Constraints on Deflation from the Equation of State of Dark Energy

In cyclic cosmology based on phantom dark energy the requirement that our universe satisfy a CBE-condition ({\it Comes Back Empty}) imposes a lower bound on the number $N_{\rm cp}$ of causal patches which separate just prior to turnaround. This bound depends on the dark energy equation of state $w = p/\rho = -1 - \phi$ with $\phi > 0$. More accurate measurement of $\phi$ will constrain $N_{\rm cp}$. The critical density $\rho_c$ in the model has a lower bound $\rho_c \ge (10^9 {\rm GeV})^4$ or $\rho_c \ge (10^{18} {\rm GeV})^4$ when the smallest bound state has size $10^{-15}$m, or $10^{-35}$m, respectively.Comment: 23 pages, 3 figures, typos fixe

A Simplest A4 Model for Tri-Bimaximal Neutrino Mixing

We present a see-saw $A_4$ model for Tri-Bimaximal mixing which is based on a very economical flavour symmetry and field content and still possesses all the good features of $A_4$ models. In particular the charged lepton mass hierarchies are determined by the $A_4\times Z_4$ flavour symmetry itself without invoking a Froggatt-Nielsen U(1) symmetry. Tri-Bimaximal mixing is exact in leading order while all the mixing angles receive corrections of the same order in next-to-the-leading approximation. As a consequence the predicted value of $\theta_{13}$ is within the sensitivity of the experiments which will take data in the near future. The light neutrino spectrum, typical of $A_4$ see-saw models, with its phenomenological implications, also including leptoproduction, is studied in detail.Comment: 20 pages, 2 figure

A See-Saw S4S_4 model for fermion masses and mixings

We present a supersymmetric see-saw $S_4$ model giving rise to the most general neutrino mass matrix compatible with Tri-Bimaximal mixing. We adopt the $S_4\times Z_5$ flavour symmetry, broken by suitable vacuum expectation values of a small number of flavon fields. We show that the vacuum alignment is a natural solution of the most general superpotential allowed by the flavour symmetry, without introducing any soft breaking terms. In the charged lepton sector, mass hierarchies are controlled by the spontaneous breaking of the flavour symmetry caused by the vevs of one doublet and one triplet flavon fields instead of using the Froggatt-Nielsen U(1) mechanism. The next to leading order corrections to both charged lepton mass matrix and flavon vevs generate corrections to the mixing angles as large as ${\cal O}(\lambda_C^2)$. Applied to the quark sector, the symmetry group $S_4\times Z_5$ can give a leading order $V_{CKM}$ proportional to the identity as well as a matrix with ${\cal O}(1)$ coefficients in the Cabibbo $2\times 2$ submatrix. Higher order corrections produce non vanishing entries in the other $V_{CKM}$ entries which are generically of ${\cal O}(\lambda_C^2)$.Comment: 30 pages, 3 figures, minor changes to match the published versio