Can noncommutative effects account for the present speed up of the cosmic expansion?

In this paper we investigate to which extent noncommutativity, an intrinsically quantum property, may influence the Friedmann-Robertson-Walker cosmological dynamics at late times/large scales. To our purpose it will be enough to explore the asymptotic properties of the cosmological model in the phase space. Our recipe to build noncommutativity into our model is based in the approach of Ref. [4] and can be summarized in the following steps: i) the Hamiltonian is derived from the Einstein-Hilbert action (plus a self-interacting scalar field action) for a Friedmann-Robertson-Walker space-time with flat spatial sections, ii) canonical quantization recipe is applied, i.e., the mini-superspace variables are promoted to operators, and the WDW equation is written in terms of these variables, iii) noncommutativity in the mini-superspace is achieved through the replacement of the standard product of functions by the Moyal star product in the WDW equation, and, finally, iv) semiclassical cosmological equations are obtained by means of the WKB approximation applied to the (equivalent) modified Hamilton-Jacobi equation. We demonstrate, indeed, that noncommutative effects of the kind considered here can be those responsible for the present speed up of the cosmic expansion.

Figure 1

Phase portraits for the exponential potential case (subsection 33) for different choices of the exponential parameter $\lambda$: i) $\lambda =0$ (top left), ii) $\lambda =0.5$ (top right), and iii) $\lambda =3$ (bottom). In this, as well as in the subsequent figures, the $x$, $y$ (and $z$) axes span the compact phase space variables $x$, $\overline{y}$ (and $\overline{z}$), respectively. Several orbits corresponding to different sets of initial conditions are shown. In each case the initial conditions on the noncommutative coordinate $\overline{y}$ correspond to either $\overline{y}(0)=0.75$ (upper groups of orbits), or to $\overline{y}(0)=0.25$ (lower groups of orbits) only. The attractor structure of the scalar field/noncommutative fluid-scaling solution (critical point $P_{\phi /\theta }$) when ${\lambda }^2<6$ (upper panels), is evident.

Figure 2

Standard GR-limit for the coshlike potential. The following values of the free parameters have been chosen: $\lambda =5$, $p=0.25$. Several orbits corresponding to different sets of initial conditions are drawn. The SF potential energy-dominated solution $P_V=(0,1)$ is the late-time attractor. It is seen that the orbits of the autonomous system of ODE spiral down to the future attractor, which is associated with coherent (damped) oscillations of the cosmological scalar field around the minimum of the potential.

Figure 3

General case for the coshlike potential (includes noncommutative effects). The flux in time $\alpha$ of the corresponding autonomous system of ODE is shown for $\lambda =5$, $p=0.25$ (top). The projections of the phase space onto the different phase planes are also shown (bottom). While the existence of the future attractor—critical point $P_{\mathrm{NC}}=(0,0,0.36)$—corresponding to the solution dominated by the noncommutative effects (see Table ), is evident, there are not found past attractors. Because of our choice of the free parameters above, $p{\lambda }^2>3$, the late-time attractor is a spiral critical point. For $p>1/2$ no future attractor can be found in the phase space either.