arXiv subjects

Zhongxu Zhai

Publications and source records attributed to Zhongxu Zhai.

At least 19 recordsLinked to original sources

arXiv2026

Correlation between baryonic process and galaxy assembly bias

Galaxy assembly bias (GAB) is the dependence of galaxy clustering on secondary properties beyond halo mass. In this work, we study the connections between GAB and baryonic processes using the Galacticus semianalytic model (SAM) for galaxy formation and evolution applied to the UNIT simulation. By generating hundreds of galaxy mocks with varying parameters governing gas cooling, star formation, stellar feedback, and active galactic nucleus (AGN) feedback, we quantify the GAB signal using a shuffling technique that compares the clustering of the original galaxy samples with that of shuffled samples, in which galaxies are randomly reassigned among halos of the same mass. We then compare the GAB signals associated with halo concentration and smoothed environmental density, and use the random forest (RF) algorithm to identify which baryonic processes are most correlated with GAB within the explored SAM parameter space. We find that for stellar-mass-selected galaxies, the dominant baryonic processes are gas cooling and stellar feedback, and the result does not change significantly with the number density; for star formation rate (SFR)-selected galaxies, the primary process shifts from star formation to gas cooling as the number density increases. These results reveal a correlational dependence of GAB on baryonic processes within the explored Galacticus SAM parameter space.

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arXiv2026

A graph-based Neural Network surrogate model for accelerating semi-analytical model of galaxy formation and evolution

Understanding how galaxy populations emerge and evolve from the growth of dark matter structure is a central challenge in galaxy formation theory. Semi-analytic models (SAMs) provide an efficient framework to address this problem, but exploring large ensembles of merger trees across broad parameter spaces remains computationally demanding. We develop a conditional graph neural network surrogate model that combines merger tree information with SAM parameters to predict galaxy properties across cosmic time. Using merger trees of dark matter halos from the Uchuu simulation and the Galacticus SAM, the model predicts stellar mass, luminosity, angular momentum, gas metal mass, and specific star formation rate across the wide redshift range of 0 <= z <= 5. For instance, the model can predict stellar mass at 0 <= z <= 3 with a scatter of 0.19-0.28 dex and coefficient of determination R^2 of 0.946-0.973 (R^2 close to 1 indicates prediction closely matching the truth). The results show that a single graph based model can reproduce these galaxy properties with good accuracy over multiple SAM realizations, merger trees and redshifts. This catalog-level model provides a practical route for accelerating SAM based studies of galaxy formation to enable a more detailed investigation of the model parameter space. The inference code, trained models, and example data products are publicly available at https://github.com/MutongCat/sam2galaxy-gnn.

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arXiv2026

ELUCID-DESI II. Revealing dark matter mass, tidal, and velocity (MTV) fields using galaxy group phase information

We introduce a novel method for reconstructing the cosmic mass, tidal, and velocity (MTV) fields over the redshift range $0 < z < 0.6$ using the phase information of galaxy groups. This approach replaces the explicit theoretical bias correction typically needed to relate galaxy groups to the underlying dark matter density field with a simulation-calibrated statistical mapping, reducing a major source of systematic uncertainty and making the method directly applicable to spectroscopic redshift surveys such as the DESI Bright Galaxy Survey (BGS). We evaluate the performance of our MTV reconstruction pipeline with mock redshift surveys that include a comprehensive set of observational selection effects. The galaxy groups used as tracers are identified with an extended halo-based group finder applied to the DESI mock galaxy catalogue with an apparent magnitude limit of $m_z < 19.65$, yielding a galaxy number comparable to that of the DESI BGS faint sample ($m_r < 20.175$). Our tests show that the reconstructed velocities are accurate and unbiased, with a residual dispersion of $\sim 120\ \mathrm{km\,s^{-1}}$ across the redshift bins. The recovered velocity field allows us to shift galaxy groups to their real-space positions, thereby correcting for the Kaiser effect. By iteratively applying this Kaiser correction to the galaxy groups, we further reconstruct the tidal field and the mass-density distribution. The reconstruction is stable with respect to the grid resolution. Overall, our results demonstrate that this group-based phase-space reconstruction provides a robust pathway to recovering the dark matter MTV fields, with strong prospects for application to DESI BGS data.

