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.

"radius_1" Depiction, Acceleration Scenario, multiple plots

RADIUS_1 DEPICTION

The next figure in the following post describes the case wherein the universe is expanding at an accelerating rate in the present era. Note that the Hubble parameter curve does not intersect t=1 at a reasonable point and that there are few points of multiple intersection.

By the way, the equations for the derivative curves were found by using the Mathematica website that features a free utility that finds derivatives. Of course, the first derivative represents the speed of expansion and the second derivative stands for the acceleration or deceleration.

Graphically, this scenario displays less symmetry and does not much illustrate Noether's theorem. Remember, these equations are semi-theoretical as well as based on phenomenology. So, being theoretical in part, they should be potentially able to respect Noether's theorem.

"radius_1c", Radius vs Time, Acceleration Scenario

"Radius 1c", Depiction of the Derivative of Exponential Deceleration

DEPICTION OF THE DERIVATIVE OF EXPONENTIAL DECELERATION

In the next post, I give the last figure. There, the plot represents the result of taking the first derivative of the exponential expansion equation that shows deceleration in the modern epoch. Looking at a short interval of time very near the beginning, we see that the speed of expansion runs through a minimum.

This may be important because these exponential curves show no induction period which is essential to serve their purpose of providing an early time for equilibration of the universe resulting in the eventual thermal homogeneity that we see now. The speed of expansion temporarily drops far below the speed of light, almost to zero, to give a minimum at a very early time. The case wherein the expansion accelerates in the modern era does not display this minimum. So, this is another reason to favor the deceleration scenario if one requires a time in the early history of the universe when equilibration could occur.

I wonder why the equilibration requirement is present in the first place. The Big Bang was not an explosion. It was a sudden expansion of space-time. There was not necessarily any turbulence to act as a source of inhomogeneity that would need to be smoothed out. Indeed, if the universe did in fact originate in a singularity, expansion from a single point should automatically result in a universe that is homogeneous and isotropic, satisfying the cosmological principle. To suppose otherwise requires a mechanism to produce turbulent expansion and none is forthcoming.

So, in order to favor the deceleration scenario, one would need to explain the supernova 1a results that indicate that the universe may be undergoing accelerating expansion in the present epoch. I would say that perhaps these results are a sort of optical illusion stemming from the possibility that when the universe is observed at such early times, it is observed as a more and more purely relativistic object. The tiny residual positive curvature of space-time may become sufficient to produce a kind of distortion of our measurements. Our gross difference in perspective results in what naively looks as if the universe is expanding at an accelerating rate. In other words, acceleration is an artifact.

The apparent development of galaxy clusters being slower in the very early universe than they are in the more recent epoch can be explained in similar fashion in that this is an artifact; this time, of the universe being very young. Youth, per se, must have some consequence, after all.

All plots herein were produced using the mathematical analysis program TK Solver Plus 5.0 by UTS.

Figure 2, Radius vs Time, radius_1c