Cosmological constant in a model with superstring-inspired E 6 unification and shadow θ-particles

We have developed the concept of parallel existence of the ordinary (O-) and mirror (M-), or shadow (Sh-) worlds. In the first part of the paper we consider a mirror world with broken mirror parity and the breaking E (6)-> SU(3)(3) in both worlds. We show that in this case the evolutions of coupling constants in the O- and M-worlds are not identical, having different parameters for similar evolutions. E (6) unification, inspired by superstring theory, restores the broken mirror parity at the scale similar to 10(18) GeV. With the aim to explain the tiny cosmological constant, in the second part we consider the breakings: E (6)-> SO(10)xU(1) (Z) in the O-world, and E'(6)-> SU(6)'xSU(2)' (theta) in the Sh-world. We assume the existence of shadow theta-particles and the low-energy symmetry group SU(3)' (C) xSU(2)' (L) xSU(2)' (theta) xU(1)' (Y) in the shadow world, instead of the Standard Model. The additional non-Abelian SU(2)' (theta) group with massless gauge fields, thetons, has a macroscopic confinement radius 1/I >' (theta) . The assumption that I >' (theta) a parts per thousand 2.3a <...10(-3) eV explains the tiny cosmological constant given by recent astrophysical measurements. In this way the present work opens the possibility to specify a grand unification group, such as E (6), from cosmology.