Oscillating universe in the DGP braneworld

With a method in which the Friedmann equation is written in a form such that evolution of the scale factor can be treated as that of a particle in a "potential", we classify all possible cosmic evolutions in the DGP braneworld scenario with the dark radiation term retained. By assuming that the energy component is pressureless matter, radiation or vacuum energy, respectively, we find that in the matter or vacuum energy dominated case, the scale factor has a minimum value $a_0$. In the matter dominated case, the big bang singularity can be avoided in some special circumstances, and there may exist an oscillating universe or a bouncing one. If the cosmic scale factor is in the oscillating region initially, the universe may undergo an oscillation. After a number of oscillations, it may evolve to the bounce point through quantum tunneling and then expand. However, if the universe contracts initially from an infinite scale, it can turn around and then expand forever. In the vacuum energy dominated case, there exists a stable Einstein static state to avoid the big bang singularity. However, in certain circumstances in the matter or vacuum energy dominated case, a new kind of singularity may occur at $a_0$ as a result of the discontinuity of the scale factor. In the radiation dominated case, the universe may originate from the big bang singularity, but a bouncing universe which avoids this singularity is also possible.Comment: 25 pages, 24 figures. To appear in PR

Small-Deviation Inequalities for Sums of Random Matrices

Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few of works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to the high-dimensional and even the infinite-dimensional cases

Emergent universe in spatially flat cosmological model

The scenario of an emergent universe provides a promising resolution to the big bang singularity in universes with positive or negative spatial curvature. It however remains unclear whether the scenario can be successfully implemented in a spatially flat universe which seems to be favored by present cosmological observations. In this paper, we study the stability of Einstein static state solutions in a spatially flat Shtanov-Sahni braneworld scenario. With a negative dark radiation term included and assuming a scalar field as the only matter energy component, we find that the universe can stay at an Einstein static state past eternally and then evolve to an inflation phase naturally as the scalar field climbs up its potential slowly. In addition, we also propose a concrete potential of the scalar field that realizes this scenario.Comment: 16 pages, 8 figure