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 ( \Omega _ { tot } ) , dark energy ( \rho _ { \Lambda } , 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 and the choice of measure underlying the calculation .
We find that unless the characteristic scale \Delta \phi 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 _ { tot } \approx 1 \pm 10 ^ { -5 } , w = -1 and \rho _ { \Lambda } in the observed ballpark but uncomfortably high .
The high energy limit predicts { n _ { s } } \approx 0.96 , d { n _ { s } } / d \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 5 parameters but prefers 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 behavior 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 favoring the detectable level r \sim 0.03 .