Massive gravity is a modified theory of general relativity. In this paper, we study, using a method in which the scale factor changes as a particle in a "potential", all possible cosmic evolutions in a ghost-free massive gravity. We find that there exists, in certain circumstances, an oscillating universe or a bouncing one. If the universe starts at the oscillating region, it may undergo a number of oscillations before it quantum mechanically tunnels to the bounce point and then expand forever. But going back to the singularity from the oscillating region is physically not allowed. So, the big bang singularity can be successfully resolved. At the same time, we also find that there exists a stable Einstein static state in some cases. However, the universe can not stay at this stable state past-eternally since it is allowed to quantum mechanically tunnel to a big-bang-to-big-crunch region and end with a big crunch. Thus, a stable Einstein static state universe can not be used to avoid the big bang singularity in massive gravity.Comment: 36 pages, 30 figures, accepted for publication in PRD. Two references adde
The stability of an Einstein static universe in the DGP braneworld scenario
is studied in this paper. Two separate branches denoted by $\epsilon=\pm1$ of
the DGP model are analyzed. Assuming the existence of a perfect fluid with a
constant equation of state, $w$, in the universe, we find that, for the branch
with $\epsilon=1$, there is no a stable Einstein static solution, while, for
the case with $\epsilon=-1$, the Einstein static universe exists and it is
stable when $-1
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
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.
A dimeric fluorescent macrocycle m-TPE Di-EtP5 (meso-tetraphenylethylene dimeric ethoxypillar[5]arene) is synthesized based on the meso-functionalized ethoxy pillar[5]arene. Through the connectivity of two pillar[5]arenes by C=C double bond, the central tetraphenylethylene (TPE) moiety is simultaneously formed. The resultant bicyclic molecule not only retains the host-guest properties of pillararenes but also introduces the interesting aggregation-induced emission properties inherent in the embedded TPE structure. Three dinitrile derivatives with various linkers are designed as guests (G1, G2, and G3) to form host-guest assemblies with m-TPE Di-EtP5. The morphological control and fluorescence properties of the assemblies are successfully realized. G1 with a shorter alkyl chain as the linker completely threads into the cavities of the host. G2, due to its longer chain length, forms a linear supramolecular polymer upon binding to m-TPE Di-EtP5. G3 differs from G2 by possessing a bulky phenyl group in the middle of the chain, which can be further assembled with m-TPE Di-EtP5 to form supramolecular layered polymer and precipitated out in solution, and can be efficiently applied to photocatalytic reactions.
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