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. . . LIKE AN
INFINITE NUMBER OF GALACTIC SUPERCLUSTERS IN FLATLAND
SCENARIO 1: EXPONENTIAL EXPANSION
WITH DECELERATING GROWTH
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part 1A: Graphical Model of the Universe (begin)
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Below is presented a figure the computer image
file for which is entitled "radius 1d", hence the title component. The green
locus of points or trace depicts exponential expansion with deceleration in the
modern epoch (equation 3). The red trace depicts the first derivative of this
curve (equation 4), velocity of expansion. The yellow curve depicts the Hubble
parameter trace (of equation 6). The relativistic time form is also shown (sky
blue, defining e according to equation 1), as would be the second
derivative of the expansion curve (equation 5). The trace of equation 5,
acceleration or deceleration, does not appear in this plot because it is
negative for this case except when t is very very small.
Note that the abscissa denotes values for other variables besides the labeled
scale factor or light horizon radius of the universe. It also denotes H0,
H, speed or velocity of expansion and the acceleration of that speed, all in
natural units.
Edwin Hubble himself
referred to recession “velocity”. This is obviously the correct term for we are
living in a hyper-dimensional universe according to the relativity paradigm.
Even though the abscissa may be taken to represent a non-dimensional scale
factor, its dimensionality connotations are paramount. The universe is expanding
radially, so its speed of expansion is directed along a radius and is thus a
vector. So, it should be referred to as a velocity.
FIGURE 2
THE CASE OF DECELERATING GROWTH
Change the label of the abscissa to match the
nature of the curve being viewed, all plots being on the same scale
Linear expansion is also
shown, describing expansion of the universe as if it had always occurred at the
speed of light (equation 2). This is included as a reference, as is the black
linear regression "Hubble Data, H0 Line" (equation 7) that follows
the path set by some of the best empirical values for the Hubble constant that
have been determined to date. See Tony Smith's homepage under the keyword
“cosconsensus” for these values3. Smith’s website is a good one for
compilations of data from disparate sources, that is, as a reference resource,
but no endorsement is given herein for his personal theories.
Equation 7 has been determined by linear regression from the above cited
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
using Rpresent = 13.72 gigalightyears.
At first glance, it seems a
bit extreme to represent the true value for H0 to be over 80% of the
speed of light for nearby objects at t = 0, as is shown in Fig. 2. But the
conversions or transformations of coordinates that were done to give this result
were double checked. Also, the converted value of H0 got from the CMB
data, the data point at the far left on the black, Hubble Data, H0
linear regression line, is credible since all visible light must be red-shifted
all the way down to the microwave region of the electromagnetic spectrum. So, an
H0 of around 95% of c is perfectly consistent. If this CMB red-shift
value is correct, the others must also be right. By the way, the second point
from the left was obtained from deep space supernova 1a data.
Besides, when applying H0,
one must remember that it is a rate per unit of distance from Earth and it
cannot be compared directly with c. Here, the units are "fraction of a universe
radius per second per fractional universe distance". So, to find recession
velocity, one must multiply the value of H0 that one wishes to apply
by the fraction of the universe radius from t = 1. Thus, if H0 for
nearby objects is 0.85, then the recession velocity for nearby objects is about
0.85 x 0.1 = 0.085 or about 8% of the speed of light. Still, H0 can
serve as a signal for whether expansion is accelerating or decelerating. The so
called "deceleration parameter" could easily be computed and plotted but, it
would be redundant.
Again, note that this set of curves depicts deceleration of expansion in the
present era. Significantly and amazingly enough, the semi-theoretical Hubble
parameter trace intersects the vertical time
=
1 line (denoting the present) at the same
point as the black, empirical, linear regression Hubble constant line, as it
most certainly should. It also intersects the t = 1 line at the same point as
does the first derivative or velocity of recession curve to give a multiple
intersect.
There are possible
situations imaginable wherein the velocity curve and the other derivatives might
be decoupled from the expansion curve. But the fact that any potential
perturbation must be set at zero is significant in itself. So, this fact must be
represented by admitting the derivatives as if they were independent equations.
Besides, this degree of
consistency suggests that these equations with this particular selection of
parameter values may correctly describe what is actually happening in the
universe today because it is so true to obvious empirical reality.
Note further that the
Hubble constant regression line intersection with the t = 0 line estimates the
value of H0 that would be measured for very nearby objects if there
was no local gravitational field and their proper motion could be eliminated. It
is the H0 of HERE as well as of NOW. H0 determinations
using ranges of very distant objects are the expansion rates for far over THERE
and for way back THEN. General relativity notwithstanding, this is a crucial
common sense notion and more will be said about it.
Some might argue
that if there was acceleration in distant space or deep time, then there is
acceleration now because motion is relative. But, acceleration is not the same
as mere motion, Einstein teaches. And, there cannot be any semblance of
simultaneity so, the inconsistency of this notion with the data of H0
determinations is real.
