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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 figures

We apply a series of null diagnostics based on the statefinder hierarchy to diagnose different holographic dark energy models including the original holographic dark energy, the new holographic dark energy, the new agegraphic dark energy, and the Ricci dark energy models. We plot the curves of statefinders $S^{(1)}_3$ and $S^{(1)}_4$ versus redshift $z$ and the evolutionary trajectories of $\{S^{(1)}_3, \epsilon\}$ and $\{S^{(1)}_4, \epsilon\}$ for these models, where $\epsilon$ is the fractional growth parameter. Combining the evolution curves with the current values of $S^{(1)}_3$, $S^{(1)}_4$, and $\epsilon$, we find that the statefinder $S^{(1)}_4$ performs better than $S^{(1)}_3$ for diagnosing the holographic dark energy models. In addition, the conjunction of the statefinder hierarchy and the fractional growth parameter is proven to be a useful method to diagnose the holographic dark energy models, especially for breaking the degeneracy of the new agegraphic dark energy model with different parameter values., Comment: 9 pages, 8 figures; published version

We propose an alternative model for the holographic dark energy in a non-flat universe. This new model differs from the previous one in that the IR length cutoff $L$ is taken to be exactly the event horizon size in a non-flat universe, which is more natural and theoretically/conceptually concordant with the model of holographic dark energy in a flat universe. We constrain the model using the recent observational data including the type Ia supernova data from SNLS3, the baryon acoustic oscillation data from 6dF, SDSS-DR7, BOSS-DR11, and WiggleZ, the cosmic microwave background data from Planck, and the Hubble constant measurement from HST. In particular, since some previous studies have shown that the color-luminosity parameter $\beta$ of supernovae is likely to vary during the cosmic evolution, we also consider such a case that $\beta$ in SNLS3 is time-varying in our data fitting. Compared to the constant $\beta$ case, the time-varying $\beta$ case reduces the value of $\chi^2$ by about 35 and results in that $\beta$ deviates from a constant at about 5$\sigma$ level, well consistent with the previous studies. For the parameter $c$ of the holographic dark energy, the constant $\beta$ fit gives $c=0.65\pm 0.05$ and the time-varying $\beta$ fit yields $c=0.72\pm 0.06$. In addition, an open universe is favored (at about 2$\sigma$) for the model by the current data., Comment: 8 pages, 4 figures

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 B

The detection of the B-mode polarization of the cosmic microwave background (CMB) by the BICEP2 experiment implies that the tensor-to-scalar ratio $r$ should be involved in the base standard cosmology. In this paper, we extend the $\Lambda$CDM+$r$+neutrino/dark radiation models by replacing the cosmological constant with the dynamical dark energy with constant $w$. Four neutrino plus dark energy models are considered, i.e., the $w$CDM+$r+\sum m_\nu$, $w$CDM+r + $N_{\rm eff}$, $w$CDM+r + $\sum m_\nu$ + $N_{\rm eff}$, and $w$CDM+r + $N_{\rm eff}$ + $m_{\nu,{\rm sterile}}^{\rm eff}$ models. The current observational data considered in this paper include the Planck temperature data, the WMAP 9-year polarization data, the baryon acoustic oscillation data, the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck CMB lensing data, the cosmic shear data, and the BICEP2 polarization data. We test the data consistency in the four cosmological models, and then combine the consistent data sets to perform joint constraints on the models. We focus on the constraints on the parameters $w$, $\sum m_\nu$, $N_{\rm eff}$, and $m_{\nu,{\rm sterile}}^{\rm eff}$., Comment: 22 pages, 8 figures, 5 tables

