Search

The recent local measurement of the Hubble constant based on the distance-ladder method has reached a $1.91\%$ precision, but this result is in tension with the early-universe measurements at the more than 4$\sigma$ level, bringing a crisis to the contemporary cosmology. In addition to the end-to-end test of the $\Lambda$CDM model in the late universe, it is also of great interest to see how the local $H_0$ measurement affects the determination of the primordial power spectra, and further to test the influences for the inflation model selection. Here, we constrain the primordial power spectra of scalar and tensor perturbations by using a series of observational data, including the Planck 2015 cosmic microwave background (CMB) temperature and polarization power spectra data, the Planck 2018 lensing power spectrum data, the BICEP2/Keck Array CMB B-mode data, and also the prior of optical depth $\tau=0.054\pm0.007$, as well as the late-universe measurements (baryon acoustic oscillations and type Ia supernovae). In particular, we use the latest $1.91\%$ measurement of the Hubble constant, $H_{0}=74.03\pm1.42$ km s$^{-1}$ Mpc$^{-1}$, in this cosmological test. We find that considering the latest local $H_{0}$ measurement in the data combination will lead to a larger fit value of $n_{\rm s}$. With the addition of the latest local measurement of $H_{0}$, it is found that the natural inflation model is totally excluded at the $2\sigma$ level, the Starobinsky $R^{2}$ inflation model is marginally favored at around the $2\sigma$ level, and the spontaneously broken SUSY inflation model is the most favored model., Comment: 2 pages, 1 figure

Applying the distance sum rule in strong gravitational lensing (SGL) and type Ia supernova (SN Ia) observations, one can provide an interesting cosmological model-independent method to determine the cosmic curvature parameter $\Omega_k$. In this paper, with the newly compiled data sets including 161 galactic-scale SGL systems and 1048 SN Ia data, we place constraints on $\Omega_k$ within the framework of three types of lens models extensively used in SGL studies. Moreover, to investigate the effect of different mass lens samples on the results, we divide the SGL sample into three sub-samples based on the center velocity dispersion of intervening galaxies. In the singular isothermal sphere (SIS) and extended power-law lens models, a flat universe is supported with the uncertainty about 0.2, while a closed universe is preferred in the power-law lens model. We find that the choice of lens models and the classification of SGL data actually can influence the constraints on $\Omega_k$ significantly., Comment: 10 pages, 4 figures; accepted for publication in The Astrophysical Journal

We investigate the impacts of dark energy on constraining massive (active/sterile) neutrinos in interacting dark energy (IDE) models by using the current observations. We employ two typical IDE models, the interacting $w$ cold dark matter (I$w$CDM) model and the interacting holographic dark energy (IHDE) model, to make an analysis. To avoid large-scale instability, we use the parameterized post-Friedmann approach to calculate the cosmological perturbations in the IDE models. The cosmological observational data used in this work include the Planck cosmic microwave background (CMB) anisotropies data, the baryon acoustic oscillation data, the type Ia supernovae data, the direct measurement of the Hubble constant, the weak lensing data, the redshift-space distortion data, and the CMB lensing data. We find that the dark energy properties could influence the constraint limits of active neutrino mass and sterile neutrino parameters in the IDE models. We also find that the dark energy properties could influence the constraints on the coupling strength parameter $\beta$, and a positive coupling constant, $\beta>0$, can be detected at the $2.5\sigma$ statistical significance for the IHDE+$\nu_s$ model by using the all-data combination. In addition, we also discuss the "Hubble tension" issue in these scenarios. We find that the $H_0$ tension can be effectively relieved by considering massive sterile neutrinos, and in particular in the IHDE+$\nu_s$ model the $H_0$ tension can be reduced to be at the $1.28\sigma$ level., Comment: 14 pages, 9 figures

Future observations of 21 cm emission from neutral hydrogen survey will become a promising approach to probe the large scale structure of Universe. In this paper, we investigate the impacts of Square Kilometer Array (SKA) 21 cm observation on the estimation of cosmological parameters. We use the simulated data of the baryonic acoustic oscillation (BAO) measurements based on the future SKA experiment with the intensity mapping (IM) technique to do the analysis. For the current observations, we use the latest cosmic microwave background (CMB) observation from {\it Planck} 2018, the optical BAO measurements, and the Type Ia supernovae (SN) observation (Pantheon compilation). We find that the SKA mock data could break the degeneracy between the matter density and the Hubble constant, further improving the cosmological constraints to a great extent. We also find that the constraint on the equation of state parameters of dark energy could be significantly improved by including the SKA mock data into the cosmological global fit., Comment: 8 pages, 3 figures, accepted for publication in PLB

In this work, we investigate the impacts of the future gravitational-wave (GW) standard siren observation by the Einstein Telescope (ET) on constraining the interacting dark energy (IDE) models. We simulate 1000 GW events in the redshift range of $0\lesssim z \lesssim 5$ based on the 10-year observation of the ET. We combine the simulated GW data with the current mainstream cosmological electromagnetic observations including the cosmic microwave background anisotropies, the baryon acoustic oscillations, and the type Ia supernovae to constrain the IDE models. We consider typical IDE models in the context of a perturbed universe. To avoid the large-scale instability problem for IDE models, we apply the extended parameterized post-Friedmann approach to calculate the cosmological perturbations. We find that the addition of the GW standard siren data could significantly improve the constraint accuracies for most of the cosmological parameters (e.g., $H_{0}$, $w$, and $\Omega_{\rm m}$). For the coupling parameter $\beta$, the constraint errors could also be slightly improved when adding the GW data in the cosmological fit., Comment: 22 pages, 5 figures; accepted for publication in JCAP

