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We show that spontaneous breaking of the translational symmetry (SBTS) ofstrongly interacting electron and phonon fields acknowledged in lightly dopedcuprates to interpret broad bands in their ARPES and optical conductivityspectra influences the carrier distribution function as charge carrier momentumis no longer certain. We develop appropriate distribution and use it tocalculate the carrier concentration in different states in essentially dopedsystems with SBTS. This allows studying doping and temperature evolution ofARPES spectra of such systems. We reveal that at increasing both temperature ordoping polarons concede dominance to bipolarons that results in a shift of thebroad band maximum to higher binding energy as it occurs in ARPES spectra ofcuprates. At further doping delocalized carriers appear and coexist withbipolarons. We show that intimate property of systems with SBTS is partition ofthe momentum space between autolocalized and delocalized carriers caused byPauli exclusion rule. We demonstrate that in ARPES spectra such partitionmanifests itself as `vertical dispersion` or waterfalls universally observed inall the cuprates at essential doping and in undoped parent compounds. Thus,studying the carriers distribution in systems where electron-phonon interactionbreaks the translation symmetry leads simultaneously to both types ofwaterfalls observed in significantly doped and undoped cuprates. The articlealso represents the methods developed to calculate the band in ARPES spectrumcaused by bipolaron photodissociation and the bipolaron binding energy andradius using coherent states for the phonon field characterization.

Distribution function of charge carriers in electron-phonon systems with  spontaneously broken translational symmetry and 'vertical dispersion' in  ARPES spectra of cuprates