The Sandage-Loeb (SL) test directly measures the expansion rate of the universe in the redshift range of $2\lesssim z\lesssim 5$ by detecting redshift drift in the spectra of Lyman-$\alpha$ forest of distant quasars. We discuss the impact of the future SL test data on parameter estimation for the $\Lambda$CDM, the $w$CDM, and the $w_0w_a$CDM models. To avoid the potential inconsistency with other observational data, we take the best-fitting dark energy model constrained by the current observations as the fiducial model to produce 30 mock SL test data. The SL test data provide an important supplement to the other dark energy probes, since they are extremely helpful in breaking the existing parameter degeneracies. We show that the strong degeneracy between $\Omega_m$ and $H_0$ in all the three dark energy models is well broken by the SL test. Compared to the current combined data of type Ia supernovae, baryon acoustic oscillation, cosmic microwave background, and Hubble constant, the 30-yr observation of SL test could improve the constraints on $\Omega_m$ and $H_0$ by more than 60\% for all the three models. But the SL test can only moderately improve the constraint on the equation of state of dark energy. We show that a 30-yr observation of SL test could help improve the constraint on constant $w$ by about 25\%, and improve the constraints on $w_0$ and $w_a$ by about 20\% and 15\%, respectively. We also quantify the constraining power of the SL test in the future high-precision joint geometric constraints on dark energy. The mock future supernova and baryon acoustic oscillation data are simulated based on the space-based project JDEM. We find that the 30-yr observation of SL test would help improve the measurement precision of $\Omega_m$, $H_0$, and $w_a$ by more than 70\%, 20\%, and 60\%, respectively, for the $w_0w_a$CDM model., Comment: 16 pages, 9 figures, 3 tables; adding a new section to address future SN and BAO observations; accepted for publication in JCAP

In this work, we constrain the spectral index $n_t$ of the primordial gravitational wave power spectrum in a universe with sterile neutrinos by using the Planck temperature data, the WMAP 9-year polarization data, the baryon acoustic oscillation data, and the BICEP2 data. We call this model the $\Lambda$CDM+$r$+$\nu_s$+$n_t$ model. The additional massive sterile neutrino species can significantly relieve the tension between the Planck and BICEP2 data, and thus can reduce the possible effects of this tension on the fit results of $n_t$. To constrain the parameters of sterile neutrino, we also utilize the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck CMB lensing data, and the cosmic shear data. We find that due to the fact that the BICEP2 data are most sensitive to the multipole $\ell\sim150$ corresponding to $k\sim0.01$ Mpc$^{-1}$, there exists a strong anticorrelation between $n_t$ and $r_{0.002}$ in the BICEP2 data, and this further results in a strongly blue-tilt spectrum. However, a slightly red-tilt tensor power spectrum is also allowed by the BICEP2 data in the region with larger value of $r_{0.002}$. By using the full data sets, we obtain $m_{\nu,{\rm{sterile}}}^{\rm{eff}}=0.48^{+0.11}_{-0.13}$ eV, $N_{\rm{eff}}=3.73^{+0.34}_{-0.37}$, and $n_t=0.96^{+0.48}_{-0.63}$ for the $\Lambda$CDM+$r$+$\nu_s$+$n_t$ model., Comment: 5 pages, 3 figures, 1 table; published in Sci China-Phys Mech Astron (Special Topic: BICEP2 and Beyond)

The Sandage-Loeb (SL) test is a unique method to probe dark energy in the "redshift desert" of $2\lesssim z\lesssim 5$, and thus it provides an important supplement to the other dark energy probes. Therefore, it is of great importance to quantify how the future SL test data impact on the dark energy constraints. To avoid the potential inconsistency in data, we use the best-fitting model based on the other geometric measurements as the fiducial model to produce 30 mock SL test data. The 10-yr, 20-yr, and 30-yr observations of SL test are analyzed and compared in detail. We show that compared to the current combined data of type Ia supernovae, baryon acoustic oscillation, cosmic microwave background, and Hubble constant, the 30-yr observation of SL test could improve the constraint on $\Omega_m$ by about $80%$ and the constraint on $w$ by about $25%$. Furthermore, the SL test can also improve the measurement of the possible direct interaction between dark energy and dark matter. We show that the SL test 30-yr data could improve the constraint on $\gamma$ by about $30%$ and $10%$ for the $Q=\gamma H\rho_c$ and $Q=\gamma H\rho_{de}$ models, respectively., Comment: 10 pages, 3 figures

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parametrized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models., Comment: 5 pages, 2 figures, 1 table; published version

