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Graphical Hot Big Bang Model of

 Inflationary Exponential

 Accelerating

 or Decelerating Growth of the

 Universe

PLUS A SINGLE, COMPLETE SEMITHEORETICAL EQUATION DEPICTING THE PHENOMENON INCLUDING INITIAL INFLATION AT THE EARLIEST TIMES

abstract

Alan Guth was really the first to propose in a detailed manner that all the major problems with the so called "Hot Big Bang" scenario for the continuous expansion of the universe could be solved easily. He proposed that if there was initial very brief hyper-accelerated growth followed by more sedate expansion according to Edwin Hubble's discovery, only some tweaking would be needed to result in a completely satisfactory hypothesis. The model presented here is based on Guth's initial proposal and it contains an idealized exponential equation for the expansion of the universe from time, t = 10^- 43 (approximately) second up to t₀ = 13.72 gigayears and even well beyond.

The model is based on natural units such that the speed of light, c = 1, the current age of the universe, t naught = 1 and the radius, radius of curvature or scale factor of the universe is Ra = 1. By implication, the mass/energy in the universe must be Ma = 1 as well. This means that plots can be superposed on the same ordinate and abscissa with only a change in labeling the axes.

This also means that transformations or translations can be done without changing the numerical value of critical variables. When this can be done, it implies significant symmetry and the existence of a conserved quantity(1). One of the cases treated by the model shows certain exceptional degrees of such symmetry. In physics, at least, a model or equation that shows much more symmetry is strongly favored.

Note that these variables are extensive in nature. The result is a model that is far more intuitive than models based on intensive variables like temperature, pressure and density. These models are useful, but they cannot tell the whole story and interpretation from these alone may be dangerous.

The author believes that sole reliance on intensive variables has led to conclusions that require the virtual abandonment of the scientific method for their support. Science cannot tolerate hypotheses that are intrinsically unfalsifiable. To do so will set a precedent for the admission of superstition and the supernatural into the body of science. This will grow like a cancer and it may kill science altogether. That is, some people who have a particular agenda will kill it.

To be falsifiable, an hypothesis must be subject to its statement in the negative or opposite sense whereupon a critical experiment must be devised that has an unequivocal result. If the experimental result or observation proves the negation of the hypothesis, it is falsified. It is not enough to devise experiments that tend to indicate only that the hypothesis is true. It must be feasible and concerted efforts must be made to critically (incontrovertibly) prove the negative or null hypothesis. It is not enough to try to prove the positive statement of the hypothesis.

This model is called the "modulated time model" or "the e-model" after the hyperinflationary parameter, e, used in the idealized Guth equation. This algebraic form can precisely reproduce the data for initial brief hyperinflation, then more relaxed expansion and later either 1.) deceleration, 2.) acceleration, or 3.) constant growth in the observable radius of our domain of the universe or its scale factor. There are many additional reasons why only one of these three possibilities is strongly favored.

(1) Noether E "Invariante Variationsprobleme". Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse 1918: 235-257. http:// arxiv.org/abs/physics/0503066v1 . Substitute correct characters to use this link.

To be addressed later, some current flaws:

How the observed H_o data points in the plots were converted to natural units and their perspective translated to the origin should be explicitly shown.

Paragraphs after "In other words...", etc. in Part 3 are still a bit clumsy and should be rephrased, but they do make sense in the way intended.

Undoubtedly, there are other bugs. The fact that the author takes the liberty to add some conjecture at appropriate points to the discussion is not a bug. Part of the reason that this paper is not published in a refereed journal is so that this may be done.

 

Part 1A

The author does not believe that anyone has ever published a phenomenologically derived semitheoretical equation encoding details of modern cosmogony. This equation purports to graphically describe, using extensive variables (not intensively), the expansion of the universe according to the current standard model. Its time-frame extends 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 presented here. It is considered to be semitheoretical but with strong empirical input from observations of the phenomenon. It is very simple. It is hoped and assumed that it will be derived from first principles eventually.

Starting with Allan Guth’s concept of initial exponential very rapid inflationary expansion¹ supported by Hubble’s idea of more moderate expansion² beyond this preparatory process, a purely exponential form is 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 hyper-exponential inflationary expansion. This Guthian inflation, according to these equations, has a few unexpected twists, as shall be seen.

