Perspectives on Constraining a Cosmological Constant-Type Parameter with Pulsar Timing in the Galactic Center

Independent tests aiming to constrain the value of the cosmological constant $\Lambda$ are usually difficult because of its extreme smallness $(\Lambda \simeq 1\times {10}^{-52}{m}^{-2},\mathrm{or}2.89\times {10}^{-122}\mathrm{in}\mathrm{Planck}\mathrm{units})$. Bounds on it from Solar System orbital motions determined with spacecraft tracking are currently at the $\simeq {10}^{-43}$–${10}^{-44}{m}^{-2}$$(5$–$1\times {10}^{-113}\mathrm{in}\mathrm{Planck}\mathrm{units})$ level, but they may turn out to be optimistic since $\Lambda$ has not yet been explicitly modeled in the planetary data reductions. Accurate $({\sigma }_{{\tau }_p}\simeq 1$–$10\mu s)$ timing of expected pulsars orbiting the Black Hole at the Galactic Center, preferably along highly eccentric and wide orbits, might, at least in principle, improve the planetary constraints by several orders of magnitude. By looking at the average time shift per orbit ${\overline{\Delta \delta \tau }}_p^{\Lambda }$, an S2-like orbital configuration with $e=0.8839,P_b=16\mathrm{yr}$ would permit a preliminarily upper bound of the order of $\left(\Lambda \right)\lesssim 9\times {10}^{-47}{m}^{-2}\left(\lesssim ,2,\times ,{10}^{-116},,\mathrm{in}\mathrm{Planck}\mathrm{units}\right)$ if only ${\sigma }_{{\tau }_p}$ were to be considered. Our results can be easily extended to modified models of gravity using $\Lambda$-type parameters.