In this work, we investigate what role the redshift drift data of Square Kilometre Array (SKA) will play in the cosmological parameter estimation in the future. To test the constraint capability of the redshift drift data of SKA-only, the $\Lambda$CDM model is chosen as a reference model. We find that using the SKA1 mock data, the $\Lambda$CDM model can be loosely constrained, while the model can be well constrained when the SKA2 mock data are used. When the mock data of SKA are combined with the data of the European Extremely Large Telescope (E-ELT), the constraints can be significantly improved almost as good as the data combination of the type Ia supernovae observation (SN), the cosmic microwave background observation (CMB), and the baryon acoustic oscillations observation (BAO). Furthermore, we explore the impact of the redshift drift data of SKA on the basis of SN+CMB+BAO+E-ELT in the $\Lambda$CDM model, the $w$CDM model, the CPL model, and the HDE model. We find that the redshift drift measurement of SKA could help to significantly improve the constraints on dark energy and could break the degeneracy between the cosmological parameters. Therefore, we conclude that redshift-drift observation of SKA would provide a good improvement in the cosmological parameter estimation in the future and has the enormous potential to be one of the most competitive cosmological probes in constraining dark energy., Comment: 12 pages, 5 figures; accepted for publication in the European Physical Journal C

In this work, we use the simulated gravitational wave (GW) standard siren data from the future observation of the Einstein Telescope (ET) to constrain various dark energy cosmological models, including the $\Lambda$CDM, $w$CDM, CPL, $\alpha$DE, GCG, and NGCG models. We also use the current mainstream cosmological electromagnetic observations, i.e., the cosmic microwave background anisotropies data, the baryon acoustic oscillations data, and the type Ia supernovae data, to constrain these models. We find that the GW standard siren data could tremendously improve the constraints on the cosmological parameters for all these dark energy models. For all the cases, the GW standard siren data can be used to break the parameter degeneracies generated by the current cosmological electromagnetic observational data. Therefore, it is expected that the future GW standard siren observation from the ET would play a crucial role in the cosmological parameter estimation in the future. The conclusion of this work is quite solid because it is based on the analysis for various dark energy models., Comment: 17 pages, 6 figures

TianQin is a space-based gravitational-wave observatory scheduled to be launched in the 2030s. In this work, we make a preliminary forecast for the cosmological parameter estimation with the gravitational-wave standard siren observation from TianQin. We simulate the standard siren data of TianQin based on its 5-year observation after the completion of construction. In the simulation, three models for the population of massive black hole binary (MBHB), i.e., pop III, Q3nod, and Q3d, are considered to predict the event numbers of MBHB mergers. We find that: (i) among the three MBHB models, the Q3nod model can provide the tightest constraints on the cosmological parameters; (ii) TianQin's standard siren observation can effectively break the parameter degeneracies inherent in the cosmic microwave background observation; and (iii) the future standard siren observation from TianQin can significantly improve the cosmological parameter estimation under the current mainstream electromagnetic observations., Comment: 17 pages, 6 figures

We investigate what role the SKA neutral hydrogen (HI) intensity mapping (IM) sky survey observation will play in weighing neutrinos in cosmology. We use the simulated data of the baryon acoustic oscillation (BAO) measurements from the HI IM survey based on SKA1 and SKA2 to do the analysis. For the current observations, we use the Planck 2015 cosmic microwave background (CMB) anisotropies observation, the optical BAO measurements, the type Ia supernovae (SN) observation (Pantheon compilation), and the latest $H_0$ measurement. We consider three mass ordering cases for massive neutrinos, i.e., the normal hierarchy (NH), inverted hierarchy (IH), and degenerate hierarchy (DH) cases. It is found that the SKA observation can significantly improve the constraints on $\Omega_{\rm m}$ and $H_0$. Compared to the current observation, the SKA1 data can improve the constraints on $\Omega_{\rm m}$ by about 33%, and on $H_0$ by about 36%; the SKA2 data can improve the constraints on $\Omega_{\rm m}$ by about 58%, and on $H_0$ by about 66%. It is also found that the SKA observation can only slightly improve the constraints on $\sum m_\nu$. Compared to the current observation, the SKA1 data can improve the constraints on $\sum m_\nu$ by about 4%, 3%, and 10%, for the NH, IH, and DH cases, respectively; the SKA2 data can improve the constraints on $\sum m_\nu$ by about 7%, 7%, and 16%, for the NH, IH, and DH cases, respectively., Comment: 8 pages, 3 figures

We study the holographic dark energy (HDE) model by using the future gravitational wave (GW) standard siren data observed from the Einstein Telescope (ET) in this work. We simulate 1000 GW standard siren data based on a 10-year observation of the ET to make this analysis. We find that all the cosmological parameters in the HDE model can be tremendously improved by including the GW standard siren data in the cosmological fit. The GW data combined with the current cosmic microwave background anisotropies, baryon acoustic oscillations, and type Ia supernovae data will measure the cosmological parameters $\Omega_{\rm m}$, $H_0$, and $c$ in the HDE model to be at the accuracies of 1.28%, 0.59%, and 3.69%, respectively. A comparison with the cosmological constant model and the constant-$w$ dark energy model shows that, compared to the standard model, the parameter degeneracies will be broken more thoroughly in a dynamical dark energy model. We find that the GW data alone can provide a fairly good measurement for $H_0$, but for other cosmological parameters the GW data alone can only provide rather weak measurements. However, due to the fact that the parameter degeneracies can be broken by the GW data, the standard sirens can play an essential role in improving the parameter estimation., Comment: 13 pages, 4 figures

