arXiv subjects

Jun-Qing Xia

Publications and source records attributed to Jun-Qing Xia.

At least 19 recordsLinked to original sources

arXiv2026

Measuring peculiar velocity and tomographic redshift dipole with DESI DR1 catalogs

The so-called ``cosmic dipole tension'' challenges the Cosmological Principle by positing a discrepancy between the Solar System's peculiar velocity inferred from the Cosmic Microwave Background (CMB) dipole and that derived from large-scale structure number-count dipoles. Here we provide a high-precision determination of the kinematic dipole using the redshift-dipole method applied to the first data release (DR1) of the Dark Energy Spectroscopic Instrument (DESI). By exploiting the Doppler-induced modulation of observed redshifts, this estimator is intrinsically less sensitive to imaging systematics and selection-function uncertainties that can bias traditional number-count measurements. We conduct a tomographic analysis of four tracer populations, Bright Galaxy Sample, Luminous Red Galaxies, Emission Line Galaxies, and quasars, spanning $0.1<z<2.1$. Survey geometry and statistical uncertainties are quantified using 1,000 \texttt{EZmock} realizations. We find that the high-redshift QSO sample implies a peculiar velocity of $v = 357.95_{-48.47}^{+55.05}\,\mathrm{km\,s^{-1}}$, in excellent agreement with the CMB-inferred value of $369.82 \pm 0.11\,\mathrm{km\,s^{-1}}$. By contrast, a complementary number-count analysis yields a significantly enhanced dipole amplitude, which we attribute to leakage of large-scale power and to incompleteness within the DESI DR1 footprint. These results indicate that the redshift dipole provides a cleaner and more reliable probe of the kinematic rest frame, offering strong support for the standard kinematic interpretation at high redshift and helping to resolve the apparent dipole anomaly.

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arXiv2026

Testing cosmic anisotropy with the Combo correlation of gamma-ray bursts

We employ the sample of 244 gamma-ray bursts (GRBs; i.e., C244) with the Combo correlation to test cosmic anisotropy. Meanwhile, the Pantheon sample is introduced to verify whether the GRB sample can suppress the fake anisotropic signals induced by inhomogeneous spatial distributions. In the dipole fitting (DF) method, under the dipole-modulated $\Lambda$CDM model, the C244 sample shifts the best-fitting longitude $l$ derived from the Pantheon sample by $54.09^\circ$ and reduces the uncertainty in $l$ by approximately $40\%$. Compared to the 118 GRBs (i.e., A118) with the $E_\mathrm{p}$-$E_\mathrm{iso}$ correlation, the shift in longitude $l$ increases by additional $21.35^\circ$. In the hemisphere comparison (HC) method, the preferred direction derived from the C244+Pantheon sample deviates from that of the Pantheon-only sample by more than $1\sigma$. In contrast, the preferred direction from the A118+Pantheon sample is consistent with the Pantheon-only result within the $1\sigma$ uncertainty. The preferred direction changes significantly as the number of GRBs increases from 118 to 244. Our results show that a larger GRB sample can reduce the fake anisotropic signals caused by inhomogeneous spatial distributions. Accordingly, we suggest that GRBs have the potential to provide a reliable probe of cosmic anisotropy.

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arXiv2026

Search for quasar pairs with Gaia astrometric data. II. Photometric redshift prediction with machine learning for the MGQPC catalogue

The identification of physically associated kiloparsec-scale quasar pairs is important for understanding galaxy evolution, the growth of supermassive black holes, and their co-evolution with host galaxies. However, their rarity and the high contamination from stellar superpositions and projected alignments require efficient pre-selection methods. We develop a machine-learning framework to produce photometric-redshift point estimates and redshift probability density functions for quasars, with the main goal of identifying high-probability quasar pair candidates in the MGQPC catalogue. We construct two large spectroscopically confirmed quasar samples with multi-wavelength photometry, based on SDSS and DESI Legacy Imaging Surveys data. CatBoost is used for point-estimate photometric-redshift regression, and FlexZBoost is used for full redshift-PDF estimation. The workflow achieves robust performance, with a normalised median absolute deviation of 0.036 and an outlier fraction of 5.6% on the test sample. Applying the trained model to the MGQPC catalogue, we identify 185 high-probability quasar pair candidates based on photometric-redshift consistency. Among them, 20 systems have been subsequently confirmed as genuine physical pairs by independent spectroscopic observations. The resulting MGQPC photometric-redshift catalogue provides a useful resource for future spectroscopic follow-up of quasar pairs and dual supermassive black holes.

