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 S？S？ -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 S？S？ -1 around the present epoch t 0, a non-vanishing shift per orbit (Δr) 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 (Δr)？≈ 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？(Δr)？≈ 2–4 pc. Our result has a general validity since it holds in any cosmological model admitting the Hubble law and a slowly varying S？S -1 (t). More generally, it is valid for an arbitrary Hooke-like extra-acceleration whose “elastic” parameter κ is slowly time-dependent, irrespectively of the physical mechanism which may lead to it. The coefficient κ 1 of the first-order term of the power expansion of κ (t) can be preliminarily constrained in a model-independent way down to a κ 1 ？ 2 x 10 -13 year -3 level from latest Solar System’s planetary observations. The radial velocities of the double lined spectroscopic binary ALPHA Cen AB yield？κ1 ？ 10 -8 year -3.