We investigate how the dark energy properties impact the constraints on the total neutrino mass in interacting dark energy (IDE) models. In this study, we focus on two typical interacting dynamical dark energy models, i.e., the interacting $w$ cold dark matter (I$w$CDM) model and the interacting holographic dark energy (IHDE) model. To avoid the large-scale instability problem in IDE models, we apply the parameterized post-Friedmann approach to calculate the perturbation of dark energy. We employ the Planck 2015 cosmic microwave background temperature and polarization data, combined with low-redshift measurements on baryon acoustic oscillation distance scales, type Ia supernovae, and the Hubble constant, to constrain the cosmological parameters. We find that the dark energy properties could influence the constraint limits on the total neutrino mass. Once dynamical dark energy is considered in the IDE models, the upper bounds of $\sum m_\nu$ will be changed. By considering the values of $\chi^2_{\rm min}$, we find that in these IDE models the normal hierarchy case is slightly preferred over the inverted hierarchy case; for example, $\Delta\chi^2=2.720$ is given in the IHDE+$\sum m_\nu$ model. In addition, we also find that in the I$w$CDM+$\sum m_\nu$ model $\beta=0$ is consistent with current observational data inside the 1$\sigma$ range, and in the IHDE+$\sum m_\nu$ model $\beta>0$ is favored at more than 2$\sigma$ level., Comment: 9 pages, 3 figures

In this paper, we investigate the observational constraints on the scenario of vacuum energy interacting with cold dark matter. We consider eight typical interaction forms in such an interacting vacuum energy scenario. The observational data used in this work to constrain these models include the JLA sample of type Ia supernovae observation, the Planck 2015 distance priors data of cosmic microwave background anisotropies observation, the baryon acoustic oscillations data, and the Hubble constant direct measurement. We find that the current observational data almost equally favor these interacting vacuum energy models. We also find that for all these models of vacuum energy interacting with cold dark matter the case of no interaction is actually well consistent with the current observational data within 1$\sigma$ range., Comment: 13 pages, 6 figures

We investigate whether the current Hubble parameter $H(z)$ measurements could help improve the constraints on dark energy on the basis of the mainstream cosmological probes including the type Ia supernovae (SN) observation, the cosmic microwave background anisotropies (CMB) observation, and the baryon acoustic oscillations (BAO) observation. For the current $H(z)$ data, we use 30 data points measured by using a differential age method. Furthermore, we also consider the future $H(z)$ measurements based on the Sandage-Loeb (SL) test by means of the E-ELT in construction, and thus we also use 30 simulated $H(z)$ data according to a 10-year SL test observation. In this work, we choose four typical dark energy models as examples, i.e., the $\Lambda$CDM model, the $w$CDM model, the $\alpha$DE model, and the GCG model, to complete the analysis. We find that, when only the current $H(z)$ data are added, the constraints on these models are not improved compared to the cases using the SN+CMB+BAO data; but when further adding the 10-year SL test data, the constraint results are tremendously improved for all the four models. Therefore, we conclude that, although the current $H(z)$ measurements could not provide an evident improvement on the basis of the current mainstream cosmological probes, the future $H(z)$ measurements from the SL test would have enormous potential to change the status of the Hubble parameter measurements in constraining dark energy., Comment: 17 pages, 2 figures

In this paper, we make a comparison for the impacts of smooth dynamical dark energy, modified gravity, and interacting dark energy on the cosmological constraints on the total mass of active neutrinos. For definiteness, we consider the $\Lambda$CDM model, the $w$CDM model, the $f(R)$ model, and two typical interacting vacuum energy models, i.e., the I$\Lambda$CDM1 model with $Q=\beta H\rho_{\rm c}$ and the I$\Lambda$CDM2 model with $Q=\beta H\rho_{\Lambda}$. In the cosmological fits, we use the Planck 2015 temperature and polarization data, in combination with other low-redshift observations including the baryon acoustic oscillations, the type Ia supernovae, the Hubble constant measurement, and the large-scale structure observations, such as the weak lensing as well as the redshift-space distortion. Besides, the Planck lensing measurement is also employed in this work. We find that, the $w$CDM model favors a higher upper limit on the neutrino mass compared to the $\Lambda$CDM model, while the upper limit in the $f(R)$ model is similar with that of $\Lambda$CDM model. For the interacting vacuum energy models, the I$\Lambda$CDM1 model favors a higher upper limit on neutrino mass, while the I$\Lambda$CDM2 model favors an identical neutrino mass with the case of $\Lambda$CDM., Comment: 10 pages, 4 figures

We wish to investigate whether there is an extension to the base $\Lambda$CDM cosmology that can resolve the tension between the Planck observation of the cosmic microwave background anisotropies and the local measurement of the Hubble constant. We consider various plausible extended models in this work, and we use the Planck 2015 observation, combined with the baryon acoustic oscillation data, the JLA type Ia supernovae data, and the local measurement of the Hubble constant (by Riess et al. in 2016), to make an analysis. We find that the holographic dark energy plus sterile neutrino model can reduce the tension to be at the 1.11$\sigma$ level, but this model is obviously not favored by the current observations. Among these extended models, the $\Lambda$CDM+$N_{\rm eff}$ model is most favored by the current observations, and this model can reduce the tension to be at the 1.87$\sigma$ level. By a careful test, we conclude that none of these extended models can convincingly resolve the $H_0$ tension., Comment: 10 pages, 1 figure

