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Cosmological perturbations and structure formation in nonlocal infrared modifications of general relativity

We study the cosmological consequences of a recently proposed nonlocal modification of general relativity, obtained by adding a term $m^2R\,\Box^{-2}R$ to the Einstein-Hilbert action. The model has the same number of parameters as $Λ$CDM, with $m$ replacing $Ω_Λ$, and is very predictive. At the background level, after fixing $m$ so as to reproduce the observed value of $Ω_M$, we get a pure prediction for the equation of state of dark energy as a function of redshift, $w_{\rm DE}(z)$, with $w_{\rm DE}(0)$ in the range $[-1.165,-1.135]$ as $Ω_M$ varies over the broad range $Ω_M\in [0.20,0.36]$. We find that the cosmological perturbations are well-behaved, and the model fully fixes the dark energy perturbations as a function of redshift $z$ and wavenumber $k$. The nonlocal model provides a good fit to supernova data and predicts deviations from General Relativity in structure formation and in weak lensing at the level of 3-4%, therefore consistent with existing data but readily detectable by future surveys. For the logarithmic growth factor we obtain $γ\simeq 0.53$, to be compared with $γ\simeq 0.55$ in $Λ$CDM. For the Newtonian potential on subhorizon scales our results are well fitted by $Ψ(a;k)=[1+μ_s a^s]Ψ_{\rm GR}(a;k)$ with a scale-independent $μ_s\simeq 0.09$ and $s\simeq 2$, while the anisotropic stress is negligibly small.