The history of the universe is an elliptic curve

Abstract

Friedmann-Lemaitre equations with contributions coming from matter, curvature, cosmological constant, and radiation, when written in terms of conformal time u rather than in terms of cosmic time t, can be solved explicitly in terms of standard Weierstrass elliptic functions. The spatial scale factor, the temperature, the densities, the Hubble function, and almost all quantities of cosmological interest (with the exception of t itself) are elliptic functions of u, in particular they are bi-periodic with respect to a lattice of the complex plane, when one takes u complex. After recalling the basics of the theory, we use these explicit expressions, as well as the experimental constraints on the present values of density parameters (we choose for the curvature density a small value in agreement with experimental bounds) to display the evolution of the main cosmological quantities for one real period 2 omega_r of conformal time (the cosmic time t never end but it goes to infinity for a finite value u_f < 2 omega_r of u). A given history of the universe, specified by the measured values of present-day densities, is associated with a lattice in the complex plane, or with an elliptic curve, and therefore with two Weierstrass invariants g2, g3. Using the same experimental data we calculate the values of these invariants, as well as the associated modular parameter and the corresponding Klein j-invariant. If one takes the flat case k=0, the lattice is only defined up to homotheties, and if one, moreover, neglects the radiation contribution, the j-invariant vanishes and the corresponding modular parameter tau can be chosen in one corner of the standard fundamental domain of the modular group (equihanharmonic case: tau = exp(2 i pi/3)). Several exact, ie non-numerical, results of independent interest are obtained in that case.