On the accuracy of slow-roll inflation given current observational   constraints
  We investigate the accuracy of slow-roll inflation in light of current observational constraints, which do not allow for a large deviation from scale invariance. We investigate the applicability of the first and second order slow-roll approximations for inflationary models, including those with large running of the scalar spectral index. We compare the full numerical solutions with those given by the first and second order slow-roll formulae. We find that even first order slow-roll is generally accurate; the largest deviations arise in models with large running where the error in the power spectrum can be at the level of 1-2%. Most of this error comes from inaccuracy in the calculation of the slope and not of the running or higher order terms. Second order slow-roll does not improve the accuracy over first order. We also argue that in the basis $\epsilon_0=1/H$, $\epsilon_{n+1}={d\ln|\epsilon_n|}/{dN}$, introduced by Schwarz et al. (2001), slow-roll does not require all of the parameters to be small. For example, even a divergent $\epsilon_3$ leads to finite solutions which are accurately described by a slow-roll approximation. Finally, we argue that power spectrum parametrization recently introduced by Abazaijan, Kadota and Stewart does not work for models where spectral index changes from red to blue, while the usual Taylor expansion remains a good approximation. 
Alexey Makarov (Princeton University)