The role of r-modes in pulsar spin-down, pulsar timing, and gravitational waves

We investigate the role of r-mode oscillations in pulsar spin-down and their implications for gravitational wave emission and pulsar timing analysis. Using a non-linear differential framework that includes r-mode contributions, we derive time-dependent solutions for rotational frequency and period evolution. These expressions are validated using observational data from the Crab pulsar with high precision. By analytically fitting braking indices and spin-down coefficients, we link measurable pulsar properties to gravitational wave signatures. Furthermore, we present closed-form expressions for neutron star compactness and tidal deformability using Lambert W and Lambert–Tsallis functions, enabling model-independent inferences from r-mode gravitational wave frequencies. Our results show that incorporating r-modes significantly improves the accuracy of spin-down models and continuous wave detectability, particularly through the inclusion of high-order frequency terms. This framework supports the modeling of timing residuals, glitch quantification, and gravitational wave constraints. Our findings have direct relevance for data analysis in ongoing and future gravitational wave observatories.