Presumably, when the universe formed from Alan Guth's inflaton, its graphical hyperbolic (proportional to 1/r*) gravitational field profile began to collapse into a parabolic (proportional to 1/r^{2}) one (see posts elsewhere). This collapse or transition continues to this day. But surely, the process is almost done. There cannot be an infinite amount of gravitational energy sequestered in the hyperbolic 1/r* field that would be available to fuel acceleration of the expansion rate via such a transformation. That is, transition to a lower energy parabolic 1/r^{2}field must provide a distinctly limited supply of extra impetus. Surely, after 13.72 billion years, the mainspring has almost run down by now.

If the expansion rate is called h, and its present value is called P, then h = P at any given time, t, including the present. The simplest equation for the expansion rate’s effect on P would be an exponential decay expression,

P = h_{0}e^{-rt}, where h_{0}is an initial value for the expansion rate, h, r is the rate of increase in this expansion.

We can get an estimate of a value for h_{0}from Alan Guth’s formulation of the theory of simple inflation. The present values of both the expansion rate, P_{1}, and acceleration rate, r, are observable. We can set t = 1, for the present value of t. So, we can summarize all relevant observations with this simple equation or the associated exponential expansion equation, R = R_{0}e^{rt}, where R is the putative “radius” or scale factor of the universe.

The current value (at t = 1) of the expansion rate is H_{0}, the Hubble “constant”, so P_{1}= H_{0}.

Exponential decay equations exhibit what is called a “dormancy” period or final plateau region. The hyperbolic 1/r* graphical curve levels off near zero and continues to subside gently almost linearly for an indefinite time. The current state of the universe may be consistent with this dormant period. The conclusion here is that acceleration may continue for a long time while slowly decreasing nearer to zero. In other words, even with acceleration of the expansion rate, there does not necessarily have to be a “Big Rip” wherein the fabric of the cosmos is irreparably torn apart as expansion proceeds beyond a certain point.

The essential detail made about point masses and singularities engendering a hyperbolic 1/r* gravitational field is a mathematical necessity. Consider what a point mass as a singularity actually means. If it does not mean that they generate a hyperbolic gravitational field, then the words point-mass (remember Guth) and "singularity" are meaningless. Karl Schwartzchild would not agree with this negation of his analysis of general relativity.

Some say that general relativity predicts that black-hole singularities must possess a gravitational field that falls off as 1/r^{2}with no difference from other Newtonian entities. I don’t believe general relativity says this under the rule that a black-hole must contain or “be” a singularity. If treated as a real singularity (see Schwartzchild metric, Wikipedia), black-holes must have hyperbolic 1/r* gravitational potentials. This is a geometric necessity.

Such a gravitational potential falls off as 1/r, or more accurately, as 1/r*. The symbol r* is the radial distance of a given star multiplied by the dimensioned value of the unit vector associated with the vector quantity of r,which makes dimensional analysis valid in, say, F = GmM/r*. So, this is a hyperbolic equation.r

The super-massive central black-hole, in association with the galactic disk which has a coincident and coaxial gravitational field equivalent to a couple of hundred billion sols, the residual hyperbolic 1/r* gravitational field at large r coincides with Milgrom’s tiny extra gravitational acceleration seen near the periphery of galaxies. The periphery is a self defining precise zone the relative location of which is responsible for Milgrom’s leftover acceleration “constant” that he wants to tack onto Newton’s Law. These observations mean that Milgrom’s MOND is unnecessary. And, Dark Matter as "quintessence" is superfluous too because all the characteristics associated with some deeply embedded Dark Matter phenomenon are explained equally well by the hyperbolic 1/r* galactic black-hole gravitational field effect. The “MOND effect” itself is evidence for the hyperbolic field.

The symbol r* is equal to r times the dimensioned value of the unit vector associated with r,. It keeps the dimensional analysis of F = GmM/r* sensible. But, there are other fundamental reasons to use it.r