arXiv subjects

Xiangkun Liu

Publications and source records attributed to Xiangkun Liu.

At least 19 recordsLinked to original sources

arXiv2026

The Agentic Observer Hackathon: Autonomous Decision-making in Simulated Astronomical Surveys

Astronomical surveys must allocate observing time while responding to changing conditions and competing scientific priorities. We present the Agentic Observer Hackathon (AOH), a community challenge for studying autonomous survey operations. Within a synthetic multi-object survey with prescribed exposure responses and assignment constraints, decision systems select pointings, assign targets and choose exposure configurations while responding to observing requests, instrumental faults and data loss. We evaluate systems submitted by 102 teams on campaigns reserved for evaluation and analyse target completion in a reference campaign. The leading systems combine numerical exposure planning with language models for interpreting operational information and supporting longer-term decisions. Exploratory tests show improved fault reporting with model access in several systems, while revised scientific priorities produce substantial changes in ranking. The reference campaign traces target completion and diagnoses errors in the baseline's inference of prescribed quality bands. The released simulator and reference strategies provide common survey conditions for comparing exposure allocation, target selection and recovery from interruptions.

astro-ph.IM↗

arXiv2026

SN 2025dys: A luminous supernova explosion from a partially stripped helium star

We present well-sampled multi-band imaging and low- to medium-resolution spectroscopic observations of the early-discovered Type~II supernova (SN)~2025dys. The photometric evolution exhibits a rise time of $\sim$\,18 days ($r$ band) and a rapid decline transitioning to the radioactive decay tail at $\sim$\,70 days. SN~2025dys is among the high-luminosity events with a peak pseudo-bolometric luminosity of $L_{\rm opt}=8.57\times10^{42}\,\mathrm{erg\,s^{-1}}$. Early X-ray emission was identified, displaying a luminosity of $7.55\times10^{40}\,\mathrm{erg\,s^{-1}}$. The observational investigation points to a dense, hydrogen-rich circumstellar medium with a mass of $M_{\rm CSM}\approx0.5$--$0.7\,\mathrm{M_\odot}$, suggesting a partially stripped, hydrogen-poor inner ejecta with a relatively low mass of $M_{\rm ej}\approx2.5\,\mathrm{M_\odot}$. The explosion possessed a kinetic energy of $E_{\rm k}\approx0.3\text{--}0.5$~foe and yielded a synthesized nickel mass of $M_{\rm Ni}\approx0.042_{-0.011}^{+0.012}\,\mathrm{M_\odot}$. These physical parameters indicate an extreme mass-loss episode before the explosion, characterized by a high mass-loss rate of $\dot{M} \gtrsim 0.01\,\mathrm{M_\odot\,yr^{-1}}$. SN~2025dys represents a stripped progenitor, intrinsically resembling a Type~IIb or even a Type~Ib SN, that exploded within a dense, hydrogen-rich circumstellar environment.

astro-ph.HE↗

arXiv2026

Simultaneous Real-Color Time-Domain Survey of the Coma Field with the Multi-channel Photometric Survey Telescope (Mephisto)

We present the results from a high-cadence, multi-channel time-domain survey targeting the Coma galaxy cluster, utilizing the newly commissioned 1.6-m Multi-channel Photometric Survey Telescope (Mephisto). Mephisto's pioneering mosaic focal-plane design enables strictly simultaneous imaging in three distinct optical bands ($u/v, g/r, i/z$). This unique capability allows for the extraction of true, instantaneous spectral energy distributions (SEDs) of fast-evolving transients. Our survey continuously monitors a 20-square-degree footprint centered on the Coma cluster. By strategically targeting this massive local structure, we exploit cosmic anisotropy to significantly amplify the volumetric discovery rate of stellar explosions. During the initial six-month pilot operation, we successfully detected and tracked nine new supernovae (SNe). Remarkably, our high-cadence, multi-band strategy allowed us to capture ultra-early rising light curves and pre-discovery signals (e.g., SN 2026dyr) hours to days prior to formal discovery alerts. These instantaneous early-phase colors allow us to reconstruct intrinsic SEDs without temporal interpolation, providing reliable constraints on the physical parameters of early supernova (SN) evolution. Furthermore, our observed supernova yield shows good agreement with theoretical forecasts based on local stellar mass and star-formation rate overdensities. These pilot results validate the efficiency of the Mephisto Coma survey, demonstrating its objective capability to systematically explore the local transient population and provide rapid optical localizations for future multi-messenger events.

