Noncommutative approach to the cosmological constant problem

In this paper, we study the cosmological constant emerging from the Wheeler-DeWitt equation as an eigenvalue of the related Sturm-Liouville problem. We employ Gaussian trial functionals and we perform a mode decomposition to extract the transverse-traceless component, namely, the graviton contribution, at one loop. We implement a noncommutative-geometry–induced minimal length to calculate the number of graviton modes. As a result, we find regular graviton fluctuation energies for the Schwarzschild, de Sitter, and anti-de Sitter backgrounds. No renormalization scheme is necessary to remove infinities, in contrast to what happens in conventional approaches.

Figure 1

Plot of $\Lambda /8\pi G$ in Planck units as a function of the scale-invariant $y$, which depends on the background choice. For dS and AdS backgrounds, the variable $y$ is replaced by $z$.