Probing the gauge symmetry breaking of the early universe in 3-3-1 models and beyond by gravitational waves

Taking the 3-3-1 models (with $SU(3)_c \otimes SU(3)_L \otimes U(1)_Y$ gauge group) as examples, we study that a class of new physics models with extended gauge group could undergo one or several first-order phase transitions associated with the spontaneously symmetry breaking processes during the evolution of the universe, which can produce detectable phase transition gravitational wave (GW) signals at future GW experiments, such as LISA, BBO, DECIGO, SKA and aLIGO. These GW signals can provide new sources of GWs with different peak frequencies, and can be used to probe the evolution history of the universe.Comment: Published version for Physics Letters

Mixed principal eigenvalues in dimension one

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the $L^{2}$-Poincar\'e inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.Comment: 45 pages; Front. Math. China, 201