Numerical Coincidences of the Universe

Science is an experimental, factual, measurement driven endeavor incorporating high degree of precision. There are  wide ranges of constants that form the crux of our physical theories and in many a cases translate them from theoretical assumptions to practical measuring mechanisms of numerical data in observations. So physical constants have deep implications on the strength of different phenomena we observe and experience and their values have been established with extreme precision and accuracy. 

For example, the age old Newtonian theory on gravity tells us that gravitational force between two point masses is directly proportional to the product of the masses and inversely proportional to the square of the distance separating them. However this simple statement is not enough to get the actual value of the force between them. There comes in the Universal Gravitational Constant(G) without which measurement of the actual magnitude of force was quite impossible. Some other such constants are Planck's Constant(h), speed of light(c), Boltzmann's Constant(k) etc.

However, these numerical constants have been observed to be subject to certain strange coincidences and relations. Some of such coincidences even implicate that our so called constants perhaps aren't even constant. So in this article we have a look at such ideas. There are mainly two main divisions of such coincidences and we will take them up one by one.

Classical views - Dirac Large Number Hypothesis

Paul Dirac was one of the early observers of such ideas and had proposed what is called the Dirac large Number Hypothesis. The hypothesis concerns dimensionless numbers which result from ratios of certain quantities. What he observed was that many of these dimension less constants taken from ratios of order of magnitudes of some extreme quantities have near about the same value. For example:

1. At his time, ratio of longest to shortest length in the universe( distance to the boundary of observable universe = 10²⁶ m / radius of an electron = 10^-15m)  = 10⁴¹
2. The ratio of longest to shortest time span relating to events and occurrences in our universe( age of the universe= 10¹⁷ s / lifespan of most unstable nuclei = 10^-24 s ) = 10⁴¹
3. The ratio of largest mass to the smallest mass during his time in our Universe( mass of universe = 10⁵⁰ tonnes / mass of electron =10^-30g) = near about  (10⁴¹)²
4. Ratio of electromagnetic force to gravitational force between two electrons placed at same distance = 10⁴² . Considering a proton-electron pair gives a value of 10³⁹ and a proton proton pair gives a value of 10³⁶.
5. Total number of elementary particles in the universe = (10³⁹)² = 10⁷⁸
6. The maximum number of degrees of freedom allowed within a sphere whose radius is the Compton wavelength of the nucleon is defined to be j₁ and isThe term j₁ in equationis also roughly the ratio of the rest energy of the nucleon to its characteristic gravitational energy, .
7. The gravitational potential energy of a nucleon due to the mass distribution of the cosmos is roughly . The characteristic gravitational binding energy of the nucleon is roughly . The ratio of the two is defined to be j₂ and is
8. The maximum number of logical operations that could have been performed by a single nucleon over the age of the Universe is, according to the Margolus–Levitin theorem,The term j₄ is proportional to the ratio of the age of the Universe to the time required for light to traverse the Compton wavelength of the nucleon.
9. The ratio of the current particle horizon to the Compton wavelength of the nucleon is
10. The ratio of the nucleon mass to the smallest possible mass, being the Wesson mass, is

11. Age of the universe in atomic units = 10³⁹.

These array of coincidences led him to believe that all natural dimensionless constants have some relation among themselves through which a few of the constants can yield the value of some other constant. Using pure mathematical constructs he did arrive to such a relation wherein he used the value of speed of light(c), mass of electron(mE), mass of proton(mP), Universal gravitational Constant(G), elementary charge(e) to yield a number that has the dimensions of time. The relation is: . The number, unbelievable as it seems has an order of 10³⁹; that of the age of universe in atomic units.

Modern Views - Enter Scott Funkhouser

The modern views on this topic with much contribution from Funkhouser has given much focus on yet another number 10¹²². The clauses mentioned above under the classical tag do not sharply apply anymore because of the discovery of smaller and yet more elementary particles like quark, neutrinos which have set new low standards of mass and length. But even then the speculations haven't grown obsolete because of new developments in this area with the new number 10¹²² which also like its ancestor pops up in many important calculations. Some instances of such appearances are:

1. Physicists think that some unknown force nearly perfectly cancels out the vacuum energy, leaving only the amount seen as dark energy to push things apart. This cancellation is imperfect to an absurdly fine margin: the unknown 'energy' differs from the vacuum energy by just one part in 10¹²². It seems incredible that any physical mechanism could be so finely poised as to reduce the vacuum energy to within a whisker of zero, but it seems to be so.
2. In the modern times, the ratio of the mass of the observable Universe to that of the smallest possible ‘quantum’ of mass(Wesson mass = 1.5×10⁻⁶⁸ kg) is about 6x10¹²¹.
3. The number of ways in which the particles of the current Universe can be arranged throughout space (a measure of entropy) is 2.5x10¹²².
4. The vacuum energy density due to the cosmological constant Λ is , where c is the vacuum speed of light and G is the Newtonian gravitational constant. With Λ≈1.2×10⁻³⁵ s⁻², the vacuum density is roughly 6.2×10⁻¹⁰ J m⁻³. The Planck energy density is defined as , where m_P is the Planck mass and l_P is the Planck length, and is roughly 4.5×10¹¹³ J m⁻³. The ratio of the Planck density to the vacuum density is defined here as n₁ and is
5. According to the Bekenstein–Hawking entropy bound, the maximum number of degrees of freedom available to the Universe is one-quarter of the surface area of the sphere whose radius is the cosmic event horizon R_e, measured in Planck units. In this current age the maximum number of degrees of freedom, defined here as n₃, is                                                    where R_e0 is the current event horizon.
6. According to the Margolus–Levitin theorem, the maximum rate at which logical operations could be performed by a physical system with energy E is 2E/(πℏ). Thus, the maximum number of operations that could have been performed by the mass of the observable Universe iswhere T₀ is the age of the Universe.
7. The characteristic size of the nucleon is of the order of 10⁻¹⁵ m, which is of the order of the Compton wavelength of the nucleon l_n≡h/(m_nc). The number of nucleon volumes  contained within the volume  of the sphere whose radius in the current event horizon is
8. The gravitational potential energy of the cosmic mass is roughly . The scale of gravitational binding energy associated with a nucleon is roughly . The ratio of the two, defined as n₆, is
9. Finally, there are at least two other pure numbers of the order of 10¹²² that may be produced naturally from the parameters of the Universe,andThe terms in above do not have any known physical significance and therefore should not be considered to contribute strongly to the new coincidence problem.

Something striking here again is that 10¹²²= almost (10⁴¹)³ which perhaps here again is another striking coincidence. 

Explanations

Unlike the classical arguments to this problem, the modern set of coincidences do have a satisfactory explanation. We take here a non-numerical approach to solve the 10¹²² coincidences.

At the root of the issue, Funkhouser says, is the fact that the current density of matter in the Universe is about the same as the observed vacuum energy density — a puzzling fact that he calls the ‘cosmic coincidence’. The vacuum energy density is thought to be constant, but the mass density changes as the Universe expands. Why they happen to be equal right now — a balance that helps us to detect dark energy amidst matter — is not known. Some have suggested explanations based on the anthropic principle: basically, it’s only under these conditions that life becomes possible, so if things were otherwise, we wouldn’t be here to see it.

Given this single coincidence, Funkhouser shows, all the 10¹²² ratios inevitably follow from the standard laws of physics and cosmology. “The major parameters of the Universe are the cosmological constant and the total mass and radius,” he says. “Due to the cosmic coincidence, they are related.” He has shown previously that similar reasoning accounts for the Eddington-Dirac large-number coincidence.

For a numerical approach to the problem, one may visit  http://rspa.royalsocietypublishing.org/content/464/2093/1345.full#sec-3 for a satisfactory explanation.

Funkhouser also notes that if the 10⁴⁰ and 10¹²² coincidences are interconnected, it can pave the way to an elegant mathematical relationship linking the mass of a nucleon with the speed of light, the gravitational constant, Planck’s constant and the cosmological constant.  Russian physicist Yakov Zel’dovich did already come to such a relationship like many other scientists, each based on some separate argument. Funkhouser himself has his own explanation where a Universe with 10 spatial dimensions will naturally yield such a relation if the seven dimensions shrank and the rest 3 formed the reality we see around us today.

