6.2 The interpolating function

The basis of the MOND paradigm is to reproduce Milgrom’s law, Eq. 7View Equation, in highly symmetrical systems, with an interpolating function asymptotically obeying the conditions of Eq. 8View Equation, i.e., μ(x) → 1 for x ≫ 1 and μ(x) → x for x ≪ 1. Obviously, in order for the relation between g and g N to be univocally determined, another constraint is that x μ(x) must be a monotonically increasing function of x, or equivalently
μ(x) + xμ ′(x) > 0, (37 )
or equivalently
d-ln-μ-> − 1. (38 ) d ln x
Even though this leaves some freedom for the exact shape of the interpolating function, leading to the various families of functions hereafter, let us insist that it is already extremely surprising, from the dark matter point of view, that the MOND prescriptions for the asymptotic behavior of the interpolating function did predict all the aspects of the dynamics of galaxies listed in Section 5.

As we have seen in Section 6.1, an alternative formulation of the MOND paradigm relies on Eq. 10View Equation, based on an interpolating function

ν (y ) = 1 ∕μ(x) where y = x μ(x). (39 )
In that case, we also have that yν(y ) must be a monotonically increasing function of y.

Finally, as we shall see in detail in Section 7, many MOND relativistic theories boil down to multifield theories where the weak-field limit can be represented by a potential ∑ Φ = iϕi, where each ϕi obeys a generalized Poisson equation, the most common case being

Φ = ΦN + ϕ, (40 )
where ΦN obeys the Newtonian Poisson equation and the scalar field ϕ (with dimensions of a potential) plays the role of the phantom dark matter potential and obeys an equation of either the type of Eq. 17View Equation or of Eq. 30View Equation. When it obeys a QUMOND type of equation (Eq. 30View Equation), the ν-function must be replaced by the &tidle;ν-function of Eq. 32View Equation. When it obeys a BM-like equation (Eq. 17View Equation), the classical interpolating function μ(x ) acting on x = |∇ Φ |∕a0 must be replaced by another interpolating function &tidle;μ (s ) acting on s = |∇ ϕ|∕a0, in order for the total potential Φ to conform to Milgrom’s law28. In the absence of a renormalization of the gravitational constant, the two functions are related through [145Jump To The Next Citation Point]
&tidle;μ(s) = (x − s)s−1 where s = x[1 − μ (x)]. (41 )
For x ≪ 1 (the deep-MOND regime), one has s = x(1 − x) ≪ 1 and x ∼ s(1 + s), yielding &tidle;μ (s ) ∼ s, i.e., although it is generally different, &tidle;μ has the same low-gravity asymptotic behavior as μ.

In spherical symmetry, all these different formulations can be made equivalent by choosing equivalent interpolating functions, but the theories will typically differ slightly outside of spherical symmetry (i.e., the curl field will be slightly different). As an example, let us consider a widely-used interpolating function [141Jump To The Next Citation Point, 166Jump To The Next Citation Point, 402Jump To The Next Citation Point, 508Jump To The Next Citation Point] yielding excellent fits in the intermediate to weak gravity regime of galaxies (but not in the strong gravity regime of the Solar system), known as the “simple” μ-function (see Figure 19View Image):

--x--- μ (x ) = 1 + x. (42 )
This yields 2 y = x ∕(1 + x), and thus ∘ -2------ x = (y + y + 4y)∕2 and ν = (1 + x)∕x yields the “simple” ν-function:
−1 1∕2 ν(y) = 1-+-(1 +-4y--)---. (43 ) 2
It also yields s = x [1 − μ (x)] = x ∕(1 + x ) = μ, and hence x = s∕ (1 − s), yielding for the “simple” &tidle;μ-function:
s μ&tidle;(s) = -----. (44 ) 1 − s
A more general family of &tidle;μ-functions is known as the α-family [15Jump To The Next Citation Point], valid for 0 ≤ α ≤ 1 and including the simple function of the α = 1 case 29:
---s--- μ&tidle;α(s) = 1 − αs (45 )
corresponding to the following family of μ-functions:
μα(x) = ----------------2x----------------. (46 ) 1 + (2 − α)x + [(1 − αx)2 + 4x]1∕2
The α = 0 case is sometimes referred to as “Bekenstein’s μ-function” (see Figure 19View Image) as it was used in [33Jump To The Next Citation Point]. The problem here is that all these μ-functions approach 1 quite slowly, with ζ ≤ 1 in their asymptotic expansion for x → ∞, μ(x) ∼ 1 − Ax −ζ. Indeed, since s = x[1 − μ(x)], its asymptotic behavior is −ζ+1 s ∼ Ax. So, if ζ > 1, s → 0 for x → ∞ as well as for x → 0, which would imply that x(s) = s&tidle;μ(s) + s would be a multivalued function, and that the gravity would be ill-defined. This is problematic because even for the extreme case ζ = 1, the anomalous acceleration does not go to zero in the strong gravity regime: there is still a constant anomalous “Pioneer-like” acceleration x [1 − μ(x )] → A, which is observationally excluded30 from very accurate planetary ephemerides [154Jump To The Next Citation Point]. What is more, these &tidle;μ-functions, defined only in the domain 0 < s < α −1, would need very-carefully–chosen boundary conditions to avoid covering values of s outside of the allowed domain when solving for the Poisson equation for the scalar field.

