Cosmological Coincidence (abstract)

The shift of the energy density ratio of total matter (dark + baryonic) Ω_M(z) from 1 to 0 – or so far to ~ 0.26 at least – is so abrupt a transition from the matter dominated era from the current era and according to the standard model of cosmology, that the quite null probability that it coincides roughly with ‘now’ – at cosmological scale – has become known as ‘cosmological coincidence’.

Another type of abrupt shift occurs when computing the k-SAT problem about the ‘satisfiability’ of a set of M Boolean clauses linked through a Conjunctive Normal Form (CNF) formula:  each clause is an OR combination of Boolean variables taking value True or False – translated into 1 or 0 – and each clause is AND linked to the others, all clauses or constraints using variables randomly taken from a pool of size N.

The ratio of clauses to variables is usually denoted a(k) = M(k)/N and experimentation have revealed that the satisfiability abruptly shifts from 1 to 0 when a(k) gets above a measured threshold when N goes to infinity .

Both issues seem a priori unrelated. However, while a(2) = 1 has been proven, regularly improved experimentation have found a_s(3) ~ 4.267 and even detected a “clustering phase transition” a_d(3) observed at a value [1] circa 3.86 with then even another, intermediary “condensation phase transition” a_c(k) for k>3.

We have previously [2] introduced the clause density ratio β = (1-N/M) = (1-1/a) and proposed that β_d = π/√(18) ~0.74048 ~ (1 – 1/3.853), in connection to the Hales-Kepler theorem about maximal sphere density ratio, giving a_d(3) = 1/(1 – π/√(18)) ~3.853, very close to the observed value.

It is useful to observe a relationship between geometrical and logical worlds so established and to consider extending it into the physical one, using the fact that any logical proposition or sentence can be written as a CNF formula, of which k-SAT only bring an indeed important restriction from the defined dimensionality of the clauses.

The logical world has no spatiality alone, as opposed to the other extreme, non-framed Euclidean ‘flat’ space alone, no less theoretical. Energy functions have however been introduced, relating randomly allocated variables in 3-SAT to spin glass [3] physics and then recently modeling a type of spheres stacking [4].

At cosmological level another type of link interpreted as formal sphere staking has been introduced [5], where the cosmological energy density ratio, written as Ω_Λ to encompass the dark energy role, whether caused by a cosmological constant or a somehow equivalent phenomenon, was proposed to take this specific value, i.e. π/√(18).

Scheme [5] also referred to the ‘no-boundary’ proposal of a universe [6] in transition from compact geometry, “part of Euclidean four-sphere of radius 1/H”, to a mostly Lorentzian, unlimited volume one.

Recent observations [7, 8] seem to suggest a universe lately, at the least, departing from ΛCDM consensus but where energy density ratios today, or whatever take their roles, fit well with values proposed in [5].

The mechanism sketched in [5] is therefore more detailed in the new, full paper now to follow and where, for these roles, the contributions of the vacuum energy and of degrees of freedom able to drive the future through the present era, are integrated in hopefully consistent and surely unusual an angle.

[1] A. Montanari, F. Ricci-Tersenghi, G. Semerjian, Clusters of solutions and replica symmetry breaking in random k-satisfiability. arXiv:0802.3627, 2008

[2] Conferences IEEE CCCA’12, Marseille 2012 & ISC-Complex Systems, Orleans University, June 2013, Future = Complexity

[3] M. Mezard, Optimization and Physics: On the satisfiability of random Boolean formulae, arXiv:cond-math/0212448, 2002

[4] M. Mezard, G. Parisi, M. Tarzia, F. Zamponi, On the solution of a ‘solvable’ model of an ideal glass of hard spheres displaying a jamming transition, Journal of Statistical Mechanics : Theory and Experiment, stacks.iop.org/JSTAT/2011/P03002

[5] P. Journeau, New Concepts of Dimensions and Consequences, AIP n°1018, http://dx.doi.org/10.1063/1.4728011, 2008

[6] J.B. Hartle, S. Hawking, Wave function of the Universe, Physical Review D, vol. 18, n°12, APS, 1983

[7] A.G. Riess et al, A 2.4% Determination of the Local Value of the Hubble Constant, arXiv1604.01424 astro-ph, 2016

[8] J.L. Bernal, L. Verde, A.G. Riess, The trouble with H₀, arXiv:1607.05617, astro-ph, 2016