Graphical Model for the Inflationary

 Accelerating or Decelerating

Growth of the Universe

A SINGLE, COMPLETE SEMITHEORETICAL EQUATION

 FOR THE EXPONENTIAL ACCELERATING, DECELERATING

 OR CONSTANT EXPANSION OF THE UNIVERSE,

INCLUDING INITIAL INFLATION

 

The author does not believe that anyone has ever published a phenomenologically derived semitheoretical equation that purports to graphically describe, using extensive variables (not intensively), the expansion of the universe according to the current standard model from near the beginning of time, including inflation, to the present and beyond. By means of a system of extremely grueling Edisonian (i.e. nonrandom or intuitive, intelligently designed) trial and error guided by theory, boundary conditions and empirical evidence, such an equation is produced. It is considered it to be semi-theoretical but with strong empirical input from observation of the phenomenon. It is very simple. It is hoped and assumed that it will be derived from first principles eventually.

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Starting with Allan Guth’s concept of initial exponential inflationary expansion[1] supported by Hubble’s idea of more moderate expansion[2] beyond this preparatory process a purely exponential form may be proposed. The necessary performance is obtained by varying parameters. All of decelerating growth, constant expansion or acceleration in the modern era and beyond can be reproduced as well as initial exponential inflationary expansion. Guthian inflation, according to these equations, has a few twists, as shall be seen.

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In other words, once the indubitable mathematical boundary conditions are satisfied, the variable “e” controls only the curvature. The main equation in the model, equation 3, must have the proper inflationary initial behavior as well, and it certainly does. If one tries to use this equation using TK Solver Plus or Mathematica, she can easily follow its behavior all the way down to t = 10^-52 second and even less, rather far below the Planck time. In deceleration mode, nothing but monotonic expansion occurs until about t = 10^-14 when expansion pauses for a while. But this very significant pause occurs only for the case of decelerating expansion in the modern era.

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TABLES OF EQUATIONS IN THE MODEL,

DEFINITIONS OF VARIABLES

AND VALUES FOR PARAMETERS

Please click on the title above to view the Tables. Use backarrow <== to return to this forum. Do not stray far or else you may not be able to return by means of the backarrow. Please do not close out of this new web page or you might close out of the forum.

SCENARIO 1: EXPONENTIAL EXPANSION WITH DECELERATING GROWTH

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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 trace depicts exponential expansion with deceleration in the modern epoch (equation 3). The red trace depicts the first derivative of this curve (equation 4), speed or velocity of expansion. The yellow curve depicts the Hubble parameter trace (of equation 6). The relativistic time form is also shown (defining e according to equation 1), as is 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.

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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.

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Note that this set of curves depicts deceleration of expansion in the present era. Significantly, the Hubble parameter trace intersects the vertical time = 1 line (denoting the present) at the same point as the 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. 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.

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For this first scenario, in the plot describing the decelerating scenario, there are several other points where there are multiple intersections on the time = 1 or radius = 1 line or both. We must use natural units to represent the traces of the equations involved and when we do so, the multiple intersections are seen to represent points where a mere transformation of coordinates or relabeling the axes can switch us between curves. The change occurs 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 invariances[5]. It 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 as is shown by inspection.

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Physicists generally prefer the alternative that shows the highest symmetry, that is, invariance under mathematical symmetry operations which also implies graphical symmetry. Mathematical versus graphical representation denotes certain axes, categories, dimensions or types of symmetry, after all. So, graphical symmetry should count just as strongly.

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EXPONENTIAL EXPANSION WITH DECELERATING GROWTH

THE ACCELERATION SCENARIO

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The H₀ linear regression line in this figure is not included for the sake of simplicity. Its intersection with the t = 1 line occurs at about R = 0.85

SCENARIO 2: THE CASE OF ACCELERATING GROWTH

 

Please click on the title above to view the model depicting deceleration of growth in the modern epoch. Use backarrow <== to return to this forum. Do not stray far or else you may not be able to return by means of the backarrow. Please do not close out of this new web page or you might close out of the forum.

 

DEPICTION OF THE DERIVATIVE

OF THE EXPONENTIAL EXPANSION (VELOCITY OF EXPANSION)

DECELERATING EXEMPLAR

AT A VERY EARLY TIME

 

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Next, presented below is one final figure. Herein, the plot represents the result of taking the first derivative of the exponential expansion equation that shows a fall-off or deceleration in the velocity of expansion in the modern epoch. But, this plot looks at a short interval of time very near the beginning and we see that the speed or velocity of expansion drops from an exceedingly high level and runs through a minimum that approaches zero. It remains far below the speed of light and this behavior lasts for several orders of magnitude of time.

