# Steep Decay Phase Shaped by the Curvature Effect. II. Spectral Evolution

## TL;DR

This paper derives an analytical formula to describe the spectral index evolution during the steep decay phase of gamma-ray bursts, considering different intrinsic spectra and energy bands, enhancing understanding of spectral behavior influenced by the curvature effect.

## Contribution

It introduces a simple analytical model for spectral evolution in the steep decay phase, accounting for various intrinsic spectra and energy regimes, which was not previously available.

## Key findings

- Spectral index evolution can be linear or logarithmic depending on the spectrum and energy band.
- When the break energy exceeds 10 keV, spectral evolution resembles that of a CPL spectrum.
- If the break energy is below 0.3 keV, the spectral index remains constant.

## Abstract

We derive a simple analytical formula to describe the evolution of spectral index $\beta$ in the steep decay phase shaped by the curvature effect with assumption that the spectral parameters and Lorentz factor of jet shell is the same for different latitude. Here, the value of $\beta$ is estimated in 0.3$-$10keV energy band. For a spherical thin shell with a cutoff power law (CPL) intrinsic radiation spectrum, the spectral evolution can be read as a linear function of observer time. For the situation with Band function intrinsic radiation spectrum, the spectral evolution may be complex. If the observed break energy of radiation spectrum is larger than 10keV, the spectral evolution is the same as that shaped by jet shells with a CPL spectrum. If the observed break energy is less than 0.3keV, the value of $\beta$ would be a constant. Others, the spectral evolution can be approximated as a logarithmal function of the observer time in generally.