Realization of the N(odd)-dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space R_{q}^{N} is a kind of noncommutative space which is obtained from ordinary Euclidean space R^{N} by deformation with parameter q. When N is odd, the structure of this space is similar to R_{q}^{3}. Motivated by realization of R_{q}^{3} by differential operators in R^{3}, we give such realization for R_{q}^{5} and R_{q}^{7} cases and generalize our results to R_{q}^{N} (N odd) in this paper, that is, we show that the algebra of R_{q}^{N} can be realized by differential operators acting on C^{infinite} functions on undeformed space R^{N}.Comment: 10 pages, LaTe

Properties of 2×22\times 2 h-deformed quantum (super)matrices

We investigate the $h$-deformed quantum (super)group of $2\times 2$ matrices and use a kind of contraction procedure to prove that the $n$-th power of this deformed quantum (super)matrix is quantum (super)matrix with the deformation parameter $nh$.Comment: Accepted by International Journal of Theoretical Physic

Exploring the full parameter space for an interacting dark energy model with recent observations including redshift-space distortions: Application of the parametrized post-Friedmann approach

Dark energy can modify the dynamics of dark matter if there exists a direct interaction between them. Thus a measurement of the structure growth, e.g., redshift-space distortions (RSD), can provide a powerful tool to constrain the interacting dark energy (IDE) models. For the widely studied $Q=3\beta H\rho_{de}$ model, previous works showed that only a very small coupling ($\beta\sim\mathcal{O}(10^{-3})$) can survive in current RSD data. However, all these analyses had to assume $w>-1$ and $\beta>0$ due to the existence of the large-scale instability in the IDE scenario. In our recent work [Phys. Rev. D 90, 063005 (2014)], we successfully solved this large-scale instability problem by establishing a parametrized post-Friedmann (PPF) framework for the IDE scenario. So we, for the first time, have the ability to explore the full parameter space of the IDE models. In this work, we reexamine the observational constraints on the $Q=3\beta H\rho_{de}$ model within the PPF framework. By using the Planck data, the baryon acoustic oscillation data, the JLA sample of supernovae, and the Hubble constant measurement, we get $\beta=-0.010^{+0.037}_{-0.033}$ ($1\sigma$). The fit result becomes $\beta=-0.0148^{+0.0100}_{-0.0089}$ ($1\sigma$) once we further incorporate the RSD data in the analysis. The error of $\beta$ is substantially reduced with the help of the RSD data. Compared with the previous results, our results show that a negative $\beta$ is favored by current observations, and a relatively larger interaction rate is permitted by current RSD data.Comment: 12 pages, 3 figure

Measuring growth index in a universe with sterile neutrinos

Consistency tests for the general relativity (GR) can be performed by constraining the growth index $\gamma$ using the measurements of redshift-space distortions (RSD) in conjunction with other observations. In previous studies, deviations from the GR expected value of $\gamma\approx 0.55$ at the 2--3$\sigma$ level were found. In this work, we reconsider the measurement of $\gamma$ in a universe with sterile neutrinos. We constrain the sterile neutrino cosmological model using the RSD measurements combined with the cosmic microwave background data (Planck temperature data plus WMAP 9-yr polarization data), the baryon acoustic oscillation data, the Hubble constant direct measurement, the Planck Sunyaev-Zeldovich cluster counts data, and the galaxy shear data. We obtain the constraint result of the growth index, $\gamma=0.584^{+0.047}_{-0.048}$, well consistent with the GR expected value (the consistency is at the 0.6$\sigma$ level). For the parameters of sterile neutrino, we obtain $N_{\rm{eff}}=3.62^{+0.26}_{-0.42}$ and $m_{\nu,{\rm{sterile}}}^{\rm{eff}}=0.48^{+0.11}_{-0.14}$ eV. We also consider the BICEP2 data and perform an analysis on the model with tensor modes. Similar fit results are obtained, showing that once light sterile neutrino is considered in the universe, GR will become well consistent with the current observations.Comment: 5 pages, 3 figures; accepted for publication in Physics Letters

Probing f(R)f(R) cosmology with sterile neutrinos via measurements of scale-dependent growth rate of structure

In this paper, we constrain the dimensionless Compton wavelength parameter $B_0$ of $f(R)$ gravity as well as the mass of sterile neutrino by using the cosmic microwave background observations, the baryon acoustic oscillation surveys, and the linear growth rate measurements. Since both the $f(R)$ model and the sterile neutrino generally predict scale-dependent growth rates, we utilize the growth rate data measured in different wavenumber bins with the theoretical growth rate approximatively scale-independent in each bin. The employed growth rate data come from the peculiar velocity measurements at $z=0$ in five wavenumber bins, and the redshift space distortions measurements at $z=0.25$ and $z=0.37$ in one wavenumber bin. By constraining the $f(R)$ model alone, we get a tight 95\% upper limit of $\log_{10}B_0<-4.1$. This result is slightly weakened to $\log_{10}B_0<-3.8$ (at 2$\sigma$ level) once we simultaneously constrain the $f(R)$ model and the sterile neutrino mass, due to the degeneracy between the parameters of the two. For the massive sterile neutrino parameters, we get the effective sterile neutrino mass $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.62$ eV (2$\sigma$) and the effective number of relativistic species $N_{\rm eff}<3.90$ (2$\sigma$) in the $f(R)$ model. As a comparison, we also obtain $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.56$ eV (2$\sigma$) and $N_{\rm eff}<3.92$ (2$\sigma$) in the standard $\Lambda$CDM model.Comment: 6 pages, 3 figures; revised version accepted for publication in Phys. Lett.

