arXiv subjects

Jing-Zhao Qi

Publications and source records attributed to Jing-Zhao Qi.

At least 19 recordsLinked to original sources

arXiv2026

HIcosmo: a differentiable JAX-based framework for cosmology inference

The Stage IV cosmological surveys, such as Euclid, LSST, DESI, and SKA, will deliver observational data of unprecedented volume, calling for efficient and reliable inference tools. This paper presents HIcosmo (High-performance Inference for Cosmology), an open-source JAX-based framework for cosmology inference. In HIcosmo, the forward model, distance integrals, likelihood evaluations, posterior sampling, and Fisher forecasts are all built from JAX primitives, so that gradients and Hessians of the log-likelihood are obtained directly by automatic differentiation, without any finite-difference approximation. The framework implements the $Λ$CDM, $w$CDM, $w_0 w_a$CDM, and interacting dark-energy models, and provides likelihoods for Type Ia supernovae (Pantheon+, DES-SN5YR, Union3), baryon acoustic oscillations (DESI DR1/DR2, SDSS), Planck 2018 distance priors, local $H_0$ measurements, and strong-lensing time delays. Its scope is restricted to background cosmology, with Boltzmann solvers and full perturbation-level likelihoods left to external tools. We validate HIcosmo against the reference implementation of each likelihood and against Cobaya. $χ^2$ values agree to absolute differences of $10^{-6}$-$10^{-2}$, and the marginalized constraints from the two codes differ by less than $0.2σ$ in every analysis tested. Leveraging just-in-time compilation and automatic differentiation, HIcosmo achieves about $8.7\times$ the end-to-end sampling throughput of Cobaya on CPU. As the dataset grows to survey scale, GPU acceleration over CPU reaches up to $20\times$. As applications, we present multi-probe $Λ$CDM joint constraints, dark-energy equation-of-state constraints, and Fisher forecasts for six 21 cm intensity-mapping surveys, including SKA1, MeerKAT, BINGO, Tianlai, and CHIME.

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arXiv2026

Gravitational wave standard sirens from GWTC-3 combined with DESI DR2 and DESY5: A late-universe probe of the Hubble constant and dark energy

Recently, the combination of the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) baryon acoustic oscillation (BAO) data and the Planck cosmic microwave background (CMB) measurements has shown a $\sim$3$σ$ preference for a dynamical dark energy model with a phantom-crossing behavior. However, such a phantom-crossing dark energy evolution further exacerbates the already severe Hubble tension in the $Λ$CDM model. Moreover, there exists a $\sim2σ$ tension between the DESI DR2 BAO and CMB datasets. Therefore, it is essential to measure the Hubble constant and dark-energy equation-of-state (EoS) parameters using only late-universe observations. In this work, we investigate a novel late-universe data combination: gravitational-wave (GW) standard sirens, BAO, and Type Ia supernovae (SNe Ia). This combination provides a fully distance-ladder- and CMB-independent determination of the Hubble constant and the dark-energy EoS. Using 47 GW standard sirens from the third Gravitational-Wave Transient Catalog, the DESI DR2 BAO data, and DESY5 SNe Ia data, in the $w_0w_a$CDM model, we obtain $H_0=74.8^{+6.3}_{-8.9}$ km s$^{-1}$ Mpc$^{-1}$, $Ω_{\rm m}=0.320^{+0.015}_{-0.012}$, $w_0=-0.775^{+0.072}_{-0.074}$, and $w_a=-0.80\pm0.47$, indicating a mild phantom-crossing behavior within the $1σ$ credible interval with an $H_0$ value consistent with the distance ladder measurements. Our analysis demonstrates the power of GW standard sirens in breaking parameter degeneracies, and this novel data combination provides joint constraints on the Hubble constant and the dark-energy EoS parameters.

