arXiv subjects

Zunli Yuan

Publications and source records attributed to Zunli Yuan.

At least 19 recordsLinked to original sources

arXiv2026

The Steep-spectrum Radio-loud AGN Luminosity Function and Its Implications for Black Hole Growth and Star Formation

We study the cosmic evolution of radio-loud active galactic nuclei (AGNs) using a beaming-minimized sample of 4{,}555 steep-spectrum sources over $0<z\lesssim4$, compiled from the XXL survey, VLA-COSMOS, and other wide-field data sets. We model the rest-frame 1.4 GHz radio luminosity function (RLF) with a luminosity-and-density evolution (LADE; DE+LE) framework coupled to a flexible local LF family. Among the tested parameterizations, Model~C is statistically preferred and provides a globally consistent description of the binned RLFs while remaining compatible with local RLF measurements and Euclidean-normalized source counts. In the fiducial solution, the LE term rises toward cosmic noon ($z\sim2$--3) and then flattens or mildly declines, whereas the DE term decreases monotonically with redshift. This combined evolution naturally reproduces the observed luminosity-dependent turnover redshift $z_{\rm peak}(L)$ (often termed ``cosmic downsizing'') without imposing \emph{a priori} distinct evolutionary laws for low- and high-power sources. We further show that the same LADE functional family calibrated for star-forming galaxies also describes radio-loud AGNs when fitted independently, enabling a unified two-component (SFG+AGN) model consistent with both the local RLF and source-count statistics. Finally, converting the AGN RLF to a kinetic luminosity function yields a radio-mode black hole accretion rate density (BHAD) whose redshift dependence closely tracks the radio-based cosmic star formation rate density (after a conventional rescaling), with both histories peaking near $z\sim2$.

astro-ph.GA↗

arXiv2026

Tracing cosmic star formation history through radio continuum spectral energy distribution and non-thermal emission

As a tracer of massive star formation unaffected by dust, the radio continuum emission provides a unique window into the formation of the first stars and galaxies in the Universe. Recent observations show that the integrated rest-frame mid-radio (~1-10 GHz) luminosity of galaxies serves as one of the most robust tracers of the star formation rate (SFR). These studies further demonstrate that the synchrotron spectral index and the shape of the radio spectral energy distribution (SED) evolves with redshift as a consequence of the cosmic evolution of star formation activity. These findings underscore the importance of deep multi-band radio continuum observations in calibrating the SFR of early galaxies and understanding the astrophysical processes governing their assembly and evolution over cosmic time. This chapter presents recent progress in radio SFR calibrations for star-forming galaxies (SFGs) and reviews radio-continuum studies of the cosmic star formation history (SFH). We highlight the transformative potential of SKA AA4, whose broad frequency coverage and high sensitivity will enable well-constrained radio SEDs for SFGs across a wide redshift range.

astro-ph.GA↗

arXiv2026

Improving Radio Source Count Estimation Using Kernel Density Estimation

Radio source counts provide a fundamental census of cosmic radio emission, yet their estimation is usually based on coarse histograms that suffer from bin-choice bias, boundary effects, and survey incompleteness. We apply and rigorously evaluate kernel density estimation (KDE) as a anonparametric alternative to the conventional binned method for estimating differential radio source counts. Using simulated flux-limited samples derived from an input luminosity function model, we compare the performance of standard KDE, adaptive KDE, and traditional binning methods. Our results show that KDE-based approaches yield more accurate and stable estimates, particularly in the high-flux regime where data are sparse and conventional methods struggle. We also apply the adaptive KDE method to real observational data from the LOFAR Two-Metre Sky Survey Deep Fields. Our analysis robustly confirms the pronounced ``drop and bump" feature at sub-mJy flux densities, but also reveals that a secondary, modest bump seen in the binned data at ~ $\sim 10$ mJy is likely a binning artifact. We also demonstrate the flexibility of KDE in addressing observational incompleteness through weighted estimation, which applies weights continuously at the level of individual sources rather than averaging them in discrete bins. These strengths make KDE a powerful tool for source-count analyses in current and future radio surveys and, more broadly, in analogous studies at other wavelengths. All computations in this study are implemented with \texttt{AstroKDE}, a Python package we have developed for astronomical applications.

astro-ph.IM↗

arXiv2026

Revisiting the 150 MHz Radio Luminosity Function of Star-Forming Galaxies with LOFAR Deep Fields through a Refined Statistical Framework

