Solar System constraints on massless scalar-tensor gravity with positive coupling constant upon cosmological evolution of the scalar field

Scalar-tensor theories of gravity modify general relativity by introducing a scalar field that couples nonminimally to the metric tensor, while satisfying the weak-equivalence principle. These theories are interesting because they have the potential to simultaneously suppress modifications to Einstein’s theory on Solar System scales, while introducing large deviations in the strong field of neutron stars. Scalar-tensor theories can be classified through the choice of conformal factor, a scalar that regulates the coupling between matter and the metric in the Einstein frame. The class defined by a Gaussian conformal factor with a negative exponent has been studied the most because it leads to spontaneous scalarization (i.e. the sudden activation of the scalar field in neutron stars), which consequently leads to large deviations from general relativity in the strong field. This class, however, has recently been shown to be in conflict with Solar System observations when accounting for the cosmological evolution of the scalar field. We here study whether this remains the case when the exponent of the conformal factor is positive, as well as in another class of theories defined by a hyperbolic conformal factor. We find that in both of these scalar-tensor theories, Solar System tests are passed only in a very small subset of coupling parameter space, for a large set of initial conditions compatible with big bang nucleosynthesis. However, while we find that it is possible for neutron stars to scalarize, one must carefully select the coupling parameter to do so, and even then, the scalar charge is typically 2 orders of magnitude smaller than in the negative-exponent case. Our study suggests that future work on scalar-tensor gravity, for example in the context of tests of general relativity with gravitational waves from neutron star binaries, should be carried out within the positive coupling parameter class.

Figure 1

Maximum value of scalar charge ${\alpha }_{\mathrm{sc}}$ inside stable NSs as a function of the coupling parameter $\beta$ in hyperbolic STTs with five different EoSs. In the background, blue (red) regions correspond to values of $\beta$ that are consistent (inconsistent) with Solar System observations after cosmological evolution. Observe that the blue regions become thinner and more sparse as $\beta$ increases. The bounds shown here are the most stringent ones one can place with initial conditions that deviate from GR as much as possible by saturating BBN constraints.

Figure 2

Schematic diagrams for both the conformal coupling potential $V_{\alpha }$ (left) and the conformal coupling $\alpha (\phi )$ (right). The exponential STT is represented by solid red curves and the hyperbolic theory by dashed blue curves. The horizontal gray lines at $\pm 1/\sqrt{3}$ in the right panel correspond to the limiting values of $\alpha (\phi )$ in the hyperbolic theory. These figures were constructed with $\beta =5$, but the qualitative features remain the same for all positive values of $\beta$.

Figure 3

Left: $1-{\gamma }_{\mathrm{PPN}}$ evaluated during the evolution of the scalar field, with $\zeta =1$ here, to show that an increase in $\beta$ makes it harder to satisfy the Cassini bound today at ${\tau }_0=0$. The dips here physically go to zero but due to finite numerical precision this is not reflected here. Right: $1-{\gamma }_{\mathrm{PPN}}$ (red) and $1-{\beta }_{\mathrm{PPN}}$ (blue) evaluated today at ${\tau }_0=0$ for a range of $\beta$.

Figure 4

Left: Value of ${\beta }_{\mathrm{crit}}$, as defined in the text, for every value of $\zeta$ which corresponds to the entire range of possible initial scalar positions at the time of BBN. Right: The total fraction out of all $\beta$ values sampled ($1<\beta <1000$) that allow Solar System tests to be passed at $\tau =0$, as a function of the initial conditions determined by $\zeta$. The general trend of the top right panel tells us that this fraction would continue to decrease as we increase the upper bound of our sample size.

Figure 5

Same as Figs. 3 and 4 but for the hyperbolic class of STTs.

Figure 6

Left: Baryonic mass of the NS as a function of central baryonic density. Deviations from GR occur above some critical density which depends on $\beta$. A more negative $\beta$ leads to larger deviations from GR, but notice that the exponential theory has more of an effect than the hyperbolic theory does. Right: Scalar charge as a function of the total baryonic mass. The “spontaneous” activation of the scalar field occurs at a particular mass, while the dependence on $\beta$ appears to be independent of the theory. Again we see that the exponential theory causes a larger activation of the scalar field than the hyperbolic theory. The circles in both panels correspond to the solutions that have maximum scalar charge and the crosses indicate the maximum mass solutions, which also mark an instability in these branches of solutions [24, 43].

Figure 7

A rough schematic representation of the instability regions from the linear stability analysis carried out in Ref. [19]. Red/blue regions are unstable to scalar-field perturbations and the black region corresponds to solutions that are unstable to gravitational collapse. The horizontal line marks the density above which $T_{\mathrm{mat}}^{*}>0$ at the center of the NS. We mark a vertical line at $\beta =-6$ (corresponding to an example scenario we used in Fig. 6) to help demonstrate the behavior of the instability.

Figure 8

Top: Baryonic mass as a function of central baryonic density for a wide range of $\beta$. There exists an asymptotic solution as $\beta \to \infty$ (as $\beta$ increases the branches only experience small changes from one to the next). Solid points indicate the maximum mass that is stable and the solid red line connects all maximum mass solutions for all $\beta$. Bottom: The scalar charge as a function of the central baryonic density for the same set of $\beta$’s as in the top panel. Solid points here correspond to the same solutions as the solid points in the top panel, with the solid red curve showing how these values change as a function of $\beta$. Even though the scalar charge grows monotonically there exists a maximum value that it can take for stable NS solutions. For large $\beta$, increasing $\beta$ continues to decrease the maximum stable scalar charge, though it does not cause the charge to vanish.

Figure 9

Rate of change of the scalar charge evaluated at the maximum mass solutions (solutions identified by points in Fig. 8) as a function of $\beta$ for the ENG EoS. There are no values for $\beta \lesssim 19$ because there are no stable solutions with scalar charge, cf. the discussion in the text. Notice that the “rapidity” of the increase of scalar charge monotonically decreases with increasing $\beta$. The apparent kink near $\beta =25$ is a result of numerical error due to the fact that we only have limited precision and must interpolate between different solution branches.