A system of natural units is used here because no other basis will allow exponentials of the type that are employed to work. Therefore, the current "radius" or radius of the light horizon of the universe, sometimes referred to as the scale factor, plus the speed of light and the current age of the universe are all set equal to 1. The mass/energy of the universe is also implied to be M = 1, at least in the present. So M can be another one of the possible vertical axes that might be implied in the resulting set of graphs.

In the tables below, equation 3 is an idealized representation of Guth’s exponential inflation. It depends on the "relativistic" time form given in equation 1, which contains the parameter e. This is called “relativistic” because, in order for the model to work, the time variable must be strongly modulated. The obvious way time may be altered under the general theory of relativity is by means of either acceleration or deceleration to high relative speed or immersion to greater or lesser relative depth within a strong gravitational field. The first moments of existence of this universe may have experienced an incredibly deep and intense gravitational field, perhaps much higher than has commonly been anticipated. It should have enormously affected time itself, especially if the initial mass of the universe was infinite. 

A suggestion of the possible initial depth of the gravitational field for the “whole” universe (U) is graphically depicted by the trace of equation 1, if the model has any good theoretical implications. The time dilation effect could indeed be far greater than one might expect, for observers are now, after 13.72 gigayears, at a far different depth within this universe’s gravitational field than this parcel of space would have been in at times so much nearer time = t = 0. The total mass of this universe, operationally effective at the time of the Big Bang and the emission of the CMB, and its apparent effective size in the deep past, may be so much larger than has been thought that there could be additional profound theoretical implications.  

Time speeds up in a weaker gravitational field. Therefore, when looking with hindsight progressively in time forward in a universe immersed within a much stronger gravitational field, backward looking observers should see what appears like a time series or video in slow motion. This perspective affects light itself. This is the origin of one component of the relativistic redshift. There is little proof that sufficient overall compensation has been applied for this gravitational effect. If one may compensate a little, one may compensate a lot. The goal must be to compensate enough for this to whatever degree of compensation that may be needed to bring observations into concordance.

To force the model to describe deceleration in the present era, a value for the (1/e)^th root of t with e ≥ 136 should be chosen. To produce an equation that describes acceleration in the present era, a value lower than e = 133 should be chosen. Flat or constant rate expansion requires a value of e = 134 or 135, approximately. The values of A and B are arbitrary values for the initial conditions. But, they must be chosen to meet the known boundary conditions, so they are actually fixed. The U must have begun at t = 0 and current conditions must be in force at t = 1.

The General Theory of Relativity lacks a root higher than the square root. Time can easily be isolated on the left side using the equation E = mc² by dimensional analysis, as can the equation itself. One can isolate time this way, but how would one produce a root of time of, say, t^(1/161) as listed in the table below? If one iterates t^(1/2) to (t^(1/2))^(1/2) to ((t^(1/2))^(1/2))^(1/2) etc.  n times until t^(1/e) ≈ t^(1/161) one obtains n somewhere between 7 and 8. In a fractal quantum U, the current number of iterations could be important. Then, perhaps n might amount to a sort of quantum number as well, for which a kind of Schroedinger equation could become solvable. Any other value for n might refer to another U or a different energy state in this one.

What if U may still be in a kind of excited energy state? Then, in a time dependent sense, we might be transitioning between states with principle quantum number n = 7 and n = 8. Would we be counting “down” or “up”? It is thought that maybe the count direction should be “down” since at t = 0, it would have to have been that n = ∞ or else it would have to have been very high. This countdown may account for the arrow of time.

But, quantum is a statistical science so, for this to be a quantum U, it MUST be an element in a statistical ensemble of Us in a multiverse. “Many Worlds” should be true. Then MOND, modified Newtonian dynamics, would make more sense. The residual MOND constant that modifies Newton’s Law of Gravity may be a result of the smearing of superpositions of Us. Then, galaxies could still exist about where they do in this U and in others but with slightly different positions and orientations. These would be in whatever statistical distribution that may be required. Relativity itself need not be adjusted, just augmented with a bit of an addendum.

