  Astronomical (or Milankovi\'c) forcing of the Earth system is key to
understanding rhythmic climate change on time scales >~ 10 kyr.
Paleoceanographic and paleoclimatological applications concerned with past
astronomical forcing rely on astronomical calculations (solutions), which
represent the backbone of cyclostratigraphy and astrochronology. Here we
present state-of-the-art astronomical solutions over the past 3.5 Gyr. Our goal
is to provide tuning targets and templates for interpreting deep-time
cyclostratigraphic records and designing external forcing functions in climate
models. Our approach yields internally consistent orbital and precession-tilt
solutions, including fundamental solar system frequencies, orbital eccentricity
and inclination, lunar distance, luni-solar precession rate, Earth's obliquity,
and climatic precession. Contrary to expectations, we find that the long
eccentricity cycle (previously assumed stable and labeled ''metronome'', recent
period ~405 kyr), can become unstable on long time scales. Our results reveal
episodes during which the long eccentricity cycle is very weak or absent and
Earth's orbital eccentricity and climate-forcing spectrum are unrecognizable
compared to the recent past. For the ratio of eccentricity-to-inclination
amplitude modulation (frequently observable in paleorecords) we find a wide
distribution around the recent 2:1 ratio, i.e., the system is not restricted to
a 2:1 or 1:1 resonance state. Our computations show that Earth's obliquity was
lower and its amplitude (variation around the mean) significantly reduced in
the past. We therefore predict weaker climate forcing at obliquity frequencies
in deep time and a trend toward reduced obliquity power with age in
stratigraphic records. For deep-time stratigraphic and modeling applications,
the orbital parameters of our 3.5-Gyr integrations are made available at
400-year resolution.
