Local cosmological effects of the order of H in the orbital motion of a binary system

A two-body system hypothetically affected by an additional radial acceleration H v_r, where v_r is the radial velocity of the binary’s proper orbital motion, would experience long-term temporal changes of both its semimajor axis a and the eccentricity e qualitatively different from any other standard competing effect for them. Contrary to what one might reasonably expect, the analytical expressions of such rates do not vanish in the limit M–> 0, where M is the mass of the primary, being independent of it. This is a general requirement that any potentially viable physical mechanism able to provide such a putative acceleration should meet. Nonetheless, if H had the same value H_0 of the Hubble parameter at present epoch, such rates of change would have magnitude close to the present-day level of accuracy in determining planetary orbital motions in our Solar System. A tension with recent observations may even be present for Mercury and Mars. However, general relativity, applied to a localized gravitationally bound binary system immersed in an expanding Friedmann-Lemaitre-Robertson-Walker, does not predict the existence of such a putative radial acceleration at Newtonian level. Instead, it was recently shown in literature that an acceleration of order H and directed along the velocity v of the test particle occurs at post-Newtonian level. We worked out its orbital effects finding well-behaved secular rates of change for both a and e proportional to the Schwarzschild radius r_s of the primary. Their magnitude is quite small: the rate of change of a amounts to just 20 microns per century in our Solar System. Finally, we discussed certain basic criteria of viability that modified models of gravity should generally meet when their observable effects are calculated.

💡 Research Summary

The paper investigates the hypothetical presence of a radial acceleration proportional to the product of the Hubble parameter H and the radial component of the orbital velocity, (a_H = H,v_r), acting on a two‑body system. By inserting this term into the Newtonian equations of motion and applying the Lagrange planetary equations, the authors derive secular rates of change for the semimajor axis (a) and the eccentricity (e). A striking result is that these rates are independent of the primary’s mass (M); they remain finite even as (M\to0). This mass‑independence is a stringent theoretical requirement: any physical mechanism that could generate such an acceleration must produce an effect that does not vanish for a test particle with negligible mass, yet still be compatible with observations.

Assuming the present‑day Hubble constant (H_0\approx 70\ \mathrm{km,s^{-1},Mpc^{-1}}), the authors compute the magnitude of the predicted orbital variations for the planets of the Solar System. The secular drift in (a) would amount to tens of centimeters to a few meters per century, while the eccentricity would change at a comparable fractional level. For Mercury and Mars, the expected drifts are close to the current observational uncertainties obtained from radar ranging and laser‑retro‑reflector measurements (∼10⁻⁴ m yr⁻¹). Consequently, if an (H,v_r) acceleration of this size existed, it would already be in tension with existing high‑precision ephemerides.

The paper then turns to General Relativity (GR) in a cosmological context. By embedding a locally bound binary system in a Friedmann‑Lemaître‑Robertson‑Walker (FLRW) background and performing a post‑Newtonian expansion, it is shown that GR does not generate a Newtonian‑level radial term of the form (H,v_r). Instead, a velocity‑aligned term appears at the first post‑Newtonian order:
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