Cumulative Burst Luminosity Distribution

• Cumulative burst luminosity distribution is a statistical tool that quantifies the frequency of astrophysical transients exceeding a set luminosity threshold.
• It employs various models like log-normal, power-law, and Schechter functions while incorporating corrections for survey completeness and instrumental biases.
• This method underpins population synthesis studies by constraining progenitor properties and physical processes in GRBs, SFXTs, and FRBs.

The cumulative burst luminosity distribution quantifies the integrated number or probability of astrophysical transient events (bursts, flares, pulses) with luminosity exceeding a given threshold. It is a central tool for characterizing populations of phenomena such as gamma-ray bursts (GRBs), supergiant fast X-ray transients (SFXTs), and fast radio bursts (FRBs), and for constraining underlying physical processes and selection biases. The precise functional form and physical origin of these cumulative distributions vary with source class, survey completeness, instrumental sensitivity, and population synthesis assumptions.

1. Mathematical Formalism and Principal Models

The cumulative luminosity distribution is defined as

$N(>L) = \int_L^\infty \phi(L')\,dL'$

where $\phi(L)$ is the differential luminosity function (LF), representing the number per unit luminosity. The LF can be log-normal, Schechter-like (power law with exponential cutoff), broken/triple power-law, or a nonparametric empirical fit. For example, in long GRBs as modeled by a multivariate log-normal world model (Shahmoradi, 2012), the marginal distribution of isotropic peak luminosity is

$$p(L_\mathrm{iso}) = \frac{1}{L_\mathrm{iso} \sqrt{2\pi\sigma_L^2}} \exp\left[ -\frac{(\ln L_\mathrm{iso} - \mu_L)^2}{2\sigma_L^2} \right]$$

with cumulative distribution

$$F(L) = \frac{1}{2} + \frac{1}{2} \,\mathrm{erf}\left( \frac{\ln L - \mu_L}{\sqrt{2}\,\sigma_L} \right)$$

For pure power-law LF, $\phi(L) \propto L^{-\alpha-1}$, so $N(>L) \propto L^{-\alpha}$.

Broken power-law models, Schechter-type functions, and nonparametric product estimators are often adopted to capture observed curvature and the presence of "knees" (breaks), as in SFXTs (Bozzo et al., 2014, Paizis et al., 2014), Swift/GRB pulses (Amaral-Rogers et al., 2016), and optical LFs (Cui et al., 2014).

Model Type	Analytic CDF Example	Key Parameters
Log-normal	$F(L) = ½ + ½\,\mathrm{erf}(...)$	$\mu_L, \sigma_L$
Power-law	$N(>L) ∝ L^{-α}$	$\alpha$
Schechter	$N(>L) ∝ \gamma(α+1, L/L_*)$	$\alpha, L_*$
Broken PL	Piecewise $N(>L)$ (see below)	$\alpha_1,\alpha_2,L_b$

2. Population Synthesis and Survey Corrections

Astrophysical burst populations require accounting for cosmological effects, survey completeness, and instrumental selection biases. For GRBs, the world model integrates over redshift, star formation rate history $\rho(z)$, comoving volume element $dV/dz$, and detection efficiency $\eta(L, z)$ (Shahmoradi, 2012). For FRBs, corrections for beam pattern, $V/V_\mathrm{max}$ weighting, and dispersion-measure-inferred redshift uncertainties are essential (Arcus et al., 18 Aug 2024, Cui et al., 2022). Properly constructed, the observed cumulative distribution, $N_\mathrm{obs}(>L)$, can be compared directly to theoretical $N_\mathrm{model}(>L)$ after folding in survey selection functions.

In SFXTs and HMXBs, cumulative luminosity distributions depend critically on soft-X versus hard-X survey thresholds, with soft-X monitoring (Swift/XRT) revealing sub-luminous knees missed by hard-X instruments (Bozzo et al., 2014).

3. Empirical Fits and Best-Fit Parameters

Numerous studies have performed maximum-likelihood or nonparametric fits to cumulative burst luminosity functions for different populations and surveys:

• Long GRBs: log-normal CDF with $\mu_L=52.53\pm0.22$, $\sigma_L=0.50\pm0.06$ (Shahmoradi, 2012); nonparametric broken power law with break $L_b\sim10^{51}$ erg s$^{-1}$ (Dong et al., 2021); triple/broken power-law with $\delta\sim1.9$ evolution index (Lan et al., 2021).
• SFXTs: Power-law CDFs with slopes $\alpha\sim1.3$–$2.7$ and knees $L_\mathrm{knee}\sim10^{34}$–$10^{35}$ erg s$^{-1}$ (Bozzo et al., 2014, Paizis et al., 2014).
• FRBs: Power-law slope $\gamma\approx-1.9$ for population energy distribution, Schechter cutoff $E_\mathrm{max}$; log-normal CDF for CHIME Catalog with different $\mu, \sigma$ for repeater/non-repeater populations (Arcus et al., 18 Aug 2024, Cui et al., 2022).

