Pseudo-rip: Cosmological models intermediate between the cosmological constant and the little rip

If we assume that the cosmic energy density will remain constant or strictly increase in the future, then the possible fates for the universe can be divided into four categories based on the time asymptotics of the Hubble parameter $H(t)$: the cosmological constant, for which $H(t)=\mathrm{\text{constant}}$, the big rip, for which $H(t)\to \infty$ at finite time, the little rip, for which $H(t)\to \infty$ as time goes to infinity, and the pseudo-rip, for which $H(t)\to \mathrm{\text{constant}}$ as time goes to infinity. Here we examine the last of these possibilities in more detail. We provide models that exemplify the pseudo-rip, which is an intermediate case between the cosmological constant and the little rip. Structure disintegration in the pseudo-rip depends on the model parameters. We show that pseudo-rip models for which the density and Hubble parameter increase monotonically can produce an inertial force which does not increase monotonically, but instead peaks at a particular future time and then decreases.

Figure 1

Plotted from the innermost to the outermost curve is the scaled $F_{\mathrm{inert}}$ for ${\rho }_1(a,0.29,10,48)$, ${\rho }_1(a,0.0108,0.003,48)$, ${\rho }_1(a,0.01078,2\times {10}^{-7},48)$, ${\rho }_1(a,0.01078,{10}^{-23},48)$, and ${\rho }_1(a,0.0108,{10}^{-92},48)$ respectively, for Model 1, given by Eq. (8). Each curve was fitted to supernova data with the extra constraint that $B\ge 0$. The values necessary for structural disintegration are indicated. From the innermost curve to the outermost: failed rip; $t_{\mathrm{MW}}-t_0=9.2\times {10}^4\mathrm{Gyr}$; $t_{\mathrm{ES}}-t_0=1.6\times {10}^9\mathrm{Gyr}$; $t_{\mathrm{Hatom}}-t_0=1.3\times {10}^{33}\mathrm{Gyr}$; $t_{\mathrm{proton}}-t_0=3.8\times {10}^{117}\mathrm{Gyr}$.

Figure 2

Plotted from the innermost to the outermost curve is the scaled $F_{\mathrm{inert}}$ for ${\rho }_2(a,17481.3,200,0.5)$, ${\rho }_2(a,3601.31,180,2)$, ${\rho }_2(a,571.1,160,10)$, ${\rho }_2(a,171.045,130,22)$, and ${\rho }_2(a,55.45,100,40)$ respectively, for Model 2, given by Eq. (9). Each curve was fitted to supernova data. The values necessary for structural disintegration are indicated. As explained in the text, disintegration times from different models should not be directly compared. From the innermost curve to the outermost: failed rip; $t_{\mathrm{MW}}-t_0=45\mathrm{Gyr}$; $t_{\mathrm{ES}}-t_0=40\mathrm{Gyr}$; $t_{\mathrm{Hatom}}-t_0=39\mathrm{Gyr}$; $t_{\mathrm{proton}}-t_0=38\mathrm{Gyr}$.