Highly compact neutron stars in scalar-tensor theories of gravity: Spontaneous scalarization versus gravitational collapse

Scalar-tensor theories of gravity are extensions of General Relativity (GR) including an extra, nonminimally coupled scalar degree of freedom. A wide class of these theories, albeit indistinguishable from GR in the weak field regime, predicts a radically different phenomenology for neutron stars, due to a nonperturbative, strong-field effect referred to as spontaneous scalarization. This effect is known to occur in theories where the effective linear coupling $\beta_0$ between the scalar and matter fields is sufficiently negative, i.e. $\beta_0 \lesssim -4.35$, and has been strongly constrained by pulsar timing observations. In the test-field approximation, spontaneous scalarization manifests itself as a tachyonic-like instability. Recently, it was argued that, in theories where $\beta_0>0$, a similar instability would be triggered by sufficiently compact neutron stars obeying realistic equations of state. In this work we investigate the endstate of this instability for some representative coupling functions with $\beta_0>0$. This is done both through an energy balance analysis of the existing equilibrium configurations, and by numerically determining the nonlinear Cauchy development of unstable initial data. We find that, contrary to the $\beta_0<0$ case, the final state of the instability is highly sensitive to the details of the coupling function, varying from gravitational collapse to spontaneous scalarization. In particular, we show, for the first time, that spontaneous scalarization can happen in theories with $\beta_0>0$, which could give rise to novel astrophysical tests of the theory of gravity.