astro-ph.HE↗

arXiv2026

Atmospheric Parameter Determination of RR Lyrae Stars With Mephisto Multi-band Photometry

Phase-resolved atmospheric parameters of pulsating stars, including RR Lyrae stars and Cepheids, are essential for studying stellar structure and pulsation physics and for establishing accurate stellar benchmarks for Galactic structure and assembly history. We present a photometric framework that simultaneously infers Teff, log g, and [Fe/H] from Mephisto multi-epoch, multi-band (u, v, g, r, i, z) photometry. A supervised neural-network forward model is built by training on synthetic colors derived from theoretical stellar atmosphere models. It reproduces the synthetic colors to a precision of 0.01 per cent. The model is applied to observed multi-epoch colors to derive phase-dependent atmospheric parameters. We apply the method to six RRab and four RRc stars in three globular clusters observed with Mephisto. We find typical per-phase uncertainties of about 62 K in effective temperature (Teff), 0.10 dex in surface gravity (log g), and 0.16 dex in metallicity ([Fe/H]). Repeated measurements across the pulsation cycle reduce the typical formal uncertainty of the final [Fe/H] estimates to roughly 0.03 dex. A sensitivity analysis indicates that the five Mephisto colors provide distinct and complementary constraints on the three stellar atmospheric parameters: (u-v) provides robust constraints on log g, (v-g) is informative for estimating [Fe/H], and (u-v), (v-g), (g-r), (r-i), and (i-z) all provide information on Teff. The results demonstrate that Mephisto multi-epoch, multi-band photometry can deliver precise, cost-effective phase-resolved atmospheric parameters for RR Lyrae stars, offering an efficient way to construct large samples of stellar benchmarks for studies of stellar pulsation physics and Galactic structure and evolution.

astro-ph.SR↗

arXiv2026

A Shell Collision Scenario for EP251222b/GRB 251222A: Connecting Prompt Emission and Double-Peaked Afterglow

GRB 251222A is a long-duration gamma-ray burst at a spectroscopic redshift of $z=3.171$. It was detected by the EP as the soft X-ray transient EP251222b before and during the gamma-ray emission. We performed a multi-wavelength (EP/WXT, fermi/GBM, swift/XRT and optical/NIR) investigation of this event. Its prompt soft X-ray and gamma-ray features, together with two afterglow rebrightenings, allow us to examine a possible connection between the prompt ejecta and the afterglow. We consider a multiple-shell scenario in which a faster, narrower shell catches up with a slower, wider shell. We assume that the selected pulse groups trace different ejecta and that the late X-ray flare marks their collision. Under these assumptions, we represent the post-collision outflow by an effective merged core and an outer annular component. A joint fit to the multi-band afterglow gives a Lorentz factor of $Γ_c\approx293$ for the merged core and $Γ_{1,w}\approx228$ for the outer component. The sample-wise collision mapping gives an energy release of $E_{\rm diss}=(1.6^{+2.1}_{-0.9})\times10^{52}$ erg and a dynamical dissipation efficiency of $0.37^{+0.07}_{-0.14}\%$. EP251222b can therefore be used to test a prompt--afterglow connection through shell collisions, although it does not uniquely establish this interpretation.

astro-ph.HE↗

arXiv2026

Mapping Dark Matter in Merging Clusters with Weak Lensing for CSST: Morphology and Filament Recovery

Weak gravitational lensing offers a direct probe of the projected dark matter distribution in cluster-scale structures and the surrounding cosmic web. We investigate the capability of the China Space Station Telescope (CSST) to recover non-spherical dark matter morphologies using mock reduced-shear maps generated from cosmological N-body simulations. We apply the deep-learning-based MIU2Net mass-reconstruction framework to CSST-like wide- and deep-field source densities, focusing on two representative systems: filament-connected halo pairs and elongated halos. MIU2Net accurately recovers halo centers, elongated mass distributions, and the filamentary structures connecting halo pairs in both survey configurations. Halo centers are localized within about 0.1 h^(-1) Mpc, and elongated-halo shapes and orientations are reliably recovered. Filament paths are stable against shape noise, with mean normalized instability parameters I_F, defined as the mean deviation from the true filament locus in units of R_200, decreasing from 0.041 to 0.037 in the low-redshift bin and from 0.051 to 0.045 in the high-redshift bin from the wide to deep field. The recovered filament shapes remain consistent with the true maps at the percent level. Compared with conventional Kaiser-Squires reconstruction, MIU2Net better preserves low-contrast anisotropic structures while suppressing small-scale noise. These results demonstrate that CSST weak-lensing data, combined with machine-learning-based mass reconstruction, will enable statistical studies of cluster assembly, anisotropic matter accretion, and the connection between massive halos and the filamentary cosmic web.