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arXiv2026

ELUCID-DESI I: A Parallel MPI Implementation of the Initial Condition Solver for Large-Scale Reconstruction Simulations

We present a highly scalable, MPI-parallelized framework for reconstructing the initial cosmic density field, designed to meet the computational demands of next-generation cosmological simulations, particularly the upcoming ELUCID-DESI simulation based on DESI BGS data. Building upon the Hamiltonian Monte Carlo approach and the FastPM solver, our code employs domain decomposition to efficiently distribute memory between nodes. Although communication overhead increases the per-step runtime of the MPI version by roughly a factor of eight relative to the shared-memory implementation, our scaling tests-spanning different particle numbers, core counts, and node layouts-show nearly linear scaling with respect to both the number of particles and the number of CPU cores. Furthermore, to significantly reduce computational costs during the initial burn-in phase, we introduce a novel ``guess'' module that rapidly generates a high-quality initial density field. The results of the simulation test confirm substantial efficiency gains: for $256^3$ particles, 53 steps ($\sim$ 54 core hours) are saved, accelerating convergence by a factor of $\sim$ 18; for $1024^3$, 106 steps ($\sim$7500 core hours), achieving a speedup factor of $\sim$ 3. The total core hour gain grows with the number of particles, rendering large-volume reconstructions computationally practical for upcoming surveys, including our planned ELUCID-DESI reconstruction simulation with $4096^3$ particles. We estimate that achieving convergence for this scale (targeting DESI-BGS data) requires about 800 HMCMC steps ($\sim$ 5 million core hours). Our initial guess module will save approximately 360 steps ($\sim$2.3 million core hours), reducing the total computational time by about 45\%.

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arXiv2026

MUltiplexed Survey Telescope (MUST) Science White Paper I: Overview of Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Surveys

The MUltiplexed Survey Telescope (MUST) is a 6.5-meter telescope under development. Dedicated to highly-multiplexed, wide-field spectroscopic surveys, MUST observes over 20,000 targets simultaneously using 6.2-mm pitch positioning robots within a ~5 deg$^2$ field of view. MUST aims to conduct the first Stage-V spectroscopic survey in the 2030s, mapping the 3D Universe with over 100 million galaxies and quasars, spanning from the nearby Universe to a redshift of z ~ 5.5, corresponding to approximately 1 billion years after the Big Bang. To cover this extensive redshift range, we present an initial conceptual target selection algorithm for different types of galaxies, ranging from local bright galaxies and luminous red galaxies to emission-line galaxies, and high-redshift (2 < z < 5.5) Lyman-break galaxies. Using Fisher forecasts, we demonstrate that MUST can address fundamental questions in cosmology, including the nature of dark energy, tests of gravity theories, and investigations into primordial physics. This is the first paper in the series of science white papers for MUST, with subsequent developments focusing on additional scientific cases such as galaxy and quasar evolution, Milky Way physics, and dynamic phenomena in the time-domain Universe.

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arXiv2026

What can galaxy clustering really tell us about the galaxy-halo connections?

Subhalo abundance matching (SHAM) is a commonly used framework for modeling the galaxy-halo connection. Yet, its standard implementation has difficulty reproducing the observed galaxy clustering with high accuracy (e.g., $χ^2/\mathrm{dof} \approx 1$). To overcome this issue, we propose a novel CS-SHAM framework, in which central and satellite galaxies are independently matched to main and satellite subhalos in simulations. Within this scheme, we introduce three free parameters to explicitly characterize the satellite fraction, $f_{\mathrm{sat}}$, as a function of stellar mass or absolute magnitude. To evaluate the performance of CS-SHAM, we apply it to two sets of mock galaxy catalogs built with the conventional SHAM method but using different subhalo mass proxies, $M_{\mathrm{peak}}$ and $V_{\mathrm{peak}}$, as well as two additional galaxy samples generated from a SAM and from TNG-300. We demonstrate that CS-SHAM reliably reproduces galaxy clustering whether $M_{\mathrm{peak}}$ or $V_{\mathrm{peak}}$ is used as the subhalo mass proxy. We also find that the models are unable to place robust constraints on $f_{\mathrm{sat}}$ if different mass proxies are employed. Indeed, within the CS-SHAM framework the halo occupation distribution (HOD) and conditional luminosity or stellar mass function (CLF/CSMF) are accurately recovered. Furthermore, we demonstrate for the first time that galaxy clustering constrains the HOD and CLF/CSMF primarily for relatively massive halos. Because the halo bias is nearly constant for low-mass halos, galaxy clustering is generally not very sensitive to the satellite population residing in these low-mass systems.

