Despite their importance for determining the evolution of the Earths atmosphere and surface conditions, the evolutionary histories of the Earths atmospheric CO$_2$ abundance during the Archean eon and the Suns activity are poorly constrained. In this study, we apply a state-of-the-art physical model for the upper atmosphere of the Archean Earth to study the effects of different atmospheric CO$_2$/N$_2$ mixing ratios and solar activity levels on the escape of the atmosphere to space. We find that unless CO$_2$ was a major constituent of the atmosphere during the Archean eon, enhanced heating of the thermosphere by the Suns strong X-ray and ultraviolet radiation would have caused rapid escape to space. We derive lower limits on the atmospheric CO$_2$ abundance of approximately 40% at 3.8~billion years ago, which is likely enough to counteract the faint young Sun and keep the Earth from being completely frozen. Furthermore, our results indicate that the Sun was most likely born as a slow to moderate {rotating young G-star} to prevent rapid escape, putting essential constraints on the Suns activity evolution throughout the solar systems history. In case that there were yet unknown cooling mechanisms present in the Archean atmosphere, this could reduce our CO$_2$ stability limits, and it would allow a more active Sun.
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