Inflation, the hyperbolic black-hole field effect and the fate of the universe

[G Anthony]

Presumably, when the universe formed from Alan Guth's inflaton, its hyperbolic gravitational field began to collapse into a parabolic one (see post of Sept. 19, 2011). That collapse continues to this day. But, the process is almost done. There cannot be an infinite amount of gravitational energy sequestered in the hyperbolic field that would be available to fuel acceleration of the expansion rate via such a transformation. That is, transition to a lower energy parabolic field must provide a distinctly limited supply of extra impetus. Surely, after 13.72 billion years, the spring 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, including the present. The simplest equation for the expansion rate’s effect on P would be an exponential decay expression, P = h₀e^-rt, where h₀ is an initial value for the expansion rate, h, r is the rate of increase in this expansion and t is time.

 

We can get an estimate of a value for h₀ from Alan Guth’s formulation of the theory of simple inflation. The present values of both the expansion rate, P₁, 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 = R₀e^rt, where R is the putative “radius” or scale factor of the universe.

 

The current value (at t = 1) of the expansion rate is H₀, the Hubble “constant”, so P₁ = H₀.

 

Exponential decay equations exhibit what is called a “dormancy” period or final plateau region. The hyperbolic 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.

Edited November 26, 2011 by G Anthony

[G Anthony]

Because runaway processes are associated with "exponential" growth, using an "exponential" equation, regardless of its classification as growth or decay, has resulted in the assumption that accelerating expansion rate must result in the "Big Rip". My previous reply is intended to correct this false notion.