Spherically Symmetric Solutions in Massive Gravity and Constraints from Galaxies

TL;DR

The work analyzes static, spherically symmetric solutions in the decoupling limit of massive gravity to understand the Vainshtein mechanism and deviations from GR. By deriving the coupled equations for the metric potentials and the scalar mode $\pi$, it identifies parameter regions where GR is recovered at short distances and where lensing observables remain close to GR. Using gravitational lensing and stellar velocity-dispersion data from the SLACS galaxy sample, it constrains the inverse graviton mass scale to $\lambda_g/r_H \gtrsim 0.01-0.02$ (95% CL), with results largely insensitive to the specific cubic/quartic couplings within the viable region. The study demonstrates how galaxy kinematics and lensing can jointly probe modified gravity theories and pushes graviton mass bounds toward the Hubble scale, while highlighting the role of the Vainshtein mechanism in screening deviations.

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.

Figures (3)

• Figure 1: Left: Results for $C=0$ and $B<0$. Right: Results for $B=0$ and $C>0$. We plot the deviations in the force laws $\varepsilon_\Phi = \Phi'/\Phi'_\text{N} - 1$ and $\varepsilon_\Psi = 1 - \Psi'/\Phi'_\text{N}$, together with deviation in the lensing potential $\Phi'_+/\Phi'_\text{N} = 1 + \frac{\varepsilon_\Phi - \varepsilon_\Psi}{2}$ and the ratio $\Phi'/\Psi' - 1 = \frac{1+\varepsilon_\Phi}{1-\varepsilon_\Psi}-1$. The solutions exhibit the Vainshtein mechanism, eq. (\ref{['boundaryconditions']}), i.e. $\lim_{r/r_V \to 0} \varepsilon_\Phi(r),\varepsilon_\Psi(r) = 0$, $\lim_{r/r_V \to \infty} \varepsilon_\Phi(r), \varepsilon_\Psi(r) = 1/3$. Furthermore, the lensing potential is essentially that of GR everywhere with small corrections at $r \simeq r_V$, i.e. $1 + \frac{\varepsilon_\Phi - \varepsilon_\Psi}{2} \simeq 1$ almost everywhere.
• Figure 2: Cosmological constraints using strong lensing data with $B=-1, C=0$ [upper left panel], $B=0, C=1$ [upper right panel], $B=-2, C=1$ [lower left panel] and $B=1, C=2$ [lower right panel]. The limits are rather insensitive to the values of $B$ and $C$ and are on the order of $\lambda_g/r_H\gtrsim 0.01-0.02$ at 95 % confidence level.
• Figure 3: As we vary the model parameter $B \equiv \frac{\beta}{\alpha^2}$, while keeping $C \equiv \frac{\gamma}{\alpha^3}$ fixed, the curve $\varepsilon = \varepsilon(r/r_V)$ develops a singular behavior at $B = \sqrt{3C}$ where the curve 'opens up', see pink dotted line above. Already before this happens the curve becomes 'multiple valued' at $B =\sqrt{5-\sqrt{13}}\sqrt{2C}$, see the dashed curve which acquires three values for $\varepsilon(r/r_V)$ in the region $r/r_V \simeq 1.2$. For $B<\sqrt{5-\sqrt{13}}\sqrt{2C}$ the curve is well-behaved, see orange solid curve. The above example highlights Vainshtein's original idea: Models with a well-behaved short and long-distances behavior might or might not analytically connect the two regions.