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arXiv2025

Testing the cosmic distance duality relation with baryon acoustic oscillations and supernovae data

One of the most fundamental relationships in modern cosmology is the cosmic distance duality relation (CDDR), which describes the relationship between the angular diameter distance ($D_{\rm A}$) and the luminosity distance ($D_{\rm L}$), and is expressed as: $η(z)=D_{\rm L}(z)(1+z)^{-2}/D_{\rm A}(z)=1$. In this work, we conduct a comprehensive test of the CDDR by combining baryon acoustic oscillation (BAO) data from the SDSS and DESI surveys with type Ia supernova (SN) data from PantheonPlus and DESY5. We utilize an artificial neural network approach to match the SN and BAO data at the same redshift. To explore potential violations of the CDDR, we consider three different parameterizations: (i) $η(z)=1+η_0z$; (ii) $η(z)=1+η_0z/(1+z)$; (iii) $η(z)=1+η_0\ln(1+z)$. Our results indicate that the calibration of the SN absolute magnitude $M_{\rm B}$ plays a crucial role in testing potential deviations from the CDDR, as there exists a significant negative correlation between $η_0$ and $M_{\rm B}$. For PantheonPlus analysis, when $M_{\rm B}$ is treated as a free parameter, no evidence of CDDR violation is found. In contrast, fixing $M_{\rm B}$ to the $M_{\rm B}^{\rm D20}$ prior with $-19.230\pm0.040$ mag leads to a deviation at approximately the $2σ$ level, while fixing $M_{\rm B}$ to the $M_{\rm B}^{\rm B23}$ prior with $-19.396\pm0.016$ mag remains in agreement with the CDDR. Furthermore, overall analyses based on the SDSS+DESY5 and DESI+DESY5 data consistently show no evidence of the deviation from the CDDR.

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arXiv2025

Combining strongly lensed and unlensed fast radio bursts: To be a more precise late-universe probe

The Macquart relation and time-delay cosmography are now two promising ways to fast radio burst (FRB) cosmology. In this work, we propose a joint method that combines strongly lensed and unlensed FRBs for improving cosmological parameter estimation by using simulated FRB data from the future sensitive coherent all-sky monitor survey, which is expected to detect a large number of FRBs including galaxy-galaxy strongly lensed events. We find that using a detectable sample of 100,000 localized FRBs including $40$ lensed events can simultaneously constrain the Hubble constant and the equation of state of dark energy, with high precision of $\varepsilon(H_0)=0.4\%$ and $\varepsilon(w)=4.5\%$ in the simplest dynamical dark energy model. The joint analysis of unlensed and lensed FRBs significantly improves the constraint on $H_0$, which could be more effective than combining either the unlensed FRBs with future gravitational wave (GW) standard sirens or the lensed FRBs with CMB. Furthermore, combining the full FRB sample with the CMB+BAO+SNe data yields $σ(H_0)=0.29~{\rm km~s^{-1}~Mpc^{-1}}$, $σ(w_0)=0.046$, and $σ(w_a)=0.15$ in the two-parameter dynamical dark energy model, which outperform the results from the CMB+BAO+SNe+GW data. This reinforces the cosmological implications of a multi-wavelength observational strategy in optical and radio bands. We conclude that the future FRB observations will shed light on the nature of dark energy and also the Hubble tension if enough events with long-duration lensing are incorporated.

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arXiv2025

Model-independent $H_0$ within FLRW: Joint constraints from GWTC-3 standard sirens and strong lensing time delays

We use 47 gravitational-wave (GW) standard sirens from the third Gravitational-Wave Transient Catalog to calibrate distances in the strong gravitational lensing (SGL) system RXJ1131-1231 and constrain the Hubble constant ($H_0$) via the distance sum rule, without assuming a specific cosmological model. For $Ω_K = 0$, we obtain $H_0 = 73.22^{+5.95}_{-5.43}$ ${\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$ and $H_0 = 70.40^{+8.03}_{-5.60}$ ${\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$ by breaking the mass-sheet transform using lens galaxy's mass models and stellar kinematics, respectively. Allowing $Ω_K$ to vary increases the central value of $H_0$ and reduces its precision. We find that GW dark sirens have significant potential for calibrating SGL systems, due to their relatively higher redshifts. By combining 42 binary black holes and RXJ1131-1231, we obtain an $H_0$ constraint with a precision approximately $40\%$ better than the measurement from GW170817 using the Hubble law. This suggests that high-precision, model-independent $H_0$ measurements can be achieved with this method as the redshift range of GW dark sirens expands, even without the need for GW bright sirens.

