This is not as obvious as it sounds at first.If NASA opened up a portal to a Virgin Earth and the US settled 5% of their population on it with the same laws, society, economy, and living standards we have here, you wouldn't expect the birth rate in the country to double overnight.
The key word here is the same living standard. But the living standard on the virgin Earth would be inherently different than from the original Earth. Two main differences are housing and job availability.
In the modern US, there are areas significantly lacking in jobs, for example rural Nebraska. On the other hand, the major population centers have a labor shortage. An important property of the large population centers is that housing is extremely expensive, mostly due do high population density and the need to optimize the limited land into high population capacity.
On a virgin world, land would be basically free. You could build spacious one-story houses with yards everywhere. Housing on a virgin world would inherently be cheaper with a good-enough industrial base, and thus the standard of living on a colony would increase in that regard.
On the other hand, jobs are inherently present on a virgin world. The world is virgin, it needs to be tamed. Every single capable person would be employed, leading to a combination of both job and housing availability. Thus, with the same level of technology, one might even expect that the colonists would have a better standard of living than the people staying on the homeworld.
But the primary difference is that in our society, children are basically an economic burden. They need to be fed, given shelter, kept healthy, educated... Parents get no economic returns, they have kids for emotional / biological reasons. But on a virgin world in development which is starving for new workers, people are wealth. Thus having more kids to help work on your farm, shop, whatever, is actually economically beneficial for the parents. Based on this, my assumption is that the population on the virgin world would grow higher than on the homeworld, per capita.
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