The history of the universe is an elliptic curve
  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.
