The first phenomenon is the observation of gravitationally-redshifted atomic lines during X-ray bursts from the source EXO 0748–676 . Figure 16 shows the values of the gravitational redshift from the surface of neutron stars with different masses in second-order scalar-tensor theories with different values of the parameter . In this calculation, the parameter was set to zero and the neutron-star structure was calculated using the equation of state U . The hatch-filled area corresponds to neutron-star masses that are unacceptable for each value of the parameter , while the thick curve separates the scalarized stars from their general-relativistic counterparts.
A dynamical measurement of the mass of EXO 0748–676 can rule out the possibility that the neutron star in this source is scalarized, because scalarized stars have very different surface redshifts compared to the general-relativistic stars of the same mass. The source EXO 0748–676 lies in an eclipsing binary system, which makes it a prime candidate for a dynamical mass measurement. In the absence of such a measurement, however, a limit on the parameter can be placed under the astrophysical constraint that the baryonic mass of the neutron stars is larger than . This is a reasonable assumption, given that a progenitor core of a lower mass would not have collapsed to form a neutron star. Combining this constraint with the measured redshift of z = 0.35 leads to a limit on the parameter , which depends only weakly on the assumed equation of state of neutron-star matter .
A second set of phenomena that can lead to strong-field tests of gravity are the fast quasi-periodic oscillations observed from many bright accreting neutron stars . The highest known frequency of such an oscillations is 1330 Hz, observed from the source 4U 1636–53 and corresponding to the Keplerian frequency of the innermost stable circular orbit of a slowly-spinning neutron star. Figure 17 shows the maximum Keplerian frequency outside a neutron star in the second-order scalar tensor theory for different values of the parameter . For small stellar masses, the limiting frequency is achieved at the surface of the star, whereas for large stellar masses, the limiting frequency is reached at the innermost stable circular orbit. Figure 17 shows that scalarized stars allow for higher frequencies than their general-relativistic counterparts. Therefore, requiring the observed oscillation frequency to be smaller than the highest Keplerian frequency of a stable orbit outside the compact object cannot be used to constrain the parameters of this theory. On the other hand, the correlations between the various dynamical frequencies outside the compact object depend strongly on the parameter and hence the gravity theory can be constrained given a particular model for the oscillations .
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