The possible equation of state of dark matter in low surface brightness galaxies

The observed rotation curves of low surface brightness (LSB) galaxies play an essential role in studying dark matter, and indicate the existence of a central constant density dark matter core. However, the cosmological N-body simulations of cold dark matter predict an inner cusped halo with a power-law mass density distribution, and cannot reproduce a central constant-density core. This phenomenon is called cusp-core problem. When dark matter is quiescent and satisfies the condition for hydrostatic equilibrium, the equation of state can be adopted to obtain the density profile in the static and spherically symmetric space-time. To address the cusp-core problem, we assume that the equation of state is independent of the scaling transformation. Its lower order approximation for this type of equation of state can naturally lead to a special case, i.e., begin{document}$p=zetarho+2epsilon V_{rm rot}^{2},rho$end{document}, where p and begin{document}$rho$end{document} represent the pressure and density, respectively, begin{document}$V_{rm rot}$end{document} depicts the rotation velocity of galaxy, and begin{document}$zeta$end{document} and begin{document}$ epsilon$end{document} are positive constants. It can obtain a density profile that is similar to the pseudo-isothermal halo model when begin{document}$epsilon$end{document} is approximately 0.15. To obtain a more universally used model, let the equation of state include the polytropic model, i.e. begin{document}$p= frac{zeta}{rho_{0}^{s}}rho^{1+s}+ 2epsilon V_{rm rot}^{2},rho$end{document}, from which we can obtain other types of density profiles, such as the profile that is nearly same as the Burkert profile, where s and begin{document}$rho_{0}$end{document} are positive constants.