Abstract
Internal structure and macroscopic properties of neutron stars are strongly correlated with the equation of state (EOS) of dense matter, while the EOS remains exceedingly uncertain especially at high densities. The observed massive neutron star with gravitation mass of $\ensuremath{\approx}2\phantom{\rule{4pt}{0ex}}{M}_{\ensuremath{\bigodot}}$ have imposed strong constraints on the EOS of super-dense matter. The upper limit of the speed of sound, as a crucial quantity to characterize the stiffness of the EOS, has significant influence on the maximum mass of a neutron star. The main propose of this work is to probe the possible lower bound for the upper limit of the speed of sound, where being coincident with the observations for massive neutron stars is considered. By employing a set of heterogeneous EOSs and through a constructed EOS model to describe the high-density part of the employed EOSs, we conclude that the upper limit of the speed of sound can be as low as 0.5386 $c$ (where $c$ is the speed of light). Furthermore, to eliminate the effect of the model dependence for the EOSs, we adopt a set of parameterized and model-independent EOSs to investigate this problem. As a result, a lower bound 0.5749 $c$ of the upper limit of the speed of sound is obtained, which is very close to the free-quark-matter's speed of sound, $\frac{1}{\sqrt{3}}c$. In addition, we semiempirically analyze the characteristics of the profile of neutron star $M\text{\ensuremath{-}}R$ relations and find out that the profiles of $M\text{\ensuremath{-}}R$ relations mainly depend on the speed of sound of the central density of neutron star sequences in the mass range around $0.3\phantom{\rule{0.28em}{0ex}}\text{to}\phantom{\rule{0.28em}{0ex}}1\phantom{\rule{0.16em}{0ex}}{M}_{\ensuremath{\bigodot}}$. However, the $M\text{\ensuremath{-}}R$ profile also can be explained by the universal relation proposed by Lattimer et al. Astroph. J. 550, 426 (2001), where the case with extremely soft EOS in ${\ensuremath{\rho}}_{0}\phantom{\rule{0.28em}{0ex}}\text{to}\phantom{\rule{0.28em}{0ex}}2{\ensuremath{\rho}}_{0}$ should be excluded.
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