6.1 Modified inertia or modified gravity: Non-relativistic actions

If one wants to modify dynamics in order to reproduce Milgrom’s heuristic law while still benefiting from usual conservation laws such as the conservation of momentum, one can start from the action at the classical level. Clearly such theories are only toy-models until they become the weak-field limit of a relativistic theory (see Section 7), but they are useful both as targets for such relativistic theories, and as internally consistent models allowing one to make predictions at the classical level (i.e., neither in the relativistic or quantum regime).

A set of particles of mass mi moving in a gravitational field generated by the matter density distribution ρ = ∑ mi δ(x − xi) i and described by the Newtonian potential ΦN has the following action22:

∫ 2 ∫ ∫ 2 SN = Skin + Sin + Sgrav = ρv-d3xdt − ρΦN d3xdt − |∇-ΦN-|-d3xdt. (13 ) 2 8πG
Varying this action with respect to configuration space coordinates yields the equations of motion d2x∕dt2 = − ∇ ΦN, while varying it with respect to the potential leads to Poisson equation 2 ∇ ΦN = 4πG ρ. Modifying the first (kinetic) term is generally referred to as “modified inertia” and modifying the last term as “modified gravity”23.

6.1.1 Modified inertia

The first possibility, modified inertia, has been investigated by Milgrom [300Jump To The Next Citation Point, 321Jump To The Next Citation Point], who constructed modified kinetic actions24 (the first term S kin in Eq. 13View Equation) that are functionals depending on the trajectory of the particle as well as on the acceleration constant a0. 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,

A [{x(t)},a0] = − ∇ ΦN , (14 )
where A is a functional of the whole trajectory {x(t)}, with the dimensions of acceleration. The Newtonian and MOND limits correspond to [a → 0,A → d2x∕dt2] 0 and [a → ∞, A [{x (t)},a ] → a−1Q ({x(t)})] 0 0 0 where Q has dimensions of acceleration squared.

Milgrom [300Jump To The Next Citation Point] investigated theories of this vein and rigorously showed that they always had to be time-nonlocal (see also Section 7.10) to be Galilean invariant25. 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. 14View Equation becomes for such trajectories:

( V 2 ) V 2 ∂ΦN μ --c- -c- = − -----, (15 ) Ra0 R ∂R
where, Vc and R are the orbital speed and radius, and μ (x) is universal for each theory, and is derived from the expression of the action specialized to circular trajectories. Thus, for circular trajectories, these theories recover exactly the heuristic Milgrom’s law. Interestingly, it is this law, which is used to fit galaxy rotation curves, while in the modified gravity framework of MOND (see hereafter), one should actually calculate the exact predictions of the modified Poisson formulations, which can differ a little bit from Milgrom’s law. However, for orbits other than circular, it becomes very difficult to make predictions in modified inertia, as the time non-locality can make the anomalous acceleration at any location depend on properties of the whole orbit. For instance, if the accelerations are small on some segments of a trajectory, MOND effects can also be felt on segments where the accelerations are high, and conversely [321]. This can give rise to different effects on bound and unbound orbits, as well as on circular and highly elliptic orbits, meaning that “predictions” of modified inertia in pressure-supported systems could differ significantly from those derived from Milgrom’s law per se. Let us finally note that testing modified inertia on Earth would require one to properly define an inertial reference frame, contrary to what has been done in [5, 179] where the laboratory itself was not an inertial frame. Proper set-ups for testing modified inertia on Earth have been described, e.g., in [201, 202]: under the circumstances described in these papers, modified inertia would inevitably predict a departure from Newtonian dynamics, even if the exact departure cannot be predicted at present, except for circular motion.

