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Motivated by a recent experiment of J.~Steinhauer, we reconsider the spectrumand the correlations of the phonons spontaneously emitted in stationarytransonic flows. The latter are described by "waterfall" configurations whichform a one-parameter family of stable flows. For parameters close to theirexperimental values, in spite of high gradients near the sonic horizon, thespectrum is accurately planckian in the relevant frequency domain, where thetemperature differs from the relativistic prediction by less than $10 \%$. Wethen study the density correlations across the horizon and the non-separablecharacter of the final state. We show that the relativistic expressions provideaccurate approximations when the initial temperature is not too high. We alsoshow that the phases of the scattering coefficients introduce a finite shift ofthe location of the correlations which was so far overlooked. This shift is dueto the asymmetry of the flow across the horizon, and persists in thedispersion-less regime. Finally we show how the formation of the sonic horizonmodifies both local and non-local density correlations.

Phonon spectrum and correlations in a transonic flow of an atomic Bose gas