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

In that case, we also have that must be a monotonically increasing function of .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 being

where obeys the Newtonian Poisson equation and the scalar field (with dimensions of a potential) plays the role of the phantom dark matter potential and obeys an equation of either the type of Eq. 17 or of Eq. 30. When it obeys a QUMOND type of equation (Eq. 30), the -function must be replaced by the -function of Eq. 32. When it obeys a BM-like equation (Eq. 17), the classical interpolating function acting on must be replaced by another interpolating function acting on , in order for the total potential to conform to Milgrom’s lawIn 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):

This yields , and thus and yields the “simple” -function: It also yields , and hence , yielding for the “simple” -function: A more general family of -functions is known as the -family [15], valid for and including the simple function of the caseThe way out to design -functions corresponding to acceptable -functions in the strong gravity regime is to proceed to a renormalization of the gravitational constant[145]: 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

meaning that where , , and . We then have for Milgrom’s law: In order to recover for , it is straightforward to show [145] 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:

The case is again the simple -function, while the case has been extensively used in rotation curve analysis from the very first analyses [28, 223], to this day [401], 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 [81] (which might be considered a fine-tuned shape, necessary to account for solar system constraints). On the other hand, the corresponding -function family is: As the simple -function ( or ) 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 -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 [325] in terms of their -function: and Finally, yet another family was suggested in [274], 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: To be complete, it should be noted that other -functions considered in the literature include [304, 505] (see also Section 7.10): and This simply shows the variety of shapes that the interpolating function of MOND can in principle takeTo 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 [38]. 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

In that case there is a return to a Newtonian behavior (but with a very strong renormalized gravitational constant ) at a very low acceleration scale , and rotation curves of galaxies are only approximately flat until the galactocentric radius Thus, one must have to not affect the observed phenomenology in galaxies. Note that the -function will never go to zero, even at the center of a system. Conversely, in QUMOND and the like, one can modify the -function in the same way:
Living Rev. Relativity 15, (2012), 10
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