The standard cold dark matter model with a cosmological constant ($\mathrm{\Lambda }$-CDM) predicts a growth of structures which tends to be higher than the values of redshift space distortion (RSD) measurements if the cosmological parameters are fixed by the cosmic microwave background data. In this Letter, we point out that this discrepancy can be resolved or understood if we assume that the graviton has a small but nonzero mass. In the context of the minimal theory of massive gravity (MTMG), due to infrared Lorentz violations measurable only at present cosmological scales, the graviton acquires a mass without being haunted by unwanted extra degrees of freedom. While the so-called self-accelerating branch of cosmological solutions in the MTMG has the same phenomenology for the background as well as the scalar- and vector-type linear perturbations as the $\mathrm{\Lambda }\mathrm{CDM}$ in general relativity (GR), it is possible to choose another branch so that the background is the same as that in GR, but the evolution of matter perturbations gets modified by the graviton mass. In studying the fit of such modified dynamics to the above-mentioned RSD measurements, we find that the $\mathrm{\Lambda }\mathrm{CDM}$ model is less probable than the MTMG by 2 orders of magnitude. With the help of the cross-correlation between the integrated Sachs-Wolfe effect and the large-scale structure, the data also pin down the graviton mass squared around ${\mu }^2\approx -(3\times {10}^{-33}\mathrm{eV}{)}^2$, which is consistent with the latest bound $|{\mu }^2|<(1.2\times {10}^{-22}\mathrm{eV}{)}^2$ set by the recent LIGO observation.

• Received 19 July 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.091104