Model independent results for the inflationary epoch and the breaking of the degeneracy of models of inflation

We address the problem of determining inflationary characteristics in a model independent way. We start from a recently proposed equation which allows to accurately calculate the value of the inflaton at horizon crossing ϕ k . We then use an equivalent form of this equation to write a formula that relates the number of e-folds from horizon crossing to the pivot scale N ke + N ep with the tensor-to-scalar index r , hence a general bound for N ke + N ep follows. N ke is the number of e-folds from the scale factor a k during inflation to the end of inflation at a e and N ep is the number of e-folds from a e to the pivot scale factor a p . In particular, at present r < 0.063 implies N ke + N ep < 112.5 e-folds at k = k p and 128.1 e-folds at the present scale with wavenumber mode k 0 . We also give a lower bound to the size of the universe during the inflationary epoch that gave rise to the current observable universe. We also discussed the problem of degeneracy of inflationary models and argue that this degeneration can only be resolved by studying model predictions from the reheating epoch.