Ian E. Consterdine
Horwich UK
2022, May 8th
Abstract
The pantheon of inflationary big bang cosmology includes model versions where time began as inflation started. An unresolved issue with basic quantum mechanics in all such models is highlighted that may or may not help with future developments in quantum gravity theory and/or cosmology. Any issues concerning the cause(s) of inflation are separate and take no part in the following discussion.
Introduction
Inflationary cosmology theory was invented by Alan Guth1 in an attempt to solve some outstanding problems of the then current state of ‘Big Bang’ cosmological reasoning. At that time, there was no reason to include a cosmological constant term in Einstein’s equations of General Relativity theory and the universe was observed to be flat to great distance. Indeed, in suitable units the Einstein equation relating his curvature tensor (the sum of the Ricci tensor and a fraction of the metric tensor) and the stress-energy tensor may be written as:
Gµν = Tµν ———————— 1
Observations since Penzias & Wilson2 and up to the most recent Planck telescope results show temperatures on opposite sides of the universe are the same to small variance. There has been no thermal contact between these opposites since long before the epoch of last scattering and this was a major stumbling block in models of cosmology hitherto.
The flatness of the universe and its horizon scale were additional features addressed by, and hailed as great successes of, this new cosmology.
Andre Linde3 addressed particle physics, scalar entities and spontaneous symmetry breaking under the inflationary umbrella and the subject grew to become a staple of modern cosmological thought.
Today, if one were to ask a leading cosmologist about their understanding of the big bang, one may receive in reply something along the lines of:
“The best way to picture the big bang is that the universe has no boundary in space, but has a boundary in time.” The quote is the contents of a tweet by @WKCosmo on 2021, November 17th. The twitter account is that of cosmologist, W. H. Kinney4 of Columbia University.
Discussion
Thus, we have learned that a boundary in time, t, is such that when t = 0, an infinite spatial surface exists that terminates all past light cones. Figure 1 shows a section of the past light cone of an event ending at such a surface.
Such a model implies that a theory of quantum gravity for t >= 0 must overcome, somehow, an infinite result from application of the Heisenberg Uncertainty Principle (HUP) for energy and time. This is because our best theory of gravity hitherto, Einstein’s theory of General Relativity5, does not accommodate this infinity well, as shown below.
In physics, the HUP yields the commutator for some members of the non-Abelian group and it may be formulated6 to show the product:
ΔE Δt ≥ ħ/2 ———————— 2
where ħ is the reduced Planck constant h/2π and Δ represents a change in quantity.
An immediate consequence of this HUP formulation is that at time
t = 0 + Δt
and where Δt → 0,
we see that ΔE → ∞
This state of energy uncertainty exists at every point on the infinite spatial surface of figure 1. Since infinite stress-energy leads to infinite curvature in General Relativity (1) theory then the surface would immediately crumple and collapse into infinitely many, infinitely small, gravitational black holes. In other words, such models self-destruct as soon as they start: off the page of ideas, they have neither past, present nor future.
Alternatively, when t reaches, for example 1/(Graham’s number7) seconds after the t = 0 boundary, the energy uncertainty will be close to a Graham’s number of Joules. Since there are infinitely many times closer to t = 0 to consider where the energy uncertainty is much higher, such models are in immediate danger.
It may be possible that a theory of quantum gravity will come to the rescue near t = 0 but such ideas remain unwritten to date. The title of this note may turn out to be an error.
We arrive at a state where a retired atmospheric physicist, whose dotage is spent trying to keep up with advances in as many fields as ability allows, has reached an impasse. In the recent past, questions related to the above thinking have been posed in some of the science forums and other places I visit across the internet.
First of note, Luboš Motl8 replied kindly but with nothing concrete during a discussion on the Hartle-Hawking, “no boundary” proposal9 and the Wheeler-de Witt equation, near early time, at his referenced blog.
Next, a recent study by Kinney & Stein10 shows that cyclic models suffer from geodesic incompleteness. However, any inflationary model that includes a t=0 state suffers the Uncertainty driven false start as discussed earlier: “You lose control of the calculation at that point.” – Kinney, personal communication. The paradox that both t=0 and cyclic time are forbidden by current analysis is resolved, at least partially and possibly completely, by the treatment of gravitational black hole collapse presented in a related blog post hereabouts11.