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arXiv2025

Testing the Cosmic Distance Duality Relation Using Strong Gravitational Lensing Time Delays and Type Ia Supernovae

We present a comprehensive test of the cosmic distance duality relation (DDR) using a combination of strong gravitational lensing (SGL) time delay measurements and Type Ia supernovae (SNe Ia) data. We investigate three different parameterizations of potential DDR violations. To bridge the gap between SGL and SNe Ia datasets, we implement an artificial neural network (ANN) approach to reconstruct the distance modulus of SNe Ia. Our analysis uniquely considers both scenarios where the absolute magnitude of SNe Ia ($M_B$) is treated as a free parameter and where it is fixed to a Cepheid-calibrated value. Using a sample of six SGL systems and the Pantheon+ SNe Ia dataset, we find no statistically significant evidence for DDR violations across all parameterizations. The consistency of our findings across different parameterizations not only reinforces confidence in the standard DDR but also demonstrates the robustness of our analytical approach.

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arXiv2024

Joint observations of late universe probes: cosmological parameter constraints from gravitational wave and Type Ia supernova data

The growing tensions between the early Universe and the late Universe increasingly highlight the importance of developing precise probes for late cosmology. As significant late-Universe probes, Type Ia supernovae (SNe Ia) and gravitational waves (GWs) can provide measurements of relative and absolute distances, respectively. Their complementary nature is likely to break the degeneracies among cosmological parameters, thereby yielding more precise constraints. In this study, we use 43 gravitational-wave sources from the Third LIGO-Virgo-KAGRA Gravitational-Wave Transient Catalog (GWTC-3) and 1590 SNe Ia from Pantheon+ compilation to constrain the dark energy models, as an attempt to achieve precise late-Universe cosmological constraints. For the dark siren GW event, we estimate the corresponding redshift using the binary black hole redshift distribution model. The combination of GW and SNe Ia data could provide the precision on the Hubble constant (H0) and the present matter density (Omega_m) of approximately 20% and 8% for the LambdaCDM model. If we consider the equation of state of dark energy (w), the combination sample constrains the precision of w to approximately 30%. Although the combination of GW and SNe Ia observations effectively breaks degeneracies among various cosmological parameters, yielding more stringent constraints, the precision of these constraints still does not meet the stringent standards required by precision cosmology. However, it is reasonable to anticipate that, in the near future, the joint observations of GWs and SNe Ia will become a powerful tool, particularly in the late Universe, for the precise measurement of cosmological parameters.

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arXiv2023

Testing the spatial geometry of the universe with TianQin: the prospect of using supermassive black hole binaries

The determination of the spatial geometry of the universe plays an important role in modern cosmology. Any deviation from the cosmic curvature $Ω_K=0$ would have a profound impact on the primordial inflation paradigm and fundamental physics. In this paper, we carry out a systematic study of the prospect of measuring cosmic curvature with the inspiral signal of supermassive black hole binaries (SMBHBs) that could be detected with TianQin. The study is based on a cosmological-model-independent method that extended the application of gravitational wave (GW) standard sirens in cosmology. By comparing the distances from future simulated GW events and simulated $H(z)$ data, we evaluate if TianQin would produce robust constraints on the cosmic curvature parameter $Ω_{k}$. More specifically, we consider 3-yr to 10-yr observations of supermassive black hole binaries with total masses ranging from $10^{3}M_\odot$ to $10^{7}M_\odot$. Our results show that in the future, with the synergy of 10-yr high-quality observations, we can tightly constrain the curvature parameter at the level of $1σ$ $Ω_k=-0.002\pm0.061$. Moreover, our findings indicate that the total mass of SMBHB does influence the estimation of cosmic curvature, implied by the analysis performed on different subsamples of gravitational wave data. Therefore, TianQin is expected to provide a powerful and competitive probe of the spatial geometry of the universe, compared to future spaced-based detectors such as DECIGO.

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arXiv2023

Model-independent measurement of cosmic curvature with the latest $H(z)$ and SNe Ia data: A comprehensive investigation

In the context of the discrepancies between the early and late universe, we emphasize the importance of independent measurements of the cosmic curvature in the late universe. We present an investigation of the model-independent measurement of the cosmic curvature parameter $Ω_k$ in the late universe with the latest Hubble parameter $H(z)$ measurements and type Ia supernovae (SNe Ia) data. For that, we use two reconstruction methods, the Gaussian process (GP) and artificial neural network (ANN) methods, to achieve the distance construction from $H(z)$ data. Our analysis reveals that the GP method provides the most precise constraint on $Ω_k$, with a constraint precision of $ξ(Ω_k)=0.13$, surpassing recent estimations using similar methods. The GP method consistently indicates a preference for a flat universe at the 2$σ$ confidence level. Moreover, we find that the choice of reconstruction method influences the estimation of $Ω_k$. The ANN reconstruction method exhibits higher sensitivity to the addition of BAO $H(z)$ data, resulting in comparable constraint precision to the GP method. A discrepancy exists between the best-fit values obtained by these two reconstruction methods, indicating their dependence on the reconstruction approach. However, we anticipate that with the improvement of sample size and precision of observational $H(z)$ data, the estimation of $Ω_k$ using this approach will become more robust and reliable.

