Two-Body Orbit Expansion Due to Time-Dependent Relative Acceleration Rate of the Cosmological Scale Factor

By phenomenologically assuming a slow temporal variation of the percent acceleration rate S̈S−1 of the cosmic scale factor S(t), it is shown that the orbit of a local binary undergoes a secular expansion. To first order in the power expansion of S̈S−1 around the present epoch t0, a non-vanishing shift per orbit ⟨∆r⟩ of the two-body relative distance r occurs for eccentric trajectories. A general relativistic expression, which turns out to be cubic in the Hubble parameter H0 at the present epoch, is explicitly calculated for it in the case of matter-dominated epochs with Dark Energy. For a highly eccentric Oort comet orbit with period Pb ≈ 31 Myr, the general relativistic distance shift per orbit turns out to be of the order of ⟨∆r⟩ ≈ 70 km. For the Large Magellanic Cloud, assumed on a bound elliptic orbit around the Milky Way, the shift per orbit is of the order of ⟨∆r⟩ ≈ 2–4 pc. Our result has a general validity since it holds in any cosmological model admitting the Hubble law and a slowly varying S̈S−1(t). More generally, it is valid for an arbitrary Hooke-like extra-acceleration whose “elastic” parameter K is slowly time-dependent, irrespectively of the physical mechanism which may lead to it. The coefficientK1 of the first-order term of the power expansion ofK(t) can be preliminarily constrained in a model-independent way down to a K1 ≲ 2 × 10−13 year−3 level from latest Solar System’s planetary observations. The radial velocities of the double lined spectroscopic binary α Cen AB yieldK1 ≲ 10−8 year−3.