--- title: Eddington-Ratio Distribution url: https://www.emergentmind.com/topics/eddington-ratio-distribution type: topic --- # Eddington-Ratio Distribution The Eddington-ratio distribution quantifies the probability that a supermassive black hole (SMBH) or active galactic nucleus (AGN) of given mass is accreting at a given fraction of its Eddington limit, λ ≡ L_bol / L_Edd. This distribution is a cornerstone for demography, fueling, and feedback studies of black hole–galaxy coevolution, serving as the statistical kernel that links accretion events to galaxy and black hole growth, observed luminosity functions, and the mapping between observed and relic black hole mass functions. ## 1. Formal Definition and Parametric Forms The Eddington ratio, λ = L_bol / L_Edd, normalizes the bolometric luminosity to the Eddington limit, $L_{\rm Edd}=1.26\times10^{38}(M_{\rm BH}/M_\odot)$ erg s$^{-1}$. The Eddington-ratio distribution function (ERDF), often denoted as ξ(λ) = dN/d log λ or Φ(λ), is typically parameterized in several analytic forms: - **Schechter Form:** $$\Phi(\lambda) = \Phi_*\left(\frac{\lambda}{\lambda_*}\right)^{\alpha}\exp\left(-\frac{\lambda}{\lambda_*}\right)$$ with normalization $\Phi_*$, characteristic Eddington ratio $\lambda_*$, and faint-end slope α. This form captures a power-law rise at low λ with exponential cutoff around $\lambda_*$ [1004.2671, 1605.01739, 2511.14575]. - **Broken-Power-Law:** $$\xi(\lambda) = \xi^*\left[\left(\frac{\lambda}{\lambda^*}\right)^{\delta_1}+\left(\frac{\lambda}{\lambda^*}\right)^{\delta_2}\right]^{-1}$$ with low- and high-λ slopes δ₁, δ₂ and break λ* [1707.05323, 1801.07717, 2201.05603, 2210.08401]. - **Log-normal:** $$\psi_{\rm Edd}(\lambda) = \frac{1}{\ln10\,\sqrt{2\pi}\,\sigma_\lambda}\,\exp\left[-\frac{(\log\lambda-\log\lambda^*)^2}{2\sigma_\lambda^2}\right]$$ with dispersion σλ (in dex) and mass- or redshift-dependent mean [1201.5383, 2311.08922, 1511.01092, 2510.01486]. - **Composite Forms:** Mixtures of Gaussians, Gaussians plus power-laws, or mass-dependent forms are used when reproducing broad features across populations or cosmic time [1111.3574, 1006.3561]. ## 2. Observational Inference and Correcting Systematics Observed ERDF shapes depend sensitively on selection functions, sample definitions, and emission-line or photometric proxies. Optical, X-ray, and infrared studies demonstrate that: - Without corrections, observed ERDFs often display an artificial turnover at $\lambda \sim 0.1$ due to flux limits [1004.2671, 1004.2671, 1004.2671]. - Rigorous maximum-likelihood or forward-modeling methods, accounting for individual detection thresholds and full selection biases, recover an intrinsic ERDF that generally rises as a Schechter-like power law to low λ, with a sharp cutoff near λ ~ 0.1–1 [1004.2671, 2511.14575, 1004.2671]. - Proper treatment of “outshining” by star formation and use of specific diagnostic diagrams (e.g., BPT or 3D line ratio spaces) is essential for optical/narrow-line samples [1605.01739, 2511.14575]. ## 3. Physical Drivers, Galaxy Properties, and Universality The functional form and parameters of the ERDF are physically motivated by models of AGN fueling, feedback, and galaxy environment: - **Self-Regulated Growth:** Feedback expels fuel post-peak and enforces a power-law decay in accretion, predicting a power-law ERDF [1010.0046, 1006.3561]. - **Accretion Modes:** Radiatively efficient (thin-disk; X-ray/optical; blue/green hosts) and inefficient (ADAF/radio-mode; red quiescent hosts) populations are distinguished by broken-power-law