Thermal RG Flow of AS Quantum Gravity

We perform the thermal Renormalization Group (RG) study of the Asymptotically Safe (AS) quantum gravity in the Einstein-Hilbert truncation by relating the temperature parameter to the running RG scale as $T \equiv k_T = \tau k$ (in natural units) in order to determine its thermal evolution in terms of the dimensionless temperature $\tau$ which is associated with the temperature of the expanding Universe. Thus, $k_T$ and $k$ are understood as running cutoffs for thermal and quantum fluctuations, respectively. Quantum effects are taken into account by moving along the thermal RG trajectory with fixed value of $\tau$ producing the quantum effective action at a given dimensionless temperature. The $\tau$-evolution of the dimensionless Newton coupling $g(\tau)$ and the dimensionless cosmological constant $\lambda(\tau)$ results in a vanishing $g$-coordinate of the Reuter (i.e., non-Gaussian UV) fixed point in the high temperature limit ($\tau \to \infty$) which means that only the symmetric phase of AS gravity survives at $\tau = \infty$. Thus, in case of large temperatures the cosmological constant takes on a negative value in the limit $k\to 0$ which was also initially predicted by certain string theories, however, in our approach this is not in disagreement with observations, since during the thermal evolution of the Universe a phase transition occurs and the cosmological constant runs to the expected positive value at low temperatures.