The model of holographic dark energy in which dark energy interacts with dark matter is investigated in this paper. In particular, we consider the interacting holographic dark energy model in the context of a perturbed universe, which was never investigated in the literature. To avoid the large-scale instability problem in the interacting dark energy cosmology, we employ the generalized version of the parameterized post-Friedmann approach to treat the dark energy perturbations in the model. We use the current observational data to constrain the model. Since the cosmological perturbations are considered in the model, we can then employ the redshift-space distortions (RSD) measurements to constrain the model, in addition to the use of the measurements of expansion history, which was either never done in the literature. We find that, for both the cases with $Q=\beta H\rho_{\rm c}$ and $Q=\beta H_0\rho_{\rm c}$, the interacting holographic dark energy model is more favored by the current data, compared to the holographic dark energy model without interaction. It is also found that, with the help of the RSD data, a positive coupling $\beta$ can be detected at the $2.95\sigma$ statistical significance for the case of $Q=\beta H_0\rho_{\rm c}$., Comment: 8 pages, 4 figures

We constrain cosmological parameters using only Hubble parameter data and quantify the impact of future Hubble parameter measurements on parameter estimation for the most typical dark energy models. We first constrain cosmological parameters using 52 current Hubble parameter data including the Hubble constant measurement from the Hubble Space Telescope. Then we simulate the baryon acoustic oscillation signals from WFIRST (Wide-Field Infrared Survey Telescope) covering the redshift range of $z\in [0.5, 2]$ and the redshift drift data from E-ELT (European Extremely Large Telescope) in the redshift range of $z\in [2,5]$. It is shown that solely using the current Hubble parameter data could give fairly good constraints on cosmological parameters. Compared to the current Hubble parameter data, with the WFIRST observation the $H(z)$ constraints on dark energy would be improved slightly, while with the E-ELT observation the $H(z)$ constraints on dark energy is enormously improved., Comment: 7 pages, 2 figures

Detection of gravitational waves produced by merger of binary compact objects could provide an independent way for measuring the luminosity distance to the gravitational-wave burst source, indicating that gravitational-wave observation, combined with observation of electromagnetic counterparts, can provide "standard sirens" for investigating the expansion history of the universe in cosmology. In this work, we wish to investigate how the future gravitational-wave standard siren observations would break the parameter degeneracies existing in the conventional optical observations and how they help improve the parameter estimation in cosmology. We take the third-generation ground-based gravitational-wave detector, the Einstein Telescope, as an example to make an analysis. By simulating 1000 events data in the redshift range between 0 and 5 based on the ten-year observation of the Einstein Telescope, we find that the gravitational-wave data could largely break the degeneracy between the matter density and the Hubble constant, thus significantly improving the cosmological constraints. We further show that the constraint on the equation-of-state parameter of dark energy could also be significantly improved by including the gravitational-wave data in the cosmological fit., Comment: 7 pages, 4 figures; accepted for publication in Physical Review D

We investigate the constraints on total neutrino mass in the scenario of vacuum energy interacting with cold dark matter. We focus on two typical interaction forms, i.e., $Q=\beta H\rho_{\rm c}$ and $Q=\beta H\rho_{\Lambda}$. To avoid the occurrence of large-scale instability in interacting dark energy cosmology, we adopt the parameterized post-Friedmann approach to calculate the perturbation evolution of dark energy. We employ observational data, including the Planck cosmic microwave background temperature and polarization data, baryon acoustic oscillation data, a JLA sample of type Ia supernovae observation, direct measurement of the Hubble constant, and redshift space distortion data. We find that, compared with those in the $\Lambda$CDM model, much looser constraints on $\sum m_{\nu}$ are obtained in the $Q=\beta H\rho_{\rm c}$ model, whereas slightly tighter constraints are obtained in the $Q=\beta H\rho_{\Lambda}$ model. Consideration of the possible mass hierarchies of neutrinos reveals that the smallest upper limit of $\sum m_{\nu}$ appears in the degenerate hierarchy case. By comparing the values of $\chi^2_{\rm min}$, we find that the normal hierarchy case is favored over the inverted one. In particular, we find that the difference $\Delta \chi^2_{\rm min} \equiv \chi^2_{\rm IH; min}-\chi^2_{\rm NH; min}> 2$ in the $Q=\beta H\rho_{\rm c}$ model. In addition, we find that $\beta=0$ is consistent with the current observations in the $Q=\beta H\rho_{\rm c}$ model, and $\beta < 0$ is favored at more than the $1\sigma$ level in the $Q=\beta H\rho_{\Lambda}$ model., Comment: 10 pages, 4 figures