Leakage of gravity from our element of the multiverse has been proposed to account for the relative weakness of this force. Perhaps it could leak into superposed doppelganger galaxies and galactic clusters with but slightly different coordinates and orientations to account for the MOND constant. The elements of a hybrid multiverse that do not fit would not count, except to weaken G. This means that all observations must somehow have an essential statistical component, such as that which may be incarnate in the ensemble of hybrid mixed waveform doppelganger universes containing doppelganger galaxies and doppelganger astronomers making doppelganger redshift measurements.

The problem then becomes to prove or disprove the null hypothesis, i.e., that this is not so.

Deceleration Scenario

If e = 161, the raw expansion curve (EQ 3), the Hubble parameter curve (EQ 6) and the observed Hubble constant, H₀, linear regression line (EQ 7) and the simple velocity of recession curve (EQ 4) all intersect at the same point on the t = 1 vertical line that denotes the present, as they most certainly should. With e so very far above 136, the exponential expansion curve is distinctly one of a decelerating nature in the modern age.

Acceleration Scenario

When e is lower than 134, pronounced acceleration is denoted in the plot, but the Hubble parameter curve, the expansion curve and the Hubble constant regression line do not all meet at the same reasonable point at t = 1. But, certainly any Hubble parameter curve and the Hubble data line MUST meet at the same sensible point at t = 1, even if the model is otherwise worthless.

Worthiness

To address the worthiness of the model consider that, whatever the details of the equations that might be used, their derivatives and therefore the resultant equation for the Hubble parameter will produce curves that are of a similar position and shape in the region of easily observable space-time, as shown in Figures 2, 3 & 4. So, an accelerating scenario cannot ever give a Hubble parameter curve that shows a correct H₀ at t = 1. This will be true regardless of the form of the equations if the shapes of the curves are anything like as are shown. 

Now,  the parameter A, the initial value for the scale factor or the beginning “radius” of the light horizon of the universe and the B parameter, a practical exponential time constant, are chosen to make the equation conform to those obvious theoretical/empirical boundary conditions. That is, the exponential expansion curve should always extrapolate toward or through the points (0,0) and (1,1).

In other words, once the indubitable mathematical boundary conditions are satisfied, the parameter 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 (to illustrate that these equations can easily be carried down to the infinitesimal range, see Figure 5). 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^-42 second and even less, rather far below the Planck time.

In deceleration mode, nothing but monotonic expansion occurs until about t ≈ 4 x 10^3 seconds when expansion pauses for a while. See Figure 5. But this very significant pause occurs only for the case of decelerating expansion occurring in the modern era.

See below for two tables of the detailed equations that were used with all their variables and parameters. All the equations are presented in a single line format, that is, using nested parentheses, (  (  )  ), * for multiplication and ^ for exponentiation, etc.

FIGURE 1

TABLES OF EQUATIONS IN THE MODEL,

DEFINITIONS OF VARIABLES AND VALUES FOR PARAMETERS

To part 1B (1A continued)

Jump forward to part 2

Fast forward to part 3 (end)

(715) 529-7591     garyakent@aol.com

Dear Colleagues,                                   

The paper referred to here is intended as FYI.

Its purpose is merely to point out some facts regarding the developing new standard model of cosmogony wherein Allan Guth’s cosmic inflation and supposed “dark energy” play a role. I think investigators everywhere should be aware of these facts before they prepare new manuscripts for publication. 

I think that the putative conclusion of dark energy is a misinterpretation of those facts. This paper is supposed to explain exactly why I think so. I have expunged all blatantly opinionated statements because I can reserve frank comments for this cover letter, which I could not include if I published this paper in a refereed journal. But the more objectively phrased main paper can stand alone if you may wish to pass it on to your professional friends and acquaintances.

See http://www.lonetree-pictures.com/Graphical%20Model%20of%20the%20Universe.htm

An early version of this paper can also be viewed in the Astronomy Magazine forums under the heading “cosmology”

 http://cs.astronomy.com/asycs/forums/t/41510.aspx

 and in

 http://neocosmology.blogspot.com/  .

I hope that this paper can generate some discussion. I raise some important questions such as

1)     How can dark energy’s accelerating expansion of the universe be consistent with the curvature of any reasonable Hubble parameter equation’s trace in a graph showing this and a linear regression line obtained from several representative good values of H₀?