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arXiv2026

Nonlinear Information from DESI Luminous Red Galaxies: An Emulator-Based Analysis of Pre- and Post-Reconstruction Power Spectra

We present joint measurements of the pre- and post-reconstruction power spectra, $P_{\rm pre}$ and $P_{\rm post}$, together with their cross-power spectrum, $P_{\rm cross}$, for the Luminous Red Galaxies (LRGs) in the DESI Data Release 1 (DR1). We jointly analyse these observables with an emulator-based full-shape modeling framework, thereby, for the first time, we extract complementary nonlinear information from the galaxy density field before and after reconstruction in real survey data. Specifically, including $P_{\rm post}$ and $P_{\rm cross}$ in addition to $P_{\rm pre}$ (hereafter $P_{\rm all}$) yields an improvement of approximately $18$-$27\%$ in the $σ_8$ constraint in both $Λ$CDM and $w$CDM, depending on the redshift bin, relative to the $P_{\rm pre}$-only analysis with the cosmic microwave background distance priors (hereafter CMB). In $w$CDM, the joint CMB+$P_{\rm all}$ analysis can tighten the constraints on $w$ by approximately $5$-$15\%$ across the two LRG redshift bins, compared to the CMB+$P_{\rm pre}$ combination. Further incorporating the Type Ia supernova dataset and comparing the cosmological constraints in $w$CDM from each individual power-spectrum component with those from the full combination, we find that $P_{\rm all}$ consistently provides the tightest constraints. From the joint CMB+$P_{\rm all}$+DES-Dovekie dataset, we obtain $Ω_m = 0.314 \pm 0.0048$ and $w = -0.988 \pm 0.023$ for the \texttt{LRG1} sample, and $Ω_m = 0.318 \pm 0.0046$ and $w = -0.988 \pm 0.025$ for \texttt{LRG2}. These results demonstrate that combining pre- and post-reconstruction power spectra with their cross-correlation enables DESI to harvest additional nonlinear information, leading to tighter constraints on cosmological parameters.

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arXiv2026

The shape-velocity alignment of satellites forged by tidal locking and dynamical friction

Utilizing the TNG50 simulation, we study two types of alignments for satellites/subhalos: 1) the alignment of their major axes with the galactocentric radial directions (radial alignment), and 2) with the motion directions (orbital alignment). We find that radial alignment is substantially stronger than orbital alignment, with both signals being consistently stronger for subhalos than for satellites. Interestingly, inward- and outward-moving satellites/subhalos show contrasting orbital alignment behaviors, which can be understood in terms of their radial alignment, orbit decay due to dynamical friction and the effect of tidal stripping. The orbital alignment is stronger in more massive halos. In the end, we explore the orbital alignment measured by a mock observer, and find that the observed alignment for MW satellites is due to projection effects, as the major axes of satellites lie within their orbital planes, approximately coplanar with the observer.

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arXiv2026

Joint analysis of small-scale galaxy clustering and galaxy--galaxy lensing from BOSS galaxies

We present a joint analysis of galaxy clustering and galaxy--galaxy lensing measurements from BOSS galaxies using a simulation-based emulation method combined with a halo occupation distribution model. Our emulators are constructed with the Aemulus $ν$ simulations, a suite of $wν$CDM $N$-body simulations with massive neutrinos as independent particle species. We combine small-scale analysis of clustering from $0.1h^{-1}$Mpc to $60.2~h^{-1}$Mpc and lensing from $1.7h^{-1}$Mpc to $60.2~h^{-1}$Mpc to perform cosmological constraints. We split the BOSS galaxies into three redshift bins to measure their clustering and employ galaxies from Dark Energy Camera Legacy Survey and Hyper Suprime-Cam as source galaxies to measure lensing separately. We find that the addition of lensing significantly improves the constraining power on $S_{8}=σ_8(Ω_m/0.3)^{0.5}$, with a weak improvement for $fσ_{8}$. Our results of $fσ_{8}$ indicate tensions of around $1\sim4σ$ below the results of CMB observations of Planck. For $S_{8}$, our results are also lower than Planck, and the tension can be mitigated when considering possible systematics in lensing measurement. As a byproduct, our analysis prefers a non-zero neutrino mass but without strong significance, with the constraining power dominated by the clustering. Given the accuracy and precision of our model and the observational data, it is anticipated that larger and higher-quality spectroscopic datasets will improve the constraints on this fundamental property in the near future.

