Alan Guth’s inflationary theory redefined.*

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A Radical View of the Inflaton Field
by Bruce E. Camber (first draft)

Lemaître, Hawking, and Guth1 were the most pivotal thinkers to advance a big bang theory. Lemaître died in 1966 at the age of 71; and, Hawking died on Pi Day, March 14, 2018 at the age of 76. Alan Guth now stands alone, the last remaining prime mover, especially with his theory of inflation and its inflaton.

Given that dimensionless constants and their equations are most fundamental keys, we ask, “What ubiquitous, old equation touches the most points within science and mathematics?2 I.suggest that the answer directs us to the best equation to define the first space-time moment within.this universe. And, I believe that equation is pi.

It’s that simple. Notwithstanding, thoughtful work by many scholars has gone into defining inflation and its inflaton. All those insights still hold keys to the universe, just not the keys to the earliest instances of our universe. For those moments, the most-infinitesimal sphere should be a good redefinition of Guth’s inflaton.3

The equations immediately begin to evolve. Basic geometries emerge and continue to evolve for seconds-minutes-hours-days-a-year (and even years). At some key critical point, the definitions by Guth and his collaborators will begin to participate. To determine when will take some study because inflation is dynamically creating the laws of physics as infinitesimal spheres populate the universe.

There are several current disparities to determine an expansion rate so a range is provided. I suspect in reality there has always been a range. Using either the PlanckStoney-or-ISO base units, and by assuming one infinitesimal sphere per unit of PlanckTime, StoneyTime, or a new ISO basetime, there would be a range from around 539-to-4605 tredecillion spheres per second.4 Of course, these are the most infinitesimal spheres possible, at least 50-to-64 base-2 orders of magnitude smaller than the neutrino and are obviously a good candidate for dark energy and dark matter.

That range or rate of expansion, considered a new definition of a cosmological constant, expanding even today, right now, creates a penultimate grid, literally to include everything, everywhere for all time. There are just 202 base-2 notations that outline our universe from the smallest duration of time, doubling each step, to the current time, the Now.5 I believe here is the starting place for group theory, systems theory and at least nine major studies currently not on any grid!

The first infinitesimal sphere has been likened to Lemaître’s primeval atom and Guth’s inflaton. Both the primeval atom and the inflaton field have always been hypothetical. I.believe our very first infinitesimal sphere is a little less hypothetical. The universe has to start with something to create space-time. Yet, our simple postulation goes much further. It is difficult to conceive of a universe totally populated by such infinitesimal spheres. Notwithstanding, even this hypothetical penultimate grid warrants inspection.6

Finite-infinite transformations between the faces of continuity-symmetry-harmony (CSH)7 are assumed. Our focus is on the finite. Defined by CSH, it is the Fourier transform,8 and then configurations of the Poincaré sphere, Poincaré homology sphere, and any-and-all types of spheres as mathematics naturally extends to include each. Only when it becomes possible to “come out” mathematically, building on natural functional dependencies, do these infinitesimal spheres manifest. This would necessarily include studies by Smale and Milnor of spheres as attractors and repellers.

Those new to this site will quickly ask about indeterminacy and quantum physics. Over the years we have struggled with the geometries of indeterminacy.9 It’s a working challenge and very much part of the discussion. So, per usual, I ask, “Where have we gone wrong with our charts, interpretations and prognostications?”10

Thank you. Thanks indeed. -BEC



Endnotes & Footnotes
All these points already have pages within this website.

[*] Alan Guth. With this footnote I have asked Alan Guth to consider a new thrust whereby his hypothetical inflaton (and even Frank Wilczek‘s hypothetical axion that Guth adopted) are redefined in the light of infinitesimal spheres and groups of infinitesimal spheres. I’ve proposed that that we first analyse the processes involved by organizing the data about spheres using a base-2 natural expansion. Every law of physics is derivative within a notation between 0 and 202. Base-2 exponentiation is the most-simple way to organize the numbers of spheres with a timestamp and to begin to grasp the unfolding functionalities of spheres and groups of spheres.

