New SubmissionFrom Moonshine to Hubble One-Parameter Family-- Theoretical Framework of Quantum Modularity

The term 'Moonshine' refers to webs of connections between Number Theory, Group Theory and String Theory. The starting point of monstrous moonshine was McKay and Thompson's observation invariant J(τ)= q^-1 + 196884q + 21493760q² + ...

Here, we found that there exist a Quantum Modularity Version: j(τ) = τ(8-τ³)/(1+ τ³), and let τ= H₀= √3 -1 then j(H₀) = 4, where H₀ is Hubble One Parameter H₀= 0.7320..., which is a dimensionless monodromy independent on any physical measurement. As fundamental elements of Quantum Modularity, other elements include:

(i) reduced Planck One Parameter h=2π√3;

(ii) Boltzmann One Parameter k_B= 8√3;

(iii) Newton's Gravitational Parameter G=2/3;

(iv) Quantum Speed Limit (uniruledness) = 4, which is greater than Light Speed One Parameter c=3.

We also found that there exists an important relationship gcd(h, k_B)= gcd(2π√3, 8√3) = 2√3. We surprisingly found that H₀²= (√3 -1)²= 4 - 2√3 =QSL - gcd(h, k_B). It is clear that Hubble One Parameter is more mysterious than Moonshine. For example, H₀²= 4 - 2√3 is just explicit exact solutions of Einstein- Friedmann equation:

H₀²= (a^'/a)² = 8/3 π (e + e_Λ) - κ/a2. (1)

and

κ/a² = H₀² (((e + e_Λ)/e_c) - 1) = H₀²( Ω_m + Ω_Λ- 1) (2)

where e_c:= 3 H₀²/(8π); Ω_m:= e/e_c; Ω_Λ = e_Λ/ e_c; e_c is the critical energy density, Ω_m and Ω_Λ are the normalized energy density of "matter" and of the cosmological constant.

We finally obtain that Ω_Λ+ Ω_m ≈ 1. It means that the dark energy Ω_Λ = 3+ 2√3 = 6.46410... or 64.5%, recently (2013) physical measurement is 68.5%, which shows that about 65-70% of the energetic content of the universe is in a form which cannot associated to ordinary matter and is for tis reason commonly referred to as "dark energy".

We also note that the invariant j(τ) function gives a maximum value QSL = 4 at τ = √3 -1 = H₀.

In Quantum Modularity Theory, we found another relationship k_B/h = 8√3/(2π√3) = 4/π, which shows an important fact: "the Sky is circle and the Earth is square.

In this paper, we cast a framework of Quantum Modularity Theory, and illustrate it has more rich and deep contents on Physics and Science.

This result proves that the classical experimental physics will enter a new era of quantum theoretical physics, where Quantum Speed Limit 4 plays a main role in the Universe Dilation:

(i) r < SQL=4: inner sheaf universe as Solar System;

(ii) r > QSL =4: outer sheaf universe as Black-Hole, Higgs, Tachyon Particles.