We investigate the impacts of the gravitational-wave (GW) standard siren observation of the Einstein Telescope (ET) on constraining the total neutrino mass. We simulate 1000 GW events that would be observed by the ET in its 10-year observation by taking the standard $\Lambda$CDM cosmology as a fiducial model. We combine the simulated GW data with other cosmological observations including cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and type Ia supernovae (SN). We consider three mass hierarchy cases for the neutrino mass, i.e., normal hierarchy (NH), inverted hierarchy (IH), and degenerate hierarchy (DH). Using Planck+BAO+SN, we obtain $\sum m_\nu<0.175$ eV for the NH case, $\sum m_\nu<0.200$ eV for the IH case, and $\sum m_\nu<0.136$ eV for the DH case. After considering the GW data, i.e., using Planck+BAO+SN+GW, the constraint results become $\sum m_\nu<0.151$ eV for the NH case, $\sum m_\nu<0.185$ eV for the IH case, and $\sum m_\nu<0.122$ eV for the DH case. We find that the GW data can help reduce the upper limits of $\sum m_\nu$ by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. In addition, we find that the GW data can also help break the degeneracies between $\sum m_{\nu}$ and other parameters. We show that the GW data of the ET could greatly improve the constraint accuracies of cosmological parameters., Comment: 8 pages, 4 figures

We investigate the observational constraints on three typical brane inflation models by considering the latest local measurement of the Hubble constant in the global fit. We also employ other observational data, including the Planck 2015 CMB data, the BICEP2/Keck Array B-mode data, and the baryon acoustic oscillations data, in our analysis. Previous studies have shown that the addition of the latest local $H_{0}$ measurement favors a larger spectral index, and can exert a significant influence on the model selection of inflation. In this work, we investigate its impacts on the status of brane inflation models. We find that, when the direct $H_{0}$ measurement is considered, the prototype model of brane inflation is still in good agreement with the current observational data within the $2\sigma$ level range. For the KKLMMT model, the consideration of the $H_{0}$ measurement allows the range of the parameter $\beta$ to be amplified to ${\cal O}(10^{-2})$, which slightly alleviates the fine-tuning problem. For the IR DBI model, the addition of the $H_{0}$ measurement does not provide a better fit. These results show that the consideration of the new $H_{0}$ prior can exert a considerable influence on the brane inflation models. At last, we show that, when $\beta \lesssim 1.1$, the equilateral non-Gaussianity in the IR DBI inflation model is compatible with the current CMB data at the 1$\sigma$ level., Comment: 9 pages, 4 figures

We investigate the cosmological constraints on sterile neutrinos in a universe in which vacuum energy interacts with cold dark matter by using latest observational data. We focus on two specific interaction models, $Q=\beta H\rho_{\rm v}$ and $Q=\beta H\rho_{\rm c}$. To overcome the problem of large-scale instability in the interacting dark energy scenario, we employ the parametrized post-Friedmann (PPF) approach for interacting dark energy to do the calculation of perturbation evolution. The observational data sets used in this work include the Planck 2015 temperature and polarization data, the baryon acoustic oscillation measurements, the type-Ia supernova data, the Hubble constant direct measurement, the galaxy weak lensing data, the redshift space distortion data, and the Planck lensing data. Using the all-data combination, we obtain $N_{\rm eff}<3.522$ and $m_{\nu,{\rm sterile}}^{\rm eff}<0.576$ eV for the $Q=\beta H\rho_{\rm v}$ model, and $N_{\rm eff}=3.204^{+0.049}_{-0.135}$ and $m_{\nu,{\rm sterile}}^{\rm eff}=0.410^{+0.150}_{-0.330}$ eV for the $Q=\beta H\rho_{\rm c}$ model. The latter indicates $\Delta N_{\rm eff}>0$ at the 1.17$\sigma$ level and a nonzero mass of sterile neutrino at the 1.24$\sigma$ level. In addition, for the $Q=\beta H\rho_{\rm v}$ model, we find that $\beta=0$ is consistent with the current data, and for the $Q=\beta H\rho_{\rm c}$ model, we find that $\beta>0$ is obtained at more than 1$\sigma$ level., Comment: 14 pages, 10 figures; new tables and figures added

In this paper, we make a deep analysis for the five typical interacting holographic dark energy models with the interaction terms $Q=3\beta H_{0}\rho_{\rm{de}}$, $Q=3\beta H_{0}\rho_{\rm{c}}$, $Q=3\beta H_{0}(\rho_{\rm{de}}+\rho_{\rm c})$, $Q=3\beta H_{0}\sqrt{\rho_{\rm{de}}\rho_{\rm c}}$, and $Q=3\beta H_{0}\frac{\rho_{\rm{de}}\rho_{c}}{\rho_{\rm{de}}+\rho_{\rm c}}$, respectively. We obtain observational constraints on these models by using the type Ia supernova data (the Joint Light-curve Analysis sample), the cosmic microwave background data (Planck 2015 distance priors), the baryon acoustic oscillations data, and the direct measurement of the Hubble constant. We find that the values of $\chi_{\rm min}^2$ for all the five models are almost equal (around~699), indicating that the current observational data equally favor these IHDE models. In addition, a comparison with the cases of interaction term involving the Hubble parameter $H$ is also made., Comment: 14 pages, 6 figures. arXiv admin note: text overlap with arXiv:1710.03068

We consider the models of vacuum energy interacting with cold dark matter in this study, in which the coupling can change sigh during the cosmological evolution. We parameterize the running coupling $b$ by the form $b(a)=b_0a+b_e(1-a)$, where at the early-time the coupling is given by a constant $b_{e}$ and today the coupling is described by another constant $b_{0}$. We explore six specific models with (i) $Q(a)=b(a)H_0\rho_0$, (ii) $Q(a)=b(a)H_0\rho_{\rm de}$, (iii) $Q(a)=b(a)H_0\rho_{\rm c}$, (iv) $Q(a)=b(a)H\rho_0$, (v) $Q(a)=b(a)H\rho_{\rm de}$, and (vi) $Q(a)=b(a)H\rho_{\rm c}$. The current observational data sets we use to constrain the models include the JLA compilation of type Ia supernova data, the Planck 2015 distance priors data of cosmic microwave background observation, the baryon acoustic oscillations measurements, and the Hubble constant direct measurement. We find that, for all the models, we have $b_0<0$ and $b_e>0$ at around the 1$\sigma$ level, and $b_0$ and $b_e$ are in extremely strong anti-correlation. Our results show that the coupling changes sign during the evolution at about the 1$\sigma$ level, i.e., the energy transfer is from dark matter to dark energy when dark matter dominates the universe and the energy transfer is from dark energy to dark matter when dark energy dominates the universe., Comment: 10 pages, 3 figures

