A large bulk flow, which is in tension with the Lambda Cold Dark Matter ($\Lambda$CDM) cosmological model, has been observed. In this paper, we provide a physically plausible explanation of this bulk flow, based on the assumption that some fraction of the observed dipole in the cosmic microwave background is due to an intrinsic fluctuation, so that the subtraction of the observed dipole leads to a mismatch between the cosmic microwave background (CMB) defined rest frame and the matter rest frame. We investigate a model that takes into account the relative velocity (hereafter the tilted velocity) between the two frames, and develop a Bayesian statistic to explore the likelihood of this tilted velocity. By studying various independent peculiar velocity catalogs, we find that: (1) the magnitude of the tilted velocity $u$ is around 400 km/s, and its direction is close to what is found from previous bulk flow analyses; for most catalogs analysed, u=0 is excluded at about the $2.5 \sigma$ level;(2) constraints on the magnitude of the tilted velocity can result in constraints on the duration of inflation, due to the fact that inflation can neither be too long (no dipole effect) nor too short (very large dipole effect); (3) Under the assumption of a super-horizon isocurvature fluctuation, the constraints on the tilted velocity require that inflation lasts at least 6 e-folds longer (at the 95% confidence interval) than that required to solve the horizon problem. This opens a new window for testing inflation and models of the early Universe from observations of large scale structure. Show Partial Abstract