All by itself, the former
paragraph cripples the conclusions of acceleration and dark energy. Plus, this
is a very strong point, as is the fact that the Hubble constant is most
certainly NOT constant. Yet some still speak as if it is, especially
journalists. Now, presented here is a graphically, algebraically and faithfully
constructed representation of the Big Bang phenomenon using extensive variables
from 10-59 universe lifetime [ x 13.72 gigayears/universe lifetime x
3.12 x 107 seconds/year = 4.28 x 10-42 second ] to about
31 gigayears. (Time, t = 1, is at 13.72 gigayears.)
The criticism leveled by
George Ellis is very cogent4. The Cosmological Principle is the key.
If the CP is not quite right, redshifts could be enhanced by a steeper than
expected climb up a gravitational well since light may have to travel into a
relative void to get here. The symmetry argument against the Ellis Phenomenon is
not valid. Observers need not be near the center of the void and may thereby not
be held to be violating the Copernican Principle by honoring Ellis. They need
only to be deep enough into an irregularity in the void.
We cannot map the whole sky
for the CMB signal. A huge swath is blocked by the galactic plane. The Milky Way
Galaxy has a large recession velocity in space-time that greatly distorts the
CMB signal too. Much information may have been lost in this distortion. There is
no way to make perfect correction for it.
Another thing is that there
is no proof nor even evidence that it is acceptible to assume that the universe
behaves at all times like an ideal gas as long as the Cosmological Principle
holds true. There is good enough reason to suppose it may be true during the
inflationary period, but not in more recent times. It is, in fact, most
certainly not an ideal gas. It satisfies none of the characteristics of any kind
of gas at all, much less an ideal gas. Space-time itself cannot truly be like an
ideal gas in all respects either. But density, pressure and temperature changes
must be applied in some way to a fluid like an ideal gas in order to explain the
putative “dark energy” result of acceleration.
And, if relativity may
treat the space-time continuum like an ideal fluid out of the necessity to use
some simple analogy in order to make computations tractable and comprehensible,
we must still keep in the back of our minds the notion that relativistic
extensions of fluid dynamics protocol to the whole universe may be a bit of a
stretch.
The space-time continuum is
still just the aether by another name. While it does not have the hydrodynamic
characteristics that used to be assumed, it may still have characteristics that
we have not yet even imagined. We don’t know. Many seem afraid to publicly admit
it. The quip that “If we knew what we were doing, it wouldn’t be research!” is
all too true, but the implications are too often disregarded.
For this first scenario:
in the plot describing the decelerating scenario, there are several other points
where there are multiple intersections on and with the time = 1 or radius = 1
line or both. Just as derivative equations are considered to be independent for
this purpose, the boundary lines are treated as equations in their own right.
This is absolutely necessary if one is to be able to derive the total number of
extant dimension of the universe (eleven) from this diagram.
Natural units must be used
to represent the traces of the equations involved and when this is done, the
multiple intersections are seen to represent points where a mere transformation
of coordinates or relabeling the axes can switch between them. The changes occur
with no effect on the form of the equations or the values for the parameters or
variables. Therefore, it is believed that these points represent invariances5.
It can be seen that they would be graphically arrayed symmetrically if they were
plotted on a different grid, say, with semi-log or log-log scales. And, they are
mathematically symmetrical as is shown by inspection.
The most important multiple
intersections occur at t = 1, the only time that really makes any difference.
So, according to Noether’s theorem6, these invariances and the
symmetries they directly imply always indicate some kind of conservation law and
implicate, at least indirectly, a fundamental physical constant.
Note that there is no
contradiction in showing the instantaneous expansion of the universe progressing
initially faster than the speed of light (Fig. 2 and Eq. 3) and an H0
determined to be rather less than 1.0 or significantly less than the speed of
light. H0 determinations measure only the average velocity of
expansion between HERE and NOW and THERE and THEN. But, if the universe’s
expansion is accelerating, “measured” or extrapolated H0 must be
greater than 1.0 in the present era for nearby objects, but it is not. Also, an
extrapolated or corrected H0 > 1.0 should mean that the observer must
be catching up to light that was emitted previously. Thus, the CMB would not be
precisely what it is supposed to be.
Furthermore, besides
mathematical symmetry, there are many more such graphical confluences of such
intersecting curves and boundary lines in the case of deceleration than in the
case of acceleration. (See below.)
Again, yes, some may
quibble that derivative equations are not independent of the original form. But,
it is thought that since it is the entire universe that is being dealt with
here, and even the foundations of existence and logic itself, each equation, in
this instance, may be considered to be independent. There may be a perturbing
complex force field or sweeping warp in the space-time continuum, or even a meta
continuum that could make these equations truly independent. That there is
apparently no perturbation is itself significant and needs to be taken into
account by admitting derivatives as independent. Besides, the empirical linear
regression line is truly independent and should qualify as part of the
mathematical model.
Physicists generally prefer
the alternative that shows the highest symmetry, that is, invariances under
mathematical symmetry operations, which also implies graphical symmetry.
Mathematical versus graphical representation denotes certain axes, categories,
dimensions or other types of symmetry. So, graphical symmetry should count just
as strongly.
Continued in part 2
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part 1A: Graphical Model of the Universe (begin)
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