Cosmological attractors to general relativity and spontaneous scalarization with disformal coupling

The canonical scalar-tensor theory model which exhibits spontaneous scalarization in the strong-gravity regime of neutron stars has long been known to predict a cosmological evolution for the scalar field which generically results in severe violations of present-day Solar System constraints on deviations from general relativity. We study if this tension can be alleviated by generalizing this model to include a disformal coupling between the scalar field $\varphi$ and matter, where the Jordan frame metric ${\tilde g}_{\mu\nu}$ is related to the Einstein frame one $g_{\mu\nu}$ by ${\tilde g}_{\mu\nu}=A(\varphi)^2 (g_{\mu\nu}+\Lambda\, \partial_\mu \varphi \, \partial_\nu\varphi)$. We find that this broader theory admits a late-time attractor mechanism towards general relativity. However, the existence of this attractor requires a value of disformal scale of the order $\Lambda\gtrsim H_0^{-2}$, where $H_0$ is the Hubble parameter of today, which is much larger than the scale relevant for spontaneous scalarization of neutron stars $\Lambda \sim R_s^{2}$ with $R_s (\sim 10^{-22} H_0^{-1})$ being the typical radius of these stars. The large values of $\Lambda$ necessary for the attractor mechanism (i) suppress spontaneous scalarization altogether inside neutron stars and (ii) induce ghost instabilities on scalar field fluctuations, thus preventing a resolution of the tension. We argue that the problem arises because our disformal coupling involves a dimensionful parameter.