  Observations of spiral galaxies strongly support a one-to-one analytical
relation between the inferred gravity of dark matter at any radius and the
enclosed baryonic mass. It is baffling that baryons manage to settle the dark
matter gravitational potential in such a precise way, leaving no "messy"
fingerprints of the merging events and "gastrophysical" feedbacks expected in
the history of a galaxy in a concordance Universe. This correlation of gravity
with baryonic mass can be interpreted from several non-standard angles,
especially as a modification of gravity called TeVeS, in which no galactic dark
matter is needed. In this theory, the baryon-gravity relation is captured by
the dieletric-like function mu of Modified Newtonian Dynamics (MOND),
controlling the transition from 1/r^2 attraction in the strong gravity regime
to 1/r attraction in the weak regime. Here, we study this mu-function in
detail. We investigate the observational constraints upon it from fitting
galaxy rotation curves, unveiling the degeneracy between the stellar
mass-to-light ratio and the mu-function as well as the importance of the
sharpness of transition from the strong to weak gravity regimes. We also
numerically address the effects of non-spherical baryon geometry in the
framework of non-linear TeVeS, and exhaustively examine how the mu-function
connects with the free function of that theory. In that regard, we exhibit the
subtle effects and wide implications of renormalizing the gravitational
constant. We finally present a discontinuity-free transition between
quasi-static galaxies and the evolving Universe for the free function of TeVeS,
inevitably leading to a return to 1/r^2 attraction at very low accelerations in
isolated galaxies.
