Uniqueness of current cosmic acceleration

One of the strongest arguments against the cosmological constant as an explanation of the current epoch of accelerated cosmic expansion is the existence of an earlier, dynamical acceleration, i.e. inflation. We examine the likelihood that acceleration is an occasional phenomenon, putting stringent limits on the length of any accelerating epoch due to minimally coupled dark energy between recombination and the recent acceleration; such an epoch must last less than 0.05 e-fold (at $z>2$) or the matter power spectrum is modified by more than 20%.

Figure 1

The superaccelerating scenario has a period of strongly negative dark energy equation of state $w_j<-1$, which drives $w_{\mathrm{tot}}<-1/3$ and so causes acceleration. The top panel plots the logs of the dark energy density $8\pi G{\rho }_w(a)/(3H_0^2)$, dark matter density $8\pi G{\rho }_m(a)/(3H_0^2)$, and Hubble parameter $H^2/H_0^2$ vs the log of the scale factor $a$. The $w_j<-1$ epoch is taken to last for $N=0.6$ or 1 e-folds, ending at $z_u=2$. However, the actual acceleration lasts for a much shorter time, shown by the thick portions of the ${\rho }_w$ curves (including for the current epoch of acceleration). In particular, note the early acceleration only lasts for 0.034 e-folds. The bottom panel plots $w(a)$ on the same horizontal scale so one can see directly the effect of $w(a)$ on the densities. The accelerated epochs are again indicated by the thick parts of the curves.

Figure 2

As Fig. 1, but for the superdecelerating scenario. Here the acceleration occurs because of relatively large ${\rho }_w(a)$, rather than supernegative $w$. To bring the dark energy density to the current level then requires a period of superdeceleration, where $w>0$ to redshift away the excess density. Note that in the $N=0.6$ case the dark energy never dominates the matter density during the early acceleration.

Figure 3

The superaccelerating scenario can only support a very small number of e-folds of acceleration because for a supernegative equation of state $w_j$ the dark energy density must start very small to end up at the observed level today. The plot shows the number of e-folds of acceleration as a function of $w_j$ and the redshift $z_u$ when superacceleration ends. The red, long-dashed curve shows the loci of the maximum number of e-folds for each choice of $z_u$ and its equivalent $w_j(N_{\max })$. For an early period of acceleration sufficiently distinct from the present acceleration, e.g. $z_u<2$, less than 0.034 e-folds of acceleration are allowed.

Figure 4

The fractional dark energy density ${\Omega }_w(a)$ (dotted, positive only curves) and total equation of state $w_{\mathrm{tot}}(a)$ (solid curves) are plotted for the superdecelerating scenario with three different values of $N_{\mathrm{dec}}$. Note that the $N_{\mathrm{dec}}=0.1$ case never achieves early acceleration ($w_{\mathrm{tot}}<-1/3$, shown by the horizontal dashed line), the $N_{\mathrm{dec}}=0.6$ case barely does, and even the $N_{\mathrm{dec}}=1$ model has only a short period of early acceleration. All models shown have $w_j=+1$ and return at $z=2$ to $\Lambda \mathrm{CDM}$ with ${\Omega }_w(a=1)=0.72$. The smooth, green curves correspond to $\Lambda \mathrm{CDM}$ without any transition.

Figure 5

Early acceleration due to a high early dark energy density can only occur, and still deliver the same dark energy density today, if it is followed by sufficient superdeceleration (here $w_j=+1$). Only the space above the solid, black curve, giving $N_{\mathrm{acc}}>0$, achieves this. Too much superdeceleration, however, strongly suppresses the total growth to the present. Only the space below the dashed, blue curve diminishes the growth relative to $\Lambda \mathrm{CDM}$ by less than 10%. The only surviving region is the thin crescent between the curves at $a_j>0.27$, or $z_j<2.7$, which hardly qualifies as early acceleration.