### 6.2 Second-order scalar-tensor gravity and radio pulsars

As discussed in the previous section, observations of strong-field phenomena provide constraints on
Brans–Dicke scalar-tensor gravity, which are, however, at most comparable to those of solar system tests.
This is true because the fractional deviation of a Brans–Dicke theory from general relativity is of order
, both for weak and strong gravitational fields, and the solar-system tests have superb accuracy. On
the other hand, a scalar-tensor theory with a second-order coupling (e.g., the one arising from the
action (22) with the coupling (23)) allows for large deviations in the strong-field regime, while being
consistent with the weak-field limits [39, 40].
In the case of neutron stars, the second-order scalar-tensor theory described by Damour and
Esposito-Farèse [39] 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.