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arXiv2026

Photometric Redshift PDFs via Neural Network Classification for DESI Legacy Imaging Surveys and Pan-STARRS

We present a neural network classification (NNC) method for photometric redshift estimation that produces well-calibrated redshift probability density functions (PDFs). The method discretizes the redshift space into ordered bins and optimizes the Continuous Ranked Probability Score (CRPS), which respects the ordinal nature of redshift and naturally provides uncertainty quantification. Unlike traditional regression approaches that output single point estimates, our method can capture multi-modal posterior distributions arising from color-redshift degeneracies. We apply this method to the DESI Legacy Imaging Surveys Data Release 10 (LSDR10) and Pan-STARRS Data Release 2 (PS1DR2), using an unprecedented spectroscopic training sample from DESI DR1 and SDSS DR19. Our method achieves $\sigma_{\mathrm{NMAD}} = 0.0153$ and $\eta = 0.50\%$ on LSDR10, and $\sigma_{\mathrm{NMAD}} = 0.0222$ and $\eta = 0.34\%$ on PS1DR2 combined with unWISE infrared photometry. The NNC method outperforms Random Forest, XGBoost, and standard neural network regression. We demonstrate that DESI DR1 significantly improves photo-$z$ performance at $z > 1$, while the combination of deep optical photometry and mid-infrared coverage is essential for achieving high precision across the full redshift range. We provide a unified photometric redshift catalog combining LSDR10 and PS1DR2 with a hierarchical model selection strategy based on available photometry. The well-calibrated PDFs produced by our method are valuable for cosmological studies and can be extended to next-generation surveys such as CSST, Euclid, and LSST.

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arXiv2025

Testing $f(R)$ Gravity from Cosmic Shear Measurements

In this work, we perform a detailed analysis to constrain the Hu-Sawicki $f(R)$ gravity model, using cosmic shear data from three prominent Stage-III weak lensing surveys: DES-Y3, KiDS-1000, and HSC-Y3. To accurately model the nonlinear matter clustering in the analysis of cosmic shear signals, we employ \texttt{FREmu}, a recently developed power spectrum emulator for the $f(R)$ gravity trained on the Quijote-MG simulations. This emulator achieves precise predictions, limiting the errors to 5\% on scales of $0.009h\,{\rm Mpc}^{-1} < k < 0.5h\,{\rm Mpc}^{-1}$. Our findings reveal that cosmic shear data alone impose only weak constraints on the $f(R)$ parameter $\log_{10}|f_{R_0}|$. To improve these constraints, we incorporate state-of-the-art external observations, including data from the cosmic microwave background and baryon acoustic oscillations. The inclusion of these external datasets significantly enhances the constraints, yielding an upper limit of $\log_{10}|f_{R_0}| < -4.98$ at the 95\% confidence level.

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arXiv2025

Foreground Removal in Ground-Based CMB Observations Using a Transformer Model

We present a novel method for Cosmic Microwave Background (CMB) foreground removal based on deep learning techniques. This method employs a Transformer model, referred to as \texttt{TCMB}, which is specifically designed to effectively process HEALPix-format spherical sky maps. \texttt{TCMB} represents an innovative application in CMB data analysis, as it is an image-based technique that has rarely been utilized in this field. Using simulated data with noise levels representative of current ground-based CMB polarization observations, the \texttt{TCMB} method demonstrates robust performance in removing foreground contamination. The mean absolute variance for the reconstruction of the noisy CMB Q/U map is significantly less than the CMB polarization signal. To mitigate biases caused by instrumental noise, a cross-correlation approach using two half-mission maps was employed, successfully recovering CMB EE and BB power spectra that align closely with the true values, and these results validate the effectiveness of the \texttt{TCMB} method. Compared to the previously employed convolutional neural network (CNN)-based approach, the \texttt{TCMB} method offers two significant advantages: (1) It demonstrates superior effectiveness in reconstructing CMB polarization maps, outperforming CNN-based methods. (2) It can directly process HEALPix spherical sky maps without requiring rectangular region division, a step necessary for CNN-based approaches that often introduces uncertainties such as boundary effects. This study highlights the potential of Transformer-based models as a powerful tool for CMB data analysis, offering a substantial improvement over traditional CNN-based techniques.