We present a comprehensive analysis of the 150~MHz radio luminosity function (LF) of star-forming galaxies (SFGs) using deep observations from the LOFAR Two-metre Sky Survey in the ELAIS-N1, Boötes, and Lockman Hole fields. Our sample comprises $\sim$56,000 SFGs over $0 < z < 5.7$. We first analyze the deepest field (ELAIS-N1), then jointly model all three fields while accounting for their distinct flux limits and selection functions. Using adaptive kernel density estimation (KDE), we reconstruct the LF continuously across redshift and luminosity without binning or parametric assumptions. The KDE results reveal clear signatures of joint luminosity and density evolution (LADE). Motivated by this, we construct and fit three parametric models--pure luminosity evolution (PLE) and two LADE variants--using a full maximum-likelihood method that includes completeness corrections and constraints from the local radio LF and Euclidean-normalized source counts (SCs). Model selection using Akaike and Bayesian Information Criteria strongly favors LADE over PLE. For ELAIS-N1, the more flexible LADE model (Model C) provides the best fit, while for the combined fields, the simpler Model B balances fit quality and complexity more effectively. Both LADE models reproduce the observed LFs and SCs across luminosity and flux density ranges, whereas PLE underperforms. We also identify a mild excess at the bright end of the LF, likely due to residual AGN contamination. This study demonstrates that combining KDE with parametric modeling offers a robust framework for quantifying the evolving radio LF of SFGs, paving the way for future work with next-generation surveys like the SKA.

astro-ph.GA↗

arXiv2026

A flexible method for estimating luminosity functions via Kernel Density Estimation -- III. Extending to Multiple Flux-Limited Samples

As the third paper in a series regarding the estimation of luminosity functions (LFs) via kernel density estimation (KDE), we present a further generalization of our framework by extending its applicability to multiple flux-limited samples. While our previous works addressed single flux-limited datasets, many practical applications involve surveys that cover disjoint fields of view with different flux limits. We introduce a piecewise estimation framework that partitions the luminosity-redshift plane into disjoint regions according to the staggered flux limits of the sub-samples. Within each region, we integrate data from all surveys capable of detecting sources into a combined sample and apply the transformation-reflection KDE method using the corresponding local flux threshold as the truncation boundary. This strategy allows for the full utilization of all available sources while maintaining rigorous statistical consistency. The robustness of this approach is validated through Monte Carlo simulations. Furthermore, application to SDSS DR7 and 2SLAQ quasar data shows overall agreement with parametric models, while a small residual discontinuity near a survey-transition boundary is discussed as a diagnostic of independent piecewise estimation and possible inter-survey systematics. The KDE calculations in each piecewise region are performed using our previously developed public Python package \texttt{kdeLF}.

astro-ph.IM↗

arXiv2026

Shape-Preserving Evolution of the Global Ultraviolet Quasar Luminosity Function to $z\simeq7.5$

We present a global unbinned-likelihood analysis of the rest-frame ultraviolet (UV) quasar luminosity function (QLF) over $0.1\le z\le7.5$, using a final analysis sample of 70,960 Type~1 quasars selected from the homogenized compilation of Kulkarni et al. (2019). We test a shape-preserving luminosity and density evolution (LADE) framework, in which the local luminosity function (LF) shape is fixed and the redshift evolution is described by separate density- and luminosity-evolution factors. For a double-power-law (DPL) local LF, we search 81 LADE models built from nine density-evolution and nine luminosity-evolution functions, and compare them with two flexible double-power-law (FDPL) reference models. The fiducial shape-preserving LADE model gives the lowest Akaike information criterion (AIC) and Bayesian information criterion (BIC) values among the main models considered, and 12 DPL-based LADE models have lower AIC and BIC values than both FDPL reference models. Repeating the model-grid analysis with a modified-Schechter local LF gives the same preferred evolutionary structure, indicating that the result is not driven only by the assumed local LF shape. The fitted evolution functions further show that the luminosity-evolution component is the more stable part of the LADE decomposition: it rises rapidly to $z\simeq2$--3 and then flattens or slowly declines. The density-evolution component is more model dependent, but the preferred LADE models consistently require a declining effective normalization toward high redshift. Taken together, we conclude that a shape-preserving framework offers a statistically efficient and compact empirical description of the global UV QLF.

astro-ph.CO↗

arXiv2026

Joint Multi-Period Fermi-LAT and LHAASO Constraints on Axion-Like Particles from Mrk 421 Using Profile Likelihood with Gaussian Copula Correlation

