Islands in the $\Lambda$-sea: An alternative cosmological model

We propose an alternative cosmological model in which our observable Universe is an island in a cosmological constant sea. Initially the Universe is filled with cosmological constant of the currently observed value but is otherwise empty. In this eternal or semieternal de Sitter spacetime, we show that local quantum fluctuations (upheavals) can violate the null energy condition and create islands of matter. The perturbation spectra of quantum fields other than that responsible for the upheaval, are shown to be scale invariant. With further cosmic evolution the island disappears and the local Universe returns to its initial cosmological constant dominated state.

Figure 1

Sketch of the behavior of the Hubble length scale with cosmic time in inflationary cosmology, and the evolution of fluctuation modes. At early times, the vacuum energy is large and so the Hubble length scale is small and constant. Once the vacuum energy decays, inflation ends, the Universe reheats and enters FRW expansion. Then the Hubble length scale grows linearly with time. The physical wavelength of a fluctuation mode starts out less than $H_{\inf }^{-1}$ at some early time $t_i$. The mode is stretched to scales larger than the horizon at some time $t_H(k)$. The mode later reenters the horizon at some epoch $t_0(k)$. For illustrative purposes we have depicted that the wavelength grows linearly with cosmic time. In reality, the wavelength grows in proportion to the scale factor.

Figure 2

Sketch of the behavior of the Hubble length scale with conformal time $\eta$ in our cosmological model, and the evolution of fluctuation modes. At early times, inflation is driven by the presently observed dark energy, assumed to be a cosmological constant. As the cosmological constant is very small, the Hubble length scale is very large—of order the present horizon size. Exponential inflation in some horizon volume ends not due to the decay of the vacuum energy as in inflationary scenarios but due to a quantum fluctuation in the time interval (${\eta }_1,{\eta }_2$) that violates the null energy condition (NEC). The NEC violating quantum fluctuation causes the Hubble length scale to decrease. After the fluctuation is over, the Universe enters radiation dominated FRW expansion, and the Hubble length scale grows with time. The physical wavelength of a quantum fluctuation mode starts out less than $H_{\Lambda }^{-1}$ at some early time ${\eta }_i$. The mode exits the cosmological horizon during the NEC violating fluctuation (${\eta }_H$) and then reenters the horizon at some later epoch ${\eta }_e$ during the FRW epoch.

Figure 3

We show a classical de Sitter spacetime for conformal time $\eta <{\eta }_P$, that transitions to a faster expanding classical de Sitter spacetime for $\eta >{\eta }_Q$. The inverse Hubble size is shown by the white region. A bundle of ingoing null rays originating at point $a$ is convergent initially but becomes divergent in the superhorizon region at point $b$. This can only occur if the NEC is violated in the region $\eta \in ({\eta }_P,{\eta }_Q)$. In the quantum domain, a classical picture of spacetime may not be valid and this is made explicit by the question marks.

Figure 4

A spacetime diagram similar to that in Fig. 3 but one in which the NEC violation occurs over a subhorizon region (shaded region in the diagram). Now the null ray bundle from $a$ to $b$ goes from being converging (within the horizon) to diverging (outside the horizon). However, it does not encounter any NEC violation along its path, and this is not possible as can be seen from the Raychaudhuri equation. Since the ingoing null rays are convergent as far out as the point $P$, the size of the quantum domain has to extend out to at least the inverse Hubble size of the initial de Sitter space. Therefore the NEC violating patch has to extend beyond the initial horizon.