The Flatness Problem and $\Lambda$

By way of a complete integration of the Friedmann equations, in terms of observables, it is shown that for the cosmological constant $\Lambda >0$ there exist nonflat Friedmann-Lemaître-Robertson-Walker models for which the total density parameter $\Omega$ remains $\sim 1$ throughout the entire history of the Universe. Further, it is shown that in a precise quantitative sense these models are not finely tuned. When observations are brought to bear on the theory, and, in particular, the Wilkinson microwave anisotropy probe observations, they confirm that we live in just such a universe. The conclusion holds when the classical notion of $\Lambda$ is extended to dark energy.

Figure 1

The state space for a single fluid FLRW model with $p=(\gamma -1)\rho$ and $\gamma >2/3$. $\mathbf{M}$ is the Milne Universe (Minkowski space). $\mathbf{B}\mathbf{b}$ is the spatially flat FLRW model and the big bang. Turning points ($H=0$) are sent to infinity by the definition (1). The diagram reflects the evolution for expanding universes.

Figure 2

The state space for ${\Omega }_M=0$ (here rotated left $\pi /2$ to save space). Again $\mathbf{M}$ is the Milne Universe (Minkowski space [22]). $\mathbf{d}\mathbf{S}$ is the spatially flat representation of de Sitter space and acts as an attractor for the two other representations of de Sitter space.

Figure 3

The loci $\alpha =\mathrm{const}$ and $Ht=\mathrm{const}$ [23] in the ${\Omega }_{\Lambda }-{\Omega }_M$ plane. The loci shown are (bottom up) $Ht=0.6$, $2/3$, $3/4$, 0.826, 1, 1.5, $\infty (\equiv \alpha =1)$ (thick) and $|\alpha |=0.01$ 0.05, 0.1, $1/4$, $1/2$, 1 (thick), 2, 3, 4, 6, 8, 12, 20, 40, 100, 500. Note that for $\Lambda >0$ the delimiter ${\Omega }_{\Lambda }=1-{\Omega }_M$ ($k=0$) is approached on either side ($k=\pm 1$) as $\alpha \to \infty$.

Figure 4

WMAP [14] and HST results [15] superimposed on part of the phase portrait of Fig. 3. In the former case the “WMAP only” results are used and in the latter case earlier confidence levels have been removed for clarity. The values of $H_0{t}_0$ shown are (top down) 1.5, 1, 0.826, $3/4$, and $2/3$. Above the delimiter ${\Omega }_{{\Lambda }_0}=1-{\Omega }_{M_0}$, $k=1$, and the values of $\alpha$ shown are (bottom up) 500, 100, 40, 20, 12, 8, 6, and 5.4. Below the delimiter $k=-1$ and the values of $\alpha$ shown are (top down) 500, 100, 40, 20, 12, 8, 6, and 5.4.