In my post number 8 in a discussion of colonizing other star systems with slower than light travel: https://historum.com/threads/generation-or-sleeper-ships-which-would-be-the-better-more-realistic-option-for-space-travel.181701/1
It is possible to build artificial space habitats with materials from small solar system objects like asteroids and comets.
So eventually there could be tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, etc. of such space habitats with populations of thousands of persons each.
And of course adding an engine and fuel supply, etc., to such a space habitat would turn it into a vast generation ship.
So I can imagine that a fleet of several such generation ships might be sent to colonize some region in the outer cometary halo of our solar system. At a speed of one percent of the speed of light, it would take such a fleet 100 years to reach a part of the cometary halo 1 light year from the Sun. At a speed of 2 percent of the speed of light it would take such a fleet 50 years to make the journey, at a speed of 3 percent it would take 33.333 years, at a speed of 4 percent it would take 25 years, at a speed of 5 percent it would take 20 years.
If a fleet of several such generation ships is sent to colonize some region in the outer cometary halo of our solar system. At a speed of two percent of the speed of light, it would take such a fleet 100 years to reach a part of the cometary halo 2 light years from the Sun. At a speed of 3 percent of the speed of light it would take such a fleet 66.666 years to make the journey, at a speed of 4 percent it would take 50 years, at a speed of 5 percent it would take 40 years, at a speed of 6 percent it would take 33.333 years.
So possibly several fleets would be sent from the inner solar system to colonize various regions in the cometary halo. And after expanding in the cometary halo for a period, perhaps centuries, each such colony there might send out one or more fleets of generation ships to colonize another, and farther, region of the cometary halo.
The average distance between a star and its nearest neighbor is about five light years in our part of the galaxy. At a speed of about 1 percent to 10 percent of the speed of light, it would take about 50 to 500 years for a generation ship to travel straight from the inner solar system of one star to the inner solar system of the other star, and about 1,000,000 to 10,000,000 years to go from side to sie of the disc of the Milky Way Galaxy.. But if generation ships make voyages of 1 light year each to colonize regions of the cometary halos, and take 100 to 300 years to each the stage to send out long distance colonizing expeditions of their own, the colonizers could reach the inner solar system of a star 5 light years away in about 450 to 1,700 years.
And travelling at about 100 to 300 years between jumps, and jumps of 1 light year each at a speed of 1 percent to 10 percent of the speed of light, a society could expand in all directions at an average speed of about 110 to 400 years per light year of distance. Thus such a society could eventually colonize the entire galactic disc of the Milky way Galaxy, 100,000 light years in diameter, in about 11,000,000 to 40,000,000 years, if they started from an outer rim of the galactic disc.
If the inhabitants of such space habitats expect that they and their descendants will live in space habitats forever, and have no desire to land on any habitable planets they might possibly find, Generation ships made of such space habitats could make much longer distances.
Possibly some generation ship fleets might travel 10 light years in a single voyage, taking 100 to 1,000 years. And if such voyages are successful, some generation ship fleets might later travel 100 light years in a single voyage taking 1,000 to 10,000 years. And eventually generation ship fleets might make voyages of 50,000 light years, taking 500,000 to 5,000,000 years, reaching and beginning to colonize distant regions of the galaxy.
With a faster than light (FTL) space drive the galaxy could theoretically be colonized much faster than with slower than light (STL) methods of propulsion.
I stated that colonists with STL travel might take 100 to 300 years after colonizing a region to start sending out colonizing expeditions of their own, and that their ships might travel at 1 percent to 10 percent of the speed of light, thus traveling at about 10 to 100 years per light year, thus expanding at a rate of about 110 to 400 years of time per light year of distance.
With a FTL drive capable of instantaneous travel to any distance, the travel time part of the equation would drop to zero, so it would all depend on the average time a colony planet would take to send out colonies of its own and how many colony planets each colony planet founds on the average.
Assume that each and every planet, including Earth, sends out two and only two colony expeditions, 100 to 1,000 years after being founded by a colony expedition.
If the average interval is only 100 years, the planets or solar systems inhabited by humans will number:
3 including Earth at first.
5 after 100 years.
9 after 200 years.
17 after 300 years.
33 after 400 years.
65 after 500 years.
129 after 600 years.
257 after 700 years.
513 after 800 years.
1,025 after 900 years.
2,049 after 1,000 years.
4,097 after 1,100 years.
8,193 after 1,200 years.
16,385 after 1,300 years.
32,769 after 1,400 years.
65,537 after 1,500 years.
131,073 after 1,600 years.
262,145 after 1,700 years.
524,289 after 1,800 years.
1,048,577 after 1,900 years.
2,097,153 after 2,000 years.
And if the average colony world takes a multiple of 100 years to send out its colony expeditions, one can multiply that by the amounts of time in the list to get the necessary time.
For example, if it takes 1,000 years for the average colony to send out it colony expeditions, the times to reach a specific number of colony worlds will be ten times longer.
And of course the average colony world might send out three or four, or some other number, of colony expeditions instead of two. And the average colony world might send out one colony expedition every 127.5 years, for example, for as many centuries or millennia as that colony world exists, instead of sending out all its colony expeditions at once.
But it seems simple to make reasonable assumptions about the average rate of colonization that will make it simple to calculate how long it will take for a specific number of worlds and solar systems to be colonized.