Everpresent $\Lambda$. II. Structural stability

Ideas from causal set theory lead to a fluctuating, time-dependent cosmological constant of the right order of magnitude to match currently quoted “dark energy” values. Although this effect was predicted some time ago [R. D. Sorkin, in Relativity and Gravitation: Classical and Quantum, Proceedings of the SILARG VII Conference, Cocoyoc, Mexico, 1990, edited by J. C. D’Olivo, E. Nahmad-Achar, M. Rosenbaum, M. P. Ryan, L. F. Urrutia, and F. Zertuche (World Scientific, Singapore, 1991), pp. 150–173; Rafael D. Sorkin, Int. J. Theor. Phys. 36, 2759 (1997).], it is only more recently that a more detailed phenomenological model of a fluctuating $\Lambda$ was introduced and simulated numerically [M. Ahmed, S. Dodelson, P. Greene, and R. D. Sorkin, Phys. Rev. D 69, 103523 (2004).]. In this paper we continue the investigation by studying the sensitivity of the model to some of the ad hoc choices made in setting it up.

Figure 1

Tracking behavior exhibited by $|\Lambda |$ for various values of the model parameters $\mu /\nu$ and $\alpha$.

Figure 2

Recollapse is typical of evolution with MixedEq (or TrEq), and prevents the cosmos from reaching its present size when $\alpha$ is too large. In the example shown $\alpha =1/5$. The energy density in matter and radiation (solid line $\rho$) moves from left to right as the cosmos expands, then retraces its path, heading toward the final singularity. In contrast, $|\Lambda |$ (dashed line) tracks $\rho$ while the cosmos expands, but remains essentially frozen as it contracts.

Figure 3

The fraction of runs that reach the present time, plotted as a function of $\mu /\nu$, for ${\alpha }^{-2}=50$. The HCEq corresponds to $\mu /\nu =0$, the TrEq to $\mu /\nu =3$.

Figure 4

Same as Fig. 3 but for ${\alpha }^{-2}=30$.

Figure 5

The end part of the curve in Fig. 3 magnified about 20 times.

Figure 6

Histograms of the final values of ${\Omega }_{\Lambda }$ for $\mu /\nu =1.5$, 2.7 and 3 ($\alpha =\sqrt{1/30}$). Also shown for comparison is a histogram for $\mu =0$ and $\alpha =\sqrt{1/60}$.