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.

New Empirical Numeric Model of the Evolution of the Universe

Phenomenological Mathematical Model of the Universe

I think that once galaxies have begun to form, the origin of the universe was quite done and its fate was sealed. Undoubtedly, there will be more progress in the physicodynamics of galaxy formation. Once discrete large scale structures begin to form in the universe, the assumption of the cosmological principle that the universe is homogeneous and isotropic begins to break down. But, before this time, it was probably perfectly appropriate.

I have never seen a mathematical model of the "origin" of the universe and its evolution, growth and development carried through to the present era and beyond. Such a model has been inferred from the data, but I don't think it has been explicitly expressed in analytic mathematical terms. Using Guth's inflationary scenario, specifically his premise of an exponential burst of growth near the very beginning, I have developed a mathematical model that is capable of depicting the evolution of the universe from 10 ^ -60 sec to 30 gigayears and beyond.

By using various choices for parameter values, it is possible with this model to represent a universe that putatively follows a trajectory true to empirical data and transitions to a new era of "neo-inflation" or dominance by dark energy. Or, it can move toward a period of deflation toward collapse in the modern epoch. Or else, it may evolve toward "flat" expansion at a constant rate. Adjustable parameters make it possible for my model to mimic any empirically determined behavior with but very few input values. The plots that I can produce may have significance beyond their graphic utility for visualization.

I use Einstein's natural units to represent the "radius" or radius of curvature of the universe, time and the speed of light. This is the only way that Guth-like equations make any sense and are definable. A plot of these equations, their 1st and 2nd derivatives, H and the expansion of the universe if it had expanded at the speed of light, seems to show a sort of symmetry.

The existence and positions of multiple intersection points where these curves cross in interesting ways portrays an array that might be interpreted as a representation of several kinds of invariance. I am tempted to interpret this array of intersections as a highly symmetric matrix of invariances, also invariant and symmetric in their superimposable, rotational and other transforms of coordinate systems, bespeaking various fundamental constants according to Noether's Theorem.

Revealed here is an interesting problem. I refer to my equations as empirical or phenomenological models that are based fundamentally on theory so that it should be possible to extrapolate successfully into the past and future alike. Using parameters that give a result consistent with the onset of a new epoch of inflation in "recent" times, this form of the model does not provide an early time when the nascent universe could have equilibrated.

Essentially, Guth's idea fixed the major criticisms of the Big Bang Theory by postulating that equilibration occurred during the exponential induction period when the universe was very small and that the transition to exponential growth then locked in the homogeneity that had been achieved. But, with the choice of parametric values giving the required result of that new inflation attributed to dark energy, there is no such induction period.

And, because there are fewer multiple intersection points displayed by this type of plot, it displays less of the above described symmetry than a plot of a type that displays developmental deceleration. Plots showing deceleration show higher "symmetry" and they provide a type of induction period in the traces of their curves. Mathematical physicists prefer theories that possess more of this kind of symmetry.

I want to combine the two types of equations by merely adding them together with variable weighting factors as is done with hybrid solutions to a quantum wave equation. The weights might themselves follow a simple equation that emphasizes the deceleration scenario in the first few moments of the universe's existence and the acceleration scenario later in its evolutionary trajectory. Then, the early induction period would be preserved while the transition to new inflation would be allowed. I want to see if my special symmetry is preserved and see what my being able to do this might imply. I will expand upon this in a future post that will be entitled "The Quantum Universe".

Of course, if the universe really did originate with a singularity, no induction period would be required. Guth's inflation might not be needed after all. If the trajectories of all waves and particles originated at the same point, they would progress throughout the whole with at least some of the same "original" intensive properties, like temperature.

The Big Bang, the origin of the universe, was not an explosion. If space-time expanded from a single point, it is unnecessary to postulate any form of unevenness or turbulence that might give rise to inhomogeneities that would need to be suppressed or quenched. Precisely what cause could be cited that would result in an uneven expansion of space-time or a turbulent beginning with consequent temperature variations? Are not temperature variations simply assumed? Why? The failure of the explosion metaphor must have a consequence.

Gary Kent