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arXiv2025

Cosmology with voids from the Nancy Grace Roman Space Telescope

We provide an accurate forecast of the expected constraining power from the main void statistics -- the void size function and the void-galaxy cross-correlation function -- to be measured by the Roman reference High Latitude Spectroscopic Survey from the Nancy Grace Roman Space Telescope. Relying on a realistic galaxy mock lightcone, covering 2000 square degrees, we find more than $8\times 10^4 $ voids and explore their constraining power in the framework of three different cosmological models: $Λ$CDM, $w$CDM, and $w_0 w_{\rm a}$CDM. This work confirms the strong complementarity of different void statistics and showcases the constraining power to be expected from Roman voids thanks to the combination of its high tracer density and large observed volume.

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arXiv2025

Direct Measurement of Galaxy Assembly Bias using DESI DR1 Data

We report the first direct measurement of galaxy assembly bias, a critical systematic in cosmology, from the Dark Energy Spectroscopic Instrument (DESI) Bright Galaxy Survey. We introduce a novel, cosmology-independent method to measure the halo occupation distribution (HOD) by combining a state-of-the-art group catalog with weak gravitational lensing. For groups binned by total luminosity, we determine the galaxy occupation number $N_{\rm gal}$ from group-galaxy cross-correlations, while weak lensing constrains the average halo mass $M_h$. Applying this to a volume-limited sample at $z{\in}[0.05,0.2]$, we measure the dependence of HOD, $N_{\rm gal}(M_h)$, on large-scale overdensity $δ_{g}$. Focusing on the satellite galaxies, we find an assembly bias parameter of $Q_{\rm sat}{=}0.05{\pm}0.14$, a result consistent with zero and in tension with many empirical galaxy formation models. Our method provides a robust approach for characterizing galaxy assembly bias to achieve precision cosmology with DESI and future Stage-V surveys.

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arXiv2025

Clustering analysis of BOSS-CMASS galaxies with semi-analytical model for galaxy formation and halo occupation distribution

The spatial distribution of massive and luminous galaxies have provided important constraints on the fundamental cosmological parameters and physical processes governing galaxy formation. In this work, we construct and compare independent galaxy-halo connection models in the application of clustering measurement at non-linear scales of BOSS-CMASS galaxies. In particular, we adopt a halo occupation distribution (HOD) model with 11 parameters and a semi-analytical model (SAM) with 16 parameters to describe the galaxy two point correlation function. With an empirical parameterization for the velocity field to model the redshift space distortion effect and the emulator technique, we can explore the parameter space of both models. We find that the HOD model is able to recover the underlying velocity field of SAM with an accuracy of 3\%, and can be improved to 1\% when the analysis is restricted to scales above 1$h^{-1}$Mpc. The comparison is based on multiple samplings in the parameter space which can verify the convergence of the models. Then we perform constraints on the model parameters using clustering measurement of CMASS galaxies. Although limited by the emulator accuracy and the flexibility of the model, we find that the clustering measurement is capable of constraining a subset of the SAM parameters, especially for components sensitive to the star formation rate. This result leads us to anticipate that a joint analysis of both clustering and abundance measurements can significantly constrain the parameters of galaxy formation physics, which requires further investigation from both theoretical and observational aspects.

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arXiv2025

Exploring the signature of assembly bias and modified gravity using small-scale clusterings of galaxies

We apply a halo velocity bias model, $γ_{f}$, within the Aemulus simulation suite for General Relativity (GR) to investigate its efficacy in identifying the signature of assembly bias and Modified Gravity (MG). In the investigation of assembly bias, utilizing galaxy clustering data ranging from scales of $0.1 - 60 \text{Mpc}\,h^{-1}$, we discover that our emulator model accurately recreates the cosmological parameters, $Ω_m$ and $σ_8$, along with the velocity bias $γ_{f}$, staying well within the 1-$σ$ error margins, provided that assembly bias is considered. Ignoring assembly bias can considerably alter our model constraints on parameters $Ω_m$ and $σ_8$ if the test sample includes assembly bias. Using our emulator for MG simulations, which encompasses two Dvali-Gabadadze-Porrati models (DGP; N1, N5) and two $f(R)$ models (F4, F6), we can effectively identify a robust signature of modified gravity, for models such as DGP-N1 and $f(R)$-F4, as indicated by a noticeable deviation of $γ_{f}$ from unity. Using the velocity dispersion of dark matter halos to effectively represent the underlying strength of the velocity field of these MG simulations, we find that the simple $γ_{f}$ model can recover the truth with minimal bias. These evaluations indicate that our simple halo-velocity bias model is capable of detecting significant MG characteristics, although additional methodologies should be pursued to improve model constraints.