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arXiv2023

Prospects for constraining interacting dark energy models from gravitational wave and gamma ray burst joint observation

With the measurement of the electromagnetic (EM) counterpart, a gravitational wave (GW) event could be treated as a standard siren. As a novel cosmological probe, GW standard sirens will bring significant implications for cosmology. In this paper, by considering the coincident detections of GW and associated $γ$ ray burst (GRB), we find that only about 400 GW bright standard sirens from binary neutron star mergers could be detected in a 10-year observation of the Einstein Telescope and the THESEUS satellite mission. Based on this mock sample, we investigate the implications of GW standard sirens on the interaction between dark energy and dark matter. In our analysis, four viable interacting dark energy (IDE) models, with interaction forms $Q=3βH ρ_{\mathrm{de}}$ and $Q=3 βH ρ_{\mathrm{c}}$, are considered. Compared with the traditional EM observational data such as CMB, BAO, and SN Ia, the combination of both GW and EM observations could effectively break the degeneracies between different cosmological parameters and provide more stringent cosmological fits. We find that the GW data could play a more important role for determining the interaction in the models with $Q=3 βH ρ_{\mathrm{c}}$, compared with the models with $Q=3βH ρ_{\mathrm{de}}$. We also show that constraining IDE models with mock GW data based on different fiducial $H_0$ values yield different results, indicating that accurate determination of $H_0$ is significant for exploring the interaction between dark energy and dark matter.

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arXiv2023

Null test for cosmic curvature using Gaussian process

The cosmic curvature $Ω_{K,0}$, which determines the spatial geometry of the universe, is an important parameter in modern cosmology. Any deviation from $Ω_{K,0}=0$ would have a profound impact on primordial inflation paradigm and fundamental physics. In this work, we adopt a cosmological model-independent method to test whether $Ω_{K,0}$ deviates from zero. We use the Gaussian process to reconstruct the reduced Hubble parameter $E(z)$ and the derivative of distance $D'(z)$ from observational data, and then determine $Ω_{K,0}$ with a null test relation. The cosmic chronometer (CC) Hubble data, baryon acoustic oscillation (BAO) Hubble data, and supernovae Pantheon sample are considered. Our result is consistent with a spatially flat universe within the domain of reconstruction $0 1$, it tends to favor a closed universe. In this sense, there is still a possibility for a closed universe. We also carry out the null test of the cosmic curvature at $0<z<4.5$ using the simulated gravitational wave standard sirens, CC+BAO and redshift drift Hubble data. The result shows that in the future, with the synergy of multiple high-quality observations, we can tightly constrain the spatial geometry or exclude the flat universe.

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arXiv2022

Cosmological model-independent measurement of cosmic curvature using distance sum rule with the help of gravitational waves

Although the cosmic curvature has been tightly constrained in the standard cosmological model using observations of cosmic microwave background anisotropies, it is still of great importance to independently measure this key parameter using only late-universe observations in a cosmological model-independent way. The distance sum rule in strong gravitational lensing (SGL) provides such a way, provided that the three distances in the sum rule can be calibrated by other observations. In this paper, we propose that gravitational waves (GWs) can be used to provide the distance calibration in the SGL method, which can avoid the dependence on distance ladder and cover a wider redshift range. Using the simulated GW standard siren observation by the Einstein Telescope as an example, we show that this scheme is feasible and advantageous. We find that $ΔΩ_k\simeq 0.17$ with the current SGL data, which is slightly more precise than the case of using SN to calibrate. Furthermore, we consider the forthcoming LSST survey that is expected to observe many SGL systems, and we find that about $10^4$ SGL data could provide the precise measurement of $ΔΩ_k\simeq 10^{-2}$ with the help of GWs. In addition, our results confirm that this method of constraining $Ω_k$ is strongly dependent on lens models. However, obtaining a more accurate phenomenological model for lens galaxies is highly predictable as future massive surveys observe more and more SGL samples, which will significantly improve the constraint of cosmic curvature.

