We investigate in this work two different types of instabilities that set limits on the rotation rates of neutron (compact) stars. The first one is that caused by rotation at the Kepler frequency, at which mass shedding at the star's equator sets in. The second limit is set by instabilities driven by the growth of gravitational radiation-reaction (GRR) driven f -modes of order m (= 2, 3, … ), which are moderated by shear and bulk viscosity. The calculations are performed for two relativistic models for the nuclear equation of state, DD2 and ACB4. The latter accounts for a phase transition that gives rise to the existence of so-called mass-twin compact stars. Our results confirm that the stable rotation periods of cold neutron stars are determined by the m = 2 modes and that these modes are excited at rotation periods between 1 and 1.4 ms (20-30% above the Kepler periods of these stars). The situation is reversed in hot neutron stars where bulk viscosity damps the GRR modes, pushing the excitation period of the f -mode instability to values below the Kepler period. For cold mass-twin compact stars, we find that the m = 2 instability sets in at rotation periods between 0.8 and 1 ms (25-30% below the Kepler period). This feature may allow one to distinguish conventional neutron stars from their possibly existing mass-twin counterparts observationally, provided the f -mode instability, which is expected to compete with the r-mode instability, sets the limit on stable rotation of compact stars.

It is generally accepted that the limit on the stable rotation of neutron stars is set by gravitational-radiation reaction (GRR) driven instabilities, which cause the stars to emit gravitational waves that carry angular momentum away from them. The instability modes are moderated by the shear viscosity and the bulk viscosity of neutron star matter. Among the GRR instabilities, the f-mode instability plays a historically predominant role. In this work, we determine the instability periods of this mode for three different relativistic models for the nuclear equation of state (EoS) named DD2, ACB4, and GM1L. The ACB4 model for the EoS accounts for a strong first-order phase transition that predicts a new branch of compact objects known as mass-twin stars. DD2 and GM1L are relativistic mean field (RMF) models that describe the meson-baryon coupling constants to be dependent on the local baryon number density. Our results show that the f-mode instability associated with m=2 sets the limit of stable rotation for cold neutron stars (T≲1010 K) with masses between 1M⊙ and 2M⊙. This mode is excited at rotation periods between 1 and 1.4 ms (∼20% to ∼40% higher than the Kepler periods of these stars). For cold hypothetical mass-twin compact stars with masses between 1.96M⊙ and 2.10M⊙, the m=2 instability sets in at rotational stellar periods between 0.8 and 1 millisecond (i.e., ∼25% to ∼30% above the Kepler period).