Structural response of neutron stars to rapid rotation and its impact on the braking index

Pulsars are rotating neutron stars that are observed to be slowing down, implying a loss of their kinetic energy. There can be several different physical mechanisms involved in their spin-down process. The properties of fast-rotating pulsars depend on the nature of the neutron star matter, which can also affect the spin-down mechanisms. In this work, we examine three different physical phenomena contributing to the spin-down: magnetic dipole radiation, gravitational mass quadrupole radiation due to the ``mountain" formation, gravitational mass current quadrupole radiation or the r-modes, and calculate the expressions for the braking indices due to all of them. We have also considered jointly the implications of the uncertainties of the equation of the state of neutron star matter and rapid rotation on the braking indices corresponding to the aforementioned processes and their combinations. In all cases, the rapid rotation results in a departure from the standard values in the literature for the braking index when the rotational effects are ignored. If generated with a saturation amplitude within the range of $10^{-4} - 10^{-1}$, the r-mode oscillations dominate the spin-down of millisecond pulsars. We also explore the braking index in the context of millisecond magnetars. This study examines two braking index measurements in the context of newly born millisecond magnetars from two observed short $\gamma$-ray bursts. The measured braking indices for these objects are consistent with our estimation, which allows us to conclude that the spin frequency of the remnants is within the range of $\sim 550-850$ Hz.