de Sitter duality and logarithmic decay of dark energy
  We investigate infrared dynamics of four-dimensional Einstein gravity in de
Sitter space. We set up a general framework to investigate dynamical scaling
relations in quantum/classical gravitational theories. The conformal mode
dependence of Einstein gravity is renormalized to the extent that general
covariance is not manifest. We point out that the introduction of an inflaton
is necessary as a counterterm. We observe and postulate a duality between
quantum effects in Einstein gravity and classical evolutions in an inflation
(or quintessence) model. The effective action of Einstein gravity can be
constructed as an inflation model with manifest general covariance. We show
that $g=G_N H^2/\pi$: the only dimensionless coupling of the Hubble parameter
$H^2$ and the Newton's coupling $G_N$ in Einstein gravity is screened by the
infrared fluctuations of the conformal mode. We evaluate the one-loop $\beta$
function of $g$ with respect to the cosmic time $\log Ht$ as
$\beta(g)=-(1/2)g^2$, i.e., $g$ is asymptotically free toward the future. The
exact $\beta$ function with the backreaction of $g$ reveals the existence of
the ultraviolet fixed point. It indicates that the de Sitter expansion stared
at the Planck scale with a minimal entropy $S=2$. We have identified the de
Sitter entropy $1/g$ with the von Neumann entropy of the conformal zero mode.
The former evolves according to the screening of $g$ and the Gibbons-Hawking
formula. The latter is found to increase by diffusion in the stochastic process
at the horizon in a consistent way. Our Universe is located very close to the
fixed point $g=0$ with a large entropy. We discuss possible physical
implications of our results such as logarithmic decay of dark energy.
