Using the first principles calculations we have studied the vibrational modes and Raman spectra of a (10, 10) single-walled carbon nanotube (SWNT) bundle under hydrostatic pressure. Detailed analysis shows that the original radial breathing mode (RBM) of the SWNT bundle disappears after the structural phase transition (SPT). And significantly a RBM-like mode appears at about 509 cm^{-1}, which could be considered as a fingerprint of the SPT happened in the SWNT bundle, and further used to determine the microscopic structure of the bundle after the SPT.