There are 202 base-2 notations. The horizontally-scrolled chart of 202 base-2 notations started in December 2011 with just Planck Length. That chart had somewhere around 202 notations. Not until 2016 when we mapped it with Planck Time did we have a more definitive stopping point: 13.79 to 13.81 billion years. The 201st doubling takes Planck Time, 5.391 16(13)×10-44 seconds, out to around 173,272,944,073,600,000 seconds or 5.4908 billion years. If we add up each notation up to the 201st notation, we are one PlanckTime unit shy of 10.98 billion years. Simple math tells us that around 2.8284 billion years has passed since the beginning of the 202nd notation (Calculation: 13.81 minus 10.9816 ≈ 2.8284). It is an important perspective. Our calculation for the UniverseClock helped us along this path. We are now challenged to exegete each notation! We quickly discovered how difficult that notational analysis can be. Here is an introductory pass at Notations 0, 31, 64, 67, 101, 137, 143, 167, 197, 199 and 202.

It all pushes us to redefine time because (1) All the notations are always active. (2) Time does not “pass.” It is. (3) Aging is real. Death is real. Sleep is real. Memory is real. And, all four must be included. Quite a challenge.

[1] Lemaître, Hawking, and Guth. Only Alan Guth has had the advantage to see the 2022 results from the James Webb Space Telescope (JWST). Although Stephen Hawking had access to the results of the Wilkinson Microwave Anisotropy probe from 2001 to 2010 (See: Katherine Freese) and results of the ESA Planck Telescope from 2009 to 2013 (See: George Efstathiou), that data is still a challenge. It has needed the JWST data 330 million years later to provide some perspective. Yet, it is all still a giant jigsaw puzzle… all to discern the very nature of the pieces of that puzzle within the first few minutes of our universe.

Guth is still confident that his group is on the right path, “…that the observational evidence for inflation just keeps mounting up and as things get measured more and more precisely it just gets better and better; so for example, one of the predictions that inflation makes is for the average mass-density of the universe and now that’s been confirmed within a half a percent.” That quote is taken from How Did the Universe Begin? (:18-:31 seconds), Closer to the Truth, 2020. Guth’s inflation is like a chameleon because it is based on just part of the overall construct. Also, see: Why Is There Anything At All?, 2021.

The earliest moments of the universe in the light of big bang theories were generally introduced to the public by Steven Weinberg within his 1977 book, The First Three Minutes. He says on page 5, that “…one-hundredth of a second (is) the earliest we can speak with any confidence” and then adds with aplomb that “the universe was about a hundred thousand million (1011) degrees Centigrade” as if he had observed it in the laboratory. Much later a group of 27 scholars from around the world titled their article, The First Three-Seconds (2020). They had trouble getting behind that first second. Within our chart the first seconds are from Notation-143 and the first hundredth of a second is from Notation-138.

It is all such a blindspot. And the big bang has been blocking a deeper analysis. Inflation has been called into question as has the big bang as a theory itself. It has become entirely obvious that we all need to be thinking more about the results from the ESA Planck Space Telescope in light of the JWST.

Notation-143. By the 143rd doubling of Planck Time (1.202 seconds), we are well into the last third of our chart. Notations 1-67 are virtually unexplored and, to the best of our knowledge, have never been cited as such within scholastic literature. We’ve called it the small-scale universe; perhaps “the infinitesimal universe” would be more appropriate. Notations 67-to-134 have been cited as the human-scale universe while Notations 135-to-202 have been cited as the large-scale universe. By that 143rd notation, on each “first pass” through, the core geometries, mathematics, and physics are being shaped by efficiencies and densities. It seems that there is no time or space for quantum indeterminacy. In that first pass, a perfection is most efficient and most simple. I can well-imagine those efficiencies become precedents and that this “perfection” — currently called smoothness — readily defines the first 330 million years right up to and within Notation-197.

[2] Ubiquitous, old equation. What comes first? The heart of the finite-infinite transformations between the faces of continuity-symmetry-harmony (CSH) is pi. Much of our classic scholarship touches it but has not defined it as CSH. Within this website, the discussion about the finite-infinite is part of many homepages, i.e. the prior homepage on de facto and de jure is one our many finite-infinite discussions. In many places within this website, you will find this declaration:

All other definitions of the infinite are put on hold. Most are personal definitions that come from personal experiences and family history. That is one’s own business, not ours. If those beliefs help you through life, that is great. Our goal here is to engage those principles and functions that give rise to mathematics, physics, and eventually all the other sciences.

