In the case of neutron stars, the second-order scalar-tensor theory described by Damour and Esposito-Farèse  leads to a non-perturbative effect known as spontaneous scalarization (similar to the spontaneous magnetization in ferromagnetism). For significantly large negative values of the parameter , there is a range of neutron-star masses for which it becomes energetically favorable for the scalar field to acquire high values inside the neutron star and affect its structure significantly compared to the general relativistic predictions. An example of the mass-radius relation for neutron stars in a second-order scalar-tensor theory with = –8 is shown in Figure 13.
The properties and stability of scalarized neutron stars have been studied extensively in the literature [39, 139, 70]. For the purposes of tests of strong-field gravity, the coupling of matter with the gravitational field and the external spacetimes of scalar stars are so different compared to their general relativistic counterparts that large negative values of can be firmly excluded with current observations of binary stellar systems that harbor radio pulsars (see Figure 15).
As expected, weak-field tests bound significantly the value of the parameter , leaving the parameter largely unconstrained. Between the binary systems with radio pulsars, the one with the white-dwarf companion provides the most stringent constraints because the large asymmetry between the two compact objects leads to the prediction of strong dipole gravitational radiation that can be excluded observationally. Finally, for large negative values of the parameter , the scalarization of the neutron stars makes the predictions of the theory incompatible with observations.
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