We make a consistency test for the general relativity (GR) through measuring the growth index $\gamma$ in a universe with massive (sterile/active) neutrinos. We employ the redshift space distortion measurements to do the analysis. To constrain other cosmological parameters, we also use other cosmological measurements, including the Planck 2015 cosmic microwave background temperature and polarization data, the baryon acoustic oscillation data, the type Ia supernova JLA data, the weak lensing galaxy shear data, and the Planck 2015 lensing data. In a universe with massive sterile neutrinos, we obtain $\gamma=0.624^{+0.055}_{-0.050}$, with the tension with the GR prediction $\gamma=0.55$ at the 1.48$\sigma$ level, showing that the consideration of sterile neutrinos still cannot make the true measurement of $\gamma$ be well consistent with the GR prediction. In a universe with massive active neutrinos, we obtain $\gamma=0.663\pm0.045$ for the normal hierarchy case, $\gamma=0.661^{+0.044}_{-0.050}$ for the degenerate hierarchy case, and $\gamma=0.668^{+0.045}_{-0.051}$ for the inverted hierarchy case, with the tensions with GR all at beyond the 2$\sigma$ level. We find that the consideration of massive active neutrinos (no matter what mass hierarchy is considered) almost does not influence the measurement of the growth index $\gamma$., Comment: 8 pages, 2 figures; final version published in Physics Letters B

We investigate how the dark energy properties change the cosmological limits on sterile neutrino parameters by using recent cosmological observations. We consider the simplest dynamical dark energy models, the $w$CDM model and the holographic dark energy (HDE) model, to make an analysis. The cosmological observations used in this work include the Planck 2015 CMB temperature and polarization data, the baryon acoustic oscillation data, the type Ia supernova data, the Hubble constant direct measurement data, and the Planck CMB lensing data. We find that, $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.2675$ eV and $N_{\rm eff}<3.5718$ for $\Lambda$CDM cosmology, $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.5313$ eV and $N_{\rm eff}<3.5008$ for $w$CDM cosmology, and $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.1989$ eV and $N_{\rm eff}<3.6701$ for HDE cosmology, from the constraints of the combination of these data. Thus, without the addition of measurements of growth of structure, only upper limits on both $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ and $N_{\rm eff}$ can be derived, indicating that no evidence of the existence of a sterile neutrino species with eV-scale mass is found in this analysis. Moreover, compared to the $\Lambda$CDM model, in the $w$CDM model the limit on $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ becomes much looser, but in the HDE model the limit becomes much tighter. Therefore, the dark energy properties could significantly influence the constraint limits of sterile neutrino parameters., Comment: 12 pages, 5 figures. arXiv admin note: text overlap with arXiv:1703.04884

We search for sterile neutrinos in the holographic dark energy cosmology by using the latest observational data. To perform the analysis, we employ the current cosmological observations, including the cosmic microwave background temperature power spectrum data from the Planck mission, the baryon acoustic oscillation measurements, the type Ia supernova data, the redshift space distortion measurements, the shear data of weak lensing observation, the Planck lensing measurement, and the latest direct measurement of $H_0$ as well. We show that, compared to the $\Lambda$CDM cosmology, the holographic dark energy cosmology with sterile neutrinos can relieve the tension between the Planck observation and the direct measurement of $H_0$ much better. Once we include the $H_0$ measurement in the global fit, we find that the hint of the existence of sterile neutrinos in the holographic dark energy cosmology can be given. Under the constraint of the all-data combination, we obtain $N_{\rm eff}= 3.76\pm0.26$ and $m_{\nu,\rm sterile}^{\rm eff}< 0.215\,\rm eV$, indicating that the detection of $\Delta N_{\rm eff}>0$ in the holographic dark energy cosmology is at the $2.75\sigma$ level and the massless or very light sterile neutrino is favored by the current observations., Comment: 10 pages, 4 figures; typos corrected, published in PRD

We report the result of a search for sterile neutrinos with the latest cosmological observations. Both cases of massless and massive sterile neutrinos are considered in the $\Lambda$CDM cosmology. The cosmological observations used in this work include the Planck 2015 temperature and polarization data, the baryon acoustic oscillation data, the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck lensing data, and the cosmic shear data. We find that the current observational data give a hint of the existence of massless sterile neutrino (as dark radiation) at the 1.44$\sigma$ level, and the consideration of an extra massless sterile neutrino can indeed relieve the tension between observations and improve the cosmological fit. For the case of massive sterile neutrino, the observations give a rather tight upper limit on the mass, which implies that actually a massless sterile neutrino is more favored. Our result is consistent with the recent result of neutrino oscillation experiment done by the Daya Bay and MINOS collaborations, as well as the recent result of cosmic ray experiment done by the IceCube collaboration., Comment: 16 pages, 4 figures; accepted for publication in Eur. Phys. J. C