astro-ph.CO↗

arXiv2026

Early Near-Infrared Excess and Rapid Disk-Corona Evolution in the Tidal Disruption Event 2024aepd

We present multi-wavelength observations of the tidal disruption event (TDE) 2024aepd, spanning primarily the first $\sim$300 days after discovery. A prominent near-infrared (NIR) excess is detected as early as $\sim$40 days. Its nearly flat power-law spectrum strongly deviates from the Rayleigh-Jeans tail of the UV-optical blackbody. Although a conventional dust-echo origin cannot be completely ruled out, free-free emission from a reprocessing photospheric envelope provides a more plausible explanation. The spectral break between the UV-optical and NIR components shifts to higher frequencies, while the inferred density-profile index remains nearly constant, suggesting evolving reprocessing conditions within a broadly unchanged density structure. In addition, the X-ray spectrum is initially dominated by a thermal disk component accompanied by a hard excess. From $\sim$178 days onward, the spectrum becomes power-law dominated and subsequently hardens, indicating the rapid emergence and strengthening of a hot corona. These results provide evidence for frequency-dependent reprocessing at early times and for the rapid development of a disk-corona system, placing new constraints on the structure and evolution of the reprocessing layer around supermassive black holes.

astro-ph.HE↗

arXiv2026

Rapid Variability and Broadband Spectral Modeling in the Flaring Activity of BL Lacertae

We report a multi-wavelength study of two flaring episodes of the blazar BL Lacertae during MJD 60500-60800 (9 July 2024 - 5 May 2025). The source reached a daily-averaged $γ$-ray flux of $(1.03 \pm 0.05) \times 10^{-5} \, \mathrm{ph \, cm^{-2} \, s^{-1}}$ ($E > 100$ MeV) on MJD 60588 (5 October 2024). Using orbit-binned data from the Large Area Telescope (LAT) onboard the \textit{Fermi Gamma-ray Space Telescope}, we identify a minimum flux halving timescale of $τ= 1.33 \pm 0.29$ hr. This constrains the upper limit on the $γ$-ray emitting region size to $R \le 2.0 \times 10^{15}$ cm, as well as its distance from the central supermassive black hole to $R_\mathrm{H} \le 5.9 \times 10^{16}$ cm, assuming a Doppler factor of $δ= 14.8$ derived from the spectral energy distribution (SED) modeling. We find tentative evidence for sub-minute $γ$-ray variability with a minimum doubling time of $0.7 \pm 0.2$ min ($p$-value = 0.03). This may originate from an extremely compact region with a size of $R \le 1.8 \times 10^{13}$ cm, suggesting that the emission arises from magnetohydrodynamic substructures, such as plasmoids within a magnetic reconnection zone. Spectral analysis reveals a significant ``softer-when-brighter'' trend ($r = 0.96, p = 4.5 \times 10^{-4}$) during the minute-scale flare peaks, indicating a complex interplay between particle acceleration and radiative cooling. The SED is reproduced using a one-zone leptonic model, in which synchrotron self-Compton (SSC) and external Compton (EC) scattering effectively account for the high-energy emissions. The reduced magnetic field strengths and hard electron injection spectral indices observed during the flaring states suggest enhanced particle acceleration efficiency, possibly associated with relativistic magnetic reconnection.

astro-ph.HE↗

arXiv2026

First application of weak lensing peak steepness statistics to HSC Y1 data: effectively probing halo density profiles

As a new probe, the weak lensing (WL) peak steepness statistics is sensitive to the density profile of halos that encodes important information of baryonic feedback and dark matter properties, leading to a promising means to statistically constrain these effects using WL data. In this article, we present its first application to HSC Y1 data to demonstrate the great potential of this new statistics. Within the phenomenological framework of HMcode2016 that attributes the baryonic feedback solely to the reduction of the halo concentration parameter and focusing on high peaks originated dominantly from massive clusters, our analyses by combining WL peak height and steepness statistics resulted in $S_8=0.76^{+0.08}_{-0.07}$ with the maximum-a-posteriori (MAP) of $0.79$ and low concentrations. Taking the form of the concentration-mass relation as $c(M,z)=A(1+z_{\rm f})/(1+z)$ with $z_{\rm f}$ being the formation redshift of halos with mass $M$ at redshift $z$, we obtain $A=1.93^{+1.33}_{-1.16}$ (MAP=$1.70$) in comparison with $A=3.34^{+1.52}_{-1.74}$ (MAP=3.31) from dark matter only simulated mocks. The result tends to support phenomenologically strong baryonic feedback effects at cluster scales.

