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 ($\Delta \alpha/\alpha$) based on stochastic quintessence models of dark energy, discussed by Chongchitnan and Efstathiou (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 $\Delta \alpha/\alpha$ 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: {\it (i) } $j(z)$ fluctuates due to the stochasticity of the models, reaching an amplitude of up to $5\%$ relatively to the $\Lambda$CDM model value ($j_{\Lambda {\rm CDM}}=1$); and {\it (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 $\Delta \alpha/\alpha (z)$, but can still accommodate the data for weak enough couplings.