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arXiv2025

Do JWST reionization (optical depth) puzzle, cosmological tensions, and CMB anomalies imply Harrison-Zel'dovich spectrum?

The James Webb Space Telescope (JWST) has observed massive galaxies at high redshifts, which implies an earlier epoch of reionization (EoR) compared with the cosmic microwave background (CMB) results. In this paper, based on \texttt{Planck 2020} (NPIPE release), \texttt{ACT DR4} and \texttt{SPT-3G} data, if assumed a Harrison-Zel'dovich (HZ) primordial power spectrum in the standard cosmological model, we show that the redshift or optical depth of reionization is larger than the case of a power-law (PL) primordial power spectrum. In HZ-$ Λ$CDM model, the redshift of reionization is $ z_\text{reio} = 9.11 \pm 0.61 $, which is consistent with the JWST result that $ z_\text{reio} \approx 8.9 $. Moreover, the cosmological tensions, i.e. Hubble ($H_0$) tension and $ S_8 $ tension are alleviated in HZ-$ Λ$CDM case. The Hubble constant is $ H_0 = 70.38 \pm 0.35 \, \text{km}/\text{s}/\text{Mpc}$ and the structure growth parameter is $ S_8 = 0.7645\pm 0.0094 $ in HZ-$ Λ$CDM model. We also consider two extensions of $ Λ$CDM, including $ Λ$CDM$ + A_\text{L} $ and $ Λ$CDM$ + Ω_\text{k} $ models. But the extensions of $ Λ$CDM with a HZ spectrum meet more serious CMB anomalies, i.e. lensing anomaly and spatial curvature anomaly as compared with the extensions of $ Λ$CDM with a PL spectrum. We discuss that these two CMB anomalies may come from the degeneracy of cosmological parameters.

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arXiv2025

A Quasar Pair Sample Compiled from DESI DR1

Interacting quasar pairs (QPs, either dual or binary) provide crucial insights into galaxy mergers, black hole growth, and large-scale structure formation. The current literature reports fewer than 200 spectroscopically confirmed QPs, highlighting the need for larger samples to enable statistically meaningful investigations. In this paper, we present a sample of 1,220 quasar pairs or candidates compiled from the DESI DR1 quasar sample. Among them, 145 systems have been previously reported. We visually classified the full sample using DESI Legacy Images and SPARCL spectra into three categories: QP (quasar pairs, N = 1020), QPC (quasar pair candidates, N = 142), LQC (lensed quasar candidates, N = 58). Within the LQC subset, we find an intriguing wide-separation ($\sim$7.15$^{\prime\prime}$) quadruply lensed quasar candidate. The redshift distribution of pair sample peaks at $z \sim 1$--$2.5$, with an overall pair fraction of $6.2^{+0.2}_{-0.2}\times10^{-4}$ (Poisson error) and a generally weak redshift dependence. 63.8\% of QPs have $|\Delta V_r| < 600$\,km/s, suggesting dynamical associations. This sample offers a statistically meaningful dataset for future studies of quasar pairs, lensing events, and potentially merger-triggered or merger-induced SMBH growth across the merger sequence.

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arXiv2025

Measuring Cosmic Growth Rate with CSST Spectroscopic Survey and Fast Radio Burst

The cosmic growth rate, which is related to peculiar velocity and is a primary scientific objective of galaxy spectroscopic surveys, can be inferred from the Redshift Space Distortion effect and the kinetic Sunyaev-Zel'dovich effect. However, the reconstruction noise power spectrum of the radial velocity field in kSZ is significantly dependent on the measurement of the small-scale galaxy-electron power spectrum $P_{ge}$. In this study, we thoroughly discuss the enhancement of cosmic growth rate measurements facilitated by Fast Radio Bursts, which probe the electron density of the universe along their propagation paths to provide crucial additional information on $P_{ge}$. Subsequently, we utilize future spectroscopic surveys from the Chinese Space Station Telescope and the CMB-S4 experiment, combined with FRB dispersion measures, to achieve precise measurements of the cosmic growth rate at redshifts $z_g = 0.15,0.45,0.75$. Employing Fisher matrix forecasting analysis, we anticipate that constraints on $fσ_8$ will reach a precision of 0.001 with a sample size of $10^6$ FRBs. Furthermore, we perform a global analysis using Markov Chain Monte Carlo methods to constrain key parameters of three distinct dark energy models and a modified gravity model based on cosmic growth rate measurements. The results demonstrate that these refined $fσ_8$ measurements considerably enhance the constraints on relevant cosmological parameters compared to those obtained from Planck. As the number of observed FRBs increases, alongside more precise galaxy surveys and next-generation CMB observations, new opportunities will arise for constraining cosmological models using the kSZ effect and for developing novel cosmological applications of FRBs.

