What if the neutron star maximum mass is beyond $sim2.3 M_{odot}$?

By assuming the formation of a black hole soon after the merger event of GW170817, Shibata et al. updated the constraints on the maximum mass ($M_textrm{max}$) of a stable neutron star within $lesssim$ 2.3 $M_{odot}$, but there is no solid evidence to rule out $M_textrm{max}>2.3~M_{odot}$ from the point of both microphysical and astrophysical views. In order to explain massive pulsars, it is naturally expected that the equation of state (EOS) would become stiffer beyond a specific density. In this paper, we consider the possibility of EOSs with $M_textrm{max}>2.3~M_{odot}$, investigating the stiffness and the transition density in a polytropic model. Two kinds of neutron stars are considered, i.e., normal neutron stars (the density vanishes on gravity-bound surface) and strange stars (a sharp density discontinuity on self-bound surface). The polytropic model has only two parameter inputs in both cases: ($rho_{rm t}$, $gamma$) for gravity-bound objects, while ($rho_{rm s}$, $gamma$) for self-bound ones, with $rho_{rm t}$ the transition density, $rho_{rm s}$ the surface density and $gamma$ the polytropic exponent. In the matter of $M_textrm{max}>2.3~M_{odot}$, it is found that the smallest $rho_{rm t}$ and $gamma$ should be $sim 0.50~rho_0$ and $sim 2.65$ for normal neutron stars, respectively, whereas for strange star, we have $gamma > 1.40$ if $rho_{rm s} > 1.0~rho_0$ and $rho_{rm s} < 1.58~rho_0$ if $gamma <2.0$ ($rho_0$ is the nuclear saturation density). These parametric results could guide further research of the real EOS with any foundation of microphysics if a pulsar mass higher than $2.3~M_{odot}$ is measured in the future. We also derive rough results of common neutron star radius range, which is $9.8~rm{km} < R_{1.4} < 13.8~rm{km}$ for normal neutron stars and $10.5~rm{km} < R_{1.4} < 12.5~rm{km}$ for strange stars.