Population synthesis of active galactic nuclei based on the radiation-regulated unification model

Technical Summary: Population Synthesis of Active Galactic Nuclei Based on the Radiation-Regulated Unification Model

Problem Statement
Active Galactic Nuclei (AGNs) are powered by supermassive black holes (SMBHs) whose evolution is linked to host galaxy properties. While X-ray surveys provide crucial constraints on AGN emission, obscuration, and accretion rates, previous population synthesis models have treated reflection and absorption as independent factors, often relying on phenomenological templates rather than self-consistent physical modeling. Furthermore, the intrinsic fraction of Compton-thick (CTK) AGNs ($N_H \ge 10^{24} \text{ cm}^{-2}$) remains debated, with model predictions ranging from 10% to 50%. A key challenge is reconciling the anti-correlation between obscuration and luminosity with the "unification model," which posits that orientation alone dictates AGN classification. Recent proposals suggest a radiation-regulated unification model where the covering factor of obscuring material is driven by the Eddington ratio ($\lambda_{\text{Edd}}$) rather than luminosity, as radiation pressure clears the circumnuclear environment. This work aims to test the consistency of this radiation-regulated model against a comprehensive set of X-ray observables while self-consistently constraining the geometry of the dusty torus and the evolution of the local AGN population.

Methodology
The authors employ a simulation-based inference (SBI) approach to constrain the geometrical and physical properties of the AGN population. The methodology involves the following steps:

1. Synthetic Population Generation: Instead of sampling from an X-ray luminosity function (XLF), the synthetic population is generated by sampling the intrinsic local active black hole mass function (BHMF) and Eddington ratio distribution function (ERDF) derived by Ananna et al. (2022, A22) for $z < 0.3$. A simple evolutionary prescription is applied where the number density of active SMBHs evolves with redshift ($e(z)$), while the shapes of the BHMF and ERDF remain constant.
2. Radiation-Regulated Geometry: A complex geometrical model is constructed using the ray-tracing code RefleX. This model includes:
   • A "lamp-post" X-ray source (double point source at $10 r_g$).
   • An accretion disc (flat annulus).
   • A Broad-Line Region (BLR) modeled as a cold, dust-free wedge with a fixed covering factor of 0.2.
   • A dusty torus whose inner edge is set at the outer boundary of the BLR ($R_{\text{BLR,out}}$). Crucially, the torus covering factor ($C_F$) is modeled as a sigmoid function of $\lambda_{\text{Edd}}$, decreasing as accretion rates increase, consistent with the radiation-regulated model.
   • Emission spectra (power-law with high-energy cutoff) are assigned based on empirical relations with $\lambda_{\text{Edd}}$.
3. Numerical Simulation: Using RefleX, the authors perform Monte Carlo simulations to propagate X-ray photons (0.1 keV – 1 MeV) through the circumnuclear material, accounting for Compton scattering, photoelectric absorption, and fluorescence. Over $6 \times 10^6$ spectra are generated across grids of geometric and spectral parameters.
4. Inference: The model parameters are inferred by comparing the simulated population against observational constraints using the Sequential Neural Posterior Estimation (SNPE) method. The constraints include:
   • The Cosmic X-ray Background (CXB) spectrum (2–100 keV).
   • Differential AGN number counts in three energy bands (14–195 keV, 8–24 keV, 2–10 keV).
   • Observed $N_H$ distributions of local AGNs.
   • The fraction of obscured AGNs as a function of $\lambda_{\text{Edd}}$.
   • The observed fraction of CTK AGNs as a function of limiting flux.

Key Contributions

• Self-Consistent Modeling: This is the first population synthesis study to sample the intrinsic BHMF and ERDF directly and implement an Eddington-ratio-dependent geometry and emission spectrum within a unified framework where obscuration and reflection are intrinsically linked via spatial distribution.
• Radiation-Regulated Validation: The study rigorously tests the radiation-regulated unification model, demonstrating that a geometry where the torus covering factor depends on $\lambda_{\text{Edd}}$ can simultaneously reproduce the CXB, number counts, and absorption properties.
• Intrinsic CTK Fraction: The authors derive a robust intrinsic CTK fraction of $40 \pm 3\%$, significantly higher than previous estimates, attributing this to the inclusion of BLR obscuration which sets a baseline CTK fraction of $\sim 20\%$.
• Parameter Constraints: The study provides posterior distributions for key physical parameters, including the torus column density distribution ($\mu, \sigma$), the dependence of the torus covering factor on $\lambda_{\text{Edd}}$ ($k$), and the evolution of the active SMBH fraction ($p_1, z_c$).

Results

• CXB and Number Counts: The model successfully reproduces the overall intensity and shape of the CXB spectrum, though it slightly overpredicts the flux above 60 keV. Differential number counts are reproduced within uncertainties, with minor overpredictions at the bright end of the 14–195 keV band and underpredictions at the faint end of softer bands.
• Absorption Properties: The model reproduces the bimodal $N_H$ distribution and the anti-correlation between the obscured fraction and $\lambda_{\text{Edd}}$. However, the absolute number of detected obscured and unobscured AGNs is overpredicted by approximately 15–25%, likely due to selection effects not fully captured in the synthetic population.
• Torus Geometry: The inferred inner edge of the torus spans $\sim 0.5 - 8$ pc for $-3 < \log \lambda_{\text{Edd}} < 0.5$, consistent with mid-infrared interferometric sizes and the radiation-regulated scenario. The covering factor decreases with increasing $\lambda_{\text{Edd}}$, recovering the trend observed in local AGNs.
• Reflection-Obscuration Correlation: The model qualitatively reproduces the observed trend where reflection increases with $N_H$ for obscured sources, and predicts reflection levels for unobscured AGNs that align better with observations than previous models.
• Evolution: The inferred evolution of the active SMBH fraction shows a sharp increase up to $z \approx 1$ followed by a plateau, consistent with pure density evolution models used in XLF studies.

Significance and Claims
The paper claims that the radiation-regulated unification model is consistent with a very comprehensive set of X-ray observables. By moving from phenomenological to physical modeling, the authors demonstrate that a simple evolutionary prescription controlling the active fraction of SMBHs is sufficient to reproduce the CXB when combined with a locally sampled BHMF and ERDF. The derived intrinsic CTK fraction of $40 \pm 3\%$ is in good agreement with recent NuSTAR observations of the local Universe and previous population synthesis predictions. The study concludes that the differences in the BHMF and ERDF between obscured and unobscured AGNs can be reproduced by sampling a single parent distribution, with the distinction arising solely from the physical and geometrical properties governing obscuration. The authors note limitations, such as the assumption of homogeneous components and the lack of other potential obscurers (e.g., polar dust, outflows), suggesting these as areas for future refinement.