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arXiv2024

Testing the framework of the halo occupation distribution with assembly bias modeling and empirical extensions

We investigate theoretical systematics caused by the application of the halo occupation distribution (HOD) to the study of galaxy clustering at non-linear scales. To do this, we repeat recent cosmological analyses using extended HOD models based on both the Aemulus and Aemulus $ν$ simulation suites, allowing for variations in the dark matter halo shape, incompleteness, baryonic effects and position bias of central galaxies. We fit to the galaxy correlation function including the projected correlation function, redshift space monopole and quadrupole, and consider how the changes in HOD affect the retrieval of cosmological information. These extensions can be understood as an evaluation of the impact of the secondary bias in the clustering analysis. In the application of BOSS galaxies, these changes do not have a significant impact on the measured linear growth rate. But, we do find weak to mild evidence for some of the effects modeled by the empirical parameterizations adopted. The modeling is able to make the HOD approach more complete in terms of cosmological constraints. We anticipate that the future and better data can provide tighter constraints on the new prescriptions of the HOD model.

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arXiv2024

CSST large-scale structure analysis pipeline: I. constructing reference mock galaxy redshift surveys

In this paper, we set out to construct a set of reference mock galaxy redshift surveys (MGRSs) for the future Chinese Space-station Survey Telescope (CSST) observation, where subsequent survey selection effects can be added and evaluated. This set of MGRSs is generated using the dark matter subhalos extracted from a high-resolution Jiutian $N$-body simulation of the standard $Λ$CDM cosmogony with $Ω_m=0.3111$, $Ω_Λ=0.6889$, and $σ_8=0.8102$. The simulation has a boxsize of $1~h^{-1} {\rm Gpc}$, and consists of $6144^3$ particles with mass resolution $3.723 \times 10^{8} h^{-1} M_\odot $. In order to take into account the effect of redshift evolution, we first use all 128 snapshots in the Jiutian simulation to generate a light-cone halo/subhalo catalog. Next, galaxy luminosities are assigned to the main and subhalo populations using the subhalo abundance matching (SHAM) method with the DESI $z$-band luminosity functions at different redshifts. Multi-band photometries, as well as images, are then assigned to each mock galaxy using a 3-dimensional parameter space nearest neighbor sampling of the DESI LS observational galaxies and groups. Finally, the CSST and DESI LS survey geometry and magnitude limit cuts are applied to generate the required MGRSs. As we have checked, this set of MGRSs can generally reproduce the observed galaxy luminosity/mass functions within 0.1 dex for galaxies with $L > 10^8 L_\odot$ (or $M_* > 10^{8.5} M_\odot$) and within 1-$σ$ level for galaxies with $L < 10^8L_\odot$ (or $M_* < 10^{8.5} M_\odot$). Together with the CSST slitless spectra and redshifts for our DESI LS seed galaxies that are under construction, we will set out to test various slitless observational selection effects in subsequent probes.

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arXiv2024

The Aemulus Project VI: Emulation of beyond-standard galaxy clustering statistics to improve cosmological constraints

