Dynamical System of Scalar Field from 2-Dimension to 3-D and its Cosmological Implication

We give the three-dimensional dynamical autonomous systems for most of the popular scalar field dark energy models including (phantom) quintessence, (phantom) tachyon, k-essence and general non-canonical scalar field models, change the dynamical variables from variables $(x, y, \lambda)$ to observable related variables $(w_{\phi}, \Omega_{\phi}, \lambda)$, and show the intimate relationships between those scalar fields that the three-dimensional system of k-essence can reduce to (phantom) tachyon, general non-canonical scalar field can reduce to (phantom) quintessence and k-essence can also reduce to (phantom) quintessence for some special cases. For the applications of the three-dimensional dynamical systems, we investigate several special cases and give the exactly dynamical solutions in detail. In the end of this paper, we argue that, it is more convenient and also has more physical meaning to express the differential equations of dynamical systems in $(w_{\phi}, \Omega_{\phi}, \lambda)$ instead of variables $(x, y, \lambda)$ and to investigate the dynamical system in 3-Dimension instead of 2-Dimension. We also raise a question about the possibility of the chaotic behavior in the spatially flat single scalar field FRW cosmological models in the presence of ordinary matter.Comment: 20 pages, 8 figures,some references added. Minor changes according to the suggestions from referee

A simple and natural interpretations of the DAMPE cosmic-ray electron/positron spectrum within two sigma deviations

The DArk Matter Particle Explorer (DAMPE) experiment has recently announced the first results for the measurement of total electron plus positron fluxes between 25 GeV and 4.6 TeV. A spectral break at about 0.9 TeV and a tentative peak excess around 1.4 TeV have been found. However, it is very difficult to reproduce both the peak signal and the smooth background including spectral break simultaneously. We point out that the numbers of events in the two energy ranges (bins) close to the 1.4 TeV excess have $1\sigma$ deficits. With the basic physics principles such as simplicity and naturalness, we consider the $-2\sigma$, $+2\sigma$, and $-1\sigma$ deviations due to statistical fluctuations for the 1229.3~GeV bin, 1411.4~GeV bin, and 1620.5~GeV bin. Interestingly, we show that all the DAMPE data can be explained consistently via both the continuous distributed pulsar and dark matter interpretations, which have $\chi^{2} \simeq 17.2$ and $\chi^{2} \simeq 13.9$ (for all the 38 points in DAMPE electron/positron spectrum with 3 of them revised), respectively. These results are different from the previous analyses by neglecting the 1.4 TeV excess. At the same time, we do a similar global fitting on the newly released CALET lepton data, which could also be interpreted by such configurations. Moreover, we present a $U(1)_D$ dark matter model with Breit-Wigner mechanism, which can provide the proper dark matter annihilation cross section and escape the CMB constraint. Furthermore, we suggest a few ways to test our proposal.Comment: 18 pages, 6 figures, 5 tables. Figures and Bibs update