[Paper Review] Spherically Symmetric Solutions in Massive Gravity and Constraints from Galaxies

Stefan Sjörs, Edvard Mörtsell|arXiv (Cornell University)|Nov 25, 2011

Cosmology and Gravitation Theories14 references11 citations

TL;DR

This paper derives analytical spherically symmetric solutions in massive gravity within the decoupling limit, demonstrating the viability of the Vainshtein mechanism under specific parameter constraints. Using galaxy-scale lensing and velocity dispersion data, it constrains the graviton Compton wavelength, finding λg/rH ≳ 0.01–0.02 at 95% confidence level, indicating a light but non-zero graviton mass.

ABSTRACT

In this paper, analytical solutions describing static and spherically symmetric sources in the decoupling limit of massive gravity are derived. We analyze the model parameter range and specify when a Vainshtein mechanism is possible. Furthermore, we use gravitational lensing and velocity dispersion data from galaxies to put constraints on the mass scale of the graviton. The result for the inverse graviton mass scale lambda_g = h/(2pi)/(c m_g), in units of the Hubble radius r_H=c/H_0, is of the order lambda_g/r_H > 0.01-0.02 at 95% confidence level.

Motivation & Objective

• To derive analytical, static, spherically symmetric solutions in massive gravity within the decoupling limit.
• To determine the parameter range where the Vainshtein mechanism can effectively screen deviations from general relativity near compact sources.
• To use observational data from galaxies—specifically gravitational lensing and stellar velocity dispersions—to constrain the mass scale of the graviton.
• To test the viability of massive gravity as a phenomenological model of late-time cosmic acceleration while satisfying solar system constraints.

Proposed method

• Derives exact solutions to the equations of motion in the decoupling limit of massive gravity for spherically symmetric, static sources.
• Applies a perturbative approach by solving a quintic equation approximately via cubic truncation, valid when nonlinear corrections are small.
• Uses geometric analysis of the solution curve in the ρε-plane to identify singularities and regions of multiple-valued behavior, defining the physical parameter space.
• Introduces the Vainshtein radius rV as a critical scale where nonlinear effects suppress deviations from general relativity.
• Employs observational data from galaxies: gravitational lensing (probing Φ + Ψ) and stellar velocity dispersion (probing Φ) to constrain model parameters.
• Performs a Bayesian analysis to derive confidence intervals on the graviton Compton wavelength λg = ℏ/(cmg) in units of the Hubble radius rH.

Experimental results

Research questions

• RQ1 Under what conditions in the parameter space of massive gravity can the Vainshtein mechanism successfully suppress deviations from general relativity near compact objects?
• RQ2 How do analytical solutions for spherically symmetric sources in massive gravity behave in the decoupling limit, particularly near the Vainshtein radius?
• RQ3 To what extent can galaxy-scale gravitational lensing and stellar velocity dispersion data constrain the mass of the graviton in massive gravity theories?
• RQ4 What is the upper bound on the graviton Compton wavelength λg that is consistent with current galaxy observations at 95% confidence level?

Key findings

• The Vainshtein mechanism is viable only within a restricted parameter range, specifically when B ≤ B_max ≈ √(5−√13)√(2C), beyond which no physical solutions exist.
• Solutions to the full quintic equation are well-approximated by cubic solutions when B ≲ B_max, with corrections below 10−6; beyond B_max, corrections exceed 40%, invalidating the approximation.
• The solution curve becomes multiple-valued before reaching the singularity at B = √(3C), signaling a breakdown of the physical solution regime.
• Galaxy lensing and velocity dispersion data constrain the inverse graviton mass scale to λg/rH ≳ 0.01–0.02 at 95% confidence level, implying a light but nonzero graviton mass.
• The model remains consistent with solar system tests only if nonlinear effects are strong enough to suppress deviations, which occurs within the Vainshtein radius.
• The analysis confirms that massive gravity can reproduce general relativity in high-density environments, but only for a narrow range of model parameters, limiting its phenomenological viability.