As we have seen in Section 6.1, an alternative formulation of the MOND paradigm relies on Eq. 10, based on an interpolating function
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 , where each obeys a generalized Poisson equation, the most common case being28. In the absence of a renormalization of the gravitational constant, the two functions are related through 
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 [141, 166, 402, 508] 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” -function (see Figure 19):, valid for and including the simple function of the case 29: . The problem here is that all these -functions approach quite slowly, with in their asymptotic expansion for , . Indeed, since , its asymptotic behavior is . So, if , for as well as for , which would imply that would be a multivalued function, and that the gravity would be ill-defined. This is problematic because even for the extreme case , the anomalous acceleration does not go to zero in the strong gravity regime: there is still a constant anomalous “Pioneer-like” acceleration , which is observationally excluded30 from very accurate planetary ephemerides . What is more, these -functions, defined only in the domain , would need very-carefully–chosen boundary conditions to avoid covering values of outside of the allowed domain when solving for the Poisson equation for the scalar field.
The way out to design -functions corresponding to acceptable -functions in the strong gravity regime is to proceed to a renormalization of the gravitational constant: this means that the bare value of in the Poisson and generalized Poisson equations ruling the bare Newtonian potential and the scalar field in Eq. 40 is different from the gravitational constant measured on Earth, (related to the true Newtonian potential ). One can assume that the bare gravitational constant is related to the measured one through that it suffices that for , and that . Then, if in the asymptotic expansion , one has . This second linear term allows to go to infinity for large and thus to be single-valued. On the other hand, for the deep-MOND regime, the renormalization of implies that for .
We can then use, even in multifield theories, -functions quickly asymptoting to 1. For each of these functions, there is a one-parameter family of corresponding -functions (labelled by the parameter ), obtained by inserting into and making sure that the function is increasing and thus invertible. A useful family of such -functions asymptoting more quickly towards 1 than the -family is the -family:[28, 223], to this day , and is thus known as the “standard” -function (see Figure 19). The corresponding -function for has a very peculiar shape of the type shown in Figure 3 of  (which might be considered a fine-tuned shape, necessary to account for solar system constraints). On the other hand, the corresponding -function family is:  in terms of their -function: , obtained by deleting the second term of the -family, and retaining the virtues of the -family in galaxies, but approaching one more quickly in the solar system: [304, 505] (see also Section 7.10): 31. 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 simple function  or some interpolation between and , while combined data of galaxies and the solar system (see Sections 6.4 and 6.5) rather tend to favor something like the function of Eq. 52 and Eq. 53 (which effectively interpolates between and , see Figure 19), although slightly higher exponents (i.e., or ) might still be needed in the weak gravity regime in order to pass solar system tests involving the external field from the galaxy . 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., ).
To end this section on the interpolating function, let us stress that if the -function asymptotes as for , then the energy of the gravitational field surrounding a massive body is infinite . What is more, if the 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 [145, 380] consists in assuming a modified asymptotic behavior of the -function, namely of the form
Living Rev. Relativity 15, (2012), 10
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