Since the B-mode polarization of the cosmic microwave background (CMB) was detected by the BICEP2 experiment and an unexpectedly large tensor-to-scalar ratio, $r=0.20^{+0.07}_{-0.05}$, was found, the base standard cosmology should at least be extended to the 7-parameter $\Lambda$CDM+$r$ model. In this paper, we consider the extensions to this base $\Lambda$CDM+$r$ model by including additional base parameters relevant to neutrinos and/or other neutrino-like relativistic components. Four neutrino cosmological models are considered, i.e., the $\Lambda$CDM+$r$+$\sum m_\nu$, $\Lambda$CDM+$r$+$N_{\rm eff}$, $\Lambda$CDM+$r$+$\sum m_\nu$+$N_{\rm eff}$, and $\Lambda$CDM+$r$+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$ models. We combine the current data, including the Planck temperature data, the WMAP 9-year polarization data, the baryon acoustic oscillation data, the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck CMB lensing data, the cosmic shear data, and the BICEP2 polarization data, to constrain these neutrino cosmological models. We focus on the constraints on the parameters $\sum m_\nu$, $N_{\rm eff}$, and $m_{\nu,{\rm sterile}}^{\rm eff}$. We also discuss whether the tension on $r$ between Planck and BICEP2 can be relieved in these neutrino cosmological models., Comment: 21 pages, 4 figures, 4 tables

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 figures

We apply the statefinder diagnostic to the holographic dark energy models, including the original holographic dark energy (HDE) model, the new holographic dark energy model, the new agegraphic dark energy (NADE) model, and the Ricci dark energy model. In the low-redshift region the holographic dark energy models are degenerate with each other and with the $\Lambda$CDM model in the $H(z)$ and $q(z)$ evolutions. In particular, the HDE model is highly degenerate with the $\Lambda$CDM model, and in the HDE model the cases with different parameter values are also in strong degeneracy. Since the observational data are mainly within the low-redshift region, it is very important to break this low-redshift degeneracy in the $H(z)$ and $q(z)$ diagnostics by using some quantities with higher order derivatives of the scale factor. It is shown that the statefinder diagnostic $r(z)$ is very useful in breaking the low-redshift degeneracies. By employing the statefinder diagnostic the holographic dark energy models can be differentiated efficiently in the low-redshift region. The degeneracy between the holographic dark energy models and the $\Lambda$CDM model can also be broken by this method. Especially for the HDE model, all the previous strong degeneracies appearing in the $H(z)$ and $q(z)$ diagnostics are broken effectively. But for the NADE model, the degeneracy between the cases with different parameter values cannot be broken, even though the statefinder diagnostic is used. A direct comparison of the holographic dark energy models in the $r$--$s$ plane is also made, in which the separations between the models (including the $\Lambda$CDM model) can be directly measured in the light of the current values $\{r_0,s_0\}$ of the models., Comment: 8 pages, 8 figures; accepted by European Physical Journal C; matching the publication version

In this paper, a new version of the interacting model of new agegraphic dark energy (INADE) is proposed and analyzed in detail. The interaction between dark energy and dark matter is reconsidered. The interaction term $Q=bH_0\rho_{\rm de}^\alpha\rho_{\rm dm}^{1-\alpha}$ is adopted, which abandons the Hubble expansion rate $H$ and involves both $\rho_{\rm de}$ and $\rho_{\rm dm}$. Moreover, the new initial condition for the agegraphic dark energy is used, which solves the problem of accommodating baryon matter and radiation in the model. The solution of the model can be given using an iterative algorithm. A concrete example for the calculation of the model is given. Furthermore, the model is constrained by using the combined Planck data (Planck+BAO+SNIa+$H_0$) and the combined WMAP-9 data (WMAP+BAO+SNIa+$H_0$). Three typical cases are considered: (A) $Q=bH_0\rho_{\rm de}$, (B) $Q=bH_0\sqrt{\rho_{\rm de}\rho_{\rm dm}}$, and (C) $Q=bH_0\rho_{\rm dm}$, which correspond to $\alpha=1$, 1/2, and 0, respectively. The departures of the models from the $\Lambda$CDM model are measured by the $\Delta$BIC and $\Delta$AIC values. It is shown that the INADE model is better than the NADE model in the fit, and the INADE(A) model is the best in fitting data among the three cases., Comment: 6 pages, 2 figures

We apply the statefinder diagnostic to the extended holographic Ricci dark energy (ERDE) model without and with interaction to study their behaviors. We plot the trajectories of various parameters for different cases. It is shown that the non-interacting model does not reach the LCDM point $\{1,0\}$ and the interacting one is favored, because the interaction makes the evolution of the statefinder pair $\{r,s\}$ quite different., Comment: 6 pages, 4 figures