We propose a joint multi-epoch profile-likelihood analysis of axion-like particles (ALPs) using five sets of simultaneous Fermi-LAT and LHAASO observations of the TeV blazar Mrk 421. Photon-ALP oscillations are calculated self-consistently together with EBL absorption for two representative jet emission models: a two-zone hybrid model and a single-zone hadronic model. To account for weak correlations among different observational epochs, we introduce a Gaussian copula with a conservative correlation coefficient $ρ= 0.03$ and perform a global optimization of nuisance parameters under the no-ALP hypothesis before profiling the ALP parameters. In the low-mass regime relevant to CAST ($m_a \lesssim 1$ neV), we obtain a 95\% CL upper limit of $g_{aγ} = 7.46 \times 10^{-13}\,\mathrm{GeV}^{-1}$. Over the full mass range $0.1$--$500$ neV, the most conservative 95\% CL upper limits are $g_{aγ} < 6.50 \times 10^{-12}\,\mathrm{GeV}^{-1}$ (two-zone) and $g_{aγ} < 7.34 \times 10^{-12}\,\mathrm{GeV}^{-1}$ (single-zone). These constraints benefit from the broadband VHE coverage and long-term monitoring provided by LHAASO. The analysis framework developed here offers a statistically consistent approach for future ALP searches with multi-messenger gamma-ray data.

astro-ph.HE↗

arXiv2026

Revisiting Very High Energy Gamma-Ray Absorption in Cosmic Propagation under the Combined Effects of Axion-Like Particles and Lorentz Invariance Violation

Very-high-energy (VHE; $E \gtrsim 100$ GeV) gamma rays are expected to experience strong attenuation during cosmological propagation due to electron-positron pair production on the extragalactic background light (EBL). Recent observations of GRB 221009A (z = 0.151), including photons up to $\sim 18$ detected by LHAASO and a $\sim 300\ \mathrm{TeV}$ event reported by Carpet-3, suggest a higher-than-expected transparency of the Universe at extreme energies. These observations cannot be explained by standard EBL absorption alone; moreover, neither Lorentz invariance violation (LIV) nor photon-axion-like particle (ALP) oscillations, when considered in isolation, appear sufficient to account for the survival of such photons over cosmological distances. In this work, we propose a joint propagation scenario that incorporates photon-ALP mixing in astrophysical magnetic fields together with subluminal quadratic LIV corrections to the $γγ$ pair-production threshold. Applying this framework to the broadband gamma-ray spectrum of GRB 221009A, we show that ALPs with coupling ($g_{aγ} = 1.685 \times 10^{-10}\mathrm{GeV}^{-1}$ ) and mass ($m_a = 9.545 \times 10^{-8}\mathrm{eV}$), combined with a quadratic LIV energy scale ($E_{\rm LIV,2} = 1.30 \times 10^{-7} E_{\rm Pl}$) adopted from the literature, can significantly enhance the photon survival probability in the energy range (10\text{-}300) TeV. The resulting enhancement exceeds that obtained from either ALP mixing or LIV effects alone. These results indicate that a combined ALP-LIV scenario may provide a viable interpretation of the extreme-energy gamma-ray observations of GRB 221009A and highlight the potential of VHE gamma-ray measurements as probes of physics beyond the Standard Model.

astro-ph.HE↗

arXiv2024

Constraints on cosmic star formation history via a new modeling of the radio luminosity function of star-forming galaxies

Radio wavelengths offer a unique possibility to trace the total star-formation rate (SFR) in galaxies, both obscured and unobscured. To probe the dust-unbiased star-formation history, an accurate measurement of the radio luminosity function (LF) for star-forming galaxies (SFGs) is crucial. We make use of an SFG sample (5900 sources) from the Very Large Array (VLA) COSMOS 3 GHz data to perform a new modeling of the radio LF. By integrating the analytical LF, we aim to calculate the history of the cosmic SFR density (SFRD) from $z\sim5$ onwards. For the first time, we use both models of the pure luminosity evolution (PLE) and joint luminosity+density evolution (LADE) to fit the LFs directly to the radio data using a full maximum-likelihood analysis, considering the sample completeness correction. We also incorporate updated observations of local radio LFs and radio source counts into the fitting process to obtain additional constraints. We find that the PLE model cannot be used to describe the evolution of the radio LF at high redshift ($z>2$). By construct, our LADE models can successfully fit a large amount of data on radio LFs and source counts of SFGs from recent observations. We therefore conclude that density evolution is genuinely indispensable in modeling the evolution of SFG radio LFs. Our SFRD curve shows a good fit to the SFRD points derived by previous radio estimates. In view of the fact that our radio LFs are not biased, as opposed those of previous studies performed by fitting the $1/V_{\rm max}$ LF points, our SFRD results should be an improvement on these previous estimates. Below $z\sim1.5$, our SFRD matches a published multiwavelength compilation, while our SFRD turns over at a slightly higher redshift ($2<z<2.5$) and falls more rapidly out to high redshift.