ERDFs, often nearly mass-independent [1707.05323, 2210.08401]. - **Mass and sSFR Dependence:** While some studies find nearly mass-independent ERDFs in star-forming systems (F_AGN~const above λ>10^{-3}), quiescent galaxies show a declining active fraction with increasing mass [2511.14575]. At high z, log-normal or Schechter ERDFs show negative mass dependence—massive SMBHs accreting at lower λ [2311.08922, 1605.01739]. - **Redshift Evolution:** Characteristic λ and the high-λ ERDF tail both shift positively with redshift, generating higher fractions of AGN at high λ in the early Universe (“downsizing” or anti-hierarchical growth) [1511.01092, 2311.08922, 1905.02732, 1111.3574, 1801.07717]. - **Population Bimodality:** Evidence for clear bi-modal ERDFs (e.g., in changing-look AGN) is limited to specific populations and generally absent in global samples, which are typically well modeled by unimodal forms [2510.01486]. ## 4. Impact on Black Hole and Galaxy Evolution Frameworks The ERDF is mathematically coupled to the evolution of the black hole mass function (BHMF) and AGN duty cycle: - **Continuity Equation:** Black hole growth via accretion evolves under $$ \frac{\partial n(M_{\rm bh},t)}{\partial t}+\frac{\partial}{\partial M_{\rm bh}}[n(M_{\rm bh},t)\langle\dot M(M_{\rm bh},t)\rangle]=0 $$ with $$ \langle\dot M(M_{\rm bh},z)\rangle = \int_{\lambda_{\min}(z)}^\infty \zeta(\lambda)\frac{(1-\eta_{\rm rad})\,\lambda\,L_{\rm Edd}(M_{\rm bh})}{\eta_{\rm rad}c^2}d\log\lambda $$ where $\zeta(\lambda)$ is the ERDF [1010.0046]. - **AGN Duty Cycle:** The fraction of SMBHs above given λ, $P(\lambda>\lambda_0) = \int_{\lambda_0}^{\lambda_{\max}} \xi(\lambda)\,d\,\log\lambda\,/\,\int_{M_{\rm BH,\min}}^{M_{\rm BH,\max}} \Phi_{\rm tot}(M_{\rm BH})\,d\,\log M_{\rm BH}$, is a key diagnostic of the SMBH activity timescale and fueling duty [2210.08401, 2201.05603, 2511.14575]. - **Host Coevolution:** Mapping stellar mass functions onto X-ray/AGN luminosity functions using a mass-independent or mass-dependent ERDF—convolved with empirical $M_{\rm BH}−M_*$ relations—successfully reproduces observed AGN XLFs, and constraints on the normalization, shape, and redshift evolution of the ERDF critically inform theoretical models of SMBH/galaxy growth [1707.05323, 2002.08965, 1511.01092, 1801.07717]. ## 5. Empirical Results across Population and Cosmic Time A synthesis of key empirical findings is shown in the following table, summarizing recent best-fit ERDF parameters in diverse environments and redshifts. (All λ are in units of $L_{\rm bol}/L_{\rm Edd}$.) | Study & Population / Redshift | ERDF Parametric Form | Key Parameters, Trends, and Results | |------------------------------------|-------------------------------------------------|------------------------------------------| | [1004.2671] HES BLAGN, z<0.3 | Schechter: $\Phi_*\left(\frac{\lambda}{\lambda_*}\right)^\alpha\exp(-\lambda/\lambda_*)$ | $\alpha=-1.95$, $\lambda_*=0.28$; steady power-law rise, exponential cutoff | | [2511.14575] MaNGA Seyferts | Schechter (Netzer 2019, Kormendy–Ho) | $\alpha = -0.67^{+0.15}_{-0.17}$; $F_{\rm AGN}(>\!10^{-3})=0.078$ mass-indep. in SF, declining in quenched | | [1707.05323] Swift-BAT X-ray, z~0.1 | Broken power law: $\xi=\xi^*[(\lambda/\lambda^*)^{\delta_1}+(\lambda/\lambda^*)^{\delta_2}]^{-1}$ | X-ray: $\log\lambda^*=-1.84$, $\delta_1=0.47$, $\delta_2=2.53$; mass-independent, different for radio/X-ray modes | | [1801.07717] Host mass-dependent | Broken power law, 3 stellar mass bins | Low-mass SF: suppressed at $\lambda<0.1$; high-mass: broader