from Continuity-Symmetry-Harmony (CSH), 1972

[3] Inflaton definitions. An excellent overview of the range of definitions is with a sampling of ten articles out of over 3700 within ArXiv that use the concept. The most important definition would naturally be from Alan Guth. One such article is Eternal inflation and its implications (PDF), Alan H. Guth, February 2007. Yet, the definitions by other early adopters like Steinhardt, Vilenkin, and Linde, are also key. A sampling of just ten: (1) Arrows of time and the beginning of the universe (PDF), Vilenkin, 2013, (2) Inflationary schism after Planck2013 (PDF), Anna IjjasPaul J. SteinhardtAbraham Loeb, 2014, (3) Inflationary paradigm after Planck 2013 (PDF), Alan H. GuthDavid I. KaiserYasunori Nomura, 2013, (4) The Inflaton Portal to Dark Matter, Lucien Heurtier (PDF), 2017,(5) Could the Higgs boson be the inflaton?, Phys.Lett. B697 (2011) 37-40 (arXiv:1011.4179, Remarks on Higgs Inflation, Michael AtkinsXavier Calmet, 2011, (6) Warm Little Inflaton (PDF), Mar Bastero-GilArjun BereraRudnei O. RamosJoao G. Rosa, 2016, (7) The Minimal GUT with Inflaton and Dark Matter Unification (PDF), Heng-Yu ChenIlia GogoladzeShan HuTianjun LiLina Wu, 2017, and (8) From Cosmic Inflation and Matter Creation to Dark Matter — Journey of the Inflaton? (PDF), B. S. Balakrishna, 2022, (9) The Peebles – Vilenkin quintessential inflation model revisited (PDF), Jaume HaroJaume AmorósSupriya Pan, 2019, and (10) On the behaviour of the quantum Universe anisotropies in a bouncing picture (PDF), Eleonora GiovannettiGiovanni Montani, 2023. Additionally, I include: Paul Steinhardt Disowns Inflation, the Theory He Helped Create, Scientific American, 2014, and Wikipedia’s study of the inflaton because it is a dynamic page.

Reference pages within this site: Guth, Steinhardt, Vilenkin, Linde, Ijjas, Kaiser, Loeb, and Peebles.

From all these (with many footnotes and references within each), I conclude the following:
1. Other than it is pointlike, there is no standard, widely-accepted definition of an inflaton.
2. An inflaton, like an axion, must define a space-time moment or it is not basic enough.
3. I would also add that a first principle to be a first principle, must be mathematically defined.

So, there will be more to come. This article is still a draft; it is in process; insights from all these scholars’ references (above) and most-recent articles will be added over time. -BEC

[4] Tredecillion range: A rather straightforward calculation with Planck Time renders 539.tredecillion spheres per second. With Stoney Time it’s 4605 tredecillion spheres per second. We have requested that the ISO comment on the difference. That such numbers could be an acceptable definition of a cosmological constant will be disputed, especially that it is the root cause of expansion (inflation). In that light, I think the insights of Katherine Freese and Will Kinney about a natural inflation deserve more attention. The question needs to be asked, “What is natural?” The work of Yasunori NomuraTaizan Watari, and Masahito Yamazaki (Berkeley Center for Theoretical Physics), Pure Natural Inflation, 2017 ask, “Does the model of inflation need to be significantly complicated? Is the agreement of ns of the quadratic potential with the data purely accidental?”

The simplest model of inflation V (φ) = m2φ2/2 [A. D. Linde, “Chaotic inflation,” Phys. Lett. 129B, 177 (1983)]—which gives the correct value for the scalar spectral index ns ‘ 0.96—is now excluded at about the 3σ level because of the non-observation of tensor modes.

arXiv:1706.08522v2 [hep-ph] 27 Nov 2017

Remember the little neutrino? The experts (IceCube Neutrino Observatory, the University of Wisconsin–Madison and the National Science Foundation) measure neutrinos say, “About 100 trillion neutrinos pass through your body every second.” More to come

[5] The current time, the Now. The most visited page on this website is titled UniverseClock. It was initiated for a 2017 conference at NASA’s Marshall Space Flight Center in Huntsville. Most people can’t imagine that the universe is only 436 quadrillion, 117 trillion, 76 billion, 600 million seconds old. That computes to about 13.81 billion years. In this model each notation is always active and the universe is constantly building on itself. The simplest equations of the universe become axioms, then first principles, and even laws. Yes, in this model of the universe, the laws of physics are being tested as we go.

