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Vainshtein Mechanism In $Λ_3$ - Theories

Giga Chkareuli, David Pirtskhalava

TL;DR

The paper addresses whether a broad class of ghost-free, nonlinear generalizations of Fierz-Pauli massive gravity in the Λ3 decoupling limit can realize the Vainshtein mechanism to recover General Relativity at sub-Vainshtein scales. It derives the two-parameter Λ3 action, exposes the Galileon-like structure, and reduces the problem to radial equations with a quintic equation for λ that governs multiple solution branches. Through numerical analysis, it shows that for β<0 the Vainshtein mechanism operates, producing a GR-like solution inside r_* that matches an asymptotically flat exterior, while an asymptotically non-decaying branch can exist outside; another branch yields a completely screened 1/r potential inside the Vainshtein radius, which is observationally unacceptable as it does not decay at infinity. For β>0, only the non-decaying or screened interior solutions persist, highlighting the sensitivity of the viability to parameter choices. Overall, the work demonstrates that a substantial portion of the parameter space in Λ3 theories can reproduce GR at relevant scales, while identifying problematic branches that fail to match cosmological or Solar System constraints.

Abstract

We explore the space of spherically symmetric, static solutions in the decoupling limit of a class of non-linear covariant extensions of Fierz-Pauli massive gravity obtained recently in arXiv:1007.0443. In general, several such solutions with various asymptotic limits exist. We find their approximate short and long-distance behaviour and use numerical analysis to match them at the Vainshtein radius, $r_*$. Our findings indicate, that for a broad range of parameters, the theory does possess the Vainshtein mechanism, screening the scalar contribution to the gravitational force within $r_*$. In addition, there exists a class of solutions in the literature, for which the $1/r$ gravitational potential is completely screened within the Vainshtein scale. However, numerical analysis indicates, that for this type of solutions, the gravitational potential does not decay at spatial infinity.

Vainshtein Mechanism In $Λ_3$ - Theories

TL;DR

The paper addresses whether a broad class of ghost-free, nonlinear generalizations of Fierz-Pauli massive gravity in the Λ3 decoupling limit can realize the Vainshtein mechanism to recover General Relativity at sub-Vainshtein scales. It derives the two-parameter Λ3 action, exposes the Galileon-like structure, and reduces the problem to radial equations with a quintic equation for λ that governs multiple solution branches. Through numerical analysis, it shows that for β<0 the Vainshtein mechanism operates, producing a GR-like solution inside r_* that matches an asymptotically flat exterior, while an asymptotically non-decaying branch can exist outside; another branch yields a completely screened 1/r potential inside the Vainshtein radius, which is observationally unacceptable as it does not decay at infinity. For β>0, only the non-decaying or screened interior solutions persist, highlighting the sensitivity of the viability to parameter choices. Overall, the work demonstrates that a substantial portion of the parameter space in Λ3 theories can reproduce GR at relevant scales, while identifying problematic branches that fail to match cosmological or Solar System constraints.

Abstract

We explore the space of spherically symmetric, static solutions in the decoupling limit of a class of non-linear covariant extensions of Fierz-Pauli massive gravity obtained recently in arXiv:1007.0443. In general, several such solutions with various asymptotic limits exist. We find their approximate short and long-distance behaviour and use numerical analysis to match them at the Vainshtein radius, . Our findings indicate, that for a broad range of parameters, the theory does possess the Vainshtein mechanism, screening the scalar contribution to the gravitational force within . In addition, there exists a class of solutions in the literature, for which the gravitational potential is completely screened within the Vainshtein scale. However, numerical analysis indicates, that for this type of solutions, the gravitational potential does not decay at spatial infinity.

Paper Structure

This paper contains 3 sections, 22 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. The Formalism and Technical Analysis
  3. Solutions

Figures (3)

  • Figure 1: The solution exhibiting the Vainshtein mechanism for $\alpha=-1$ and $\beta=-1$.
  • Figure 2: The asymptotically non-decaying solution for $\alpha=-1$ and $\beta=-1$.
  • Figure 3: The asymptotically non-decaying solution for $\alpha=-1$ and $\beta=1$. No asymptotically decaying solutions exist for this choice of parameters.