astro-ph.CO↗

arXiv2026

X-rays breaking out of pre-explosion ejecta mark a supernova's first light

Massive stars die as core-collapse supernovae, whose optical light emerges days after the implosion. Theory predicts that the initial collapse-driven shock, upon breaking through the star and dense circumstellar medium, emits a brief thermal flash of soft X-rays and ultraviolet. Yet these elusive first signals have remained largely undetected, owing to limited wide-field soft X-ray monitoring. Here we report the discovery of a soft X-ray flash, EP260321a, followed days later by a broad-lined supernova from an envelope-stripped progenitor. Its X-ray spectrum, best modeled with blackbody, establishes it as the long-sought archetypal shock breakout. The burst's duration and energetics place the breakout at a radius of 300 solar radii, tracing a dense surrounding shell and revealing abrupt mass ejection within the final month before collapse.

astro-ph.HE↗

arXiv2026

RAG-Match: Retrieval-Augmented Knowledge Injection and Hierarchical Reasoning for Calibrated Semantic Relevance

Semantic relevance judgment for search is particularly challenging in knowledge-intensive scenarios, where accurate ranking requires not only semantic matching but also background grounding, multi-step reasoning, and well-calibrated decision boundaries. Existing relevance models mainly rely on direct label supervision or shallow semantic similarity, which limits their ability to handle implicit intent, factual equivalence, and fine-grained relevance distinctions. To address this issue, we propose \textsc{RAG-Match}, a three-stage framework that integrates knowledge-augmented pretraining, hierarchical reasoning alignment, and preference-based decision calibration for relevance modeling. The key idea is to first strengthen query-centered semantic grounding, then align the model with structured relevance reasoning, and finally correct decision-level inconsistencies in difficult boundary cases. Experimental results on a real-world search relevance benchmark show that \textsc{RAG-Match} consistently outperforms strong LLM-based baselines across multiple ranking metrics, demonstrating the effectiveness of combining knowledge injection, reasoning supervision, and preference optimization for fine-grained relevance judgment.

cs.IR↗

arXiv2026

The Intermediate-Mass Black Hole Reverberation Mapping Project: Stable Optical Continuum Lags of an IMBH in the Dwarf Galaxy NGC 4395 Over Years

NGC 4395 is a nearby dwarf spiral galaxy hosting an active galactic nucleus (AGN) powered by an intermediate-mass black hole (IMBH, $M_{\rm BH} \sim 10^{4}$--$10^{5}\,M_\odot$). Recent optical continuum reverberation mapping studies have suggested potential lag variations between different epochs, offering important clues to the physical mechanisms governing variability in the vicinity of the central black hole. We present continuous intranight multi-band photometric monitoring of NGC 4395 based on five nights of observations, including three nights from the Faulkes Telescope North (two of which are archival) and two new nights from Mephisto. This represents the first systematic investigation of optical continuum lag stability in a galaxy hosting a robustly confirmed IMBH. By applying difference-imaging techniques to both the new observations and the reprocessed archival data, we detect statistically significant optical inter-band lags of $\sim 5$--15 minutes, which increase monotonically with increasing wavelength. No obvious $u$-band lag excess is observed, implying a negligible fractional contribution from diffuse continuum (DC) emission to the optical continuum, in agreement with our spectral decomposition results. The inter-band lags remain stable over multi-year baselines. We suggest that this long-term lag stability may be related to the minor DC contribution, a relatively steady disk-corona structure, and the unusually high X-ray-to-optical luminosity ratio characteristic of low-luminosity AGNs, which likely allows X-ray reprocessing to dominate over other potential variability mechanisms. Future facilities like Gemini/SCORPIO, with its simultaneous optical-to-near-infrared coverage, will be ideally suited to play an important role in advancing this field.

astro-ph.GA↗

arXiv2026

ResRank: Unifying Retrieval and Listwise Reranking via End-to-End Joint Training with Residual Passage Compression

