Quasinormal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi></mml:math>-modes of anisotropic quark stars in full general relativity

We investigate the fundamental mode, or $f$-mode, oscillations of anisotropic quark stars within the framework of full general relativity. We consider two different equations of state (EOS), one is the MIT bag model EOS for noninteracting quark matter, and the other is EOS-A, which incorporates interquark interactions. Our study systematically examines the impact of pressure anisotropy on the equilibrium structure, as well as on the frequencies and damping times of $f$-mode oscillations in quark stars. Our results confirm that the $f$-mode frequency scales linearly with the square root of the average density, with anisotropy influencing both the slope and intercept of this relation. This behavior is consistent with our previous findings for neutron stars. The dependence of the $f$-mode frequency on total mass reveals distinct trends based on the relative dominance of tangential and radial pressure. When the tangential pressure exceeds the radial pressure, the frequency increases monotonically with mass, exhibiting rapid growth for massive quark stars. Conversely, when the radial pressure dominates, the frequency also increases with mass; however, in cases where the radial pressure is significantly greater than the tangential pressure, the frequency decreases as mass increases. Additionally, we find that, for low-mass quark stars, stronger tangential pressure leads to an increase in frequency, while beyond a threshold mass, a further increase in tangential pressure results in a decrease in frequency. For the chosen range of anisotropic strengths, the frequency varies between 1.3 and 2.3 kHz for the MIT bag EOS and between 1.8 and 3.4 kHz for EOS-A. Furthermore, we find that the normalized damping time follows a linear trend with compactness, with anisotropic strength affecting both the slope and intercept of this relation. For a fixed stellar mass, an increase in tangential pressure relative to radial pressure reduces the damping time, whereas a decrease in tangential pressure significantly increases it. The damping time ranges from 83 to 900 ms for the MIT bag EOS and from 60 to 761 ms for EOS-A. Finally, we present semiempirical expressions for both the frequency and damping time as functions of mass, radius, and anisotropic strength. The frequency exhibits a cubic polynomial dependence on anisotropy, while the damping time follows a quartic dependence. Our results provide new insights into the role of anisotropy in quark star oscillations and may aid in constraining the equation of state of dense matter through future gravitational wave observations.