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Implications of Joint Spectral Analysis of Gamma-Ray Bursts detected by Fermi Large Area Telescope and Gamma-ray Burst Monitor on Phenomenological Correlations

Gamma-ray bursts (GRBs) have emerged as powerful cosmological probes for exploring the distant Universe, owing to their immense luminosities and detectability at high redshifts. Several empirical correlations have been established, particularly involving their energy properties. This work aims to enhance the precision of these correlations through joint spectral analysis, focusing on reducing uncertainties in both the spectral indices and the peak energy ($E_{\rm p}$) derived from spectral fitting. We extend previous studies using both traditional and novel spectral models, utilizing a sample of 37 GRBs observed by the Fermi Gamma-ray Burst Monitor (GBM) and Large Area Telescope (LAT), incorporating the LAT Low-Energy (LLE) technique, over the period 2008-2024. Our analysis compares results from joint fits (GBM-LAT-LLE) against those from GBM-only fits. The study focuses on fitting time-integrated ${\rm T}_{90}$ and peak flux in the rest frame. Among the observable phenomenological correlations, we revisit the Amati and Yonetoku relations: the Amati relation links the intrinsic peak energy ($E_{i,\rm p}$) to the total isotropic energy ($E_{\rm iso}$) emitted during ${\rm T}_{90}$, while the Yonetoku relation connects $E_{i,\rm p}$ to the isotropic luminosity ($L_{\rm iso}$). Refining these correlations aims to deepen our understanding of GRB energetics and improve the precision of cosmological parameter estimates derived from GRB observations.