Predictions & Consequences

If indeed the relations are of practical consequence and not mere coincidences then they imply that:

1. The value of the constants must change with time to always be proportional to the ever increasing age of universe, or in simple terms: the natural constants aren't constants but variable in adjustment to the age of the universe. It perhaps may also mean that the ratio of coulombic and gravitational forces for a proton-electron pair isn't constant too and increases with time.
2. The number of particles in the universe will be increasing proportionally to the square of the age of the universe. Thus new matter must be continually created.
3. It proposes an open universe that can only expand continually forever. We know that our universe is expanding at an accelerating rate and if the hypothesis is true then with increasing age of universe it must continue to expand forever meeting eventually a heat death after a long time.
4. Dirac's work on LNH led him to conclude that there are two different time evolutions in nature that are closely related to one another: one that governs the motions of galaxies, planets, and the evolution of the Universe; and one that governs the periodic vibrations of atoms, the atomic time. The former appears in the Einstein's theory of gravitation, or the general theory of relativity, and the latter appears in the theory of atoms, or the quantum theory.

Experimental Verification

Dirac Large Number Hypothesis essentially predicted that natural constants may have some variability which is a very daring prediction with tremendous consequences. Looking out for experimental verification, we do not really meet such extraordinary evidence to support such extraordinary prediction. However there are two points to note:

1. The more recent measurements of the universal gravitational constant G have created some controversies over its value. Several research teams around the world have come up with a value of G which differs from the previous accepted value by 0.1% to 0.7%. Two possibilities exist: (1) we underestimated the uncertainty in our measurement of G; (2) G actually varies with time [check point 6 under Bibliography for original source].
2. Results from spectral analysis of quasars indicates that the fine structure constant  which is a combination of electric charge, Planck constant and speed of light, magnetic permeability and measures the strength of interactions between charged particles and electromagnetic fields, may be changing with time. The fine-structure constant appears to have been larger by one part in 100,000 in the direction of the southern hemisphere constellation Ara, 10 billion years ago. Similarly, the constant appeared to have been smaller by a similar fraction in the northern direction, billions of years ago. If the fine structure constant varies with time then at least one other constant - the electric charge, the Planck constant or the speed of light - must also vary with time [check point 5 under Bibliography for original source].

Conclusion

The real nature and the truth of this dilemma may be forever a secret until a definitive evidence comes up. But that doesn't deter the enthusiasm of scientists working hard to find an answer to this incredible problem; rather it only feeds the enthusiasm because the unknown is the most lucrative and it is the job of a scientist to investigate the unknown with an epitome of patience. We can only expect some startling evidence to come up that can develop our understanding to a better depth about our universe because as we know 

Extraordinary claims require extraordinary proofs

Perhaps there may be other such special numbers that may pop up in the near future starting a new series of coincidences. The perfect quote to recall in reference to such a speculation is of the legendary scientist Nikola Tesla:

If only you would know the magnificence of 3 ,6 and 9, then you would have a key to the universe"

To end the article, here are a few words from the master himself - Paul Dirac on his Large Number Hypothesis.

On an ending note, for the extra smart readers out there for whom this average post doesn't provide much satisfaction, I present here two links(to papers on arXiv) that talk of yet more coincidences in physical laws, albeit at a very advanced level :

1. This paper presents 10 advanced examples from cosmology of striking numerical coincidences of physical quantities in the universe:  http://vixra.org/pdf/1112.0093v1.pdf
2. This paper illustrates a new set of examples based on yet another frequently appearing number 2¹²⁸ : http://www.ptep-online.com/index_files/2013/PP-33-07.PDF

Bibliography:

1. On Scott Funkhouser: http://www.nature.com/news/2008/080219/full/news.2008.610.html
2. LNH: http://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis
3. Interview with Dirac: http://www.fdavidpeat.com/interviews/dirac.htm
4. 10¹²² coincidences: http://rspa.royalsocietypublishing.org/content/464/2093/1345.full#sec-3
5. News on change in Fundamental Structure constant: http://physicsworld.com/cws/article/news/2010/sep/02/changes-spotted-in-fundamental-constant
6. On predictions and confirmations of LNH: https://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0CFIQFjAD&url=http%3A%2F%2Fresources.edb.gov.hk%2Fphysics%2FarticleDoc%2FMechanics%2F01Large%2520Number%2520Hypothesis(English).doc&ei=UFlZU--IJpS48gXtloCYDQ&usg=AFQjCNFA9XQcaCzNcswasG1RV2LYAd2AcQ&sig2=aRRlIGkk1UMCwsT_gD-LQw&bvm=bv.65397613,d.dGc