Scalar-Tensor Models of Normal and Phantom Dark Energy

TL;DR

This work analyzes dark energy in scalar-tensor gravity, showing that two free functions $F(\Phi)$ and $U(\Phi)$ can realize phantom behavior today and even cross the phantom divide $w_{DE}=-1$ without invoking ghosts. It derives a general integral form for $F(z)$ from the background expansion $H(z)$ and, for constant or growing potentials $U$, investigates the resulting cosmological and Solar System constraints, including post-Newtonian parameters $\gamma_{PN}$ and $\beta_{PN}$ and the Brans-Dicke parameter $\omega_{BD}$. The study demonstrates that while Solar System tests mildly constrain $F_1$, cosmological data, especially on the background evolution, are pivotal for identifying phantom behavior; vanishing $U$ is ruled out by SN data, and a growing $U$ can yield viable models up to high redshift with asymptotic stability. The results offer a framework for reconstructing the underlying scalar-tensor Lagrangian from cosmological measurements, highlighting the interplay between local gravity tests and cosmic evolution in assessing phantom dark energy models.

Abstract

We consider the viability of dark energy (DE) models in the framework of the scalar-tensor theory of gravity, including the possibility to have a phantom DE at small redshifts $z$ as admitted by supernova luminosity-distance data. For small $z$, the generic solution for these models is constructed in the form of a power series in $z$ without any approximation. Necessary constraints for DE to be phantom today and to cross the phantom divide line $p=-ρ$ at small $z$ are presented. Considering the Solar System constraints, we find for the post-Newtonian parameters that $γ_{PN}<1$ and $γ_{PN,0}\approx 1$ for the model to be viable, and $β_{PN,0}>1$ (but very close to 1) if the model has a significantly phantom DE today. However, prospects to establish the phantom behaviour of DE are much better with cosmological data than with Solar System experiments. Earlier obtained results for a $Λ$-dominated universe with the vanishing scalar field potential are extended to a more general DE equation of state confirming that the cosmological evolution of these models rule them out. Models of currently fantom DE which are viable for small $z$ can be easily constructed with a constant potential; however, they generically become singular at some higher $z$. With a growing potential, viable models exist up to an arbitrary high redshift.

Figures (4)

• Figure 1: Several quantities are displayed for the model with parametrization (\ref{['wCPL']}) with $\alpha\equiv 1+w_0=-0.2$ and $\beta \equiv w_1=0.4$, while $F_1=0$ and $\frac{\Omega_{U,0}}{\Omega_{DE,0}}=\frac{\Omega_{U,0}}{0.7}=0.97585$. The curves shown represent the following (rescaled) quantities in function of redshift $z$: $\frac{F}{F_0}$ (solid), $10\times \Phi'^2$ (dotted), $10\times \phi'^2$ (dashed) and $0.1\times \omega_{BD}$ (dot-dashed). It is seen that $\omega_{BD}$ and $\Phi'^2$ become negative at $z\approx 10$. The model remains valid beyond $z\approx 10$ untill $z\approx 18$ as long as $\omega_{BD}>-\frac{3}{2}$ or equivalently $\phi'^2>0$. So, we see that the model remains valid for a large interval where $\Phi'^2<0$. Of course, it is impossible to reconstruct $\Phi$ in this interval using the $Z=1$ parametrization. Note that $\phi'^2$ becomes negative before$z_m\approx 22$ where $F'(z_m)=0$, in accordance with (\ref{['ineq']}).
• Figure 2: The quantity $\frac{F(z)}{F_0}$ is displayed for the parametrization (\ref{['wCPL']}) with $\alpha\equiv 1+w_0=-0.2$ and $\beta \equiv w_1=0.4$ and $F_1=0$. We have the following values for $\frac{\Omega_{U,0}}{\Omega_{DE,0}}=\frac{\Omega_{U,0}}{0.7}$ from bottom to top: $0.9758,~0.975824492,~0.975826,~0.97587$. The second curve has its minimum at $F=0$ and is superimposed on the third curve which has its minimum at $F=2.4\times 10^{-3}$.
• Figure 3: The quantity $\phi'^2$ is shown for the same models as Figure 2. The short, resp. long, dashed curve corresponds to $\frac{\Omega_{U,0}}{\Omega_{DE,0}}=\frac{\Omega_{U,0}}{0.7} =0.975824492,~{\rm resp}~0.975826.$
• Figure 4: The function $\frac{F}{F_0}$ is shown for several models with vanishing potential $U=0$. The solid lines correspond to the initial conditions $F'_0=0$ while the dashed lines correspond to the maximal values allowed by the solar system constraint $\omega_{BD,0} > 4 \times 10^4$. We have from left to right the following equation of state parameter $w$: $-2,-1.5$, polynomial expression (\ref{['wpg']}), $-1,-0.5$. It is seen that the limit of regularity of these models corresponds to very low redshifts, $0.566\le z\le 0.663$ for $-2\le w\le -1$. Note that the polynomial expression (\ref{['wpg']}) represents crossing of the phantom divide.