6.1.2 Bekenstein–Milgrom MOND

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 [38Jump To The Next Citation Point]:

∫ 2 2 2 Sgrav BM ≡ − a0F(|∇-Φ|-∕a0)d3xdt, (16 ) 8πG
where F can be any dimensionless function. The Lagrangian being non-quadratic in |∇ Φ|, this has been dubbed by Bekenstein & Milgrom [38Jump To The Next Citation Point] Aquadratic Lagrangian theory (AQUAL). Varying the action with respect to Φ then leads to a non-linear generalization of the Newtonian Poisson equation26:
[ ( |∇ Φ |) ] ∇. μ ----- ∇ Φ = 4πG ρ (17 ) a0
where μ(x) = F ′(z) and z = x2. In order to recover the μ-function behavior of Milgrom’s law (Eq. 7View Equation), i.e., μ (x) → 1 for x ≫ 1 and μ (x) → x for x ≪ 1, one needs to choose:
F (z ) → z for z ≫ 1 and F (z) → 2z3∕2 for z ≪ 1. (18 ) 3
The general solution of the boundary value problem for Eq. 17View Equation leads to the following relation between the acceleration g = − ∇ Φ and the Newtonian one, g = − ∇ Φ N N
( g ) μ --- g = gN + S, (19 ) a0
where g = |g |, and S is a solenoidal vector field with no net flow across any closed surface (i.e., a curl field S = ∇ × A such that ∇.S = 0). This, it is equivalent to Milgrom’s law (Eq. 7View Equation) up to a curl field correction, and is precisely equal to Milgrom’s law in highly symmetric one-dimensional systems, such as spherically-symmetric systems or flattened systems for which the isopotentials are locally spherically symmetric. For instance, the Kuzmin disk [52] is an example of a flattened axisymmetric configuration for which Milgrom’s law is precisely valid, as its Newtonian potential ∘ -2-----------2- ΦN = − GM ∕ R + (b + |z|) is equivalent on both sides of the disk to that of a point mass above or below the disk respectively.

In vacuum and at very large distances from a body of mass M, the isopotentials always tend to become spherical and the curl field tends to zero, while the gravitational acceleration falls well below a0 (a regime known as the “deep-MOND” regime), so that:

∘ ------- Φ (r) ∼ GM a0 ln(r). (20 )

An important point, demonstrated by Bekenstein & Milgrom [38Jump To The Next Citation Point], 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 a0 (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:

2K + W = 0. (21 )
where K = M ⟨v2⟩∕2 is the total kinetic energy of the system, ∑ M = imi being the total mass of the system, ⟨v2⟩ the second moment of the velocity distribution, and ∫ W = − ρx.∇ Φd3x is the “virial”, proportional to the total potential energy. Milgrom [301Jump To The Next Citation Point, 302] showed that, in Bekenstein–Milgrom MOND, the virial is given by:
∘ -------- ∫ [ ] W = − 2- GM 3a0 − -1--- 3a2F (|∇ Φ |2∕a2 ) − μ (|∇ Φ |∕a0)|∇ Φ |2 d3x. (22 ) 3 4πG 2 0 0
For a system entirely in the extremely weak field limit (the “deep-MOND” limit x = g∕a0 ≪ 1) where μ (x ) = x and 3∕2 F (z) = (2∕3)z, the second term vanishes and we get √ ----3--- W = (− 2∕3) GM a0 (see [301Jump To The Next Citation Point] 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, 509Jump To The Next Citation Point, 511]. Since the kinetic energy K = K + K orb int can be separated into the orbital energy 2 Korb = m1m2v rel∕ (2M ) and the internal energy of the bodies ∑ ∘ ----3-- Kint = (1 ∕3) Gm ia0 , we get from the scalar virial theorem of a stationary system:
2 [∘ -------- ∑ ∘ -------] m1m2v--rel= 2- GM 3a0 − Gm3 a0 . (23 ) M 3 i i
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 :
√ ---- m1m2 v2rel 2 [ 3∕2 3∕2 3∕2] Ga0 F2body = ------------= -- (m1 + m2 ) − m 1 − m 2 ------. (24 ) m1 + m2 r 3 r

The latter equation is not valid for N-body configurations, for which the Bekenstein–Milgrom (BM) modified Poisson equation (Eq. 17View Equation) 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 [50Jump To The Next Citation Point, 77Jump To The Next Citation Point, 96Jump To The Next Citation Point, 147Jump To The Next Citation Point, 250Jump To The Next Citation Point, 457Jump To The Next Citation Point]. One of them [77Jump To The Next Citation Point, 457Jump To The Next Citation Point] is to use a multigrid algorithm to solve the discrete form of Eq. 17View Equation (see also Figure 17View Image):

