Semiclassical (quantum Field Theory) and Quantum (string) de Sitter Regimes:. New Results

We compute the quantum string entropy S_s(m, H) from the microscopic string density of states ρ_s(m, H) of mass m in de Sitter space-time. We find for high m (high Hm → c/α') a new phase transition at the critical string temperature T_s = (1/2πk_B)L_cl c²/α', higher than the flat space (Hagedorn) temperature t_s (L_cl = c/H, the Hubble constant H acts at the transition, producing a smaller string constant α' and thus, a higher tension). T_s is the precise quantum dual of the semiclassical (QFT Hawking-Gibbons) de Sitter temperature T_sem = ħ c/(2πk_BL_cl). By precisely identifying the semiclassical and quantum (string) de Sitter regimes, we find a new formula for the full de Sitter entropy S_sem(H), as a function of the usual Bekenstein-Hawking entropy S sem⁽⁰⁾(H). For L_cl ≫ ℓ_Planck, i.e. for low H<< c/l Planck, S sem⁽⁰⁾(H) is the leading term, but for high H near c/ℓ_Planck, a new phase transition operates and the whole entropy S_sem (H) is drastically different from the Bekenstein-Hawking entropy S sem⁽⁰⁾(H). We compute the string quantum emission cross-section σ_string by a black hole in de Sitter (or asymptotically de Sitter) space-time (bhdS). For T_{sem bhdS} ℓ T_s (early evaporation stage), it shows the QFT Hawking emission with temperature T_{sem bhdS} (semiclassical regime). For T_{sem bhdS} → T_s, σ_string exhibits a phase transition into a string de Sitter state of size L s = l s²/L cl, (l s= √ hbar α '/c}), and string de Sitter temperature T_s. Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root branch point (de Vega-Sanchez transition). New bounds on the black hole radius r_g emerge in the bhdS string regime: it can become r_g = L_s/2, or it can reach a more quantum value, r_g = 0.365 ℓ_s.