In this work, we make the first step to derive nonradial pulsation equations in extra dimensions and investigate how the $f$- and $p_1$-mode frequencies of strange quark stars, within the Cowling approximation, change with the number of dimensions. In this regard, the study is performed by solving numerically the nonradial pulsation equations, adjusted for a $d$-dimensional spacetime ($d\ge 4$). We connect the interior to a Schwarzschild-Tangherlini exterior metric and analyze the $f$- and $p_1$- mode frequencies. We found that the frequencies could become higher than those found in four-dimensional spacetime. The $f$-mode frequency is essentially constant and only for large gravitational radius values grows monotonically and fast with the gravitational radius. In a gravitational radius range, where $f$-mode frequencies are constant, they increase for spacetime dimensions $4\le d\le 6$ and decrease for $d\ge 7$. Regarding $p_1$-mode frequencies they are always larger for higher dimensions and decay monotonically with the increase of the gravitational radius. In extra dimensions, as it happens for four-dimensional spacetime, we found $p_1$-mode frequencies are always larger than the $f$-modes ones. In the Newtonian gravity, for a homogeneous star in $d$ dimensions, we observe that the $f$-mode eigenfrequencies are constant and given by the relation ${\omega }^2=lM{G}_d/R^{d-1}$, where $l$ represents the spherical harmonic index, $M{G}_d$ is the total star mass and $R$ the stellar radius. For some gravitational radius interval, we show that a homogeneous star in Newtonian gravity is a good approximation to investigate the $f$-mode frequency of strange stars in the relativistic frame. In each dimension considered, we find that the $f$-mode frequencies of strange stars are essentially constant since they depend on the average star energy density that is almost constant as a function of the total star mass. Moreover, for a fixed energy density, we also find that the $f$-mode frequency changes with the volume of the unitary sphere in the $d-1$ dimension, which attains its maximum value at $d=6$. In neutron stars in four dimensions, where the average energy density of the star increases with the central energy density, the $f$-mode frequencies will increase with the star mass. Thus, the possibility to measure in gravitational wave detectors the $f$-mode oscillation frequency coming from compact stars with different pulsar masses and observe almost constant frequency values, for $d=4$, in the range $f\sim 2–3[\mathrm{kHz}]$ with $M\le 1.8M_{\odot }$, it would be a good sign of the existence of strange quark stars that still lack an astronomical confirmation. Finally, if the $f$-mode frequencies are still constant and different from the range values of $d=4$ for larger total masses, it would be evidence that quarks can propagate in extra spacetime dimensions and strange quark stars in $d$ dimension could exist.

• Received 1 June 2020
• Accepted 16 September 2020

DOI:https://doi.org/10.1103/PhysRevD.102.084014