Large language model (LLM) based listwise reranking has emerged as the dominant paradigm for achieving state-of-the-art ranking effectiveness in information retrieval. However, its reliance on feeding full passage texts into the LLM introduces two critical bottlenecks: the "lost in the middle" phenomenon degrades ranking quality as input length grows, and the inference latency scales super-linearly with sequence length, rendering it impractical for industrial deployment. In this paper, we present ResRank, a unified retrieval-reranking framework that fundamentally addresses both challenges. Inspired by multimodal LLMs that project visual inputs into compact token representations, ResRank employs an Encoder-LLM to compress each candidate passage into a single embedding, which is then fed alongside the query text into a Reranker-LLM for listwise ranking. To alleviate the misalignment between the compressed representation space and the ranking space, we introduce a residual connection structure that combines encoder embeddings with contextualized hidden states from the reranker. Furthermore, we replace the conventional autoregressive decoding with a one-step cosine-similarity-based scoring mechanism, eliminating the generation bottleneck entirely. ResRank is trained through a carefully designed dual-stage, multi-task, end-to-end joint optimization strategy that simultaneously trains the encoder and reranker, achieving learning objective alignment between retrieval and reranking while substantially reducing training complexity. Extensive experiments on TREC Deep Learning and eight BEIR benchmark datasets demonstrate that ResRank achieves competitive or superior ranking effectiveness compared to existing approaches while requiring zero generated tokens and processing only one token per passage, yielding a fundamentally better balance between effectiveness and efficiency.

cs.IR↗

arXiv2026

MIU2Net: weak-lensing mass inversion using deep learning with nested U-structures

One of the primary goals of next-generation gravitational lensing surveys is to measure the large-scale distribution of dark matter, which requires accurate mass inversion to convert weak-lensing shear maps into convergence (kappa) fields. This work develops a mass inversion method tailored for upcoming space missions such as CSST and Euclid, aiming to recover both the mass distribution and the convergence power spectrum with high fidelity. We introduce MIU2Net, a versatile deep-learning framework for kappa-map reconstruction based on the U2-Net architecture. A new loss function is constructed to jointly estimate the convergence field and its frequency-domain energy distribution, effectively balancing optimal mean squared error and optimal power-spectrum recovery. The method incorporates realistic observational effects into shear fields, including shape noise, reduced shear, and complex masks. Under noise levels anticipated for future space-based lensing surveys, MIU2Net recovers the convergence power spectrum with 4% uncertainties up to l approximately 500, significantly outperforming Wiener filtering and MCALens. Beyond two-point statistics, the method accurately reconstructs the convergence distribution, peak centroid, and peak amplitude. Compared to other learning-based approaches such as DeepMass, MIU2Net reduces the root-mean-square error by 5% without smoothing and by 38% with a 1-arcmin smoothing scale. MIU2Net represents a substantial advancement in mass inversion methodology, offering improved accuracy in both RMSE and power-spectrum reconstruction. It provides a promising tool for mapping dark matter environments and large-scale structures in the era of next-generation space lensing surveys.

astro-ph.CO↗

arXiv2025

SN 2024aecx: A double-peaked rapidly evolving Type IIb supernova at 11 Mpc

We present the results of low-resolution spectroscopic and densely sampled multi-band photometric follow-up of supernova (SN) 2024aecx. The SN was discovered in the spiral galaxy NGC 3521 (distance $\sim$11 Mpc) within a day after the explosion. The early spectra of SN 2024aecx show a weak signature of hydrogen lines, which disappeared in $\sim$30 days after the explosion. Light curves in all bands show a distinct feature of two peaks, and the first peak is likely due to the shock cooling emission. The early phase light curve evolution of SN 2024aecx has similarity with the typical Type IIb events, but the decay rate in different bands (e.g., $\rm Δm_{15}$ = 1.60 $\pm$ 0.05 mag, $g$-band) is significantly faster in the post-peak phase. It attained the secondary maximum in $\sim$19 days ($g$-band) with a peak absolute magnitude of M$_{g}$ = -17.94 $\pm$ 0.10 mag. SN 2024aecx colors trend redder in early epochs ($<$8 days), followed by a duration in which it grows bluer, then later gets redder again $>$20 days after explosion. The analytical model fitting to the light curves reveals an envelope mass and progenitor radii in the range of $\sim$0.03 - 0.24 $M_\odot$ and $\sim$169 - 200 $R_\odot$, respectively. Modeling of the pseudo-bolometric light curve suggests that synthesized $^{56}$Ni in the explosion was $\sim$0.15 M$_{\odot}$ with ejecta mass and kinetic energy of $\sim$0.7 M$_{\odot}$ and $\sim$0.16 $\times$ 10$^{51}$ erg, respectively. The observational properties and modeling indicate that the SN~2024aecx progenitor belongs to the extended progenitor category.

