The Origin of Cross-Energy-Similar FRED Profiles in Gamma-Ray Bursts Pulses

1. Problem Statement

Gamma-ray bursts (GRBs) often exhibit a Fast-Rise-Exponential-Decay (FRED) light curve profile. While simple FRED profiles were traditionally thought to result from a single episode of energy injection from a central engine, recent observations (e.g., GRB 230307A) revealed that these profiles are composite, consisting of many short-timescale individual pulses.

This creates a theoretical dilemma:

• Energy Dependence: The overall profile exhibits "cross-energy similarity," where peak time delays ($t_p$) and pulse widths ($t_w$) evolve with energy as power laws ($t_p, t_w \propto E^{-\alpha}$), suggesting a unified, coherent mechanism rather than a random superposition of independent shocks.
• Composite Nature: The presence of individual sub-pulses suggests a complex, multi-step dissipation process, contradicting models that assume a single, instantaneous dissipation event.
• Existing Models: Traditional models (e.g., internal shocks, standard ICMART) struggle to simultaneously explain the smooth cross-energy similarity and the composite, multi-pulse nature of these bursts. Previous "toy models" could explain the similarity but lacked a self-consistent physical derivation of the radiation spectrum.

2. Methodology

The authors propose a self-consistent physical model based on the dynamics of magnetic reconnection in a magnetically dominated relativistic jet.

• Physical Framework:
  • Trigger: An impulsive energy injection from the central engine creates a localized perturbation in a magnetically dominated jet shell.
  • Propagation: This perturbation propagates outward in the jet front as Alfvén waves within a geometrically thin shell. Unlike standard models where reconnection seeds are simultaneous, here the perturbation spreads in concentric circles (polar direction), sequentially triggering local magnetic reconnection at different latitudes.
  • Radiation Mechanism: At each triggered "ring" (local radiation spot), electrons are accelerated to a power-law distribution via magnetic reconnection. These electrons cool via synchrotron radiation and adiabatic expansion as the jet moves forward.
• Mathematical Derivation:
  • The authors solve the electron cooling and radiation equations at local spots to derive the emissivity function $f(\nu'_{emt}, t; t_{inj})$.
  • They integrate the specific flux over the expanding radiation zone, accounting for the Doppler factor ($D$), Lorentz factor ($\Gamma$), and the geometric expansion of the radiation area ($R^2 \sin \theta$).
  • The observed flux $F_{\nu_{obs}}(t_{obs})$ is calculated by integrating over the injection time $t_{inj}$, considering the competition between the expanding radiation area (rising term) and the decreasing Doppler factor (decaying term).
• Simulation & Population Study:
  • Single Burst Simulation: Parameters (e.g., $\Gamma=180$, $B'_0=1$G, $p=2.8$) were chosen to simulate a single FRED GRB across multiple energy bands.
  • Monte Carlo Population: A population of 1,000 GRBs was simulated with distributed physical parameters ($\Gamma$, $r_0$, $B'_0$, electron limits $\gamma_m, \gamma_M$) to establish empirical relationships between physical and observational parameters.
  • Data Catalog: A new catalogue of 17 "general single FRED GRBs" was constructed from Fermi/GBM data to compare with simulations.

3. Key Contributions

1. Self-Consistent Physical Model: The paper moves beyond phenomenological "toy models" by deriving the light curve and spectrum from first principles (electron cooling equations and magnetohydrodynamic propagation), providing a fully self-consistent explanation for single FRED GRBs.
2. Resolution of the Dilemma: The model successfully reconciles the "composite" nature of the light curve (many sub-pulses) with the "cross-energy-similar" global profile. The global FRED shape arises from the sequential triggering of reconnection rings, while the sub-pulses correspond to discrete radiation spots within those rings.
3. Empirical Relations: The authors derived a linear empirical relation (Eq. 3) linking physical parameters (Lorentz factor, magnetic field, injection radius) to observables ($T_{90}$, $E_p$, spectral indices), allowing for the estimation of physical properties from observations with reduced computational cost.
4. Reinterpretation of Central Engine Activity: The study argues that the GRB light curve is a convolution of the central engine's impulsive activity and the jet's internal response, rather than a direct map of the engine's activity history.

4. Key Results

• Cross-Energy Similarity: The simulation naturally reproduces the power-law evolution of peak time delay ($t_p$) and pulse width ($t_w$) with energy. The fitted power indices for both were $\approx -0.37$, matching observational data and demonstrating that the profiles in different bands are time-stretched copies of each other.
• Spectral Evolution: The model reproduces the typical hard-to-soft spectral evolution. The peak energy ($E_p$) decreases over time due to the decreasing Doppler factor as the radiation zone expands to higher latitudes.
• Local Intensity Tracking: The model explains the observation that $E_p$ tracks the instantaneous intensity on short timescales (a feature seen in GRB 230307A) as a result of electron cooling in local radiation spots.
• Sub-pulse Characteristics: The individual pulses within the FRED profile are aligned across energy bands and show a systematic trend of increasing duration over time, consistent with recent high-time-resolution observations.
• Duration Dichotomy: The Monte Carlo simulations show that single FRED GRBs can have both short ($T_{90} < 2$s) and long ($T_{90} > 2$s) durations. This suggests that long-duration Type-I GRBs (like GRB 211211A and 230307A) can arise from impulsive central engine activity, challenging the strict merger-short/collapsar-long dichotomy.
• Observational Agreement: The simulated distribution of $t_p-E$ and $t_w-E$ indices clusters along the diagonal line (where indices are identical), matching the observed catalogue of single FRED GRBs.

5. Significance

• Mechanism Identification: The paper provides strong evidence that the peculiar cross-energy-similar FRED profile is a signature of magnetically dominated jets where dissipation is driven by propagating magnetic perturbations (Alfvén waves) rather than random internal shocks.
• Central Engine Constraints: By decoupling the light curve morphology from the direct activity history of the central engine, the model allows for a more accurate reconstruction of the central engine's true activity timescale. It implies that long GRBs do not necessarily require a long-lived central engine (e.g., a collapsar accretion disk) but could result from a short impulsive event in a magnetically dominated jet.
• Predictive Power: The established empirical relations provide a practical tool for future GRB studies to infer physical jet parameters (magnetic field strength, Lorentz factor) directly from light curve and spectral data.
• Unification: The model unifies the understanding of smooth FRED profiles and complex, variable light curves, suggesting that the difference lies primarily in the number of local radiation spots (smoothness) and the presence of mini-jets, rather than fundamentally different physical mechanisms.