of course be done without
modifying Newtonian gravity, by writing the Newtonian gravitational action in the following
way^{27}:

The concept of the effective density of matter that would source the MOND force field in Newtonian gravity is extremely useful for an intuitive comprehension of the MOND effect, and/or for interpreting MOND in the dark matter language: indeed, subtracting from this effective density the baryonic density yields what is called the “phantom dark matter” distribution. In AQUAL, it requires deriving the Newtonian Poisson equation after having solved for the MOND one. On the other hand, in QUMOND, knowing the Newtonian potential yields direct access to the phantom dark matter distribution even before knowing the MOND potential. After choosing a -function, one defines

and one has, for the phantom dark matter density, This -function appears naturally in an alternative formulation of QUMOND where one writes the action as a function of an auxiliary potential : leading to a potential obeying a QUMOND equation with , and .Numerically, for a given Newtonian potential discretized on a grid of step , the discretized phantom dark matter density is given on grid points by (see Figure 17 and cf. Eq. 25, see also [11]):

This means that any N-body technique (e.g., treecodes or fast multipole methods) can be adapted to QUMOND (a grid being necessary as an intermediate step). Once the Newtonian potential (or force) is locally known, the phantom dark matter density can be computed and then represented by weighted particles, whose gravitational attraction can then be computed in any traditional manner. An example is given in Figure 18, where one considers a rather typical baryonic galaxy model with a small bulge and a large disk. Applying Eq. 35 (with the -function of Eq. 43) then yields the phantom density [253]. Interestingly, this phantom density is composed of a round “dark halo” and a flattish “dark disk” (see [305] for an extensive discussion of how such a dark disk component comes about; see also [50] and Section 6.5.2 for observational considerations). Let us note that this phantom dark matter density can be slightly separated from the baryonic density distribution in non-spherical situations [226], and that it can be negative [297, 490], contrary to normal dark matter. Finding the signature of such a local negative dark matter density could be a way of exhibiting a clear signature of MOND.Finally, let us note that, as shown in [319, 509], (i) a system made of high-acceleration constituents, but with a low-acceleration center-of-mass, moves according to a low-acceleration MOND law, while (ii) the virial of a system is given by

meaning that for a system entirely in the extremely weak field limit where and , the second term vanishes and we get , precisely like in Bekenstein–Milgrom MOND. This means that, although the curl-field correction is in general different in AQUAL and QUMOND, the two-body force in the deep-MOND limit is the same [509].
Living Rev. Relativity 15, (2012), 10
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