Figure 1

The function $F_{\alpha }({\xi }_{*})$ determining the equilibrium solutions in the case of a flat universe [see Eq. (2.20)] is plotted for different values of $\alpha =[1,2,5]$. The existence of two solutions is evident by drawing horizontal lines. $F_{\alpha }$ has a minimum at ${\xi }_{*}^{\min }=-\frac{1}{2\alpha }[2W_{-1}(-\frac{1}{2\sqrt{e}})+1]\approx 1.256/\alpha$, with value $F_{\alpha }({\xi }_{*}^{\min })\approx 2.455/\alpha$. Here $W_k(z)$ denotes the $k$ branch of the Lambert $W$ function satisfying the equation $W(z)e^{W(z)}=z$, where $k$ is any nonzero integer [29]. If the variable $z$ is real, then there are two possible real values of $W(z)$ in the interval $-1/e\le z<0$. The branch satisfying $W(z)\ge -1$ is denoted by $W_0(z)$ and is referred to as the principal branch of the $W$ function, whereas the branch satisfying $W(z)\le -1$ is denoted by $W_{-1}(z)$. The existence of two roots for Eq. (2.20) is thus guaranteed if $2.455/\alpha ⪅w_{(\mathrm{in})}|g|/2(1+w_{(\mathrm{in})})$.Reuse & Permissions

Figure 2

The solutions for $\xi$ (a), $x$ (b), and ${\mathcal{P}}_{(\alpha ,g)}$ (c) are shown as functions of $\tau$ for the choice of parameters $w_{(\mathrm{in})}=1/3$, $g=-8$, ${\Omega }_{k,0}=-0.01$, ${\Omega }_{\Lambda ,0}=0$ and different values of $\alpha =[1,2,5]$, with initial conditions $\xi ({\tau }_0)=1.01$ and $x({\tau }_0)=1$. The corresponding values of ${\tau }_0$ are ${\tau }_0\approx [0.668,0.847,1.343]$, respectively. (d) shows instead the behavior of $p_{(\mathrm{sc})}$ as a function of $\rho$ in units of ${\rho }_{(\mathrm{crit}),0}$. Note that for $\alpha =5$ the integration stops approaching the equilibrium solution $\xi \sim 1$ and ${\mathcal{P}}_{(\alpha ,g)}\sim -3$, implying $w_{\mathrm{eff}}\equiv p_{(\mathrm{sc})}/\rho \sim -1$. Here (and below) a dot on each curve marks the corresponding present-day value.Reuse & Permissions

Figure 3

The evolution of dimensionless SC density, scale factor, and SC pressure are shown in (a)–(c), respectively, for the choice of parameters $w_{(\mathrm{in})}=1/3$, $g=-8$, ${\Omega }_{\Lambda ,0}=0$, ${\Omega }_{m,0}=0.266$, ${\Omega }_{k,0}=-0.01$, and different values of $\alpha =[1,2,2.7,5]$, with initial conditions $\xi ({\tau }_0)=0.734$ and $x({\tau }_0)=1$. The corresponding values of ${\tau }_0$ are ${\tau }_0\approx [0.682,0.876,1.007,1.097]$, respectively. (d) shows instead the behavior of $p_{(\mathrm{sc})}$ as a function of $\rho$ in units of ${\rho }_{(\mathrm{crit}),0}$. Note that for $\alpha =2.7$ and $\alpha =5$ the integration stops approaching the equilibrium solutions.Reuse & Permissions

Figure 4

The evolution of the effective $w_{\mathrm{eff}}\equiv p_{(\mathrm{sc})}/\rho$ is shown in (a) for the same choice of parameters and initial conditions as in Fig. 3. (b) shows instead its behavior as a function of the redshift.Reuse & Permissions

Figure 5

Distance modulus vs redshift for different values of the parameter $\alpha =[1,2,2.7,5]$. Data are taken from Ref. 22 (gold sample). The model equations (2.16, 2.24, 3.4) have been numerically integrated for the same choice of parameters and initial conditions as in Fig. 3, plus the additional condition $r({\tau }_0)=0$ (i.e., $d_c({\tau }_0)=0$). For the Hubble constant we have used the value $H_0=71\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ [21]. The curve for $\alpha =2.7$ is practically superimposed to the $\Lambda \mathrm{CDM}$ one (thick black dashed curve).Reuse & Permissions

Figure 6

Hubble parameter as a function of the redshift for different values of the parameter $\alpha =[1,2,2.7,5]$. Data are taken from Ref. 24. The prediction from the $\Lambda \mathrm{CDM}$ model is also shown (thick black dashed curve). The Hubble parameter is expressed in units of $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ (the value of $H_0$ is the same as in Fig. 5).Reuse & Permissions

Figure 7

Deceleration parameter as a function of the redshift for different values of the parameter $\alpha =[1,2,2.7,5]$. The choice of parameters as well as initial conditions is the same as in Fig. 3. The curve corresponding to the $\Lambda \mathrm{CDM}$ model is also shown for comparison (thick black dashed curve).Reuse & Permissions

Figure 8

Present-day value of the deceleration parameter $q_0$ and age of the Universe $t_0={\tau }_0/H_0$ (expressed in Gyrs) as functions of the parameter $\alpha$. The choice of parameters as well as initial conditions is the same as in Fig. 3.Reuse & Permissions

Figure 9

The evolution of the density contrast associated with the SC fluid is shown in (a) for different values of the parameter $\alpha$. The linear perturbation equations (4.2) have been numerically integrated with a representative value of $\tilde{k}={10}^{-2}$ and initial conditions $\delta \xi ({\tau }_0)=\delta \tilde{u}({\tau }_0)=\tilde{\psi }({\tau }_0)=\delta {\tilde{\rho }}_m({\tau }_0)={10}^{-3}$. The choice of remaining parameters as well as initial conditions for the background SC density $\xi$ and scale factor $x$ is the same as in Fig. 3. (b) corresponds, instead, to a simple SC model without any additional component in the cosmological fluid. The choice of parameters as well as initial conditions in this case is the same as in Fig. 2. Integrating backward in time shows that now the system becomes soon unstable for every value of $\alpha$. The presence of a pressureless matter component has then a stabilizing effect on the whole system for $\alpha \gtrsim 2$. In fact, the perturbation undergoes oscillations remaining bounded all the way up to very early times and vanishes at late times.Reuse & Permissions