Model for the Inflationary Growth of the Universe

PHENOMENOLOGICAL EQUATION
FOR THE EXPONENTIAL CONTRACTION
OR EXPANSION OF THE UNIVERSE

I do not believe that anyone has ever published an intuitive equation that purports to graphically describe, extensively (not intensively), the expansion of the universe according to the current standard model from near the beginning of time to the present and beyond. By means of a system of Edisonian trial and error guided by theory, boundary conditions and empirical evidence, I have produced such an equation. I consider it to be semi-theoretical but with strong empirical input from observation of the phenomenon. Eventually, I hope, It will be derived from first principles.

I started with Allan Guth’s concept of initial exponential inflationary expansion followed by Hubble’s idea of more moderate expansion beyond this preparatory process. By varying parameters, deceleration, constant expansion or acceleration in the modern era can be reproduced as well as initial exponential inflationary expansion. Guthian inflation, according to these equations, has a few twists, as we shall see.

I use Einstein’s natural units because no other basis will allow exponentials of the type that I employ to work. Therefore, the current "radius" or radius of the light horizon of the universe, the speed of light and the current age of the universe are all set equal to 1. Equation 3 is an idealized representation of Guth’s exponential inflation. It depends on the "relativistic" time form given in equation 1 which contains the parameter e.

To force the equation to describe deceleration in the present era, a value above about e=100 should be chosen. To produce an equation that describes acceleration in the present era, a value of rather lower than e=100 should be chosen. The values of A and B are chosen to make the equation conform to empirical boundary conditions, including the fact that the exponential expansion curve should always pass through the points (0,0) and (1,1).

In other words, once the boundary conditions are satisfied, e controls only the curvature.

See the next post for two tables of the equations I used with their variables and parameters. Sorry, the equations are written in single line format, that is, using nested parentheses, (() ), ^ for exponentiation, etc.