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 $\phi$ and matter, where the Jordan frame metric ${\tilde{g}}_{\mu \nu }$ is related to the Einstein frame one $g_{\mu \nu }$ by ${\tilde{g}}_{\mu \nu }=A(\phi {)}^2(g_{\mu \nu }+\mathrm{\Lambda }{\partial }_{\mu }\phi {\partial }_{\nu }\phi )$. 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 $\mathrm{\Lambda }\gtrsim H_0^{-2}$, where $H_0$ is the Hubble parameter of today, which is much larger than the scale relevant for spontaneous scalarization of neutron stars $\mathrm{\Lambda }\sim R_s^2$ with $R_s(\sim {10}^{-22}{H}_0^{-1})$ being the typical radius of these stars. The large values of $\mathrm{\Lambda }$ 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.

Figure 1

Scalar-tensor cosmology with only the conformal coupling. In all panels, the solid curves correspond to ${\gamma }_{\alpha }=-5$, while the dashed curves correspond to ${\gamma }_{\alpha }=5$. In both cases we used $\phi (0)=0.2$ as the initial condition, which satisfies Eq. (54). Top left: The phase space portrait of the scalar field’s evolution (left panel) clearly shows the attractor mechanism in action for ${\gamma }_{\alpha }>0$. For ${\gamma }_{\alpha }<0$ the scalar field asymptotes to infinity with constant velocity ${\phi }^{\prime }\approx 1.6$. Top right: the evolution of the ratio between matter and cosmological constant densities. For ${\gamma }_{\alpha }>0$, the present day ${\rho }_m/{\rho }_v=0.455$ ratio happens around $\tau =8.246$, which is indicated by the circle in the top-left panel and by the vertical lines in the other panels. For ${\gamma }_{\alpha }<0$, the ratio evolves to dramatically violate observations. Bottom row: the evolution of the PPN parameters ${\beta }_{\mathrm{PPN}}$ (left) and ${\gamma }_{\mathrm{PPN}}$ (right). For ${\gamma }_{\alpha }>0$, the scalar field evolves as to satisfy the PPN constraints (39) (horizontal lines), while for ${\gamma }_{\alpha }<0$ both constraints are violated at present day and in future.

Figure 2

Scalar-tensor cosmology with disformal coupling. In all panels, the solid curves correspond to $({\lambda }_x,{\gamma }_{\alpha })=({\lambda }_{0.6},-4.22)$, the dashed curves to $({\lambda }_{0.7},-5.68)$ and the dot-dashed curves to $({\lambda }_{0.8},-7.54)$. We used the initial conditions $\phi (0)=0.0503$, 0.0434, 0.0376, respectively, which satisfy the BBN constraint Eq. (55). The panels are similar to those of Fig. 1, but here we focus only on examples in which attractor mechanisms happen and therefore focus on $\lambda <0$. Top left: we show the phase space portrait of the scalar field. For ${\gamma }_{\alpha }$ we see that the attraction towards GR persists, while it can now also occur for ${\gamma }_{\alpha }<0$. Top right: we show the ratio ${\rho }_m/{\rho }_v$. In all three cases they are similar, visually indistinguishable. The present-day ratio 0.455 is reached at ${\tau }_0\approx 8.14$ in all three examples. The circles in the left panel and the horizontal line on the right panel are the present-day ${\tau }_0\approx 8.14$, at which ${\rho }_m/{\rho }_v=0.455$.