    Extrapolation of observed H₀ values obtained for distant objects corresponding to times long past to an H₀ that would be obtained for near zero distance and for the present time (if the local gravitational field and proper motion of neighboring objects could be eliminated) must be identical with the computed value of the Hubble parameter at time = the present if the computed value is any good.  But, there is no way that accelerating expansion could give an equation for the Hubble parameter, H, that comes anywhere near H₀ at t = 1. Due to the required shape of the graph of the accelerating equation and its derivatives (see the paper), the accelerating scenario cannot ever give an equation for H that intersects the time = present boundary at a reasonable point. This point is explicitly depicted in the paper. 

2)     Is it not momentous enough that the SN 1a supernova discoveries depicting a concordant brightness and redshift H₀ that is larger than expected based on H₀ determinations alone demonstrate mostly that brightness is not degraded by intergalactic “grey dust’ at these very large distances? It also debunks claims that redshifts are caused by intergalactic “plasma”.                                                                                          

A larger H₀ at great distance is an H₀ for “way over there” and “way back then”. It has been implied that H₀ determined from SN 1a redshifts should apply equally to “here and now” as well as “there and then” in a bow toward relativity.  But, what do we do with the many H₀ determinations that have been made that show a smaller H₀ at intermediate distances and times? What do we do with the fact that H₀ is not constant? Whether expansion is accelerating or decelerating, H₀ must vary, by definition. Only for linear expansion at a steady rate would H₀ actually be constant.

    Extrapolation into the future is not the same as interpolation between points running backward into the past. That is, running time in reverse toward the deep past does not necessarily imply how the universe must behave moving forward in time. If H₀ had increased in the ancient SN 1a epoch in such a way that it applies to the present, the middle determinations for H₀ would begin show this. But, the plot of empirical H₀s belies this notion (see the paper).

3)     If relativity is invoked to deduce acceleration in the present era how does this conclusion respect Einstein’s own assertion that general relativity will almost certainly not pertain to the universe as a whole?  

hold that it does so pertain generates a form of Russell’s Paradox (RP). The solution of this occurrence of RP is, briefly, that the universe, as a set of relativistic tensors, cannot contain itself since this is illogical for all sets. A set is never of the right “type” to contain itself:  http://www.utm.edu/research/iep/p/par-russ.htm#H3

4)     When we entertain the premise that there existed an “inflaton” as a Higgs-like particle representing a sort of super Higgs field, do we not begin to pile assumption upon assumption?                                                       

We do not even know if the Higgs field itself exists, much less the Higgs boson. It is dangerous to try to support a hypothesis this way.

5)     Is it not clear that Physics is not ready to accept all the implications of a purely quantum universe?               

We cannot have a quantum universe that is not really always of this nature, in the deep past as well as in the present. If there was an inflaton point-particle and an inflation field, the universe must still be showing overall quantum effects like a “Many Worlds” superposition that accounts for dark matter and MOND theory without disturbing general relativity. Indeed, the universe may be folded over onto itself according to the boundaries of our light horizon such that Many Worlds is effectively produced by our observational limitations themselves. So, gravity may be leaking into our parcel of the universe from parcels outside our range of observation, and vice versa, magnifying or modifying the effect of the universe as being rather larger than was supposed.

6)     Might not the conclusion of acceleration be due to a sort of illusion stemming from a naïve interpretation of general relativity

Certainly this would be a more salutary result than the necessity to throw away the standards of the scientific method as some have suggested.

7)     What is the implication of George Ellis’ suggestion that the cosmological principle might not be exactly true? Why should we NEVER be at or near the center of ANYTHING?                                                               

If the overall shape of the universe or if the shape of the cosmic void in which we may be embedded is not exactly spherical, we need not be at or near the center anyway for the effect of the Ellis distortion to be observed as it may. There is a strong bias in the CMB signal distribution due to our motion through space. The plane of our galaxy is also in the way. What if any innate asymmetry might be obscured by these effects? What if the universe rotates in a hyperspace beyond Einstein’s spacetime with little or no visible evidence in the CMB? In such a hyperspacetime, which is implicit in Guthian, superstring or supersymmetry paradigms, perhaps it could rotate on more than one axis at a time, obscuring any CMB influence. Would this not potentially give an effect on redshifts, especially at the largest distances?  

