The dynamic simulation model paints a picture of a Moon that drifted slowly away from an icy Earth, lit by a Sun shining around 30 percent less brightly than it does today.
As well as filling in some of the blanks about how Earth and the Moon first became neighbours in the Hadean period, this latest research also helps to explain the Moon's equatorial or fossil bulge – the way it's thicker than it really should be around the middle.
"The Moon's fossil bulge may contain secrets of Earth's early evolution that were not recorded anywhere else," says co-lead researcher Shijie Zhong, from the University of Colorado Boulder.
"Our model captures two time-dependent processes and this is the first time that anyone has been able to put timescale constraints on early lunar recession."
The Moon is currently moving away from our planet at the rate of about 4 centimetres (1.57 inches) a year, which is why Earth's rotation is gradually slowing down and our days are always getting ever-so-slightly longer.
We can log those measurements for ourselves, but what the new study does is examine how that movement might have happened billions of years ago. The model the researchers came up with shows a slow separation over several hundred million years.
For that to have been the case though, Earth would have needed to be less influenced by tidal forces that it is today, suggesting much of the planet's water was still frozen solid.
"Earth's hydrosphere, if it even existed at the Hadean time, may have been frozen all the way down, which would have all but eliminated tidal dissipation or friction," says Zhong.
That in turn would suggest the Sun was significantly weaker than it is now, leading to a colder planet Earth.
The concepts of a "snowball Earth" and a cooler Sun have both been proposed by scientists before, though not this far back in time – it's difficult to peer back 4 billion years into history, though the new research adds some useful extra data to the various possible permutations.
The final mystery the new study addresses is why the Moon is flatter at its poles and wider at its equator than it should be, given its rotation and speed. That was a problem first raised by French mathematician and physicist Pierre-Simon Laplace 200 years ago.
Most scientists think the bulge comes from a time when the Moon was hotter, bigger, and closer to Earth than it is now, with the excess material gradually getting frozen into place as the Moon spun out from our planet.
What the new research has managed to do is develop a model that fits that hypothesis, showing how an icy Earth and a slowly retreating Moon could've helped form that extra padding that we see today.
It's still just a hypothesis – albeit a very intelligently calculated one – but we might have just a little more of an idea about how the two objects split up billions of years ago.
Next up, the researchers want to refine their sums to look at the period between 3.8 and 4.5 billion years back in time.
The researchers themselves admit there's still much about this time and these events that is uncertain.
We don't have much in the way of direct evidence for any of this, but the researchers conclude that "our lunar fossil bulge formation model provides new and unique insights into studies of climate and surface environment of the early Earth".
The research has been published in Geophysical Research Letters.