We constrain the neutrino mass in the scenario of vacuum energy interacting with cold dark matter by using current cosmological observations. To avoid the large-scale instability problem in interacting dark energy models, we employ the parameterized post-Friedmann (PPF) approach to do the calculation of perturbation evolution, for the $Q=\beta H\rho_{\rm c}$ and $Q=\beta H\rho_{\Lambda}$ models. The current observational data sets used in this work include Planck (cosmic microwave background), BSH (baryon acoustic oscillations, type Ia supernovae, and Hubble constant), and LSS (redshift space distortions and weak lensing). According to the constraint results, we find that $\beta>0$ at more than $1\sigma$ level for the $Q=\beta H\rho_{\rm c}$ model, which indicates that cold dark matter decays into vacuum energy; while $\beta=0$ is consistent with the current data at $1\sigma$ level for the $Q=\beta H\rho_{\Lambda}$ model. Taking the $\Lambda$CDM model as a baseline model, we find that a smaller upper limit, $\sum m_{\nu}<0.11$ eV ($2\sigma$), is induced by the latest BAO BOSS DR12 data and the Hubble constant measurement $H_{0} = 73.00 \pm 1.75$ km~s$^{-1}$~Mpc$^{-1}$. For the $Q=\beta H\rho_{\rm c}$ model, we obtain $\sum m_{\nu}<0.20$ eV ($2\sigma$) from Planck+BSH. For the $Q=\beta H\rho_{\Lambda}$ model, $\sum m_{\nu}<0.10$ eV ($2\sigma$) and $\sum m_{\nu}<0.14$ eV ($2\sigma$) are derived from Planck+BSH and Planck+BSH+LSS, respectively. We show that these smaller upper limits on $\sum m_{\nu}$ are affected more or less by the tension between $H_{0}$ and other observational data., Comment: 20 pages, 6 figures; major revision made

We investigate how the properties of dark energy affect the cosmological measurements of neutrino mass and extra relativistic degrees of freedom. We limit ourselves to the most basic extensions of $\Lambda$ cold dark matter (CDM) model, i.e. the $w$CDM model with one additional parameter $w$, and the $w_{0}w_{a}$CDM model with two additional parameters, $w_{0}$ and $w_{a}$. In the cosmological fits, we employ the 2015 cosmic microwave background temperature and polarization data from the Planck mission, in combination with low-redshift measurements such as the baryon acoustic oscillations, Type Ia supernovae and the Hubble constant ($H_{0}$). Given effects of massive neutrinos on large-scale structure, we further include weak lensing, redshift space distortion, Sunyaev--Zeldovich cluster counts and Planck lensing data. We show that, though the cosmological constant $\Lambda$ is still consistent with the current data, a phantom dark energy ($w<-1$) or an early phantom dark energy (i.e. quintom evolving from $w<-1$ to $w>-1$) is slightly more favoured by current observations, which leads to the fact that in both $w$CDM and $w_0w_a$CDM models we obtain a larger upper limit of $\sum m_\nu$. We also show that in the three dark energy models, the constraints on $N_{\rm eff}$ are in good accordance with each other, all in favour of the standard value 3.046, which indicates that the dark energy parameters almost have no impact on constraining $N_{\rm eff}$. Therefore, we conclude that the dark energy parameters can exert a significant influence on the cosmological weighing of neutrinos, but almost cannot affect the constraint on dark radiation., Comment: 12 pages, 11 figures; accepted by MNRAS; match the publication version

The Sandage-Loeb (SL) test is a promising method for probing dark energy because it measures the redshift drift in the spectra of Lyman-$\alpha$ forest of distant quasars, covering the "redshift desert" of $2\lesssim z\lesssim5$, which is not covered by existing cosmological observations. Therefore, it could provide an important supplement to current cosmological observations. In this paper, we explore the impact of SL test on the precision of cosmological constraints for two typical holographic dark energy models, i.e., the original holographic dark energy (HDE) model and the Ricci holographic dark energy (RDE) model. To avoid data inconsistency, we use the best-fit models based on current combined observational data as the fiducial models to simulate 30 mock SL test data. The results show that SL test can effectively break the existing strong degeneracy between the present-day matter density $\Omega_{m0}$ and the Hubble constant $H_0$ in other cosmological observations. For the considered two typical dark energy models, not only can a 30-year observation of SL test improve the constraint precision of $\Omega_{m0}$ and $h$ dramatically, but can also enhance the constraint precision of the model parameters $c$ and $\alpha$ significantly., Comment: 7 pages, 2 figures; published in Sci. China-Phys. Mech. Astron. arXiv admin note: text overlap with arXiv:1511.06957

Strong gravitational lensing (SGL) has provided an important tool for probing galaxies and cosmology. In this paper, we use the SGL data to constrain the holographic dark energy model, as well as models that have the same parameter number, such as the $w$CDM and Ricci dark energy models. We find that only using SGL is difficult to effectively constrain the model parameters. However, when the SGL data are combined with CBS (CMB+BAO+SN) data, the reasonable estimations can be given and the constraint precision is improved to a certain extent, relative to the case of CBS only. Therefore, SGL is an useful way to tighten constraints on model parameters., Comment: 8 pages, 4 figures