There is untapped cosmological information in galaxy redshift surveys in the non-linear regime. In this work, we use the AEMULUS suite of cosmological $N$-body simulations to construct Gaussian process emulators of galaxy clustering statistics at small scales ($0.1-50 \: h^{-1}\,\mathrm{Mpc}$) in order to constrain cosmological and galaxy bias parameters. In addition to standard statistics -- the projected correlation function $w_\mathrm{p}(r_\mathrm{p})$, the redshift-space monopole of the correlation function $ξ_0(s)$, and the quadrupole $ξ_2(s)$ -- we emulate statistics that include information about the local environment, namely the underdensity probability function $P_\mathrm{U}(s)$ and the density-marked correlation function $M(s)$. This extends the model of AEMULUS III for redshift-space distortions by including new statistics sensitive to galaxy assembly bias. In recovery tests, we find that the beyond-standard statistics significantly increase the constraining power on cosmological parameters of interest: including $P_\mathrm{U}(s)$ and $M(s)$ improves the precision of our constraints on $Ω_m$ by 27%, $σ_8$ by 19%, and the growth of structure parameter, $f σ_8$, by 12% compared to standard statistics. We additionally find that scales below $\sim6 \: h^{-1}\,\mathrm{Mpc}$ contain as much information as larger scales. The density-sensitive statistics also contribute to constraining halo occupation distribution parameters and a flexible environment-dependent assembly bias model, which is important for extracting the small-scale cosmological information as well as understanding the galaxy-halo connection. This analysis demonstrates the potential of emulating beyond-standard clustering statistics at small scales to constrain the growth of structure as a test of cosmic acceleration.

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arXiv2024

The effective volume of supernovae samples and sample variance

The source of the tension between local SN Ia based Hubble constant measurements and those from the CMB or BAO+BBN measurements is one of the most interesting unknowns of modern cosmology. Sample variance forms a key component of the error on the local measurements, and will dominate the error budget in the future as more SNe Ia are observed. Many methods have been proposed to estimate sample variance in many contexts, and we compared results from a number of approximate methods to estimates from N-body simulations in a previous paper, confirming that sample variance for the Pantheon SNe Ia sample does not solve the Hubble tension. We now extend this analysis to include the more accurate analytic method based on calculating correlations between the radial peculiar velocities of SNe Ia, comparing this technique with results from numerical simulations. We consider the dependence of these errors on the linear power spectrum and how non-linear velocities contribute to the error. Using this technique, and matching sample variance errors from more approximate methods, we can define an effective volume for SNe Ia samples, finding that the Pantheon sample is equivalent to a top-hat sphere of radius $\sim220~h^{-1}$Mpc. We use this link between sample-variance errors to compute $ΔH_{0}$ for idealised surveys with particular angular distributions of SNe Ia. For example, a half-sky survey at the Pantheon depth has the potential to suppress the sample variance of $H_{0}$ to $\sim0.1$ km s$^{-1}$Mpc$^{-1}$, a significant improvement compared with the current result. Finally, we consider the strength of large-scale velocity power spectrum required to explain the Hubble tension using sample variance, finding it requires an extreme model well beyond that allowed by other observations.

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arXiv2024

Emulating power spectra for pre- and post-reconstructed galaxy samples

The small-scale linear information in galaxy samples typically lost during non-linear growth can be restored to a certain level by the density field reconstruction, which has been demonstrated for improving the precision of the baryon acoustic oscillations (BAO) measurements. As proposed in the literature, a joint analysis of the power spectrum before and after the reconstruction enables an efficient extraction of information carried by high-order statistics. However, the statistics of the post-reconstruction density field are difficult to model. In this work, we circumvent this issue by developing an accurate emulator for the pre-reconstructed, post-reconstructed, and cross power spectra ($P_{\rm pre}$, $P_{\rm post}$, $P_{\rm cross}$) up to $k=0.5~h~{\rm Mpc^{-1}}$ based on the \textsc{Dark Quest} N-body simulations. The accuracy of the emulator is at percent level, namely, the error of the emulated monopole and quadrupole of the power spectra is less than $1\%$ and $10\%$ of the ground truth, respectively. A fit to an example power spectra using the emulator shows that the constraints on cosmological parameters get largely improved using $P_{\rm pre}$+$P_{\rm post}$+$P_{\rm cross}$ with $k_{\rm max}=0.25~h~{\rm Mpc^{-1}}$, compared to that derived from $P_{\rm pre}$ alone, namely, the constraints on ($Ω_m$, $H_0$, $σ_8$) are tightened by $\sim41 \%-55\%$, and the uncertainties of the derived BAO and RSD parameters ($α_{\perp}$, $α_{||}$, $fσ_8$) shrink by $\sim 28\%-54\%$, respectively. This highlights the complementarity among $P_{\rm pre}$, $P_{\rm post}$ and $P_{\rm cross}$, which demonstrates the efficiency and practicability of a joint $P_{\rm pre}$, $P_{\rm post}$ and $P_{\rm cross}$ analysis for cosmological implications.

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