The nine major studies that are “not on the grid” are actually on the grid, yet below our ability to measure them. Once each discipline accepts the concept of infinitesimal spheres, I am confident that their mathematicians will quickly work out the new details for their study. Thus, there are many more studies to come.

[6] Penultimate grids open for inspection. I believe it will be helpful to re-engage our current concept of point particles to begin to see them as large aggregations of infinitesimal spheres. Even the concept of points and vertices need to be reconsidered for a very basic redefinition.

Back in high school it bothered me that there was only one definition of a point. I imagined hundreds. The differences between them were in how the ends were secured and what was allowed to pass through them. I saw them all as computing circuits. Clearly the domain of Langlands programs and string theories, this penultimate grid does warrant further study.

[7] Continuity-symmetry-harmony define pi and are defined by pi. Pi defined the finite and infinite and are defined by it as well. In our mind, the realities of pi, continuity-symmetry-harmony, are really real and the foundations of the foundations.

[8] Fourier transform. Every formula that involves pi (starting with the Fourier transform) needs to be re-reviewed in light of CSH and the 202 base-2 notations. Pi is everywhere.

[9] Geometries of indeterminacy. There is a look and feel of quantum fluctuations; the mysteries are all within the geometries. It appears that scholars were first foiled by Aristotle (384-321 BC). Five tetrahedrons create a gap that he missed; and for 1800 years his mistake was repeated by scholars. That is worth pondering. Aristotle was so great it took 1800 years to countermand his mistake. And, iIt is still untouchable. That correction was eventually forgotten until in 1926, a little-known MIT mathematician, Dirk Struik, rekindled that scholarship. Struik’s work did not receive much attention until in 2012 when two scholars, Jeffrey Lagarias and Chaunming Zong, lifted it up again. Yet, these two mathematicians were more interested in packing densities. There is no exploration of the meaning of the gap. Subsequently, in May 2022, the five octrahedral-gap was introduced within this website and questions have been asked of many scholars, “What are these gaps all about?”

[10] Our charts, interpretations and prognostications. Simple logic, simple math, and simple geometries render our charts, interpretations and prognostications. It all awaits critical review, so until then, there will be more to come.


References & Resources
As references are studied, key references and other resources will be added.

 Symmetry in QFT and Gravity (video), Hirosi Ooguri (and Nathan Seiber), 2022
  Mathematically, equations building on natural functional dependencies:
….–  Using math in physics: 5. Functional dependence (PDF), E. F. Redish, Univ. Maryland, 2022
•  Quantum Energy Inequalities along stationary worldlines,
Christopher J. FewsterJacob Thompson, 4 Jan 2023
•  ESA Group (PDF): The universe at 380,000 years,2009
•  Pure Natural Inflation, Yasunori Nomura, Taizan Watari, and Masahito Yamazaki,
Berkeley Center for Theoretical Physics, Department of Physics, 2017
The Friedmann–Lemaître–Robertson–Walker (FLRW) Metric
•  A pedagogical explanation for the non-renormalizability of gravity, (PDF), Assaf Shomer, 2008.
Path integrals and Gaussian fixed point. See Assaf Shomer’s on page 7:
“The derivation of the path integral formula in quantum mechanics of a massive particle involves chopping up the quantum evolution into very short time intervals and inserting complete sets of states between them.”
•  Doplicher S, Fredenhagen K, Roberts JE (1995) The quantum structure of spacetime at the Planck scale and quantum fields. Communications in Mathematical Physics 172(1):187–220
•  Scale invariance and conformal symmetries

Personal projections and ruminations. Standard Model of Cosmology and the Standard Model for Particle Physics: Of the 202 notations, the first 64 notations open a map to make the connections.

There is a place for the some of the big bang numbers but not until after the first few seconds.

Thrust in our universe. In September 2017, I wrote about the thrust in our universe. So now, over five years later, it is time to revisit that article and update it as much as possible. The major update would involve our understanding more about the three facets of pi and how each is a Janus-face for the finite and the infinite. How are the functions of continuity-symmetry-harmony abiding?

Major studies. I have written to Robert Langlands, Ed Frenkel, and others within Langlands programs. They have not yet acknowledged the 202 mathematical notations. Why not? It’s just math and logic. There is no philosophy. There are no agendas. It is what it is, simple math.

I have also written to people within string theory. None have acknowledged the 202 notations.

