Integrated Sachs-Wolfe-galaxy cross-correlation bounds on the two branches of the minimal theory of massive gravity

The minimal theory of massive gravity (MTMG) has two branches of stable cosmological solutions: a self-accelerating branch, which, except for the mass of tensor modes has exactly the same behavior of linear perturbations as $\Lambda$CDM in general relativity (GR), and a normal branch with nontrivial behavior. We explore the influence of the integrated Sachs-Wolfe-galaxy correlation constraints on the normal branch of MTMG, which, in its simplest implementation, has one free parameter more than $\Lambda$CDM in GR (or the self-accelerating branch of MTMG): $\theta$. This parameter is related to the graviton mass and only affects the behavior of the cosmological linear perturbation dynamics. Using 2d-mass and SDSS data, we check which values of $\theta$ lead to a positive or negative cross-correlation. We find that positive cross-correlation is achieved for a large parameter-space interval. Within this allowed region of parameter space, we perform a $\chi^2$ analysis in terms of the parameter $\theta$, while keeping the other background parameters fixed to the best-fit values of Planck. We then infer that the normal branch of MTMG fits the data well in a nontrivial portion of the parameter space, and future experiments should be able to distinguish such a model from $\Lambda$CDM in GR (or the self-accelerating branch of MTMG).