Characterising Vainshtein Solutions in Massive Gravity

TL;DR

This work analyzes static, spherically symmetric solutions in a ghost-free non-linear massive gravity model, focusing below the gravitational Compton wavelength where nonlinear helicity-0 dynamics can induce the Vainshtein mechanism. By reducing the problem to a decoupled quintic equation for the helicity-0 mode $h$ with $A(\rho)=(\rho_v/\rho)^3$, the authors classify asymptotic and inner solutions and map their possible global matchings across the two-parameter phase space $(\alpha,\beta)$. They show distinct regimes: some yield asymptotically flat spacetimes with GR recovery via Vainshtein, others are asymptotically non-flat yet still display Vainshtein, and some exhibit self-shielding or lack global solutions altogether. Overall, the paper provides a comprehensive phase-space atlas of solutions, clarifying when GR is recovered on intermediate scales and how the Vainshtein mechanism depends on the theory's parameters.

Abstract

We study static, spherically symmetric solutions in a recently proposed ghost-free model of non-linear massive gravity. We focus on a branch of solutions where the helicity-0 mode can be strongly coupled within certain radial regions, giving rise to the Vainshtein effect. We truncate the analysis to scales below the gravitational Compton wavelength, and consider the weak field limit for the gravitational potentials, while keeping all non-linearities of the helicity-0 mode. We determine analytically the number and properties of local solutions which exist asymptotically on large scales, and of local (inner) solutions which exist on small scales. We find two kinds of asymptotic solutions, one of which is asymptotically flat, while the other one is not, and also two types of inner solutions, one of which displays the Vainshtein mechanism, while the other exhibits a self-shielding behaviour of the gravitational field. We analyse in detail in which cases the solutions match in an intermediate region. The asymptotically flat solutions connect only to inner configurations displaying the Vainshtein mechanism, while the non asymptotically flat solutions can connect with both kinds of inner solutions. We show furthermore that there are some regions in the parameter space where global solutions do not exist, and characterise precisely in which regions of the phase space the Vainshtein mechanism takes place.

Figures (7)

• Figure 1: Phase space diagram in $(\alpha,\beta)$ for the solutions to the quintic equation (\ref{['quintic']}) in $h$, where the different regions show different matching of inner solutions to asymptotic ones. The lines splitting the regions are half parabolas ($\beta \propto \alpha^{2}$, with $\alpha >0$ or $\alpha <0$) due to rescaling symmetry of eq. (\ref{['quintic']}).
• Figure 2: Numerical solutions for the case $\textbf{F}_{-} \leftrightarrow \textbf{L}$.
• Figure 3: Numerical solutions for the case $\textbf{F}_{+} \leftrightarrow \textbf{C}_{+}$.
• Figure 4: Numerical solutions for the case $\textbf{D} \leftrightarrow \textbf{C}_{+}$.
• Figure 5: Numerical solutions for the gravitational potentials, for the case $\textbf{D} \leftrightarrow \textbf{C}_{+}$.