We explore the impact of the Sandage-Loeb (SL) test on the precision of cosmological constraints for $f(T)$ gravity theories. The SL test is an important supplement to current cosmological observations because it measures the redshift drift in the Lyman-$\alpha$ forest in the spectra of distant quasars, covering the "redshift desert" of $2 \lesssim z \lesssim5$. To avoid data inconsistency, we use the best-fit models based on current combined observational data as fiducial models to simulate 30 mock SL test data. We quantify the impact of these SL test data on parameter estimation for $f(T)$ gravity theories. Two typical $f(T)$ models are considered, the power-law model $f(T)_{PL}$ and the exponential-form model $f(T)_{EXP}$. The results show that the SL test can effectively break the existing strong degeneracy between the present-day matter density $\Omega_m$ and the Hubble constant $H_0$ in other cosmological observations. For the considered $f(T)$ models, a 30-year observation of the SL test can improve the constraint precision of $\Omega_m$ and $H_0$ enormously but cannot effectively improve the constraint precision of the model parameters., Comment: 5 pages, 2 figures

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 D

We apply the statefinder hierarchy plus the fractional growth parameter to explore the extended Ricci dark energy (ERDE) model, in which there are two independent coefficients $\alpha$ and $\beta$. By adjusting them, we plot evolution trajectories of some typical parameters, including Hubble expansion rate $E$, deceleration parameter $q$, the third and fourth order hierarchy $S_3^{(1)}$ and $S_4^{(1)}$ and fractional growth parameter $\epsilon$, respectively, as well as several combinations of them. For the case of variable $\alpha$ and constant $\beta$, in the low-redshift region the evolution trajectories of $E$ are in high degeneracy and that of $q$ separate somewhat. However, the $\Lambda$CDM model is confounded with ERDE in both of these two cases. $S_3^{(1)}$ and $S_4^{(1)}$, especially the former, perform much better. They can differentiate well only varieties of cases within ERDE except $\Lambda$CDM in the low-redshift region. For high-redshift region, combinations $\{S_n^{(1)},\epsilon\}$ can break the degeneracy. Both of $\{S_3^{(1)},\epsilon\}$ and $\{S_4^{(1)},\epsilon\}$ have the ability to discriminate ERDE with $\alpha=1$ from $\Lambda$CDM, of which the degeneracy cannot be broken by all the before-mentioned parameters. For the case of variable $\beta$ and constant $\alpha$, $S_3^{(1)}(z)$ and $S_4^{(1)}(z)$ can only discriminate ERDE from $\Lambda$CDM. Nothing but pairs $\{S_3^{(1)},\epsilon\}$ and $\{S_4^{(1)},\epsilon\}$ can discriminate not only within ERDE but also ERDE from $\Lambda$CDM. Finally we find that $S_3^{(1)}$ is surprisingly a better choice to discriminate within ERDE itself, and ERDE from $\Lambda$CDM as well, rather than $S_4^{(1)}$., Comment: 8 pages, 14 figures; published version

The model of holographic dark energy (HDE) with massive neutrinos and/or dark radiation is investigated in detail. The background and perturbation evolutions in the HDE model are calculated. We employ the PPF approach to overcome the gravity instability difficulty (perturbation divergence of dark energy) led by the equation-of-state parameter $w$ evolving across the phantom divide $w=-1$ in the HDE model with $c<1$. We thus derive the evolutions of density perturbations of various components and metric fluctuations in the HDE model. The impacts of massive neutrino and dark radiation on the CMB anisotropy power spectrum and the matter power spectrum in the HDE scenario are discussed. Furthermore, we constrain the models of HDE with massive neutrinos and/or dark radiation by using the latest measurements of expansion history and growth of structure, including the Planck CMB temperature data, the baryon acoustic oscillation data, the JLA supernova data, the Hubble constant direct measurement, the cosmic shear data of weak lensing, the Planck CMB lensing data, and the redshift space distortions data. We find that $\sum m_\nu<0.186$ eV (95\% CL) and $N_{\rm eff}=3.75^{+0.28}_{-0.32}$ in the HDE model from the constraints of these data., Comment: 18 pages, 5 figures; revised version accepted for publication in JCAP

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. B

By detecting redshift drift in the spectra of Lyman-$\alpha$ forest of distant quasars, Sandage-Loeb (SL) test directly measures the expansion of the universe, covering the "redshift desert" of $2 \lesssim z \lesssim5$. Thus this method is definitely an important supplement to the other geometric measurements and will play a crucial role in cosmological constraints. In this paper, we quantify the ability of SL test signal by a CODEX-like spectrograph for constraining interacting dark energy. Four typical interacting dark energy models are considered: (i) $Q=\gamma H\rho_c$, (ii) $Q=\gamma H\rho_{de}$, (iii) $Q=\gamma H_0\rho_c$, and (iv) $Q=\gamma H_0\rho_{de}$. The results show that for all the considered interacting dark energy models, relative to the current joint SN+BAO+CMB+$H_0$ observations, the constraints on $\Omega_m$ and $H_0$ would be improved by about 60\% and 30--40\%, while the constraints on $w$ and $\gamma$ would be slightly improved, with a 30-yr observation of SL test. We also explore the impact of SL test on future joint geometric observations. In this analysis, we take the model with $Q=\gamma H\rho_c$ as an example, and simulate future SN and BAO data based on the space-based project WFIRST. We find that in the future geometric constraints, the redshift drift observations would help break the geometric degeneracies in a meaningful way, thus the measurement precisions of $\Omega_m$, $H_0$, $w$, and $\gamma$ could be substantially improved using future probes., Comment: 6 pages, 5 figures; accepted for publication in EPJC. arXiv admin note: text overlap with arXiv:1407.7123

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)