The Deceleration Scenario

THE CASE OF EXPONENTIAL DECELERATION

In this post, I prepare to present a figure in the next post entitled "radius 1d" wherein the green trace depicts exponential deceleration (equation 3), the red trace depicts the first derivative of this curve (equation 4) and the greenish yellow curve depicts the Hubble parameter curve (equation 6). The relativistic time form is also shown (equation 1), as is the second derivative (equation 5).

Linear expansion is also shown, describing expansion of the universe if it had occurred at the speed of light (equation 2). This is included as a reference, as is the black linear regression "Hubble Data Line" (equation 7) that follows the path set by the best empirical values for the Hubble constant that have been determined. See Tony Smith's homepage under the keyword "cosconsensus" for these values.

Equation 7 has been determined from empirical values for the Hubble constant converted to natural units using the average value for the range of distances that were used in each determination and transformed into distances from the origin instead of distances from Earth.

Note that this set of curves depict deceleration in the present era, the Hubble parameter trace intersects the vertical time=1 line at the same point as the empirical linear regression Hubble constant line, as it should. It also intersects the t=1 line at the same point as does the first derivative curve to give a multiple intersect. This suggests to me that these equations with this particular selection of parameter values may correctly describe what is actually happening in the universe today.

There are several other points where there are multiple intersections on the time=1 or radius=1 line. We must use natural units to represent the traces of these equations and when we do so, the multiple intersections represent points where a mere transformation of coordinates can switch us between curves. Therefore, I believe that these points represent invariances and it is can be seen that they would be graphically arrayed symmetrically if they were plotted on a different grid, say, with log-log scales. And, they are mathematically symmetrical.

So, according to Noether’s theorem, they define one or another of the truly fundamental physical constants and conservation laws. Furthermore, besides mathematical symmetry, there are more such confluences of intersecting curves in this depiction than there are in the one displaying acceleration (see below). Physicists generally prefer the alternative that shows the highest symmetry, mathematical and graphical. Mathematical versus graphical representation denotes one of the axes or types of symmetry, after all.