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arXiv2025

Constraints on Evolutions of Fundamental Constants from Clustering of Fast Radio Burst Dispersion Measure

Constrained measurements of fundamental physical constants using astronomical observational data represent a powerful method for investigating potential new physics. In particular, the dispersion measure (DM) of fast radio bursts (FRBs), which probes the electron density along their propagation paths, may be influenced by the space-time variation of the fine-structure constant $α$. In this study, we analyze the cross-correlation signal between foreground galaxies and the DM of background FRBs to constrain the evolution of $α$. Assuming large-scale structure (LSS) galaxy surveys with the capabilities of the China Space Station Telescope (CSST) at $z=0.15$ and { a mock FRB survey with $N_{\text{FRB}}=10^5$ at $z=0.4$, we test how well $α$ variation can be constrained}, with a standard deviation of $σ(Δα/ α) = 0.0007$ at $z=0.15$. Furthermore, taking into account the nonminimal coupling between the scalar field and the electromagnetic field, the variation in $α$ can lead to the non-conservation of photon number along geodesics. This would result in a violation of the CDDR and affect the evolution of the Cosmic Microwave Background (CMB) temperature. In this work, we { obtain constraints results} on the CDDR parameter $η$ and the parameter $β$ governing CMB temperature evolution at $z=0.15$, yielding $σ(η) = 0.0004$ and $σ(β) = 0.0006$, respectively. Finally, we relate the variation in $α$ to the time evolution of the proton-to-electron mass ratio, { reporting a standard deviation} of $σ(Δμ/μ) = 0.002$ at $z=0.15$. Future FRB surveys hold significant potential for advancing our understanding of the evolution of fundamental physical constants.

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arXiv2025

Testing General Relativity using Large Scale Structures Photometric Redshift Surveys and Cosmic Microwave Background Lensing Effect

The $E_G$ statistic provides a valuable tool for evaluating predictions of General Relativity (GR) by probing the relationship between gravitational potential and galaxy clustering on cosmological scales within the observable universe. In this study, we constrain the $E_G$ statistic using photometric redshift data from the Dark Energy Survey (DES) MagLim sample in combination with the Planck 2018 Cosmic Microwave Background (CMB) lensing map. Unlike spectroscopic redshift surveys, photometric redshift measurements are subject to significant redshift uncertainties, making it challenging to constrain the redshift distortion parameter $β$ with high precision. We adopt a new definition for this parameter, $β(z) = {fσ_8(z)}/{bσ_8(z)}$. In this formulation, we reconstruct the growth rate of structure, $fσ_8(z)$, using Artificial Neural Networks (ANN) method, while simultaneously utilizing model-independent constraints on the parameter $bσ_8(z)$, directly obtained from the DES collaboration. After obtaining the angular power spectra $C_\ell^{gg}$ (galaxy-galaxy) and $C_\ell^{gκ}$ (galaxy-CMB lensing) from the combination of DES photometric data and Planck lensing, we derive new measurements of the $E_G$ statistic: $E_G = 0.354 \pm 0.146$, $0.452 \pm 0.092$, $0.414 \pm 0.069$, and $0.296 \pm 0.069$ (68$\%$ C.L.) across four redshift bins: $z = 0.30, 0.47, 0.63$, and $0.80$, respectively, which are consistent with the predictions of the standard $Λ$CDM model. Finally, we forecast the $E_G$ statistic using future photometric redshift data from the China Space Station Telescope, combined with lensing measurements from the CMB-S4 project, indicating an achievable constraint on $E_G$ of approximately 1$\%$, improving the precision of tests for GR on cosmological scales.