Note that single power-law LFs for prompt/optical emission in GRBs are statistically rejected in favor of Schechter, ERPLD, or broken power-law forms (Cui et al., 2014).

Population	Preferred Model	Key Fit Parameters	Reference
LGRB (BATSE)	Multivariate log-norm	$\mu_L=52.53$, $\sigma_L=0.50$	(Shahmoradi, 2012)
LGRB (Swift)	BPL/TPL, $\delta$ evo	$\delta\approx1.9$, $L_{c}\sim10^{51.6}$	(Lan et al., 2021)
SFXT	Power-law/broken PL	$\alpha_{low}\sim0.6$, $L_\mathrm{knee}$	(Bozzo et al., 2014)
FRB (ASKAP)	Power-law/Schechter	$\gamma=-1.96$/$E_\mathrm{max}=6.3\times10^{25}\,$J Hz$^{-1}$	(Arcus et al., 18 Aug 2024)
FRB (CHIME)	Log-normal	$\mu=101.25$, $\sigma=1.38$	(Cui et al., 2022)

4. Physical Interpretation and Origin

The shape and breaks in cumulative luminosity distributions encode physical constraints:

• In GRBs, the log-normal distribution reflects multiplicative stochasticity in engine, jet collimation, and radiative processes (Shahmoradi, 2012). Internal-shock models predict a broken power-law LF, with a bright-end slope set by the energy injection distribution and a faint-end slope due to low-$\kappa$ (Lorentz factor contrast) inefficiency (Zitouni et al., 2010).
• SFXTs exhibit power-law CDFs and low duty cycles, interpreted as self-organized criticality in quasi-spherical settling accretion, magnetic gating, or wind clump instability (Bozzo et al., 2014, Paizis et al., 2014).
• FRBs display pure or broken power-law $(N(>E)\propto E^{\gamma+1})$, log-normals, and Schechter-type downturns. Magnetar-based models posit broken power-law energy distributions with $\sim$2–3 dex energy width, low-energy cutoffs, and separate channels for repeaters versus non-repeaters (Wadiasingh et al., 2019).
• The position of knees, slopes, and normalization constants encode characteristic scales such as the mean mass accretion rate (SFXTs), magnetar flare energetics (FRBs), and evolutionary history (GRB break evolution).

5. Impact of Selection Effects and Instrumental Biases

Instrumental sensitivity, energy-band coverage, and survey strategy induce substantial distortions in observed cumulative distributions:

• Hard X-ray instruments (INTEGRAL, Swift/BAT, HETE-2) skew SFXT distributions toward high luminosity, missing low-level states; only soft X-ray (XRT) resolves the full dynamic range and properly locates $L_\mathrm{knee}$ (Bozzo et al., 2014).
• Redshift incompleteness and flux limits affect GRB samples; completeness in $z$ and flux matters more than modest instrumental thresholds (Dong et al., 2021, Lan et al., 2021).
• For FRBs, beam pattern, $V/V_\mathrm{max}$, and DM-$z$ relation uncertainties must be modeled; host-galaxy identification bias can artificially enhance apparent low-$E$ excess or high-$E$ cutoff (Arcus et al., 18 Aug 2024, Cui et al., 2022).

The propagation of cosmological and instrumental error (e.g. DM-$z$ scatter), when less than $40\%$, leaves the log-normal fit parameters and cumulative distribution robust within $5\%$ variation (Cui et al., 2022).

6. Astrophysical and Observational Implications

Cumulative burst luminosity distributions serve as benchmarks for population synthesis, constraining physical parameters, progenitor populations, accretion physics, and event rates. Strong correlations between burst properties (e.g. $E_{\mathrm{iso}}$–$E_{\mathrm{pk,z}}$, $L_{\mathrm{iso}}$–$T_{90,\mathrm{z}}$) emerge naturally within multivariate distribution fitting (Shahmoradi, 2012). The detection of knees, cutoffs, and changes of slope mark transitions in physical regimes: e.g. Compton-cooling thresholds in SFXTs, magnetospheric breakdown energies for FRBs, or shell dynamics for GRBs (Bozzo et al., 2014, Wadiasingh et al., 2019). "Flattening" or roll-off at low flux/energy (rate saturation) provides an observable test for model predictions, especially in next-generation surveys with improved sensitivity (Wadiasingh et al., 2019, Arcus et al., 18 Aug 2024).

Future empirical work with larger, more complete samples will refine these cumulative distributions, break degeneracies between functional forms (e.g. power-law vs. Schechter), and constrain evolutionary effects. Comprehensive error propagation, multi-band coverage, and physically motivated population synthesis will be required to translate observed cumulative luminosity distributions into robust constraints on underlying physical processes, event rates, and progenitor demographics.