Stochastic quintessence models: Jerk and fine-structure variability constraints

We report on constraints to the cosmological jerk parameter ($j$) and to possible variability of the fine-structure constant ($\mathrm{\Delta }\alpha /\alpha$) based on stochastic quintessence models of dark energy, discussed by Chongchitnan and Efstathiou [Phys. Rev. D 76, 043508 (2007)]. We confirm the results by these authors in the sense that many viable solutions can be obtained, obeying current observational constraints in low redshifts. We add the observables $j$ and $\mathrm{\Delta }\alpha /\alpha$ to this conclusion. However, we find peculiarities that may produce, in the nearby universe, potential observational imprints in future cosmological data. We conclude, for redshifts $z\lesssim 3$, that (i) $j(z)$ fluctuates due to the stochasticity of the models, reaching an amplitude of up to 5% relatively to the $\mathrm{\Lambda }$ cold dark matter model value ($j_{\mathrm{\Lambda }\mathrm{CDM}}=1$); and (ii) by contrasting two distinct (“extreme”) types of solutions, variabilities in $\alpha (z)$, linked to a linear coupling ($\zeta$) between the dark energy and electromagnetic sectors, are weakly dependent on redshift, for couplings of the order $|\zeta |\sim 10^{-4}$, even for large variations in the equation of state parameter at relatively low redshifts. Nonlinear couplings produce an earlier and steeper onset of the evolution in $\mathrm{\Delta }\alpha /\alpha (z)$, but can still accommodate the data for weak enough couplings.

Figure 1

Illustration of the main features of the energy phase space and general characteristics of the stochastic trajectories of the proposal by [13].

Figure 2

Trajectories of viable stochastic quintessence models in the energy phase space. Left panel: the ($z_{\text{trans}}=2.5$)-ensemble models, obtained from different sets of initial conditions and $\lambda (z)$ distributions (for each of the 20 sets of the ensemble). Right panel: the ($z_{\text{trans}}=1.5$)-ensemble models, obtained a fixed set of initial conditions but different $\lambda (z)$ distributions (for each of the 5 sets of the ensemble). Different trajectory colors indicate different sets [or, equivalently, different $\lambda (z)$ distributions]. The “$+$” symbols in the phase space of the ($z_{\text{trans}}=1.5$)-ensemble models (right panel) indicate the (fixed) initial conditions taken from a set of the ($z_{\text{trans}}=2.5$) ensemble (see text).

Figure 3

Trajectories of viable stochastic quintessence models in the energy phase space, for the selected models: Early (left panels) vs late (right panels), from the ($z_{\text{trans}}=2.5$) ensemble (top panels) and the ($z_{\text{trans}}=1.5$) ensemble (bottom panels).

Figure 4

The behavior of cosmological parameters of the selected solutions as a function of redshift: the dark energy equation of state parameter $w_q(z)$ (top panel in each figure) and the dark energy density parameter $\mathrm{\Omega }(z)$ (bottom panel in each figure). Figures are ordered as in Fig. 3: Early (left panels) vs late (right panels), from the ($z_{\text{trans}}=2.5$) ensemble (top figures) and the ($z_{\text{trans}}=1.5$) ensemble (bottom figures).

Figure 5

Representative curves of type (i)–(iii) behaviors in the evolution of the dark energy equation of state parameter $w_q(z)$. The latest model obtained from the ($z_{\text{trans}}=2.5$) ensemble is indicated by the “IC” curve, and its initial conditions were fixed for all the sets generated by the ($z_{\text{trans}}=1.5$) ensemble.

Figure 6

The potential of the models and the corresponding derivatives $dV/d\varphi$ (insets). Figures are ordered as in Fig. 3: Early (left panels) vs late (right panels), from the ($z_{\text{trans}}=2.5$) ensemble (top figures) and the ($z_{\text{trans}}=1.5$) ensemble (bottom figures).

Figure 7

The jerk parameter of the models as compared to $\mathrm{\Lambda }\mathrm{CDM}$. The horizontal line at $|j(z)-1|=0$ the signals a $\mathrm{\Lambda }\mathrm{CDM}$ behavior ($\equiv j_{\mathrm{\Lambda }\mathrm{CDM}}$). Figures are ordered as in Fig. 3: Early (left panels) vs late (right panels), from the ($z_{\text{trans}}=2.5$) ensemble (top figures) and the ($z_{\text{trans}}=1.5$) ensemble (bottom figures). The “latest model” (labeled as IC in Fig. 5) is indicated in the top right panel, as a thick line.

Figure 8

Stochastic quintessence predictions for the variation $\mathrm{\Delta }\alpha /\alpha (z)$ for PC. Curves are parametrized by the coupling factor $|\zeta |\equiv |{\zeta }_{\mathrm{PC}}|$. LC models correspond to $q=1$ in Eq. (13). In each figure, the lower panel is a zoom-in of the upper panel, showing with more clarity the corresponding weakly coupling regime. Upper left panel: LC curves, using the latest model (labeled as IC in Figs. 5 and 7) as a reference for the calculations. Upper right panel: LC curves, for the type-(i) model (cf. Fig. 5). Lower left panel: PC curves (quadratic coupling, $q=2$) for the “IC model.” Lower left panel: PC curves (quadratic coupling, $q=2$) for the IC model. Lower right panel: PC curves ($q=6$) for the IC model.

Figure 9

Stochastic quintessence predictions for the variation $\mathrm{\Delta }\alpha /\alpha (z)$ for MC, adopting $q=3$, and using the latest model (labeled as IC in Figs. 5 and 7) as a reference for the calculations. Curves are parametrized by the coupling factors $|{\zeta }_{\mathrm{PC}}|$ and ${\zeta }_{\mathrm{EC}}$. In each figure, the lower panel is a zoom-in of the upper panel, showing with more clarity the corresponding weakly coupling regime. Lighter grey curves in all panels correspond to the PC curves with same coupling factors $|{\zeta }_{\mathrm{PC}}|$ but without the exponential term. Upper panel: Small deviations from the purely PC curves, for ${\zeta }_{\mathrm{EC}}=1\times {10}^{-4}$. Lower panel: Stronger deviations from the purely PC curves, for ${\zeta }_{\mathrm{EC}}=5\times {10}^{-4}$.