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Scalar-tensor theories of gravity are extensions of General Relativity (GR)including an extra, nonminimally coupled scalar degree of freedom. A wide classof these theories, albeit indistinguishable from GR in the weak field regime,predicts a radically different phenomenology for neutron stars, due to anonperturbative, 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 timingobservations.  In the test-field approximation, spontaneous scalarization manifests itselfas a tachyonic-like instability. Recently, it was argued that, in theorieswhere $\beta_0>0$, a similar instability would be triggered by sufficientlycompact neutron stars obeying realistic equations of state. In this work weinvestigate the endstate of this instability for some representative couplingfunctions with $\beta_0>0$. This is done both through an energy balanceanalysis of the existing equilibrium configurations, and by numericallydetermining the nonlinear Cauchy development of unstable initial data. We findthat, contrary to the $\beta_0<0$ case, the final state of the instability ishighly sensitive to the details of the coupling function, varying fromgravitational 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 theoryof gravity.

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