The Role of $r$-Modes in Pulsar Spindown, Pulsar Timing and Gravitational Waves (2307.11270v3)

Abstract: We investigate the role of r-modes in the spin-down of pulsars, focusing on their implications for pulsar timing and gravitational wave emissions. Our study employs a non-linear differential equation incorporating the contribution of r-modes to derive time-dependent rotational frequency and period functions. This model is validated against observational data from the Crab pulsar, demonstrating a high degree of accuracy. By fitting the braking indices and spin-down coefficients, we establish direct and analytical relationships between observable pulsar properties and weak gravitational wave signals. We also derive analytical expressions for neutron star compactness and tidal deformability using Lambert W solutions, independent of the equation of state (EoS). These solutions provide new insights into the mathematical relationships between physical quantities, constraining the parameter space for r-mode gravitational wave frequency searches. Our results show that incorporating r-modes significantly enhances our ability to measure the neutron star EoS and predict pulsar age, rotational velocity, and gravitational wave frequencies. The seventh-order approximation used in our model is essential for accurately capturing the contributions of r-modes to the spin-down process. This framework can be applied to model pulsar timing residuals, account for glitches, and improve the detection and analysis of continuous gravitational waves from pulsars. With the advent of next-generation gravitational wave detectors, our findings offer promising prospects for disentangling individual events from the stochastic gravitational wave background, advancing our understanding of neutron star interiors and their dynamic processes.