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 corrected

The holographic $\Lambda(t)$CDM model in a non-flat universe is studied in this paper. In this model, to keep the form of the stress-energy of the vacuum required by general covariance, the holographic vacuum is enforced to exchange energy with dark matter. It is demonstrated that for the holographic model the best choice for the IR cutoff of the effective quantum field theory is the event horizon size of the universe. We derive the evolution equations of the holographic $\Lambda(t)$CDM model in a non-flat universe. We constrain the model by using the current observational data, including the 557 Union2 type Ia supernovae data, the cosmic microwave background anisotropy data from the 7-yr WMAP, and the baryon acoustic oscillation data from the SDSS. Our fit results show that the holographic $\Lambda(t)$CDM model tends to favor a spatially closed universe (the best-fit value of $\Omega_{k0}$ is -0.042), and the 95% confidence level range for the spatial curvature is $-0.101<\Omega_{k0}<0.040$. We show that the interaction between the holographic vacuum and dark matter induces an energy flow of which the direction is first from vacuum to dark matter and then from dark matter to vacuum. Thus, the holographic $\Lambda(t)$CDM model is just a time-varying vacuum energy scenario in which the interaction between vacuum and dark matter changes sign during the expansion of the universe., Comment: 8 pages, 4 figures. version for publication in EPJC. arXiv admin note: text overlap with arXiv:1112.2350

The initial condition $\Omega_{\rm de}(z_{\rm ini})=n^2(1+z_{\rm ini})^{-2}/4$ at $z_{\rm ini}=2000$ widely used to solve the differential equation of the density of the new agegraphic dark energy (NADE) $\Omega_{\rm de}$, makes the NADE model be a single-parameter dark-energy cosmological model. However, we find that this initial condition is only applicable in a flat universe with only dark energy and pressureless matter. In fact, in order to obtain more information from current observational data, such as cosmic microwave background (CMB) and baryon acoustic oscillations (BAO), we need to consider the contribution of radiation. For this situation, the initial condition mentioned above becomes invalid. To overcome this shortage, we investigate the evolution of dark energy in the matter-dominated and radiation-dominated epochs, and obtain a new initial condition $\Omega_{\rm de}(z_{\rm ini})=\frac{n^2(1+z_{\rm ini})^{-2}}{4}(1+\sqrt{F(z_{\rm ini})})^2$ at $z_{\rm ini}=2000$, where $F(z)\equiv\frac{\Omega_{r0}(1+z)}{\Omega_{m0}+\Omega_{r0}(1+z)}$ with $\Omega_{r0}$ and $\Omega_{m0}$ being the current density parameters of radiation and pressureless matter, respectively. This revised initial condition is applicable for the differential equation of $\Omega_{\rm de}$ obtained in the standard Friedmann-Robertson-Walker (FRW) universe with dark energy, pressureless matter, radiation, and even spatial curvature, and can still keep the NADE model being a single-parameter model. With the revised initial condition and the observational data of type Ia supernova (SNIa), CMB and BAO, we finally constrain the NADE model. The results show that the single free parameter $n$ of the NADE model can be constrained tightly., Comment: 6 pages, 1 figure

We extend the holographic Ricci dark energy model to include some direct, non-gravitational interaction between dark energy and dark matter. We consider three phenomenological forms for the interaction term $Q$ in the model, namely, $Q$ is taken proportional to the Hubble expansion rate and the energy densities of dark sectors (taken to be $\rho_{\rm de}$, $\rho_{\rm m}$, and $\rho_{\rm de}+\rho_{\rm m}$, respectively). We obtain a uniform analytical solution to the three interacting models. Furthermore, we constrain the models by using the latest observational data, including the 557 Union2 type Ia supernovae data, the cosmic microwave background anisotropy data from the 7-yr WMAP, and the baryon acoustic oscillation data from the SDSS. We show that in the interacting models of the holographic Ricci dark energy, a more reasonable value of $\Omega_{\rm m0}$ will be obtained, and the observations favor a rather strong coupling between dark energy and dark matter., Comment: 9 pages, 4 figures; to appear in EPJC; typos corrected, published version