Light curves from rapidly rotating neutron stars

We calculate light curves produced by a hot spot of a rapidly rotating neutron star, assuming that the spot is perturbed by a core $r$-mode, which is destabilized by emitting gravitational waves. To calculate light curves, we take account of relativistic effects such as the Doppler boost due to the rapid rotation and light bending assuming the Schwarzschild metric around the neutron star. We assume that the core $r$-modes penetrate to the surface fluid ocean to have sufficiently large amplitudes to disturb the spot. For a $l'=m$ core $r$-mode, the oscillation frequency $\omega\approx2m\Omega/[l'(l'+1)]$ defined in the co-rotating frame of the star will be detected by a distant observer, where $l'$ and $m$ are respectively the spherical harmonic degree and the azimuthal wave number of the mode, and $\Omega$ is the spin frequency of the star. In a linear theory of oscillation, using a parameter $A$ we parametrize the mode amplitudes such that ${\rm max}\left(|\xi_\theta|,|\xi_\phi|\right)/R=A$ at the surface, where $\xi_\theta$ and $\xi_\phi$ are the $\theta$ and $\phi$ components of the displacement vector of the mode and $R$ is the radius of the star. For the $l'=m=2$ $r$-mode with $\omega=2\Omega/3$, we find that the fractional Fourier amplitudes at $\omega=2\Omega/3$ in light curves depend on the angular distance $\theta_s$ of the spot centre measured from the rotation axis and become comparable to or even larger than $A\sim0.001$ for small values of $\theta_s$.