I believe people are naturally incrementalists. It is more comfortable. The Planck units were ostensibly ignored until 2001 and by that time Hawking-Guth-and-family had a hold on the theory about the start of the universe. With Hawking’s death, that hold has become somewhat more relaxed. With the JWST it’s time to open up the discussions. It will include conformal-quantum-and-scalar field theories (CFT, QFT). Although John Wheeler’s sense of simplicity was a good idea, for most of the octogenarians and nonagenarians, this base-2 model of 202 notations is just too simple. It is too obvious. Yet, maybe not. Prior to Frank Wilczek’s three articles about the Planck scale, Planck’s numbers were aloofly small much like Paul Dirac’s were aloofly big.


In 1980 in Paris at the Institut Henri Poincaré, Jean-Pierre Vigier and I made a six-month study of the EPR paradox in light of the work of Alain Aspect in d’Orsay. Instead of infinitesimal spheres, Vigier had suggested that we use the metaphor of dominos falling. That action-at-a-distance is not instantaneous. Infinitesimal spheres within the packing densities suggested by the Planck-or-Stoney-or-ISO numbers, would be instantaneous.

Mathematics and physics of the finite begin here.


There will be emails to many of our scholars about the key points.

25 February 2023, Anna Ijjas, NYU
25 February 2023, Katherine Freese, University of Texas, Austin
23 February 2023, Alexander Vilenkin, Tufts University
12 February 2023, John Moffat, Toronto, Perimeter, Waterloo
10 February 2023, Vladislav Yakovlev, Max Planck Institute for Quantum Optics
8 February 2018, Ian Walmsley, Imperial College of London
3 February 2023, Alan Guth, MIT
31 January 2023, Thomas Sumner, Simons Foundation
30 January 2023, Basil Hiley, University of London (UCL)
29 January 2023, Thomas Lin, Quanta Magazine
27 January 2023, Drew Harrell, Washington Post
26 January 2023, Carl Zimmer, New York Times
25 January 2023, James Sethna, Cornell
24 January 2023, Rebecca BoyleQuanta Magazine
23 January 2023, Rohan Naidu, MIT Pappalardo Fellow


There will also be many instant messages to thought leaders about the following key questions:

  1. Is it possible that the first instance of the universe is defined by Planck’s base units?
  2. Is it possible that the first manifestation of those base units is an infinitesimal sphere?
  3. Might the characteristics of pi describe those spheres?
  4. Might the Fourier Transform impart either electromagnetism or gravitation to each sphere?
  5. Is it possible that one sphere manifests per unit of length and time?
  6. Doesn’t that compute to 539 tredecillion spheres per second using Planck units and 4605 tredecillion per second using Stoney Time?
  7. Is it possible that the densities within the earliest notations are on the order of a blackhole or neutron star?
  8. To create some sense of order with the generation of infinitesimal spheres, may we use base-2 notation?
  9. Using base-2 notation, are there 202 base-2 notations from Planck Time to the current time?
  10. Is it significant that at one second the Planck Length multiple is a very close approximation of the distance light travels?
  11. Is it significant that quantum fluctuations are measured within Notation-67? Notation-72 appears to be the limit of our abilities to measure a duration of time.
  12. Would these notations, 1-64, provide 64 possible redefinitions of a point-particle? (And, I would add a vertex.)

8:45 PM · Feb 1, 2023 @DrOsamaSiddique @Harvard_Law @UniofOxford @IGLP_HarvardLaw There is a type of natural law within pi (π) ( that also creates a mathematically-integrated view of the universe where value comes from its continuity-symmetry-harmony. March 4, 2023

11:14 AM · Jan 31, 2023 @RBReich Everyone should find their creative thing that makes them happy and brings them joy and ask, “Is there a business in there?” Millions have. We call it small business and it enriches the soul and satisfies the heart and inspires the mind.


Participate       You are always invited.


Keys to this page, inflaton

• This page became the homepage during the early morning of February 4, 2023.
• The last update was February 28, 2023.
• This page was initiated on February 3, 2023 at 11:11 AM
• The URL for this file is
• The initial headline for this article: Infinitesimal Spheres as Inflatons
• First byline: Alan Guth’s inflationary theory redefined.


Also reviewed: Could the Higgs boson be the inflaton?, Phys.Lett. B697 (2011) 37-40 (arXiv:1011.4179, Remarks on Higgs Inflation, Michael AtkinsXavier Calmet, 2011