4πG ρi,j,k = (25 ) [(Φi+1,j,k − Φi,j,k)μM1 − (Φi,j,k − Φi −1,j,k)μL1 + (Φi,j+1,k − Φi,j,k)μM2 − (Φi,j,k − Φi,j−1,k)μL2 + (Φ − Φ )μ − (Φ − Φ )μ ]∕h2 i,j,k+1 i,j,k M3 i,j,k i,j,k−1 L3

The gradient component (∂∕∂x, ∂∕∂y, ∂∕∂z), in μ(x), is approximated in the case of μMl by ([Φ(B ) − Φ(A )]∕h, [Φ (I) + Φ(H ) − Φ(K ) − Φ (J)]∕(4h ),[Φ (C ) + Φ (D ) − Φ (E ) − Φ (F )]∕(4h&#x (see Figure 17View Image).

View Image

Figure 17: Discretisation scheme of the BM modified Poisson equation (Eq. 17View Equation) and of the phantom dark matter derivation in QUMOND. The node (i,j,k) corresponds to A on the upper panel. The gradient components in μ(x) (for Eq. 25View Equation) and ν(y) (for Eq. 35View Equation) are estimated at the Li and Mi points. Image courtesy of Tiret, reproduced by permission from [457Jump To The Next Citation Point], copyright by ESO.

In [457Jump To The Next Citation Point] 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. 20View Equation 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 a0. There are several ways to implement such an idea, but for instance, Bekenstein [36Jump To The Next Citation Point] proposed in this vein a generalization of the AQUAL formalism by adding a velocity scale s 0, in order to allow for effective variations of the acceleration constant as a function of the deepness of the potential, namely:

1 ∫ 2 2 Sgrav Bek ≡ −----- a20e−2Φ∕s0F(|∇ Φ|2e2Φ∕s0∕a20)d3xdt, (26 ) 8πG
leading to
[ ( ) ] ( ) ( ) |∇-Φ| |∇-Φ|2 |∇-Φ-| a20eff |∇-Φ-|2- ∇. μ a0eff ∇ Φ − s2 μ a0eff + s2 F a2 = 4πG ρ, (27 ) 0 0 0eff
where 2 a0eff = a0e−Φ∕s0. Interestingly, with this “modified MOND”, Gauss’ theorem (or Newton’s second theorem) would no longer be valid in spherical symmetry. A suitable choice of s0 (e.g., on the order of 103 km ∕s; see [36Jump To The Next Citation Point]) could affect the dynamics of galaxy clusters (by boosting the modification with an effectively higher value of a0) 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 s0 could allow (modulo a renormalization of a 0) 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.

6.1.3 QUMOND

Another way [319Jump To The Next Citation Point] of modifying gravity in order to reproduce Milgrom’s law is to still keep the “matter action” unchanged ∫ 2 3 Skin + Sin = ρ (v ∕2 − Φ)d xdt, thus ensuring that varying the action of a test particle with respect to the particle degrees of freedom leads to d2x∕dt2 = − ∇ Φ, but to invoke an auxiliary acceleration field gN = − ∇ ΦN in the gravitational action instead of invoking an aquadratic Lagrangian in |∇ Φ |. 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 way27:

1 ∫ Sgrav N = − ----- (2∇ Φ.gN − g2N)d3xdt. (28 ) 8πG
It gives, after variation over gN (or over ΦN): gN = − ∇ Φ. And after variation of the full action over Φ: − ∇.g = 4πG ρ N, i.e., Newtonian gravity. One can then introduce a MONDian modification of gravity by modifying this action in the following way, replacing 2 gN by a non-linear function of it and assuming that it derives from an auxiliary potential gN = − ∇ ΦN, so that the new degree of freedom is this new potential:
1 ∫ Sgrav QUMOND ≡ − ----- [2∇ Φ.∇ ΦN − a20Q (|∇ΦN |2∕a20)]d3xdt. (29 ) 8πG
Varying the total action with respect to Φ yields: ∇2 ΦN = 4πG ρ. And varying it with respect to the auxiliary (Newtonian) potential ΦN yields:
[ ( |∇Φ |) ] ∇2 Φ = ∇. ν ----N-- ∇ΦN (30 ) a0
where ν(y) = Q ′(z) and z = y2. Thus, the theory requires one only to solve the Newtonian linear Poisson equation twice, with only one non-linear step in calculating the rhs term of Eq. 30View Equation. For this reason, it is called the quasi-linear formulation of MOND (QUMOND). In order to recover the ν-function behavior of Milgrom’s law (Eq. 10View Equation), i.e., ν (y) → 1 for y ≫ 1 and ν(y) → y−1∕2 for y ≪ 1, one needs to choose:
4-3∕4 Q (z ) → z for z ≫ 1 and Q (z) → 3z for z ≪ 1. (31 )
The general solution of the system of partial differential equations is equivalent to Milgrom’s law (Eq. 10View Equation) up to a curl field correction, and is precisely equal to Milgrom’s law in highly-symmetric one-dimensional systems. However, this curl-field correction is different from the one of AQUAL. This means that, outside of high symmetry, AQUAL and QUMOND cannot be precisely equivalent. An illustration of this is given in [509Jump To The Next Citation Point]: for a system with all its mass in an elliptical shell (in the sense of a squashed homogeneous spherical shell), the effective density of matter that would source the MOND force field in Newtonian gravity is uniformly zero in the void inside the shell for QUMOND, but nonzero for AQUAL.

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 ν-function, one defines

&tidle;ν(y) = ν(y) − 1, (32 )
and one has, for the phantom dark matter density,
ρ = ∇.-(&tidle;ν∇-ΦN-). (33 ) ph 4πG
This &tidle;ν-function appears naturally in an alternative formulation of QUMOND where one writes the action as a function of an auxiliary potential Φph:
1 ∫ Sgrav QUMOND = − ----- [|∇ Φ |2 − |∇ Φph|2 − a20H (|∇ Φ − ∇ Φph |2∕a20)]d3xdt, (34 ) 8πG
leading to a potential Φph obeying a QUMOND equation with &tidle;ν(y) = H ′(y2), and Φ = ΦN + Φph.

Numerically, for a given Newtonian potential discretized on a grid of step h, the discretized phantom dark matter density is given on grid points (i,j,k ) by (see Figure 17View Image and cf. Eq. 25View Equation, see also [11Jump To The Next Citation Point]):

ρph(i,j,k) = (35 ) [(ΦN (i+1,j,k) − ΦN (i,j,k))ν&tidle;M1 − (ΦN (i,j,k) − ΦN (i−1,j,k))&tidle;νL1 + (ΦN (i,j+1,k) − ΦN (i,j,k))&tidle;νM2 − (ΦN (i,j,k) − ΦN (i,j− 1,k))ν&tidle;L2 + (Φ − Φ )&tidle;ν − (Φ − Φ )&tidle;ν ]∕(4πGh2 ). N(i,j,k+1) N(i,j,k) M3 N (i,j,k) N(i,j,k−1) L3
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 18View Image, where one considers a rather typical baryonic galaxy model with a small bulge and a large disk. Applying Eq. 35View Equation (with the ν-function of Eq. 43View Equation) then yields the phantom density [253Jump To The Next Citation Point]. 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 [50Jump To The Next Citation Point] 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 [226Jump To The Next Citation Point], and that it can be negative [297, 490Jump To The Next Citation Point], 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.
View Image

Figure 18: (a) Baryonic density of a model galaxy made of a small Plummer bulge with a mass of 2 × 108M ⊙ and Plummer radius of 185 pc, and of a Miyamoto–Nagai disk of 1.1 × 1010M ⊙, a scale-length of 750 pc and a scale-height of 300 pc. (b) The derived phantom dark matter density distribution: it is composed of a spheroidal component similar to a dark matter halo, and of a thin disk-like component (Figure made by Fabian Lüghausen [253])

Finally, let us note that, as shown in [319, 509Jump To The Next Citation Point], (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

-------- ∫ [ ] W = − 2∘ GM 3a − -1--- − 3a2Q (|∇ Φ |2∕a2) + 2ν(|∇Φ |∕a )|∇ Φ |2 d3x, (36 ) 3 0 4πG 2 0 N 0 N 0 N
meaning that for a system entirely in the extremely weak field limit where −1∕2 ν (y ) = y and Q (z) = (4∕3)z3∕4, the second term vanishes and we get √ -------- W = (− 2 ∕3) GM 3a0, 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].
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