The way out to design &tidle;μ-functions corresponding to acceptable μ-functions in the strong gravity regime is to proceed to a renormalization of the gravitational constant[145Jump To The Next Citation Point]: this means that the bare value of G in the Poisson and generalized Poisson equations ruling the bare Newtonian potential ϕN and the scalar field ϕ in Eq. 40View Equation is different from the gravitational constant measured on Earth, GN (related to the true Newtonian potential Φ N). One can assume that the bare gravitational constant G is related to the measured one through

GN = ξG, (47 )
meaning that x = y + s where x = ∇ Φ∕a 0, y = ∇ ϕ ∕a = ∇ Φ ∕ (ξa ) N 0 N 0, and sμ&tidle;(s) = y. We then have for Milgrom’s law:
xμ(x ) = ξ(x − s) = ξs &tidle;μ(s). (48 )
In order to recover μ (x) → 1 for x → ∞, it is straightforward to show [145Jump To The Next Citation Point] that it suffices that &tidle;μ(s) → &tidle;μ0 for s → ∞, and that ξ = 1 + μ&tidle;−01. Then, if ζ > 1 in the asymptotic expansion μ(x) ∼ 1 − x−ζ, one has s ∼ (1 + &tidle;μ− 1)− 1x −ζ+1 + (1 + μ&tidle; )−1x 0 0. This second linear term allows s to go to infinity for large x and thus x(s) to be single-valued. On the other hand, for the deep-MOND regime, the renormalization of G implies that &tidle;μ(s) → s∕ξ for s ≪ 1.

We can then use, even in multifield theories, μ-functions quickly asymptoting to 1. For each of these functions, there is a one-parameter family of corresponding &tidle;μ-functions (labelled by the parameter &tidle;μ (∞ ) = &tidle;μ0), obtained by inserting μ(x) into − 1 s = x[1 − ξ μ(x)] and making sure that the function is increasing and thus invertible. A useful family of such μ-functions asymptoting more quickly towards 1 than the α-family is the n-family:

μ (x) = ----x-----. (49 ) n (1 + xn )1∕n
The case n = 1 is again the simple μ-function, while the case n = 2 has been extensively used in rotation curve analysis from the very first analyses [28Jump To The Next Citation Point, 223], to this day [401Jump To The Next Citation Point], and is thus known as the “standard” μ-function (see Figure 19View Image). The corresponding &tidle;μ-function for n ≥ 2 has a very peculiar shape of the type shown in Figure 3 of [81Jump To The Next Citation Point] (which might be considered a fine-tuned shape, necessary to account for solar system constraints). On the other hand, the corresponding ν-function family is:
[ 1 + (1 + 4y−n)1∕2]1∕n νn (y ) = ----------------- . (50 ) 2
As the simple μ-function (α = 1 or n = 1) fits galaxy rotation curves well (see Section 6.5.1) but is excluded in the solar system (see Section 6.4), it can be useful to define μ-functions that have a gradual transition similar to the simple function in the low to intermediate gravity regime of galaxies, but a more rapid transition towards one than the simple function. Two such families are described in [325Jump To The Next Citation Point] in terms of their ν-function:
−y −1∕2 −y νβ(y) = (1 − e ) + βe (51 )
− yγ∕2 −1∕γ − 1 − yγ∕2 νγ (y ) = (1 − e ) + (1 − γ )e . (52 )
Finally, yet another family was suggested in [274Jump To The Next Citation Point], obtained by deleting the second term of the γ-family, and retaining the virtues of the n-family in galaxies, but approaching one more quickly in the solar system:
−yδ∕2− 1∕δ νδ(y) = (1 − e ) . (53 )
To be complete, it should be noted that other μ-functions considered in the literature include [304Jump To The Next Citation Point, 505Jump To The Next Citation Point] (see also Section 7.10):
21∕2 μ(x) = (1-+-4x--)--−--1, (54 ) 2x
−3 μ (x) = 1 − (1 + x ∕3) . (55 )
This simply shows the variety of shapes that the interpolating function of MOND can in principle take31. Very precise data for rotation curves, including negligible errors on the distance and on the stellar mass-to-light ratios (or, in that case, purely gaseous galaxies) should allow one to pin down its precise form, at least in the intermediate gravity regime and for “modified inertia” theories (Section 6.1.1) where Milgrom’s law is exact for circular orbits. Nowadays, galaxy data still allow some, but not much, wiggle room: they tend to favor the α = n = 1 simple function [166Jump To The Next Citation Point] or some interpolation between n = 1 and n = 2 [141Jump To The Next Citation Point], while combined data of galaxies and the solar system (see Sections 6.4 and 6.5) rather tend to favor something like the γ = δ = 1 function of Eq. 52View Equation and Eq. 53View Equation (which effectively interpolates between n = 1 and n = 2, see Figure 19View Image), although slightly higher exponents (i.e., γ > 1 or δ > 1) might still be needed in the weak gravity regime in order to pass solar system tests involving the external field from the galaxy [62Jump To The Next Citation Point]. Again, it should be stressed that the most salient aspect of MOND is not its precise interpolating function, but rather its successful predictions on galactic scaling relations and Kepler-like laws of galactic dynamics (Section 5.2), as well as its various beneficial effects on, e.g., disk stability (see Section 6.5), all predicted from its asymptotic form. The very concept of a pre-defined interpolating function should even in principle fully disappear once a more profound parent theory of MOND is discovered (see, e.g., [22Jump To The Next Citation Point]).
View Image

Figure 19: Various μ-functions. Dotted green line: the α = 0 “Bekenstein” function of Eq. 46View Equation. Dashed red line: the α = n = 1 “simple” function of Eq. 46View Equation and Eq. 49View Equation. Dot-dashed black line: the n = 2 “standard” function of Eq. 49View Equation. Solid blue line: the γ = δ = 1 μ-function corresponding to the ν-function defined in Eq. 52View Equation and Eq. 53View Equation. The latter function closely retains the virtues of the n = 1 simple function in galaxies ( x < ∼ 10 ), but approaches 1 much more quickly and connects with the n = 2 standard function as x ≫ 10.

To end this section on the interpolating function, let us stress that if the μ-function asymptotes as μ (x ) = x for x → 0, then the energy of the gravitational field surrounding a massive body is infinite [38Jump To The Next Citation Point]. What is more, if the &tidle;μ function of relativistic multifield theories asymptotes in the same way to zero before going to negative values for time-evolution dominated systems (see Section 9.1), then a singular surface exists around each galaxy, on which the scalar degree of freedom does not propagate, and can therefore not provide a consistent picture of collapsed matter embedded into a cosmological background. A simple solution [145Jump To The Next Citation Point, 380] consists in assuming a modified asymptotic behavior of the μ-function, namely of the form

μ(x) ∼ 𝜀0 + x for x ≪ 1. (56 )
In that case there is a return to a Newtonian behavior (but with a very strong renormalized gravitational constant GN ∕𝜀0) at a very low acceleration scale x ≪ 𝜀0, and rotation curves of galaxies are only approximately flat until the galactocentric radius
∘ ------- 1 GN M R ∼ -- ------. (57 ) 𝜀0 a0
Thus, one must have 𝜀0 ≪ 1 to not affect the observed phenomenology in galaxies. Note that the μ-function will never go to zero, even at the center of a system. Conversely, in QUMOND and the like, one can modify the ν-function in the same way:
1 ν(y) ∼ 𝜀--+-y1∕2 for y ≪ 1. (58 ) 0

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