Inflation without Inflaton: A Model for Dark Energy

The interaction between two initially causally disconnected regions of the universe is studied using analogies of non-commutative quantum mechanics and deformation of Poisson manifolds. These causally disconnect regions are governed by two independent Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) metrics with scale factors $a$ and $b$ and cosmological constants $\Lambda_a$ and $\Lambda_b$, respectively. The causality is turned on by positing a non-trivial Poisson bracket $[ {\cal P}_{\alpha}, {\cal P}_{\beta} ] =\epsilon_{\alpha \beta}\frac{\kappa}{G}$, where $G$ is Newton's gravitational constant and $\kappa$ is a dimensionless parameter. The posited deformed Poisson bracket has an interpretation in terms of 3-cocycles, anomalies and Poissonian manifolds. The modified FLRW equations acquire an energy-momentum tensor from which we explicitly obtain the equation of state parameter. The modified FLRW equations are solved numerically and the solutions are inflationary or oscillating depending on the values of $\kappa$. In this model the accelerating and decelerating regime may be periodic. The analysis of the equation of state clearly shows the presence of dark energy. By completeness, the perturbative solution for $\kappa \ll1$ is also studied.