ERDF; necessary for SFR–$L_X$ flatness | | [2311.08922] HSC+SDSS $z\sim4$ BLAGN | Mass-dependent log-normal/Schechter | $\log\lambda^*=0.40$, $\sigma=0.32$, $k_\lambda=-0.19$ (peak shifts lower at high $M_{\rm BH}$) | | [1511.01092] X-ray BLAGN, 14.75$ | Log-normal, $\sigma_m\approx0.39$ | $\langle\lambda\rangle\propto (1+z)^3$, peaks at $M_{\rm BH}\sim5\times10^{7}M_\odot$ | | [2210.08401] BASS DR2 obscured/unobsc. | Broken power law | Unobsc.: $\delta_1=0.16$, $\delta_2=0.22$, $\lambda^*=1.1$; Obsc.: $\delta_1=0.27$, $\delta_2=0.32$, $\lambda^*=10^{-1.75}$ | | [1010.0046] Theoretical, z ≲ 5 | Power-law time-weighted: $\zeta(\lambda)\sim\left(\lambda/\lambda_{\rm peak}\right)^{-\beta_l}\exp(-\lambda/\lambda_{\rm peak})$ | $\beta_l=0.3$, $\lambda_{\rm peak}=2.5$; $\eta_{\rm rad,0}=0.11$; $\tau_Q\gtrsim0.5$ Gyr | | [1605.01739] SDSS AGN (optical/X-ray) | Schechter: $\lambda^{-0.4}\exp(-\lambda/1)$ | Consistent with X-ray and obscured samples; log-normal artifact in SF galaxies due to selection | Characteristic results are that ERDFs for AGN are generally broad (0.3–0.8 dex in log λ), strongly rising toward lower λ, and display redshift and mass-dependence mainly in their normalization and turnover location. Intrinsic mass-independence of ERDFs for radiatively efficient and inefficient modes is supported in the local Universe [1707.05323, 2201.05603]. ## 6. Theoretical and Simulation Perspectives - **Hydrodynamical simulations** confirm log-normal forms for $P(\lambda|M_{\rm BH},z)$, with evolution of the mean and width tied to cosmological gas fraction and feedback processes [1201.5383, 1905.02732]. - **Continuity-equation and semi-analytic models** systematically link the ERDF to black hole mass/bulge buildup, radiative efficiency, and duty cycles [1010.0046, 1111.3574]. - **Eddington ratio–driven feedback models** predict power-law or broken-power-law ERDFs reflecting the self-regulated shut-off of fueling [1010.0046, 1006.3561]. - **Host and fueling mode dichotomy:** Physical bimodality in ERDFs—reflecting hot/cold accretion modes, jet/radiative feedback, or merger/secular fueling—naturally emerges in population synthesis [1707.05323, 2210.08401]. ## 7. Astrophysical Implications and Open Directions - **Obscuration and AGN Geometry:** Sharp transitions in ERDF shape between Type 1 and Type 2 AGN, with much steeper high-λ cutoffs for obscured systems, are predicted and observed, supporting radiation-regulated unification scenarios [2210.08401, 2201.05603]. - **AGN Duty Cycle and Cosmic Growth:** The fraction of time/mass SMBHs spend above fixed λ is a direct probe of cosmic AGN “on” fraction and is sensitive to underlying ERDF parameters and their evolution [2511.14575, 1905.02732]. - **Downsizing and Cosmic History:** The CSFR/ERDF link, population-wide suppression of high-λ activity at late times, and the changing locus of black hole growth are all set by the underlying ERDF [1111.3574, 2311.08922, 2002.08965]. - **Selection Effects and Completeness:** Accurate inference of the intrinsic ERDF, especially in the low-λ regime, is fundamentally limited by survey depth and sample biases; next-generation IR/X-ray and deep emission-line surveys are critical for comprehensive mapping [1605.01739, 2511.14575, 1905.02732]. The Eddington-ratio distribution thus encodes fundamental constraints on the stochasticity, regulation, and evolution of SMBH accretion and underpins both phenomenological and theoretical models of black hole–galaxy coevolution. Source: https://www.emergentmind.com/topics/eddington-ratio-distribution