8)     Can there really be more than one effective value for H₀ in the present instant applying to nearby objects, an indirectly inferred one that indicates acceleration, and simultaneously a much lower value for H₀ for now, at t = 1, that is obtained by short extrapolation from direct observation?

    The exact logical device that reverses the intuitive conclusion from hard data that Hubble expansion is slowing down must be explicitly stated. This reversal is an unstated bias, a silent assumption or a hidden postulate that needs to be proven by uncomplicated straightforward argument and evidence. This putative reversal of common sense needs to be explained in a common sense manner, immune from mathematical or theoretical obfuscation. If relativistic argument is invoked, it must be mathematically proven by reference to the equations of relativity.

9)     Is it not an enormous logical jump to deduce intrinsic properties of the whole universe based on the approximation that is in force when the universe is treated “like” an ideal gas?                           

Even Einstein had to use the hydrodynamic simplification of an ideal or perfect fluid, with properties very different from the so-called aether, that he called a “continuum” of infinitesimally small entities having no affinity. Our super-fluid dynamical metaphor and concordant approximations dictate the final forms of our theories and facilitate the computations necessary to make sense of them. But, they are still just approximations. Neither words nor numbers can fully capture reality. To hold otherwise requires proof.

10)  If the universe’s expansion rate is accelerating in the present era and has been doing so for billions of years, then the rate must have already exceeded the speed of light, since H₀ based on distant old cosmological objects had by then already reached a large percentage of c.

If we have exceeded the speed of light, then we are catching up to light that had been emitted in the past. But, this light would be detectable only at very low frequencies at this time and would look like a bunch of crowded, overlapping emission lines that had been redshifted. The range of detectability would probably be in the microwave region of the spectrum. Therefore, the CMB would not really represent the phenomenon that it is purported to do and conclusions based on this “erroneous” deduction could be all wrong. The number of unsettling new questions raised, versus the number of questions answered by acceleration and dark energy is unfavorable to dark energy. That there have to be ad hoc patches applied in order to admit dark energy is disturbing.

11)  What is the business of science?

    We need to show that we know that we are not employed to complicate matters. Our mandate, according to Einstein, is to simplify them. There will be serious PR and political consequences if we have to reverse ourselves again on such a matter.

I hope that I have not overstated my case. But, at least a few of my questions should be valid whether my suggested answers may not.

             Gary Kent

See http://www.lonetree-pictures.com for more contact information

P.S.      I use a pseudonym because folks find it difficult to spell and pronounce my name correctly even though it is so simple. Besides being a photographer, I am basically “only” a chemist but I have had much training in physics and advanced mathematics. Due to financial constraints, I could not actually sit to write my PhD dissertation. But, my research was complete. I should sign my name with the title PhD (abd), all but dissertation. Research involved developing a mathematical model for the interaction of certain platinum coordination compounds with neoplasm DNA. I used my experimental results, which showed there was no correlation between the length of several types of alkyl substituent groups in the cis ligands of square planar platinum II compounds and their rates of substitution in a particular SN2 reaction. This reaction was supposed to be a surrogate for the reaction with DNA. I found that such rates could not correlate with clinical efficacy either.

I did, however, find that there is a strong direct correlation between the alkyl group lengths and efficacy. I used the model developed by H. Kubinyi based on differential solubility between an aqueous and a lipid phase. This quantitative structure activity relation (QSAR) shows distinct maxima as chain lengths are increased in many disparate alkyl structural variants. These compounds have been studied so thoroughly that little could be gained by this information other than to succinctly describe results, I am sure. But, in other systems, perhaps for other diseases, it suggests that the Kubinyi model could be very valuable.

Prof. Kubinyi was unable to help me pursue this and my funding ran out before I could stand for my defense of thesis.  Needless to say, I remain interested in computational chemistry and physics, because my research involved physical as well as chemical processes in a hydrodynamical milieu.

 To part 1B (1A continued)

Jump forward to part 2

Fast forward to part 3 (end)

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