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arXiv2024

Probing Cosmic Isotropy with the FAST All Sky HI Survey

This paper leverages the first released catalog from the FAST All Sky \textsc{Hi} Survey (FASHI) to examine the hypothesis of cosmic isotropy in the local Universe. Given the design of the overall FAST survey, the inhomogeneous detection sensitivity of FASHI is likely to introduce significant biases in the statistical properties of the catalog. To mitigate the potential influence of spurious clustering effects due to these sensitivity variations, we focus on extragalactic \textsc{Hi} sources within the sensitivity range of $[0.65, 1.0]$. This refined subsample is divided into ten distinct sky regions, for which we compute the two-point angular correlation functions (2PACF) over angular scales of $0.5^\circ < θ< 10^\circ$. We apply the Markov chain Monte Carlo method to fit these 2PACFs with a power-law model and assess the statistical significance of the best-fit parameters for the ten FASHI sky regions by comparing them against results from mock catalogs generated under the assumptions of homogeneity and isotropy. Our findings indicate that the local Universe, as traced by the \textsc{Hi} sources in the FASHI survey, aligns with the cosmic isotropy hypothesis within a $2σ$ confidence level. We do not detect any statistically significant deviations from cosmic isotropy in the FASHI survey data.

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arXiv2024

COSMIC: A Galaxy Cluster Finding Algorithm Using Machine Learning

Building a comprehensive catalog of galaxy clusters is a fundamental task for the studies on the structure formation and galaxy evolution. In this paper, we present COSMIC (Cluster Optical Search using Machine Intelligence in Catalogs), an algorithm utilizing machine learning techniques to efficiently detect galaxy clusters. COSMIC involves two steps, including the identification of the brightest cluster galaxies and the estimation of the cluster richness. We train our models on the galaxy data from the Sloan Digital Sky Survey and WHL galaxy cluster catalog. Validated to a test data in the region of northern Galactic cap, COSMIC algorithm demonstrates a high completeness when cross-matching with previous cluster catalogs. Richness comparison with previous optical and X-ray measurements also demonstrated a tight correlation. Our methodology showcases robust performance in galaxy cluster detection and holds promising prospects for applications in upcoming large-scale surveys.

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arXiv2024

Constraints on Primordial Magnetic Fields from High Redshift Stellar Mass Density

Primordial magnetic fields (PMFs) play a pivotal role in influencing small-scale fluctuations within the primordial density field, thereby enhancing the matter power spectrum within the context of the $Λ$CDM model at small scales. These amplified fluctuations accelerate the early formation of galactic halos and stars, which can be observed through advanced high-redshift observational techniques. Therefore, Stellar Mass Density (SMD) observations, which provide significant opportunities for detailed studies of galaxies at small scales and high redshifts, offer a novel perspective on small-scale cosmic phenomena and constrain the characteristics of PMFs. In this study, we compile 14 SMD data points at redshifts $z > 6$ and derive stringent constraints on the parameters of PMFs, which include the amplitude of the magnetic field at a characteristic scale of $λ=1\,{\rm Mpc}$, denoted as $B_0$, and the spectral index of the magnetic field power spectrum, $n_{\rm B}$. At 95\% confidence level, we establish upper limits of $B_0 < 4.44$ nG and $n_{\rm B} < -2.24$, along with a star formation efficiency of approximately $f_*^0 \sim 0.1$. If we fix $n_{\rm B}$ at specific values, such as $-2.85$, $-2.9$, and $-2.95$, the 95\% upper limits for the amplitude of the magnetic field can be constrained to 1.33 nG, 2.21 nG, and 3.90 nG, respectively. Finally, we attempt to interpret recent early observations provided by James Webb Space Telescope (JWST) using the theory of PMFs, and find that by selecting appropriate PMF parameters, it is possible to explain these results without significantly increasing the star formation efficiency.

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arXiv2024

CMBFSCNN: Cosmic Microwave Background Polarization Foreground Subtraction with Convolutional Neural Network

In our previous study, we introduced a machine-learning technique, namely CMBFSCNN, for the removal of foreground contamination in cosmic microwave background (CMB) polarization data. This method was successfully employed on actual observational data from the Planck mission. In this study, we extend our investigation by considering the CMB lensing effect in simulated data and utilizing the CMBFSCNN approach to recover the CMB lensing B-mode power spectrum from multi-frequency observational maps. Our method is first applied to simulated data with the performance of CMB-S4 experiment. We achieve reliable recovery of the noisy CMB Q (or U) maps with a mean absolute difference of $0.016\pm0.008\ μ$K (or $0.021\pm0.002\ μ$K) for the CMB-S4 experiment. To address the residual instrumental noise in the foreground-cleaned map, we employ a "half-split maps" approach, where the entire dataset is divided into two segments sharing the same sky signal but having uncorrelated noise. Using cross-correlation techniques between two recovered half-split maps, we effectively reduce instrumental noise effects at the power spectrum level. As a result, we achieve precise recovery of the CMB EE and lensing B-mode power spectra. Furthermore, we also extend our pipeline to full-sky simulated data with the performance of LiteBIRD experiment. As expected, various foregrounds are cleanly removed from the foreground contamination observational maps, and recovered EE and lensing B-mode power spectra exhibit excellent agreement with the true results. Finally, we discuss the dependency of our method on the foreground models.

