Measuring cosmic dipole with the GRB luminosity-time relation

We present a new analysis of cosmic dipole anisotropy using gamma-ray bursts (GRBs) as high-redshift standardizable candles. GRBs are ideal probes for testing the cosmological principle thanks to their high luminosity, wide redshift range, and nearly isotropic sky coverage. For the first time, we employ the luminosity-time (L-T) relation, known in the literature as the bidimensional X-ray Dainotti relation, corrected for redshift evolution, to standardize a sample of 176 long GRBs detected by \textit{Swift}. We test for dipolar modulations in the GRB Hubble diagram using both the Dipole Fit Method and a new approach introduced here, the Anisotropic Residual Analysis Method. Both methods yield consistent results: a dipole amplitude of $A_d \simeq 0.6 \pm 0.2$ pointing towards (RA, DEC) $\approx (134^\circ \pm 30^{\circ}, -36^\circ \pm 21^{\circ})$ (equatorial coordinates). As shown in the Appendix, this corresponds to a boost velocity of the observer with respect to the GRB rest-frame in the antipodal direction from the dipole direction. Extensive isotropy tests and 20,000 Monte Carlo simulations confirm that the detected signal cannot be explained by chance alignments or by the angular distribution of the GRB sample. We also show how, by incorporating a dipole term, residual correlations are eliminated, showing that the dipole model provides a better fit than standard isotropic $\Lambda$CDM.