astro-ph.HE↗

arXiv2025

SN 2024gy: Multi-epoch Spectroscopic Features Suggestive of Delayed Detonation in a Type Ia Supernova

We present photometric and spectroscopic observations of SN 2024gy, a Type Ia supernova (SN Ia) exhibiting high-velocity features (HVFs) in its early-time spectra. This SN reaches a peak $B$-band magnitude of $-19.25 \pm 0.29$ mag and subsequently declines by $Δm_{15}(B) \approx 1.12$ mag, consistent with the luminosity-width relation characteristic of normal SNe Ia. Based on the peak thermal luminosity of $(1.2 \pm 0.3) \times 10^{43}$ erg s$^{-1}$, we estimate that $0.57 \pm 0.14~\rm M_{\odot}$ of $^{56}$Ni was synthesized during the explosion. Our dense early spectral monitoring revealed significant velocity disparities within the ejecta. Notably, absorption features from the Ca II near-infrared triplet were observed at velocities exceeding 25,000 km s$^{-1}$, while the Si II $λ$6355 line velocity at the same epoch was significantly lower at $\sim$ 16,000 km s$^{-1}$. This velocity disparity likely reflects distinct ionization states of intermediate-mass elements in the outermost layers. The prominent Ca II HVFs may originate from ionization suppression within the highest-velocity ejecta, potentially indicative of minimal hydrogen mixing in a delayed-detonation explosion scenario. Additionally, the Ni/Fe ratio derived from the nebular spectrum of SN 2024gy provides further support for this model.

astro-ph.HE↗

arXiv2025

Introduction to the Chinese Space Station Survey Telescope (CSST)

The Chinese Space Station Survey Telescope (CSST) is an upcoming Stage-IV sky survey telescope, distinguished by its large field of view (FoV), high image quality, and multi-band observation capabilities. It can simultaneously conduct precise measurements of the Universe by performing multi-color photometric imaging and slitless spectroscopic surveys. The CSST is equipped with five scientific instruments, i.e. Multi-band Imaging and Slitless Spectroscopy Survey Camera (SC), Multi-Channel Imager (MCI), Integral Field Spectrograph (IFS), Cool Planet Imaging Coronagraph (CPI-C), and THz Spectrometer (TS). Using these instruments, CSST is expected to make significant contributions and discoveries across various astronomical fields, including cosmology, galaxies and active galactic nuclei (AGN), the Milky Way and nearby galaxies, stars, exoplanets, Solar System objects, astrometry, and transients and variable sources. This review aims to provide a comprehensive overview of the CSST instruments, observational capabilities, data products, and scientific potential.

astro-ph.IM↗

arXiv2025

Constraining interacting dark energy models with the halo concentration-mass relation

The interacting dark energy (IDE) model is a promising alternative cosmological model which has the potential to solve the fine-tuning and coincidence problems by considering the interaction between dark matter and dark energy. Previous studies have shown that the energy exchange between the dark sectors in this model can significantly affect the dark matter halo properties. In this study, utilising a large set of cosmological $N$-body simulations, we analyse the redshift evolution of the halo concentration-mass ($c$-$M$) relation in the IDE model, and show that the $c$-$M$ relation is a sensitive proxy of the interaction strength parameter $ξ_2$, especially at lower redshifts. Furthermore, we construct parametrized formulae to quantify the dependence of the $c$-$M$ relation on $ξ_2$ at redshifts ranging from $z=0$ to $0.6$. Our parametrized formulae provide a useful tool in constraining $ξ_2$ with the observational $c$-$M$ relation. As a first attempt, we use the data from X-ray, gravitational lensing, and galaxy rotational curve observations and obtain a tight constraint on $ξ_2$, i.e. $ξ_2 = 0.071 \pm 0.034$. Our work demonstrates that the halo $c$-$M$ relation, which reflects the halo assembly history, is a powerful probe to constrain the IDE model.

astro-ph.CO↗