Testing models of vacuum energy interacting with cold dark matter

We test the models of vacuum energy interacting with cold dark matter and try to probe the possible deviation from the $\Lambda$CDM model using current observations. We focus on two specific models, $Q=3\beta H\rho_{\Lambda}$ and $Q=3\beta H\rho_c$. The data combinations come from the Planck 2013 data, the baryon acoustic oscillations measurements, the type-Ia supernovae data, the Hubble constant measurement, the redshift space distortions data and the galaxy weak lensing data. For the $Q=3\beta H\rho_c$ model, we find that it can be tightly constrained by all the data combinations, while for the $Q=3\beta H\rho_{\Lambda}$ model, there still exist significant degeneracies between parameters. The tightest constraints for the coupling constant are $\beta=-0.026^{+0.036}_{-0.053}$ (for $Q=3\beta H\rho_{\Lambda}$) and $\beta=-0.00045\pm0.00069$ (for $Q=3\beta H\rho_c$) at the $1\sigma$ level. For all the fit results, we find that the null interaction $\beta=0$ is always consistent with data. Our work completes the discussion on the interacting dark energy model in the recent Planck 2015 papers. Considering this work together with the Planck 2015 results, it is believed that there is no evidence for the models beyond the standard $\Lambda$CDM model from the point of view of possible interaction.Comment: 7 pages, 2 figures; final version published in Physical Review

Sterile neutrinos help reconcile the observational results of primordial gravitational waves from Planck and BICEP2

We show that involving a sterile neutrino species in the $\Lambda$CDM+$r$ model can help relieve the tension about the tensor-to-scalar ratio $r$ between the Planck temperature data and the BICEP2 B-mode polarization data. Such a model is called the $\Lambda$CDM+$r$+$\nu_s$ model in this paper. Compared to the $\Lambda$CDM+$r$ model, there are two extra parameters, $N_{\rm eff}$ and $m_{\nu,{\rm sterile}}^{\rm eff}$, in the $\Lambda$CDM+$r$+$\nu_s$ model. We show that in this model the tension between Planck and BICEP2 can be greatly relieved at the cost of the increase of $n_s$. However, comparing with the $\Lambda$CDM+$r$+$dn_s/d\ln k$ model that can significantly reduce the tension between Planck and BICEP2 but also makes trouble to inflation due to the large running of the spectral index of order $10^{-2}$ produced, the $\Lambda$CDM+$r$+$\nu_s$ model is much better for inflation. By including a sterile neutrino species in the standard cosmology, besides the tension with BICEP2, the other tensions of Planck with other astrophysical data, such as the $H_0$ direct measurement, the Sunyaev-Zeldovich cluster counts, and the galaxy shear data, can all be significantly relieved. So, this model seems to be an economical choice. Combining the Planck temperature data, the WMAP-9 polarization data, and the baryon acoustic oscillation data with all these astrophysical data (including BICEP2), we find that in the $\Lambda$CDM+$r$+$\nu_s$ model $n_s=0.999\pm 0.011$, $r=0.21^{+0.04}_{-0.05}$, $N_{\rm eff}=3.95\pm 0.33$ and $m_{\nu,{\rm sterile}}^{\rm eff}=0.51^{+0.12}_{-0.13}$ eV. Thus, our results prefer $\Delta N_{\rm eff}>0$ at the 2.7$\sigma$ level and a nonzero mass of sterile neutrino at the 3.9$\sigma$ level.Comment: 5 pages, 3 figure

A global fit study on the new agegraphic dark energy model

We perform a global fit study on the new agegraphic dark energy (NADE) model in a non-flat universe by using the MCMC method with the full CMB power spectra data from the WMAP 7-yr observations, the SNIa data from Union2.1 sample, BAO data from SDSS DR7 and WiggleZ Dark Energy Survey, and the latest measurements of $H_0$ from HST. We find that the value of $\Omega_{k0}$ is greater than 0 at least at the 3$\sigma$ confidence levels (CLs), which implies that the NADE model distinctly favors an open universe. Besides, our results show that the value of the key parameter of NADE model, $n=2.673^{+0.053+0.127+0.199}_{-0.077-0.151-0.222}$, at the 1--3$\sigma$ CLs, where its best-fit value is significantly smaller than those obtained in previous works. We find that the reason leading to such a change comes from the different SNIa samples used. Our further test indicates that there is a distinct tension between the Union2 sample of SNIa and other observations, and the tension will be relieved once the Union2 sample is replaced by the Union2.1 sample. So, the new constraint result of the NADE model obtained in this work is more reasonable than before.Comment: 6 pages, 3 figures; typos correcte