Scalar-tensor quintessence with a linear potential: Avoiding the big crunch cosmic doomsday

All quintessence potentials that are either monotonic with negative interval or have a minimum at negative values of the potential, generically predict a future collapse of the scale factor to a “doomsday” singularity. We show that this doomsday is generically avoided in models with a proper nonminimal coupling of the quintessence scalar field to the curvature scalar $R$. For simplicity, we consider linear quintessence potential $V=-s\varphi$ and linear nonminimal coupling $F=1-\lambda \varphi$. However, our result is generic and is due to the fact that the nonminimal coupling modifies the effective potential that determines the dynamics of the scalar field. Thus, for each positive value of the parameter $s$, we find a critical value ${\lambda }_{\text{crit}}(s)$ such that for $\lambda >{\lambda }_{\text{crit}}(s)$ the negative potential energy does not dominate the Universe and the cosmic doomsday big crunch singularity is avoided because the scalar field eventually rolls up its potential. We find that ${\lambda }_{\text{crit}}(s)$ increases approximately linearly with $s$. For $\lambda >{\lambda }_{\text{crit}}(s)$ the potential energy of the scalar field becomes positive and it eventually dominates while the dark energy equation of state parameter tends to $w=-1$ leading to a de Sitter universe.

Figure 1

The collapse of the scale factor for representative quintessence with linear potential ($s=1$) and various values of slopes $\lambda$. The plot is logarithmic and its linear nature indicates that for $\lambda >{\lambda }_{\text{crit}}\simeq 0.24$ the Universe reaches a de Sitter evolution. The present time corresponds to $t_0=0.96$. The magnified plot shows more clearly the evolution of the scale factor towards the big crunch singularity for $\lambda <{\lambda }_{\text{crit}}$.

Figure 2

The evolution of the scalar field for $s=1$ and values of $\lambda$ above and below the critical values ${\lambda }_{\text{crit}}(s=1)=0.24\pm 0.01$. For $\lambda <{\lambda }_{\text{crit}}$ the field rolls towards larger values down its potential $V=-s\varphi$ (upper curves). Thus the potential becomes negative with strongly attractive gravity leading to a big crunch singularity. For stronger nonminimal coupling $\lambda >{\lambda }_{\text{crit}}$ (lower curves) the field evolves towards negative values (up its potential). The positive potential energy of the field eventually dominates and its repulsive gravity leads to eternal expansion avoiding the big crunch singularity. The solid lines correspond to initial conditions ${\dot{\varphi }}_i=0$ while the dashed lines correspond to ${\dot{\varphi }}_i=15$.

Figure 3

The evolution of the equation of state parameter for $s=1$ and values of $\lambda$ above and below the critical values ${\lambda }_{\text{crit}}(s=1)=0.24\pm 0.01$. The present time $t_0$ (obtained by demanding ${\mathrm{\Omega }}_m(t_0)=0.3$) corresponds to $t=t_0=0.96$. For $\lambda >{\lambda }_{\text{crit}}\simeq 0.24$ the Universe continues its expansion reaching a de Sitter phase since $w_{\mathrm{DE}}=-1$. For $\lambda <{\lambda }_{\text{crit}}$ the dark energy equation of state $w_{\mathrm{DE}}$ becomes positive and diverges leading to infinite attractive gravity and a big crunch singularity at a finite future time which depends on the values of the parameters $\lambda$ and $s$.

Figure 4

The critical value of $\lambda$ for various values of the slope $s$ of the linear potential. The dependence of ${\lambda }_{\text{crit}}$ on $s$ is approximately linear. For $\lambda >{\lambda }_{\text{crit}}(s)$ the big crunch doomsday singularity is avoided.

Figure 5

(a) The numerical evolution of the scalar field equation of state parameter $w(z)$ from $z=2$ up to the present time ($z=0$) (red dotted lines) superposed with their corresponding best fits corresponding to the CPL parametrization $w(z)=w_0+w_1\frac{z}{1+z}$ [33, 34] (blue dashed lines). This plot corresponds to parameter values $s=6$, $\lambda =0$ (best fitted by $w_0=-0.33$, $w_1=-1.04$) and $s=6$, $\lambda =1.18$ (best fitted by $w_0=-1.26$, $w_1=2.03$). These best-fit CPL parameter values are not consistent with current cosmological constraints $w_0=-1.005_{-0.17}^{+0.15}$, $w_1=-0.48_{-0.54}^{+0.77}$ [35]. (b) Similar to (a) for parameter values $s=1$, $\lambda =0$ (best fitted by $w_0=-0.967$, $w_1=-0.047$) and $s=1$, $\lambda =0.22$ (best fitted by $w_0=-1.04$, $w_1=-0.24$). These parameter values are consistent with current cosmological constraints at $1\sigma$ level.