Most recently12, Professor George Efstathiou gave the 22nd Hintze lecture entitled, “The legacy from Planck, do we have a standard model of cosmology?”. Having watched this excellent talk over the internet I learned that although the “ΛCDM” model used to describe the most recent Planck telescope observations and results13 is the best fit in almost all circumstances, there remains a few issues with some of the derived parameters used in inflationary models that lead Prof. Efstathiou to imagine that the future for cosmological inflation is not set in stone.
The “infinite uncertainty energy” is a known issue in models of cosmological inflation that gets swept under the carpet all too often. I am also aware that I’m not the smartest observer in this domain of reason by a very long chalk but the issue has niggled me long enough.
Inflation beginning at t = 0 may be an incorrect model, or a theory of quantum gravity near t = 0 will remove the infinite uncertainty energy.
Inflation having no beginning i.e. inflation is past eternal, requires no such rescue but questions concerning the end of inflation then arise. How and when did inflation stop along an infinite timeline?
Inflationary cosmology may not be the relevant mathematical description of cosmological history after all and ideas like Penrose’s Conformal Cyclic Cosmology14 and/or the recent Ijjas & Steinhardt15 proposal may turn out to be less speculative than hitherto. Century 21 is no less interesting as the debate continues and the evidence is amassed.
To end on a light hearted note – I wanted to discuss quantum gravity and the Wheeler-De Witt16 equation but I could never find the time. 🙂
References
1 Alan H. Guth, 1981, “Inflationary universe: A possible solution to the horizon and flatness problems”, Phys. Rev. D 23, 347
2 Penzias, A.A.; R. W. Wilson, 1965, “A Measurement Of Excess Antenna Temperature At 4080 Mc/s”. Astrophysical Journal Letters. 142: 419–421.
3 Andre D. Linde, 1990, “Particle physics and inflationary cosmology”, Contemporary Concepts in Physics, Chur: Harwook Academic Publication
4 W. H. Kinney, 2002, “Cosmology, Inflation and the physics of nothing”, https://ned.ipac.caltech.edu/level5/Sept02/Kinney/frames.html
5 Einstein, A., & Lawson, R. W., 1921. Relativity: The special and general theory. New York: Holt.
6 For derivations see, e.g. https://slidetodoc.com/heinsenbergs-uncertainty-principle-it-is-impossible-to-determine/
7 A definition of Grahams’ number is given at wikipedia – https://en.wikipedia.org/wiki/Graham’s_number
8 “Most laymen have a remarkable psychological problem with the universe born out of nothing”, Luboš Motl, 2019: Discussion comments https://motls.blogspot.com/2019/06/most-laymen-have-remarkable.html#comment-4494500165
9 J. B. Hartle and S. W. Hawking, 1983, “Wave function of the Universe”, Phys. Rev. D 28, 2960
10 Cyclic Cosmology and Geodesic Completeness, 2022, William H. Kinney, Nina K. Stein https://arxiv.org/abs/2110.15380
11 “Gravitational Black Holes with a Twist and Supersymmetry”, 2022, Ian E. Consterdine, https://localsymmetriesonly.home.blog/2022/04/07/shellhole/
12 Reference date: 2021, 24th November. Professor George Efstathiou of Cambridge University, UK, delivered the 22nd Hintze lecture hosted by the University of Oxford, UK, entitled, “The legacy from Planck, do we have a standard model of cosmology?”.
13 Planck Telescope publications to 2021, https://www.cosmos.esa.int/web/planck/publications
14 Sir Roger Penrose, 2012, “The basic ideas of conformal cyclic cosmology”, AIP Conference Proceedings 1446, 233; https://doi.org/10.1063/1.4727997
15 A. Ijjas, P.J. Steinhardt, 2019, A new kind of cyclic universe, Phys.Lett. B795, 666-672
16 Alexander Yu. Kamenshchik et al, 2018, “Time in quantum theory, the Wheeler-DeWitt equation and the Born-Oppenheimer approximation”, https://arxiv.org/pdf/1809.08083.pdf
Local symmetries allowed 