What does inflation really predict?

If the inflaton potential has multiple minima, as may be expected in, e.g., the string theory 'landscape', inflation predicts a probability distribution for the cosmological parameters describing spatial curvature (Ω_tot), dark energy (ρ_Λ, w etc), the primordial density fluctuations (Q, n_s, d n_s/d ln k etc) and primordial gravitational waves (r, n_t etc). We compute this multivariate probability distribution for various classes of single-field slow-roll models, exploring its dependence on the characteristic inflationary energy scales, the shape of the potential V and the choice of measure underlying the calculation. We find that unless the characteristic scale Δphgr on which V varies happens to be near the Planck scale, the only aspect of V that matters observationally is the statistical distribution of its peaks and troughs. For all energy scales and plausible measures considered, we obtain the predictions \Omega

_{\mathrm {tot}}\approx 1\pm 10^{-5} , w = -1 and ρ_Λ in the observed ballpark but uncomfortably high. The high energy limit predicts n_{\mathrm {s}}\approx 0.96 , \rmd n_{\mathrm {s}}/\rmd \ln \, k\approx-0.0006 , r\approx 0.15 and n_t\approx-0.02 , consistent with observational data and indistinguishable from eternal V\propto \phi

^2 inflation. The low energy limit predicts five parameters but favours larger Q and redder n_s than observed. We discuss the coolness problem, the smoothness problem and the pothole paradox, which severely limit the viable class of models and measures. Predictions insensitive to pre-inflationary conditions can arise either from eternal inflation attractor behaviour or from anthropic selection effects probing merely a tiny non-special part of the initial distribution. We argue that these two mechanisms are severely challenged by the coolness problem and the smoothness problem, respectively. Our findings bode well for detecting an inflationary gravitational wave signature with future CMB polarization experiments, with the arguably best-motivated single-field models favouring the detectable level r\sim 0.03 .