 

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This should be important because this kind of exponential curve shows no initial induction period. According to the Guth scenario, a latency period is thought to be essential to serve the purpose of providing an early time for rapid equilibration of the universe resulting in the isotropic thermal homogeneity that we see now in the CMB.  The case of the model wherein the velocity of expansion accelerates in the modern era does not display this minimum. So, if the model itself is credible, 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.

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Still, why is there an equilibration requirement in the first place? The Big Bang was not an explosion. It was a sudden expansion of a tiny parcel of a false vacuum, of a high-energy state or excited packet of space-time. It is supposed to be a virtual quantum point particle that came into existence at an extremely high energy level due to higher probability.

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When inflationary expansion began, 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, extremely rapid fractal growth or “automatic” expansion from a single point should naturally result in a universe that is homogeneous and isotropic, satisfying the Cosmological Principle. To suppose otherwise requires a mechanism to produce turbulent expansion and no such mechanism has been forthcoming.  .Instead of supposing that the universe is and was like a digital automaton, quantum irregularities in an excited continuum of something like a Higgs field are cited as sources of a sort of wrinkling of the high-energy state surface of the false vacuum “inflaton”. Then, we need to remember that we do not even know whether any such Higgs-like field, or the associated Higgs-like boson, the inflaton, actually exist or existed. We begin to pile assumption upon assumption. It is dangerous to try to support a hypothesis this way. Besides, it is clear that Physics is not really ready to accept all the implications of a purely quantum universe anyway.

.Nonetheless, in order to favor the deceleration scenario, one would still need to explain the distant supernova 1a results. These have been interpreted as indicating that the universe may be undergoing accelerating expansion in the present epoch , [9] . The real value of these results is to show that previous brightness measurements have not been affected by intervening “grey dust”.  .Perhaps the interpretation of results that implies acceleration is a sort of optical illusion. It may stem from the possibility that when the universe is observed at such early times, it is observed as a more and more purely relativistic object deeper inside a far more intense gravitational field than we have ever thought, because the universe may be far larger than we think.

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See the trace of equation 1 for an indication of how strongly gravity may modulate time if this model makes any sense.  Plus, a more positive trace of curvature of space-time than we think may exist may be sufficient to produce an additional distortion of our measurements when initial conditions occur much closer to t = 0. Perhaps our gross difference in perspective, and our failure to compensate completely for a possible excess distortion due to our perspective, results in what naively looks as if the universe is expanding at an accelerating rate. 

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In other words, acceleration may be an artifact. If insufficient effort was made to compensate for the fact that light must have been climbing out of a much stronger gravitational field and for the fact of time dilation in such a field, then artifacts may still persist. Then, with a “derated” red-shift, one that is reduced by sufficient compensation for gravitational effects, the cosmological distance covered per unit net red-shift will increase and there will tend to be no disparity between red-shift distances and those determined by brightness measurements. Thus, acceleration will disappear along with dark energy. 

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George Ellis has suggested that the bulk of the volume of our parcel of the universe as defined by the radius of our light horizon is, by chance, less dense than average, that is, the assumption of the Copernican or Cosmological Principle of homogeneity and isotropism may be somewhat inaccurate. Then our red-shift observations may be so skewed that this alone could mean that dark energy has been erroneously inferred[10]. .Furthermore, to repeat, it may be wrong to assume that an increased rate of expansion denoted by faster than expected recession rates for very distant, very old 1a supernovae nearer t = 0 means that the recession rate must be accelerating in the present. This is not logical.

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If logic matters, one would expect that an increased corrected value for H₀ obtained for very distant and therefore very old objects should mean that a very much lower “more modern” value for H₀ indicates that the expansion rate is decelerating. Look at the regression plot of H₀ versus time in the first figure and see the comments above.

.There cannot simultaneously be more than one effective value for H₀ for the present instant applying to nearby objects, an indirectly nonintuitively inferred one that indicates acceleration, and a much lower value for H₀ at near t = 1 that is obtained by simple straightforward extrapolation from direct observation. The exact logical device that reverses the intuitive conclusion that Hubble expansion is slowing down is nowhere to be found. 

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It is an unstated bias, a silent assumption or a hidden postulate that needs to be proven by uncomplicated direct argument. This putative reversal of common sense needs to be explained in a common sense manner, immune from mathematical or theoretical obfuscation. This is the business of science. We need to show that we know that we are not employed to complicate matters. Our mandate is to simplify them. There will be serious consequences if we have to reverse ourselves again on such issues.

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If there is no dark energy, the apparent flatness of the universe must still be explained in the absence of enough matter and energy in our matter/energy inventory to produce a flat inference from analysis of the cosmic microwave background. There may be a small enough residual positive curvature of space to account for a much larger universe than we can see, limited as we are by the light horizon. Indeed, if the universe is exactly flat, it may well be infinite or certainly large enough to account for the “missing mass”. Also, a convenient infinity could help account for the depth of the gravity well implied by the curve of equation 1.

