Regularization of fluctuations near the sonic horizon due to the quantum potential and its influence on Hawking radiation

We consider dynamics of fluctuations in transonically accelerating Bose-Einstein condensates and/or luminous fluids using the hydrodynamic approach. It is known that neglecting the quantum potential (QP) leads to a singular behavior of quantum and classical fluctuations in the vicinity of the Mach (sonic) horizon, which in turn gives rise to Hawking radiation. The neglect of the QP is well founded at not too small distances $\left|x\right|\gg l_h$ from the horizon, where $l_h$ is the healing length. Taking the QP into account, we show that a second characteristic length $l_r>l_h$ exists, such that the linear fluctuation modes become regularized for $\left|x\right|\ll l_r$. At $\left|x\right|\gg l_r$ the modes keep their singular behavior, which, however, is influenced by the QP. As a result we find a deviation of the high frequency tail of the spectrum of Hawking radiation from Planck's black-body radiation distribution, which can be described by an effective Hawking temperature decreasing with increasing frequency.