R-mode frequencies of rapidly and differentially rotating relativistic neutron stars

R-modes of neutron stars could be a source of gravitational waves for ground based detectors. If the precise r-mode frequency sigma in an inertial frame is known, guided gravitational wave searches with enhanced detectability are possible. Because of its physical importance, many authors have calculated the r-mode frequency. For the dominant mode, the associated gravitational wave frequency is 4/3 times the angular velocity of the star Omega, subject to various corrections of which relativistic and rotational corrections are the most important. This has led several authors to investigate the dependence of the r-mode frequency on factors such as the relativistic compactness parameter M/R and the angular velocity of stars with different equations of state. The results found so far, however, are almost independent of the equation of state. Here, we investigate the effect of rapid rotation and differential rotation on sigma. We work with a series of rotating neutron stars with polytropic equations of state, and we perform the time evolution of the linear perturbation equations in the relativistic Cowling approximation with a finite-difference code. We find that rotational effects in the r-mode frequency can be larger than relativistic effects for rapidly spinning stars with low compactness, reducing the observed frequency. Differential rotation also acts to decrease sigma, but its effect increases with the compactness. The results presented here are relevant to the design of gravitational-wave and electromagnetic r-mode searches.