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arXiv2022

A new way to explore cosmological tensions using gravitational waves and strong gravitational lensing

In recent years, a crisis in the standard cosmology has been caused by inconsistencies in the measurements of some key cosmological parameters, Hubble constant $H_0$ and cosmic curvature parameter $Ω_K$ for example. It is necessary to remeasure them with the cosmological model-independent methods. In this paper, based on the distance sum rule, we present such a way to constrain $H_0$ and $Ω_K$ simultaneously in the late universe from strong gravitational lensing time delay (SGLTD) data and gravitational wave (GW) standard siren data simulated from the future observation of the Einstein Telescope (ET). Based on the currently 6 observed SGLTD data, we find that the constraint precision of $H_0$ from the combined 100 GW events can be comparable with the measurement from SH0ES collaboration. As the number of GW events increases to 700, the constraint precision of $H_0$ will exceed that of the \textit{Planck} 2018 results. Considering 1000 GW events as the conservative estimation of ET in ten-year observation, we obtain $H_0=73.69\pm 0.36 \mathrm{~km~s^{-1}~Mpc^{-1}}$ with a 0.5\% uncertainty and $Ω_K=0.076^{+0.068}_{-0.087}$. In addition, we simulate 55 SGL systems with 6.6\% uncertainty for the measurement of time-delay distance. By combining with 1000 GWs, we infer that $H_0=73.65\pm0.35 \mathrm{~km~s^{-1}~Mpc^{-1}}$ and $Ω_K=0.008\pm0.048$. Our results suggest that this approach can play an important role in exploring cosmological tensions.

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arXiv2022

Strongly lensed type Ia supernovae as a precise late-universe probe of measuring the Hubble constant and cosmic curvature

Strongly lensed type Ia supernovae (SNe Ia) are expected to have some advantages in measuring time delays of multiple images, and so they have a great potential to be developed into a powerful late-universe cosmological probe. In this paper, we simulate a sample of lensed SNe Ia with time-delay measurements in the era of the Legacy Survey of Space and Time (LSST). Based on the distance sum rule, we use lensed SNe Ia to implement model-independent constraints on the Hubble constant $H_0$ and cosmic curvature parameter $Ω_K$ in the late universe. We find that if 20 lensed SNe Ia could be observed, the constraint on $H_{0}$ is better than the measurement by the SH0ES collaboration. When the event number of lensed SNe Ia increases to 100, the constraint precision of $H_{0}$ is comparable with the result from Planck 2018 data. Considering 200 lensed SNe Ia events as the optimistic estimation, we obtain $ΔH_0=0.33$ $\rm km\ s^{-1}\ Mpc^{-1}$ and $ΔΩ_K=0.053$. In addition, we also simulate lensed quasars in different scenarios to make a comparison and we find that they are still a useful cosmological probe even though the constraint precision from them is much less than that obtained from lensed SNe Ia. In the era of LSST, the measurements of time delay from both lensed SNe Ia and lensed quasars are expected to yield the results of $ΔH_0=0.26 ~\rm km\ s^{-1}\ Mpc^{-1}$ and $ΔΩ_K=0.044$.

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arXiv2022

Cosmological Parameter Estimation Using Current and Future Observations of Strong Gravitational Lensing

Remarkable development of cosmology is benefited from the increasingly improved measurements of cosmic distances including absolute distances and relative distances. In recent years, however, the emerged cosmological tensions motivate us to explore the independent and precise late-universe probes. The two observational effects of strong gravitational lensing (SGL), the velocity dispersions of lens galaxies and the time delays between multiple images, can provide measurements of relative and absolute distances respectively, and their combination is possible to break the degeneracies between cosmological parameters and enable tight constraints on cosmological parameters. In this paper, we combine the observed 130 SGL systems with velocity-dispersion measurements and 7 SGL systems with time-delay measurements to constrain dark-energy cosmological models. It is found that the combination of the two effects does not significantly break the degeneracies between cosmological parameters as expected. However, with the simulations of 8000 SGL systems with well-measured velocity dispersions and 55 SGL systems with well-measured time delays based on the forthcoming LSST survey, we find that the combination of two effects can significantly break the parameter degeneracies, and make the constraint precision of cosmological parameters meet the standard of precision cosmology. We conclude that the observations of SGL will become a useful late-universe probe for precisely measuring cosmological parameters.