astro-ph.GA↗

arXiv2022

Redshift-evolutionary X-ray and UV luminosity relation of quasars from Gaussian copula

We construct a three-dimensional and redshift-evolutionary X-ray and ultraviolet ($L_X-L_{UV}$) luminosity relation for quasars from the powerful statistic tool called copula, and find that the constructed $L_X-L_{UV}$ relation from copula is more viable than the standard one and the observations favor the redshift-evolutionary relation more than $3σ$. The Akaike and Bayes information criterions indicate that the quasar data support strongly the three-dimensional $L_X-L_{UV}$ relation. Our results show that the quasars can be regarded as a reliable indicator of the cosmic distance if the $L_X-L_{UV}$ relation from copula is used to calibrate quasar data.

astro-ph.CO↗

arXiv2022

Gamma ray burst constraints on cosmological models from the improved Amati correlation

An improved Amati correlation was constructed in (ApJ 931 (2022) 50) by us recently. In this paper, we further study constraints on the $Λ$CDM and $w$CDM models from the gamma ray bursts (GRBs) standardized with the standard and improved Amati correlations, respectively. By using the Pantheon type Ia supernova sample to calibrate the latest A220 GRB data set, the GRB Hubble diagram is obtained model-independently. We find that at the high redshift region ($z>1.4$) the GRB distance modulus from the improved Amati correlation is larger apparently than that from the standard Amati one. The GRB data from the standard Amati correlation only give a lower bound limit on the present matter density parameter $Ω_{\mathrm{m0}}$, while the GRBs from the improved Amati correlation constrain the $Ω_{\mathrm{m0}}$ with the $68\%$ confidence level to be $0.308^{+0.066}_{-0.230}$ and $0.307^{+0.057}_{-0.290}$ in the $Λ$CDM and $w$CDM models, respectively, which are consistent very well with those given by other current popular observational data including BAO, CMB and so on. Once the $H(z)$ data are added in our analysis, the constraint on the Hubble constant $H_0$ can be achieved. We find that two different correlations provide slightly different $H_0$ results but the marginalized mean values seem to be close to that from the Planck 2018 CMB observations.

astro-ph.CO↗

arXiv2022

The improved Amati correlations from Gaussian copula

In this paper, we obtain two improved Amati correlations of the Gamma-Ray burst (GRB) data via a powerful statistical tool called copula. After calibrating, with the low-redshift GRB data, the improved Amati correlations based on a fiducial $Λ$CDM model with $Ω_\mathrm{m0}=0.3$ and $H_0=70~\mathrm{km~s^{-1}Mpc^{-1}}$, and extrapolating the results to the high-redshift GRB data, we obtain the Hubble diagram of GRB data points. Applying these GRB data to constrain the $Λ$CDM model, we find that the improved Amati correlation from copula can give a result well consistent with $Ω_\mathrm{m0}=0.3$, while the standard Amati and extended Amati correlations do not. This results suggest that when the improved Amati correlation from copula is used in the low-redshift calibration method, the GRB data can be regarded as a viable cosmological explorer. However, the Bayesian information criterion indicates that the standard Amati correlation remains to be favored mildly since it has the least model parameters. Furthermore, once the simultaneous fitting method rather than the low-redshift calibration one is used, there is no apparent evidence that the improved Amati correlation is better than the standard one. Thus, more works need to be done in the future in order to compare different Amati correlations.

astro-ph.CO↗

arXiv2022

A flexible method for estimating luminosity functions via Kernel Density Estimation -- II. Generalization and Python implementation

