The Renormalizable Three-Term Polynomial Inflation with Large Tensor-to-Scalar Ratio

We systematically study the renormalizable three-term polynomial inflation in the supersymmetric and non-supersymmetric models. The supersymmetric inflaton potentials can be realized in supergravity theory, and only have two independent parameters. We show that the general renormalizable supergravity model is equivalent to one kind of our supersymmetric models. We find that the spectral index and tensor-to-scalar ratio can be consistent with the Planck and BICEP2 results, but the running of spectral index is always out of the $2\sigma$ range. If we do not consider the BICEP2 experiment, these inflationary models can be highly consistent with the Planck observations and saturate its upper bound on the tensor-to-scalar ratio ($r \le 0.11$). Thus, our models can be tested at the future Planck and QUBIC experiments.Comment: 38 pages, 40 figure

P-harmonic theory on ellipsoids with geometric applications.

In the first part of this thesis, we are interested in representing homotopy groups by p-harmonic maps with applications to minimal submanifolds of ellipsoids. In the second part, we discuss Liouville-type theorems for p-harmonic or p-stable maps into either a closed upper-half ellipsoid or a p-SSU ellipsoid. In the third part, we are interested in existence and non-existence of stable rectifiable currents on an ellipsoid. In the forth part, we study ellipsoids as geometric applications of Yang-Mills instabilities of convex hypersurfaces. In the fifth part, we verify that all of conclusions in the above topological, analytic and geometric theorems on ellipsoids are still valid on compact convex hypersurfaces. In the last part, we make sharp global integral estimates by a unified method, and find a dichotomy between constancy and infinity of weak sub- and supersolutions of a large class of degenerate and singular nonlinear partial differential equations on complete noncompact Riemannian manifolds

The Minimal GUT with Inflaton and Dark Matter Unification

Giving up the solutions to the fine-tuning problems, we propose the non-supersymmetric flipped $SU(5)\times U(1)_X$ model based on the minimal particle content principle, which can be constructed from the four-dimensional $SO(10)$ models, five-dimensional orbifold $SO(10)$ models, and local F-theory $SO(10)$ models. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form complete $SU(5)\times U(1)_X$ representation. Proton lifetime is around $5\times 10^{35}$ years, neutrino masses and mixing can be explained via seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and vacuum stability problem can be solved as well. In particular, we propose that inflaton and dark matter particle can be unified to a real scalar field with $Z_2$ symmetry, which is not an axion and does not have the non-minimal coupling to gravity. Such kind of scenarios can be applied to the generic scalar dark matter models. Also, we find that the vector-like particle corrections to the $B_s^0$ masses can be about 6.6%, while their corrections to the $K^0$ and $B_d^0$ masses are negligible.Comment: 5 pages, 4 figures;V2: published versio

Supergravity inflation on a brane

We discuss supergravity inflation in braneworld cosmology for the class of potentials $V(\phi)=\alpha \phi^n\rm{exp}(-\beta^m \phi^m)$ with $m=1,~2$. These minimal SUGRA models evade the $\eta$ problem due to a broken shift symmetry and can easily accommodate the observational constraints. Models with smaller $n$ are preferred while models with larger $n$ are out of the $2\sigma$ region. Remarkably, the field excursions required for $60$ $e$-foldings stay sub-planckian $\Delta\phi <1$.Comment: 10 pages, 4 figure

Generalizing Liouville-type Problems for Differential 1-Forms from Lq Spaces to Non-Lq Spaces

We obtain Liouville-type results for closed and p-pseudo-coclosed differential 1-forms ! with energy of lim inf r!1 1 r2 R B(x0;r) j!jqdv \u3c 1 (that is, 2-finite growth), which extends finite q-energy ( R M j!jqdv \u3c 1) in Lq spaces to infinite q-energy ( R M j!jqdv = 1) in non-Lq spaces. In particular, we recapture mathematicians\u27 vanishing results of Liouville- type theorem for ! with finite q-energy in Lq spaces. Our method in this paper provides a successful way to work on Liouville-type problems for differential forms with a variety of energy conditions in broad spaces