7.9 Dipolar dark matter

As we have seen, many relativistic MOND theories do invoke the existence of new “dark fields” (scalar or vector fields), which, if massive, can even sometimes truly be thought of as “dark matter” enjoying non-standard interactions with baryons52 (Section 7.6 and [82]). The bimetric version of MOND (Section 7.8) also invokes the existence of a new type of matter, the “twin matter”. This clearly shows that, contrary to common misconceptions, MOND is not necessarily about “getting rid of dark matter” but rather about reproducing the success of Milgrom’s law in galaxies. It might require adding new fields, but the key point is that these fields, very massive or not, would not behave simply as collisionless particles.

In a series of papers, Blanchet & Le Tiec [55, 56Jump To The Next Citation Point, 57, 58Jump To The Next Citation Point, 59, 60Jump To The Next Citation Point] have pushed further the idea that the MOND phenomenology could arise from the fundamental properties of a form of dark matter itself, by suggesting that dark matter could carry a space-like53 four-vector gravitational dipole moment ξμ, following the analogy between Milgrom’s law and Coulomb’s law in a dieletric medium proposed by [56] (see Eq. 9View Equation) or between the Bekenstein–Milgrom modified Poisson equation and Gauss’ law in terms of free charge density (see Eq. 17View Equation). The dark matter medium is described as a fluid with mass current μ μ J = ρu (where ρ is the equivalent of the mass density of the atoms in a dielectric medium, i.e., it is the ordinary mass density of a pressureless perfect fluid, and uμ is the four-velocity of the fluid54.) endowed with the dipole moment vector ξμ (which will affect the total density in addition to the above mass density ρ), with the following action [60Jump To The Next Citation Point]:

∫ S ≡ d4x √ − g-[c2(J ξ˙μ − ρ) − W (P )], (100 ) DM μ
where P is the norm of the projection perpendicular to the four-velocity (not the norm of the polarization field55) of the polarization field P μ = ρξμ, and where the dot denotes the covariant proper time derivative. The specific dynamics of this dark matter fluid will thus arise from the coupling between the current and the dipolar field (analogue to the coupling to an external polarization field in electromagnetism), as well as from the internal non-gravitational force acting on the dipolar dark particles and characterized by the potential W (P). Let us note that the normal matter action and the gravitational Einstein–Hilbert action are just the same as in GR.

The equations of motion of the dark matter fluid are then gotten by varying the action w.r.t. the dipole moment variable μ ξ and w.r.t. to the current μ J, boiling down in the non-relativistic limit to:

dv- = g − f, (101 ) dt
d2ξ 1 --2-= f + --∇ [W (P) − P W ′(P)] + (P∇ )g, (102 ) dt ρ
where v is the ordinary velocity of the fluid, g = − ∇ Φ is the gravitational field, and f = − (P ∕P )W ′∕ρ is the internal non-gravitational force field making the dark particles motion non-geodesic. What is more, the Poisson equation in the weak-field limit is recovered as:
− ∇.(g − 4 πP ) = 4 πG (ρb + ρ). (103 )
In order to then reproduce the MOND phenomenology in galaxies, the next step is the “weak-clustering hypothesis”, namely the fact that, in galaxies, the dark matter fluid does not cluster much (ρ ≪ ρb) and is essentially at rest (v = 0) because the internal force of the fluid precisely balances the gravitational force in such a way that the polarization field P is precisely aligned with the gravitational one g, and g ∝ − W ′(P ). The potential thus plays the role of the “MOND function”, and, e.g., choosing to determine it up to third order in expansion as
W (P ) ∝ Λ∕(8π ) + 2πP 2 + 16π2P 3∕(3a0) + 𝒪 (P 4) (104 )
then yields the desired MOND behavior in Eq. 103View Equation, with the n = 1 “simple” μ-function (see Eqs. 42View Equation and 49View Equation).

This model has many advantages. The monopolar density of the dipolar atoms ρ will play the role of CDM in the early universe, while the minimum of the potential W (P ) naturally adds a cosmological constant term, thus making the theory precisely equivalent to the ΛCDM model for expansion and large scale structure formation. The dark matter fluid behaves like a perfect fluid with zero pressure at first-order cosmological perturbation around a FLRW background and thus reproduces CMB anisotropies. Let us also note that, if the potential W (P ) defining the internal force of the dipolar medium is to come from a fundamental theory at the microscopic level, one expects that the dimensionless coefficients in the expansion all be of order unity after rescaling by a2 0, thus naturally leading to the coincidence 2 Λ ∼ a0.

However, while the weak clustering hypothesis and stationarity of the dark matter fluid in galaxies are suppported by an exact and stable solution in spherical symmetry [58Jump To The Next Citation Point], it remains to be seen whether such a configuration would be a natural outcome of structure formation within this model. The presence of this stationary DM fluid being necessary to reproduce Milgrom’s law in stellar systems, this theory loses a bit of the initial predictability of MOND, and inherits a bit of the flexibility of CDM, inherent to invoking the presence of a DM fluid. This DM fluid could, e.g., be absent from some systems such as the globular clusters Pal 14 or NGC 2419 (see Section 6.6.3), thereby naturally explaining their apparent Newtonian behavior. However, the weak clustering hypothesis in itself might be problematic for explaining the missing mass in galaxy clusters, due to the fact that the MOND missing mass is essentially concentrated in the central parts of these objects (see Section 6.6.4).

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