We propose a generalization of our previous KDE (kernel density estimation) method for estimating luminosity functions (LFs). This new upgrade further extend the application scope of our KDE method, making it a very flexible approach which is suitable to deal with most of bivariate LF calculation problems. From the mathematical point of view, usually the LF calculation can be abstracted as a density estimation problem in the bounded domain of $\{Z_1 f_{\mathrm{lim}}(z) \}$. We use the transformation-reflection KDE method ($\hatϕ$) to solve the problem, and introduce an approximate method ($\hatϕ_{\mathrm{1}}$) based on one-dimensional KDE to deal with the small sample size case. In practical applications, the different versions of LF estimators can be flexibly chosen according to the Kolmogorov-Smirnov test criterion. Based on 200 simulated samples, we find that for both cases of dividing or not dividing redshift bins, especially for the latter, our method performs significantly better than the traditional binning method $\hatϕ_{\mathrm{bin}}$. Moreover, with the increase of sample size $n$, our LF estimator converges to the true LF remarkably faster than $\hatϕ_{\mathrm{bin}}$. To implement our method, we have developed a public, open-source Python Toolkit, called \texttt{kdeLF}. With the support of \texttt{kdeLF}, our KDE method is expected to be a competitive alternative to existing nonparametric estimators, due to its high accuracy and excellent stability. \texttt{kdeLF} is available at \url{http://github.com/yuanzunli/kdeLF} with extensive documentation available at \url{http://kdelf.readthedocs.org/en/latest~}.

astro-ph.IM↗

arXiv2020

A flexible method for estimating luminosity functions via Kernel Density Estimation

We propose a flexible method for estimating luminosity functions (LFs) based on kernel density estimation (KDE), the most popular nonparametric density estimation approach developed in modern statistics, to overcome issues surrounding binning of LFs. One challenge in applying KDE to LFs is how to treat the boundary bias problem, since astronomical surveys usually obtain truncated samples predominantly due to the flux-density limits of surveys. We use two solutions, the transformation KDE method ($\hatϕ_{\mathrm{t}}$), and the transformation-reflection KDE method ($\hatϕ_{\mathrm{tr}}$) to reduce the boundary bias. We develop a new likelihood cross-validation criterion for selecting optimal bandwidths, based on which, the posterior probability distribution of bandwidth and transformation parameters for $\hatϕ_{\mathrm{t}}$ and $\hatϕ_{\mathrm{tr}}$ are derived within a Markov chain Monte Carlo (MCMC) sampling procedure. The simulation result shows that $\hatϕ_{\mathrm{t}}$ and $\hatϕ_{\mathrm{tr}}$ perform better than the traditional binned method, especially in the sparse data regime around the flux-limit of a survey or at the bright-end of the LF. To further improve the performance of our KDE methods, we develop the transformation-reflection adaptive KDE approach ($\hatϕ_{\mathrm{tra}}$). Monte Carlo simulations suggest that it has a good stability and reliability in performance, and is around an order of magnitude more accurate than using the binned method. By applying our adaptive KDE method to a quasar sample, we find that it achieves estimates comparable to the rigorous determination by a previous work, while making far fewer assumptions about the LF. The KDE method we develop has the advantages of both parametric and non-parametric methods.

stat.ME↗

arXiv2018

A mixture evolution scenario of the AGN radio luminosity function. II. Do low- and high-power radio-loud AGNs evolve differently?

Following previous work, we further confirm that the cosmic evolution of steep-spectrum radio-loud AGNs (active galactic nuclei) can be reproduced by a simple combination of density evolution (DE) and luminosity evolution (LE). This mixture evolution scenario can naturally explain the luminosity-dependent evolution of radio-loud AGNs. Our models successfully fitted a large amount of data on radio luminosity functions (RLFs) of steep-spectrum sources and multi-frequency source counts. The modeling indicates that the DE slowly increase as $(1+z)^{0.3 \thicksim 1.3}$ out to $z \thicksim 0.8$, and then rapidly decreases as $(1+z)^{-6.8 \thicksim -5.7}$, while the LE rapidly increase as $(1+z)^{4.8}$ out to a higher redshift (at least $z>3.5$). We find a high-redshift decline (i.e. redshift cutoff) in the number density of steep-spectrum radio sources, but we cannot conclude whether such decline is sharp or shallow. We believe that whether a redshift cutoff occurs or not depends mainly on DE, while its steepness is decided by LE, which, however, cannot be well constrained due to the lack of high-redshift samples. Most intriguingly, according to our mixture evolution scenario, there appears to be no need for different evolution for the low- and high-power radio-loud AGNs. Both types of sources experience the same combined evolution of DE and LE.