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If we are prepared to accept a singularity at t = 0 with an infinitely high density and temperature, why can we not tolerate an infinite universe? If there are as many as 11 dimensions in our universe then, might not this supposed singularity morph to a “singular surface” when viewed from a different perspective? This singular surface would act like an enormous phase boundary so that our universe would have come to exist within this surface, like a system of countless galactic superclusters in Flatland[11]. Perhaps this would serve as a rationale for the universe’s having come into being as a virtual particle which implies the participation of the surface of a black-hole (because the universe has not yet been annihilated). This singular surface could be the phase boundary of the event horizon of the ultimate black-hole, the mother of all black-holes or the MOAB.

 

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

 

VELOCITY OF EXPANSION AT A VERY EARLY TIME

 

Please click on the title above to view the graph depicting a distinct minimum in the velocity of expansion at a very early time, seen only with the decelerating scenario. Use backarrow <== to return to this forum. Do not stray far or else you may not be able to return by means of the backarrow. Please do not close out of this new web page or you might close out of the forum.

 

[1]  Allan Guth, Inflationary Universe: A possible solution to the horizon and flatness

                           problems,  Phys Rev D, 23/2, 347-356 (15ja81)

                           The Inflationary Universe,  Perseus Books, 1997

[7]   http://calc101.com/webMathematica/derivatives.jsp

Adam Reiss, Ron Gilliland, et al,  Astron. J. 116, 1009 - 1038 (1998)

[9]   Saul Perlmutter, et al,  Astrophys. J. 517, 565-586 (1999)

10]  George Ellis,  ibid

[11] Flatland – A Romance of Many Dimensions, Edwin A. Abbott 

After the rapid inflation does the total heat content of the Universe remain the same? Is there an overhang of curvature at infinite duration for any dimensions? Are the dimensions still being unraveled "just" at our horizon? If you think as time as a dimension, do you "see" any constraint on it unraveling before the initial Guthian inflation?

Well, that is certainly interesting. The term phenomenological, to me implies a theory based strickly on the phenomena and nothing else but peripheral excursions into speculation.  I like this old quote, even though some may not appreciate it.  "We have learned that we do not see directly, but mediately, and that we have no means of correcting these colored and distorting lenses which we are, or of computing the amount of their errors. Perhaps these subject-lenses have a creative power; PERHAPS THERE ARE NO OBJECTS"  (Emerson buy the way  Maybe, just maybe, the universe exists only because thinking beings perceive it to be.  How else could all the variables be so finely tuned to allow life to exist be present? If the only universe we can exist in is such an infinitesimally tiny probability, why can't it be that consciouness creates the reallity that exists around us and not the reverse of the "Big Bang" creating living being and thus consciousness? I realize some of the same questions come into play, but sometimes I need to think in a difrferent direction. Oh Well, thanks for listening!

Mark O.

Time as a dimension? Why? Isn't time just a vehicle our minds use to explain to ourselves the change that occur around us? Think of reality as a flowing thing (AKA wave function) the instant a (thing, particle, person, event, etc) is observed, it becomes a reality and not a probability, hence stopping it's motion.  I think of time as a series of instants with only the moment existing at that instant, kind of like a movie projecter, a frame at a time. Our minds are able to adjust for this by somehow acting similar to a movie projecter, being able to string things along with no zigs or zags.

Who Knows?

Thanks Guys,

Mark O.

GaryAKent : I like your presentation. I also have some questions, you may be able to to help me with. Some of my questions being, How does anti-matter play a part in your concept the universe? Do you accept Mr. Hawking's explanation of the Hawking Radiation? Do you accept Mr. Richard's Feynman's vitual pair concept? Thank you for your time.

My thought on antimatter is alluded to by my comment that physics is not ready to entertain the idea of a fully quantum universe. If Allan Guth is right and the universe began as a quantum particle materializing in a very high energy state because high energy states are more probable simply by virtue of their unlimited number, then we can apply some of the things that we know about quantum particles and waves to the present universe as well.

Quantum particles and waves exist as entities having interference counterparts to complete their mathematical descriptions. Particle or wave A has an interference wave form -A and combinations of A and -A. Each would be another manifestation of the universe but would exist entire of itself, but not unconnected. This is apart from the possibility of Wheeler's "Many Worlds" theory. It is simply the way that quantum wave forms need to be described for quantum theory to work at all.

