Constructing X-ray Spectral Models of Galaxies: Varying Contributions from X-ray Binary Populations with Host Galaxy Properties

This paper presents a procedure to generate realistic X-ray spectral models for galaxies hosting X-ray binary (XRB) populations by accounting for statistical variations and intrinsic properties of the X-ray luminosity function (XLF). The work addresses the fact that recent studies have shown that emission from XRB populations varies with stellar mass (M⋆), star formation rate (SFR), and metallicity (Z), and that as galaxies approach low SFR and low Z, the relatively shallow slope of the XRB luminosity function can yield very large stochastic variations in the total X-ray luminosity expected from the XRB population for fixed values of M⋆, SFR, and Z.

The authors created a procedure to statistically sample any XLF and model total X-ray spectra for XRB populations and their stochastic uncertainties. They demonstrate the accuracy of this procedure using data for galaxies ranging from high to low M⋆, SFR, and Z and generating X-ray spectral models consistent with Chandra observations. For galaxies that lie on the galactic main-sequence, they relate SFR and Z to M⋆ using established M⋆-SFR and M⋆-Z relations, and construct main-sequence (MS) XRB spectral models, which provide typical XRB spectral shapes, normalizations, and uncertainties as a function of M⋆. The spectral model library associated with this work is available at https://doi.org/10.5281/zenodo.20126734.

The paper adopts XLF scaling relations from Lehmer et al. (2021) for HMXBs (dependent on SFR and Z) and Lehmer et al. (2019) for LMXBs (dependent on M⋆). The HMXB XLF includes a break luminosity and a cutoff luminosity, with slopes that vary with Z. The LMXB XLF has a break and a cutoff as well. The total XLF is the sum of the HMXB and LMXB components.

The authors discuss the statistical properties of the XLF, noting that the number of sources observed in any given galaxy follows a Poisson distribution, so variations about the mean expected value are extremely common. They demonstrate that the relatively shallow, high-L slope of the HMXB XLF leads to a population-integrated luminosity that depends on the most luminous source within the population. They show that in the low-SFR regime, the LX-SFR relation is non-linear (LX ∝ SFR 1.67 for an average slope of 1.6), while in the high-SFR regime where the XLF is fully populated, the relation becomes linear.

To mitigate computational costs of simulating large numbers of sources, the authors took a hybrid approach: they sample the ≈15 brightest sources (following a Poisson distribution with λ=15) and integrate over the lower luminosity end of the XLF. This method is significantly faster than sampling every source, with speedups of 4.7 times for a Milky-Way-like galaxy and 304 times for a high-SFR galaxy.

The authors present probability density functions (PDFs) of LX/SFR for HMXB populations varying with SFR and Z, and PDFs of LX/M⋆ for LMXB populations varying with M⋆. They find that as SFR and M⋆ increase, the PDFs become narrower, more symmetric, and approach a Gaussian-like distribution, indicating that the degree of stochasticity decreases. For low SFR and low M⋆, the PDFs are wide and skewed, sometimes showing two peaks due to the nonzero probability of hosting a high-luminosity source.

For spectral modeling, the authors constructed empirical distributions of X-ray point-source spectral shapes as a function of observed luminosities using Chandra point-source catalogs from Lehmer et al. (2024) and PHANGS galaxies. They identified 765 X-ray point sources with high confidence of being associated with their host galaxies, spanning a luminosity range of log L = 36.7–40.2. They performed X-ray spectral fitting using an absorbed power-law model (tbabs×tbabs×pow) and used MCMC to sample the posterior distributions of intrinsic column density (NH,int) and photon index (Γ). They constructed covariance contours of log NH,int and Γ for several bins of log L, which are used to assign realistic spectral parameters to sampled sources.

The galaxy-integrated X-ray spectral models are constructed by sampling from the XLFs and assigning spectral models to each sampled source. The final model is given by Equation 8, which integrates over the low-luminosity end of the XLF and sums the individual spectral models of the n brightest sources. The base functions FE(E, L) are unit-normalized spectral shapes as a function of luminosity, and they show that low-luminosity sources have steeper, declining slopes at high energies, while high-luminosity sources have flatter spectra.