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arXiv2023

CoLFI: Cosmological Likelihood-free Inference with Neural Density Estimators

In previous works, we proposed to estimate cosmological parameters with the artificial neural network (ANN) and the mixture density network (MDN). In this work, we propose an improved method called the mixture neural network (MNN) to achieve parameter estimation by combining ANN and MDN, which can overcome shortcomings of the ANN and MDN methods. Besides, we propose sampling parameters in a hyper-ellipsoid for the generation of the training set, which makes the parameter estimation more efficient. A high-fidelity posterior distribution can be obtained using $\mathcal{O}(10^2)$ forward simulation samples. In addition, we develop a code-named CoLFI for parameter estimation, which incorporates the advantages of MNN, ANN, and MDN, and is suitable for any parameter estimation of complicated models in a wide range of scientific fields. CoLFI provides a more efficient way for parameter estimation, especially for cases where the likelihood function is intractable or cosmological models are complex and resource-consuming. It can learn the conditional probability density $p(\boldsymbolθ|\boldsymbol{d})$ using samples generated by models, and the posterior distribution $p(\boldsymbolθ|\boldsymbol{d}_0)$ can be obtained for a given observational data $\boldsymbol{d}_0$. We tested the MNN using power spectra of the cosmic microwave background and Type Ia supernovae and obtained almost the same result as the Markov Chain Monte Carlo method. The numerical difference only exists at the level of $\mathcal{O}(10^{-2}σ)$. The method can be extended to higher-dimensional data.

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arXiv2023

Lensing reconstruction from the cosmic microwave background polarization with machine learning

The lensing effect of the cosmic microwave background (CMB) is a powerful tool for our study of the distribution of matter in the universe. Currently, the quadratic estimator (EQ) method, which is widely used to reconstruct lensing potential, has been known to be sub-optimal for the low-noise levels polarization data from next-generation CMB experiments. To improve the performance of the reconstruction, other methods, such as the maximum likelihood estimator and machine learning algorithms are developed. In this work, we present a deep convolutional neural network model named the Residual Dense Local Feature U-net (RDLFUnet) for reconstructing the CMB lensing convergence field. By simulating lensed CMB data with different noise levels to train and test network models, we find that for noise levels less than $5μ$K-arcmin, RDLFUnet can recover the input gravitational potential with a higher signal-to-noise ratio than the previous deep learning and the traditional QE methods at almost the entire observation scales.

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arXiv2023

Delensing of Cosmic Microwave Background Polarization with machine learning

Primordial B-mode detection is one of the main goals of next-generation cosmic microwave background (CMB) experiments. Primordial B-modes are a unique signature of primordial gravitational waves (PGWs). However, the gravitational interaction of CMB photons with large-scale structures will distort the primordial E modes, adding a lensing B-mode component to the primordial B-mode signal. Removing the lensing effect (`delensing') from observed CMB polarization maps will be necessary to improve the constraint of PGWs and obtain a primordial E-mode signal. Here, we introduce a deep convolutional neural network model named multi-input multi-output U-net (MIMO-UNet) to perform CMB delensing. The networks are trained on simulated CMB maps with size $20^{\circ} \times 20^{\circ}$. We first use MIMO-UNet to reconstruct the unlensing CMB polarization ($Q$ and $U$) maps from observed CMB maps. The recovered E-mode power spectrum exhibits excellent agreement with the primordial EE power spectrum. The recovery of the primordial B-mode power spectrum for noise levels of 0, 1, and 2 $μ$K-arcmin is greater than 98\% at the angular scale of $\ell<150$. We additionally reconstruct the lensing B map from observed CMB maps. The recovery of the lensing B-mode power spectrum is greater than roughly 99\% at the scales of $\ell>200$. We delens observed B-mode power spectrum by subtracting reconstructed lensing B-mode spectrum. The recovery of tensor B-mode power spectrum for noise levels of 0, 1, 2 $μ$K-arcmin is greater than 98 \% at the angular scales of $\ell<120$. Even at $\ell=160$, the recovery of tensor B-mode power spectrum is still around 71 \%.

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