astro-ph.HE↗

arXiv2018

Determining the core radio luminosity function of radio AGNs via copula

The radio luminosity functions (RLFs) of active galactic nuclei (AGNs) are traditionally measured based on total emission, which doesn't reflect the current activity of the central black hole. The increasing interest in compact radio cores of AGNs requires determination of the RLF based on core emission (i.e., core RLF). In this work we have established a large sample (totaling 1207) of radio-loud AGNs, mainly consisting of radio galaxies (RGs) and steep-spectrum radio quasars (SSRQs). Based on the sample, we explore the relationship between core luminosity ($L_c$) and total luminosity ($L_t$) via a powerful statistical tool called "Copula". The conditional probability distribution $p(\log L_{c} \mid \log L_{t})$ is obtained. We derive the core RLF as a convolution of $p(\log L_{c} \mid \log L_{t})$ with the total RLF which was determined by previous work. We relate the separate RG and SSRQ core RLFs via a relativistic beaming model and find that SSRQs have an average Lorentz factor of $γ=9.84_{-2.50}^{+3.61}$, and that most are seen within $8^{\circ} \lesssim θ\lesssim 45^{\circ}$ of the jet axis. Compared with the total RLF which is mainly contributed by extended emission, the core RLF shows a very weak luminosity-dependent evolution, with the number density peaking around $z\thicksim 0.8$ for all luminosities. Differences between core and total RLFs can be explained in a framework involving a combination of density and luminosity evolutions where the cores have significantly weaker luminosity evolution than the extended emission.

astro-ph.GA↗

arXiv2017

Using the Markov Chain Monte Carlo method to study the physical properties GeV-TeV BL Lac objects

We fit the spectral energy distributions (SEDs) of 46 GeV - TeV BL Lac objects in the frame of leptonic one-zone synchrotron self-Compton (SSC) model and investigate the physical properties of these objects. We use the Markov Chain Monte Carlo (MCMC) method to obtain the basic parameters, such as magnetic field (B), the break energy of the relativistic electron distribution ($γ'_{\rm{b}}$) and the electron energy spectral index. Based on the modeling results, we support the following scenarios on GeV-TeV BL Lac objects: (1) Some sources have large Doppler factors, implying other radiation mechanism should be considered. (2) Comparing with FSRQs, GeV-TeV BL Lac objects have weaker magnetic field and larger Doppler factor, which cause the ineffective cooling and shift the SEDs to higher bands. Their jet powers are around $4.0\times 10^{45}~\rm{ erg\cdot s}^{-1}$, comparing with radiation power, $5.0\times 10^{42}~\rm{ erg\cdot s}^{-1}$, indicating that only a small fraction of jet power is transformed into the emission power. (3) For some BL Lacs with large Doppler factors, their jet components could have two substructures, e.g., the fast core and the slow sheath. For most GeV-TeV BL Lacs, Kelvin-Helmholtz instabilities are suppressed by their higher magnetic fields, leading few micro-variability or intro-day variability in the optical bands. (4) Combined with a sample of FSRQs, an anti-correlation between the peak luminosity $L_{\rm {pk}}$ and the peak frequency $ν_{\rm {pk}}$ is obtained, favoring the blazar sequence scenario. In addition, an anti-correlation between the jet power $P_{\rm {jet}}$ and the break Lorentz factor $γ_{\rm {b}}$ also supports the blazar sequence.

astro-ph.HE↗

arXiv2017

Constraining the redshifts of TeV BL Lac objects

We present a model-dependent method to estimate the redshifts of three TeV BL Lac objects (BL Lacs) through fltting their (quasi-) simultaneous multi-waveband spectral energy distributions (SEDs) by one-zone leptonic synchrotron self-Compton (SSC) model. Considering the impact of electron energy distributions (EEDs) on the results, we use three types of EEDs, such as the power-law EED with exponential cut-ofi (PLC), the log-parabola (PLLP) and the broken power-law (BPL) EEDs, to flt the SEDs. We also use a parameter fi to describe the uncertainties of the extragalactic background light (EBL) models, as in (Abdo et al. 2010b). We then use Markov Chain Monte Carlo (MCMC) method to explore multi-dimensional parameter space and obtain the uncertainties of the model parameters based on the observational data. We apply our method to obtain the redshifts of three TeV BL Lac objects in the marginalized 68% confldence, and flnd that the PLC EED does not flt the SEDs. For 3C66A, the redshift is 0.14 - 0.31 and 0.16 - 0.32 in the BPL and PLLP EEDs; for PKS1424+240, the redshift is 0.55 - 0.68 and 0.55 - 0.67 in the BPL and PLLP EEDs; for PG1553+113, the redshift is 0.22 - 0.48 and 0.22 - 0.39 in the BPL and PLLP EEDs. We also estimate the redshift of PKS1424+240 in the high stage to be 0.46 - 0.67 in the PLLP EED, roughly consistent with that in the low stage.

astro-ph.HE↗