The authors test their method on 5 galaxies covering a wide range of M⋆, SFR, and Z: NGC 1569, NGC 3310, the Antennae galaxies (NGC 4038/4039), NGC 3377, and NGC 3115. They plot 500 spectral models for each galaxy and compare them to Chandra data. They find general convergence between their simulated models and the observed spectra, with null hypothesis probabilities ranging from pnull = 0.079 (Antennae) to pnull = 0.81 (NGC 3115). The models show a wider distribution of probable models for galaxies with low M⋆ and low SFR due to their higher degree of intrinsic stochastic scatter.

For main-sequence galaxies, the authors use empirical M⋆-SFR and M⋆-Z relations to determine reasonable values of SFR and Z given M⋆. They use the Aird et al. (2019) M⋆-SFR relation, the Berg et al. (2012) MZR for low-M⋆ (log(M⋆/M⊙) < 8.25), the Lebouteiller et al. (2025) MZR for intermediate-M⋆ (8.25 ≤ log(M⋆/M⊙) ≤ 10.5), and the Curti et al. (2020) MZR for high-M⋆ (log(M⋆/M⊙) > 10.5). They generate PDFs of LX/SFR for MS galaxies with different M⋆, showing similar trends to the HMXB PDFs: higher M⋆ (and thus higher SFR) produces Gaussian-like PDFs with lower stochasticity, while low-mass galaxies show broad, skewed distributions with two areas of high probability.

The authors present median LX vs. SFR for MS galaxies, showing convergence between their sampling method and previous work. They also present spectral models for typical MS galaxies with M⋆ = 10 8, 10 9, 10 10, and 10 11 M⊙, showing larger areas of uncertainty at lower M⋆.

Key results summarized in the paper include:

• Stochastic scatter originates through the shallow slope of the XLF at low luminosities, and since the number of expected XRB sources follows a Poisson distribution, small variations in number of sources constitute a large possible range of log L.

• The authors account for stochasticity by sampling from the ≈15 brightest sources and integrating over the remainder of the XLF curve to get the probability distribution of log LX.

• As SFR and M⋆ increase, probability distributions of LX become narrower, more symmetric, and approach a Gaussian-like distribution.

• Using Chandra data, the authors discovered differing degrees of covariance between NH,int and Γ depending on the source's luminosity, and they bin log L and draw from these contours to contribute to more realistic spectral models.

• They developed a procedure to generate spectral models by combining the sampling process and base functions to build the final model, accounting for the low-L end of the population while saving computational time.

• They find general convergence between their multiple simulated models and Chandra spectral data for the 5 test galaxies, providing reasonable estimations of spectral contributions for galaxies with extreme values of M⋆, SFR, and Z in the 0.5-8 keV band.

• For galaxies on the MS, they use empirical parameterizations of the MS and M⋆-Z relationship to determine reasonable values of SFR and Z given M⋆, displaying MS models with higher degrees of stochasticity for low M⋆.

The paper concludes by discussing future work, including expanding to study extremely metal-poor galaxies (XMPs), adapting the sampling process to study stellar age effects using XLFs with star formation history dependence, connecting these X-ray spectral models with stellar population models using SED fitting codes, and further separating XRBs into four categories (BH binaries, magnetized and non-magnetized NS binaries, and ULXs) using NuSTAR X-ray color space. The authors also plan to replace the power-law model with more physically-motivated models (blackbody, accretion disk, thermal Comptonization, self-irradiated funnel accretion) to improve fitting and allow extrapolation outside the X-ray band.

Improvements to AI Systems:

1. Stochastic Spectral Synthesis Engine for Galaxy Evolution Models

• Implement the hybrid sampling method (Poisson-distributed bright-source sampling + analytic integration of the faint end) into AI-driven galaxy formation simulations. This allows AI to rapidly generate realistic X-ray spectral energy distributions (SEDs) for arbitrary galaxy parameters (M⋆, SFR, Z) without expensive full N-body or radiative transfer calculations.

• Capability: AI can now predict full X-ray spectra (0.5–8 keV) and their intrinsic scatter for millions of galaxies in cosmological simulations, enabling statistical comparisons with deep Chandra/XMM surveys.

2. Luminosity-Dependent Spectral Parameter Priors

• Train a neural network or Gaussian process on the paper’s binned covariance contours of log NH,int vs. photon index Γ as a function of X-ray luminosity. This replaces fixed power-law assumptions with luminosity-aware priors.

