Cosmological evolution and Solar System consistency of massive scalar-tensor gravity

The scalar-tensor theory of Damour and Esposito-Farèse recently gained some renewed interest because of its ability to suppress modifications to general relativity in the weak field, while introducing large corrections in the strong field of compact objects through a process called scalarization. A large sector of this theory that allows for scalarization, however, has been shown to be in conflict with Solar System observations when accounting for the cosmological evolution of the scalar field. We here study an extension of this theory by endowing the scalar field with a mass to determine whether this allows the theory to pass Solar System constraints upon cosmological evolution for a larger sector of coupling parameter space. We show that the cosmological scalar field goes first through a quiescent phase, similar to the behavior of a massless field, but then it enters an oscillatory phase, with an amplitude (and frequency) that decays (and grows) exponentially. We further show that after the field enters the oscillatory phase, its effective energy density and pressure are approximately those of dust, as expected from previous cosmological studies. Due to these oscillations, we show that the scalar field cannot be treated as static today on astrophysical scales, and so we use time-dependent perturbation theory to compute the scalar-field-induced modifications to Solar System observables. We find that these modifications are suppressed when the mass of the scalar field and the coupling parameter of the theory are in a wide range, allowing the theory to pass Solar System constraints, while in principle possibly still allowing for scalarization.

Figure 1

Evolution of the scalar field (left panel) and its derivative (right panel) as functions of the dimensionless conformal $p$-time in massive DEF theory with $\beta =-4.5$ and $m=5.8\times {10}^{-28}\mathrm{eV}$. In both panels, the dashed vertical lines roughly separate the different cosmological eras (radiation, matter and dark energy). Observe that after a quiescent phase, the field begins to oscillate rapidly while its amplitude decays exponentially to zero. The amplitude of the envelope of its derivative grows during radiation and matter domination, approaching the limiting value of $\sqrt{3}$ (horizontal dashed green lines on right panel) but decays during dark energy domination. The inset in the right panel shows the decay of the scalar field during the matter-dominated era and the dark-energy dominated era for $p\ge -4$.

Figure 2

Hubble parameter as a function of $p$-time for $\beta =-4.5$, $m=5.8\times {10}^{-28}\mathrm{eV}$ and ${\phi }_0=1$. Top: The Hubble parameter is computed first by solving the field equations numerically (dashed red line), and then fitting this data to the analytical approximate model of Eq. (64) (black line). Bottom: Fractional percent difference between the numerical and the analytical result. Observe that the error between the numerical solution and the analytic expression is at most approximately 1%.

Figure 3

Evolution of the equation of state of the scalar field as a function of $p$-time, obtained by numerically solving the field equation with $\beta =-4.5$, $m=5.8\times {10}^{-28}\mathrm{eV}$ and ${\phi }_0=1$. Observe the equation of state averages to zero.

Figure 4

Schematic representation of a spacelike hypersurface $t=\mathrm{cst}$ of the spacetime, the two submanifolds we consider, and the different scales at play in our perturbative scheme. The region $r\gg L_C$ is dominated by cosmology and the behavior of the field is given in Sec. 3. We are interested in computing observables in the Solar System submanifold, which is characterized by length scales of $\mathcal{O}(R_{\mathrm{SS}})$.