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Lambda Perturbations of Keplerian Orbits in the Expanding Universe

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Abstract

To estimate the influence of “dark energy” on Keplerian orbits, we solve the general-relativistic equations of motion of a test particle in the field of a pointlike mass embedded in the cosmological background formed by the cosmological constant with realistic cosmological Robertson–Walker asymptotics at infinity. It is found that under certain relations between three crucial parameters of the problem—the initial radius of the orbit, the Schwarzschild and de Sitter radii—a specific secular perturbation caused by \(\Lambda\)-term becomes significant, i.e., can reach the rate of the standard Hubble flow. This fact is interesting both by itself and may have important consequences for the long-term dynamics of planets and stellar binary systems.

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Notes

  1. It was emphasized for the first time by Balaguera-Antolínez et al. [13] that the specific interplay between \(r_{g}\) and \(r_{\Lambda}\) can result in a manifestation of the \(\Lambda\)-term effects at spatial scales much less than \(r_{\Lambda}\); but that consideration was performed for the static Kottler metric (1). Besides, “small-scale” cosmological effects were also found in the collapsing matter overdensities (see, e.g., [22] and references therein); but this analysis was performed with models of “dynamical” DE and is therefore irrelevant to the present study.

  2. For example, in the Earth–Moon system, \(r^{*}_{g}=r_{g}/R_{0}\sim 10^{-11}\) and \(r^{*}_{\Lambda}=r_{\Lambda}/R_{0}\sim 10^{18}\).

  3. A somewhat greater value, \(\dot{R}_{\textrm{rot}}=2.53\) cm/yr, used in our earlier work [11], was obtained from the data on the Earth’s deceleration rate corrected for observations of the ancient solar eclipses, which could be hardly reliable.

  4. Yet another option might be DE mediated by the quantum uncertainty relation, as suggested in our recent paper [30]. In that case, it is unclear if the effective (i.e., uncertainty-mediated) \(\Lambda\)-term will contribute to the small-scale dynamics of celestial bodies.

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ACKNOWLEDGMENTS

I am grateful to Yu. Baryshev, N. Capitaine, M.L. Fil’chenkov, S.S. Gerstein, C. Lämmerzahl, S.A. Klioner, S.M. Kopeikin, J. Müller, K. Nordtvedt, M. Nowakowski, E.V. Pitjeva, and A.V. Toporensky for valuable discussions and critical comments. I am also grateful to Wilhelm and Else Heraeus-Stiftung for the opportunity to present this work at a few workshops.

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  1. Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya str. 84/32, Moscow, 117997, Russia

    Yu. V. Dumin

  2. Sternberg Astronomical Institute (GAISh), Lomonosov Moscow State University, Universitetskii prosp. 13, Moscow, 119234, Russia

    Yu. V. Dumin

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Dumin, Y.V. Lambda Perturbations of Keplerian Orbits in the Expanding Universe. Gravit. Cosmol. 26, 307–315 (2020). https://doi.org/10.1134/S0202289320040040

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