  Scale invariance is expected in empty Universe models, while the presence of
matter tends to suppress it. As shown recently, scale invariance is certainly
absent in cosmological models with densities equal to or above the critical
value $\varrho_{\mathrm{c}} =3H^2_0/(8 \pi G)$. For models with densities below
$\varrho_{\mathrm{c}}$, the possibility of limited effects remains open. If
present, scale invariance would be a global cosmological property. Some traces
could be observable locally. For the Earth-Moon two-body system, the predicted
additional lunar recession would be increased by 0.92 cm/yr, while the tidal
interaction would also be slightly increased.
  The Earth-Moon distance is the most systematically measured distance in the
Solar System, thanks to the Lunar Laser Ranging (LLR) experiment active since
1970. The observed lunar recession from LLR amounts to 3.83 ($\pm 0.009$)
cm/yr; implying a tidal change of the length-of-the-day (LOD) by 2.395 ms/cy.
However, the observed change of the LOD since the Babylonian Antiquity is only
1.78 ms/cy, a result supported by paleontological data, and implying a lunar
recession of 2.85 cm/yr. The significant difference of (3.83-2.85) cm/yr = 0.98
cm/yr, already pointed out by several authors over the last two decades,
corresponds well to the predictions of the scale-invariant theory, which is
also supported by several other astrophysical tests.
