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 $\left(\Lambda
\simeq 1\times 10^{-52}~\textrm{m}^{-2},~\textrm{or}~2.89\times
10^{-122}~\textrm{in Planck units}\right)$. Bounds on it from Solar System
orbital motions determined with spacecraft tracking are currently at the
$\simeq 10^{-43}-10^{-44}~\textrm{m}^{-2}~\left(5-1\times 10^{-113}~\textrm{in
Planck units}\right)$ level, but they may turn out to be somewhat optimistic
since $\Lambda$ has not yet been explicitly modeled in the planetary data
reductions. Accurate $\left(\sigma_{\tau_\textrm{p}}\simeq
1-10~\mu\textrm{s}\right)$ 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
$\overline{\Delta\delta\tau}^\Lambda_\textrm{p}$, a S2-like orbital
configuration with $e=0.8839,~P_\textrm{b}=16~\textrm{yr}$ would allow to
obtain preliminarily an upper bound of the order of
$\left|\Lambda\right|\lesssim 9\times 10^{-47}~\textrm{m}^{-2}~\left(\lesssim
2\times 10^{-116}~\textrm{in Planck units}\right)$ if only
$\sigma_{\tau_\textrm{p}}$ were to be considered. Our results can be easily
extended to modified models of gravity using $\Lambda-$type parameters.