• Capability: AI spectral fitting tools (e.g., XSPEC-based auto-fitters) can now assign physically plausible, luminosity-dependent spectral shapes to unresolved X-ray sources, improving deconvolution of blended populations in low-resolution observations.

3. Uncertainty-Aware X-ray Luminosity Function (XLF) Emulator

• Build a surrogate model that maps (M⋆, SFR, Z) → full probability distribution of total LX (not just mean), using the paper’s PDFs for HMXB and LMXB populations. This can be integrated into AI-based SED fitting codes (e.g., CIGALE, Prospector) as a likelihood term.

• Capability: AI can now fit galaxy multi-wavelength data while correctly marginalizing over XRB stochasticity, yielding unbiased estimates of SFR and stellar mass for low-mass, low-metallicity galaxies where scatter is highly non-Gaussian.

4. Main-Sequence X-ray Spectral Library Generator

• Automate the paper’s procedure to generate spectral model libraries for main-sequence galaxies across M⋆ = 10 8–10 11 M⊙, using the provided M⋆-SFR and M⋆-Z relations. This library can be embedded in AI pipelines for rapid galaxy classification or anomaly detection.

• Capability: AI can instantly compare observed X-ray spectra of star-forming galaxies against a physically motivated, stochastic library to flag outliers (e.g., AGN contamination, ultraluminous X-ray sources) with quantified confidence.

5. Computational Speedup for Large-Volume X-ray Population Synthesis

• Adopt the paper’s hybrid approach (sampling 15 brightest sources + analytic faint-end integration) in AI-driven Monte Carlo codes. This yields 300× speedup for high-SFR galaxies, enabling real-time generation of mock X-ray catalogs for survey design (e.g., Athena, Lynx).

• Capability: AI can produce millions of realistic X-ray galaxy spectra in minutes, facilitating training of deep learning classifiers on synthetic data that includes realistic Poisson noise and source clustering.

6. Metallicity-Dependent Spectral Slope Correction

• Incorporate the paper’s finding that HMXB XLF slopes vary with Z into AI-based spectral template fitting. This allows AI to disentangle metallicity effects from star formation in integrated X-ray spectra.

• Capability: AI can estimate Z from X-ray spectra alone (in conjunction with optical/NIR data), improving metallicity maps of high-redshift galaxies where direct spectroscopy is infeasible.

7. Non-Linear LX–SFR Relation Calibrator

• Use the paper’s derived non-linear relation (LX ∝ SFR 1.67 at low SFR) to correct AI-driven star formation rate estimators that rely on X-ray luminosity. This prevents systematic underestimation of SFR in dwarf galaxies.

• Capability: AI-based SFR indicators become accurate across the full galaxy mass range, including starbursts and quiescent dwarfs, by applying SFR-dependent scaling corrections.

8. Anomaly Detection in X-ray Source Populations

• Train a variational autoencoder (VAE) on the paper’s simulated spectral models (including stochastic variations) to learn the normal manifold of XRB-dominated galaxy spectra.

• Capability: AI can flag galaxies whose observed X-ray spectra deviate significantly from the expected XRB contribution, automatically identifying candidates for obscured AGN, tidal disruption events, or exotic X-ray binaries for follow-up.

9. Time-Evolution Extrapolation for Star-Forming Galaxies

• Extend the paper’s static XLF sampling to include star formation history (SFH) dependence (as suggested in future work), using AI to learn the mapping from SFH to X-ray spectral evolution.

• Capability: AI can predict how a galaxy’s X-ray spectrum changes over cosmic time, enabling synthetic observations of high-redshift galaxies in JWST/Chandra deep fields.

10. Physical Model Replacement for Power-Law Assumption

• Replace the simple absorbed power-law with the paper’s planned physical models (blackbody, disk, Comptonization, funnel accretion) in AI spectral fitters. Train the AI on these physically motivated templates to extrapolate beyond 0.5–8 keV.

• Capability: AI can now predict X-ray emission in the 0.1–100 keV band (e.g., for NuSTAR, XRISM) and connect X-ray spectra to accretion physics, enabling direct constraints on black hole mass and spin in XRB populations.