Cosmological perturbations and observational constraints on nonlocal massive gravity

Nonlocal massive gravity can provide an interesting explanation for the late-time cosmic acceleration, with a dark energy equation of state $w_{\rm DE}$ smaller than $-1$ in the past. We derive the equations of linear cosmological perturbations to confront such models with the observations of large-scale structures. The effective gravitational coupling to nonrelativistic matter associated with galaxy clusterings is close to Newton's gravitational constant $G$ for a mass scale $m$ slightly smaller than today's Hubble parameter $H_0$. Taking into account the background expansion history as well as the evolution of matter perturbations $\delta_m$, we test for these models with Type Ia Supernovae (SnIa) from Union 2.1, the cosmic microwave background (CMB) measurements from Planck, a collection of baryon acoustic oscillations (BAO), and the growth rate data of $\delta_m$. Using a higher value of $H_0$ derived from its direct measurement ($H_0 \gtrsim 70$ km s$^{-1}$ Mpc$^{-1}$) the data strongly support the nonlocal massive gravity model ($-1.1 \lesssim w_{\rm DE} \lesssim -1.04$ in the past) over the $\Lambda$CDM model ($w_{\rm DE}=-1$), whereas for a lower prior (67 km s$^{-1}$ Mpc$^{-1}$ $\lesssim$ $H_0 \lesssim 70$ km s$^{-1}$ Mpc$^{-1}$) the two models are statistically comparable.