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arXiv2022

Galaxy-Scale Test of General Relativity with Strong Gravitational Lensing

Although general relativity (GR) has been precisely tested at the solar system scale, precise tests at a galactic or cosmological scale are still relatively insufficient. Here, in order to test GR at the galactic scale, we use the newly compiled galaxy-scale strong gravitational lensing (SGL) sample to constrain the parameter $γ_{PPN}$ in the parametrized post-Newtonian (PPN) formalism. We employ the Pantheon sample of type Ia supernovae observation to calibrate the distances in the SGL systems using the Gaussian Process method, which avoids the logical problem caused by assuming a cosmological model within GR to determine the distances in the SGL sample. Furthermore, we consider three typical lens models in this work to investigate the influences of the lens mass distributions on the fitting results. We find that the choice of the lens models has a significant impact on the constraints on the PPN parameter $γ_{PPN}$. We use the Bayesian information criterion as an evaluation tool to make a comparison for the fitting results of the three lens models, and we find that the most reliable lens model gives the result of $γ_{PPN}=1.065^{+0.064}_{-0.074}$, which is in good agreement with the prediction of $γ_{PPN}=1$ by GR. As far as we know, our 6.4% constraint result is the best result so far among the recent works using the SGL method.

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arXiv2021

Using a multi-messenger and multi-wavelength observational strategy to probe the nature of dark energy through direct measurements of cosmic expansion history

In the near future, the redshift drift observations in optical and radio bands will provide precise measurements on $H(z)$ covering the redshift ranges of $2<z<5$ and $0<z<0.3$. In addition, gravitational wave (GW) standard siren observations could make measurements on the dipole anisotropy of luminosity distance, which will also provide the $H(z)$ measurements in the redshift range of $0<z<3$. In this work, we propose a multi-messenger and multi-wavelength observational strategy to measure $H(z)$ based on the three next-generation projects, E-ELT, SKA, and DECIGO, and we wish to see whether the future $H(z)$ measurements could provide tight constraints on dark-energy parameters. The dark energy models we consider include $Λ$CDM, $w$CDM, CPL, HDE, and I$Λ$CDM models. It is found that E-ELT, SKA1, and DECIGO are highly complementary in constraining dark energy models. Although any one of these three data sets can only give rather weak constraints on each model we consider, the combination of them could significantly break the parameter degeneracies and give much tighter constraints on almost all the cosmological parameters. Moreover, we find that the combination of E-ELT, SKA1, DECIGO, and CMB could further improve the constraints on dark energy parameters, e.g., $σ(w_0)=0.024$ and $σ(w_a)=0.17$ in the CPL model, which means that these three promising probes will play a key role in helping reveal the nature of dark energy.

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arXiv2021

Prospects for Constraining Interacting Dark Energy Models with 21 cm Intensity Mapping Experiments

We forecast constraints on cosmological parameters in the interacting dark energy models using the mock data generated for neutral hydrogen intensity mapping (IM) experiments. In this work, we only consider the interacting dark energy models with energy transfer rate $Q=βHρ_{\rm c}$, and take BINGO, FAST, SKA1-MID, and Tianlai as typical examples of the 21 cm IM experiments. We find that the Tianlai cylinder array will play an important role in constraining the interacting dark energy model. Assuming perfect foreground removal and calibration, and using the Tianlai-alone data, we obtain $σ(H_0)=0.19$ km s$^{-1}$ Mpc$^{-1}$, $σ(Ω_{\rm m})=0.0033$ and $σ(σ_8)=0.0033$ in the I$Λ$CDM model, which are much better than the results of Planck+optical BAO (i.e. optical galaxy surveys). However, the Tianlai-alone data cannot provide a very tight constraint on the coupling parameter $β$ compared with Planck+optical BAO, while the Planck+Tianlai data can give a rather tight constraint of $σ(β)=0.00023$ due to the parameter degeneracies being well broken by the data combination. In the I$w$CDM model, we obtain $σ(β)=0.00079$ and $σ(w)=0.013$ from Planck+Tianlai. In addition, we also make a detailed comparison among BINGO, FAST, SKA1-MID, and Tianlai in constraining the interacting dark energy models. We show that future 21 cm IM experiments will provide a useful tool for exploring the nature of dark energy and play a significant role in measuring the coupling between dark energy and dark matter.

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