When the universe came into existence, A, -A, A + (-A), A - (-A) and the null form also came into existence. They each exist having mass and energy. The null form may be thought of as having imaginary matter and energy so that real quantities in it are mathematically imaginary while imaginary quantities are real or some combination of these so that the result is "null".

Antimatter would be confined to the -A components so that our parcel of the universe would not have any stable antimatter at all. The absence of antimatter in our form of the universe would be automatic. So, we do not find clouds of anti hydrogen in anti galaxies around here. If they existed, we would see annihilation and release of stupendous amounts of energy in the zones where matter and antimatter inevitably come into contact.

I have no idea what the other combinations might be like. Perhaps they would have certain physical laws that make them behave like opposite counterparts. But we could never know.

But, if this may be plausible, then the location of the "missing matter" that cosmologists moan about is revealed. They say that more than 70% of the matter and energy in the universe is not accounted for. So, they invent dark energy to make up for their lack of imagination.

Furthermore, if these superpositions of quantum waveforms are corrolated but unconnected except for gravity, some or all of them might account for dark matter if they were present in the same space as in our parcel of the universe but oriented differently. Quantum particle or waves can exist in the same space if they have different "quantum numbers", which are like essential parameters that make quantum equations solvable. 

If wave forms for galaxies and galactic clusters constituted a superposed distribution, then the discrepancies found for the gravitational force of attraction around these objects would be explainable. In fact, this could provide a mechanism for Modified Newtonian Dynamics or MOND without having to alter the theorems of Relativity.

The thing about Many Worlds is that quantum theory is statistical in nature. So, if we speak in this way we imply that statistical distributions exist which implies the Many Worlds interpretation, by definition. If other manifestations of galaxies exist in superposition with slightly different orientations according to, say, a Gaussian distribution, then the weighting factors for each contribution could be adjusted to exacly match the MOND result for a non-Newtonian modification of Newton's Laws.   

For Relativity to work, time must be regarded as a mere dimension in space-time. We perceive it differently because matter without energy is timeless and as soon as you add energy to matter, time is manifest. Time and energy are therefore synonymous. They are related according to the equation E=mc^2. This equation was not necessarily derived by Einstein through some arcane theoretical considerations either. One can get it by simple dimensional analysis.

Since the universe probably began as a particle having incredibly high energy, it has been evolving as if it were descending from a height or rolling downhill. Time proceeds in the direction that we perceive because it IS the direction of this downhill decline. The universe is running down so that at some point at infinity, it will again become a null, timeless entity. But, entropy will have risen to infinity. So, highest entropy is not chaos, but nothingness.

The form that I chose to use to fit the data happens to work and it also happens to have a theoretical basis. But, if I had never heard of Guth's inflation theory, the model would still have been a phenomenological model because it does not DEPEND on theory.

I think the universe is WYSIWYG - what you see is what you get. It really is as we perceive it. We perceive it because it is what it is. There is nothing mysterious about this.

What may be hard to accept is that there really is no mystery. What is so incomprehensible about the universe is that it is comprehensible at all. (A. Einstein)

When Allan Guth and other cosmologists speak of the coming into existence of a parcel of false vacuum in a high energy state that then proceeds to "fall" down to lower and lower energy states as time unravels implies a sort of meta-time.

They say we cannot say anything about what happened before the beginning of time nor even before the universe was about one Planck instant old (rearrange the equation for Planck's constant to get "t" alone on the left side). But everything else they say implies an existence before and beyond t = 0.

The universe seems to be unfolding as it should. It appears to have originated as a dense balled-up knot of reality that is becoming untied because it is too dense. Quantum states of nearly the same energy literally repel one another and they "break symmetry". When the universe is finally straightened out, we will cease to exist. But the membrane, sheet or strand will not. 

GaryAKent : Thank you for your reply. I find your ideas interesting, however I sort of like the concept of  space-time being a vector quanity of upper dimensions. Instead of branes colliding and rebounding they collide and merge, having some complementary dimension and some transverse, resulting in vectored relativistic dimensions, and some dimensions with differences in analog form and some in integer allowing a vectored analog quanta. The integer position of one system being relative to matrixed (digital vector) analog space-time position in another, this would allow a range of vectored analog positions between interger positions of another set of dimension. So we could actualy be two seperate universes merging with the initial collision being the big bang.

Mr Kent, I enjoyed your posts greatly.

Now I'm trying not to think backwards for fear of.....something bad happening!

I feel like the center of a hard flung frisbee. Moving rapidly, staying together but knowing I'll fly apart someday.

My professional life revolves around modifying automotive cylinder heads to produce less turbulence........to think that the initial velocity/expansion had no resistence, hence no turbulence...

......what graph lines can I intersect to make THAT happen. I could rule the (automotive) world!

ok, I'm going to bed. And think. Great post.