For context, I'm considering a story similar to the Netfilx TV show The 100 in which humanity has to recolonize the Earth following a major extinction-level event. I'm aiming for the society they recreate to be similar to our current one. Everything from technology to cities to daily life is almost identical to ours with the only major difference being geopolitics. So my question is: Is it possible, given a global catastrophic event in which humanity needs to repopulate the Earth, for technological development to "freeze" for, say, 100 years or so since human efforts would be geared towards resettling the Earth? I'm aiming for a modern dark age that doesn't involve the loss of current tech but rather a temporary slow down or halt of technological development. So that, for example, by 2150 or 2200, we've only progressed maybe 40-50 years technologically from where we are now.
There's quite a few good answers already here, so I'd like to focus on the questions itself...
Is it possible, given a global catastrophic event in which humanity needs to repopulate the Earth, for technological development to "freeze" for, say, 100 years or so since human efforts would be geared towards resettling the Earth? I'm aiming for a modern dark age that doesn't involve the loss of current tech but rather a temporary slow down or halt of technological development. So that, for example, by 2150 or 2200, we've only progressed maybe 40-50 years technologically from where we are now.
In the case of a cataclysmic event in which most of the Earth's population is wiped out, there would have to be a very good reason motivating people to continue to try to increase technologically. Since survival would become the most important issue, most of modern technological development would be useless and hence worthless.
For instance, trillions of dollars go into information technology development alone. Trying to make systems larger, more complex, more robust and more user friendly. This all would be useless in a world where there are only hundreds to thousands of people. Consider the defense industry, the space industry, the entertainment industry... All of these industries invest trillions of dollars and contribute enormously to technological development and all of these would be worthless in the scenario given. This can be true for nearly all other branches of technology...when living consists of fighting for survival why put time, energy and resources into anything but trying to rebuild and repopulate?
Thus, while it is possible to keep our current level of technology in the sense that the knowledge can be there...for instance stored in some fashion. There would not be enough people to harness this knowledge, nor enough resources to apply it, let alone resources to further this knowledge.
Thus, depending on who survives and how the rebuild society, I'd expect at first, knowledge to be increasingly lost over time, until the population has increased enough for survival to be a non-issue. Fortunately, with access to all the basic information about how the world works this should not take too long, but will still probably take time scales of hundreds of years and this is assuming no other major cataclysms or wars.
That a small population can try to resettle the Earth is possible.
That the same population can start from where the technology development went idle is highly unlikely.
First of all most of the knowledge needed to operate today's machinery and systems will be gone. With that will be gone all our logistics and supply chains. After 1700 years we still don't know how Romans made their concrete, for example.
You might say: "well, we don't have coal shipments, but we have books telling us that coal was used to feed power plants". Yes, but most of the mines are not shallow like the ones which were first used at the dawn of the industrial revolution. Again, to operate those you will need modern systems.
That would probably be a permanent hindrance on the development possibilities of this new settling culture.
With the exception of a near extinction event, there probably won't be a modern dark ages.
Even the dark ages we heard about in school is considered to be a myth in modern academia. Science still advanced throughout the Middle Ages and wasn't suppressed as much as many people assumed by the Catholic Church. Other civilizations, religious or otherwise, saw scientific advances and a rise in the average lifespan.
Basically, the only way to have a real dark age is to damage the human population to the point where scientific advancements are difficult, if not nearly impossible. There could be an order that suppresses science, but as we learn from the myth of the historical dark ages, you would have to have an order that would repress all scientific advancements for most human civilizations. Otherwise, you will have other civilizations somewhere out there advancing progress (since even if the Dark Ages were real, there would be other groups like the Chinese and Muslims still making scientific discoveries).
There is a rule in biology called the 50/500 rule: a rule that says if everything goes perfectly, you would need about 50 people to survive and carefully breed to prevent inbreeding, and 500 people to prevent genetic problems in future generations. Basically, a disaster that reduces humanity to a few hundred or thousand people rapidly could provide a dark age. This would create a scenario for humans where immediate survival and not inbreeding would take precedence over new discoveries. This temporary small population would make it believable that there are no other people on the planet who would be making massive scientific breakthroughs, and would force any future generations to have to learn basic survival skills before considering any new breakthroughs or progress. The population would also start off small enough to make it nonviable to run factories, mines, and other industrial facilities necessary to maintain modern technology, at least for a few generations.
A near-extinction event such as you propose will have wide-reaching social effects. What caused the near extinction? If the perception of the survivors is that some aspect of the pre-extinction society was the primary cause for all of the deaths, then we can expect that the survivors will not be quick to continue those "deadly" aspects.
If science and technology are thought to have caused the problem, then it does not make sense to ramp up science and technology again. Substitute "lack of religious faith" or "vegan diets" or "too much partisan politics" or "whatever" and watch how society switches to different forms and beliefs.
We humans are quite adept at making up stories to explain the inexpiable. A quick review of mythology, the typical political campaign, and the content of certain cable news channels will disabuse you of the notion that they have to make "rationale" sense. In the situation that you propose, the survivors would not have the time or energy to work through the detailed analysis to get to the "truth."
Enlightened reason, such as we associate with science and technology, might re-emerge after a time, but I would not be surprised if that time was measured in millennia rather than decades.
A catastrophe such as you describe would produce a rapid reordering of priorities. Many forms of technology would be cast aside as useless for survival, and often the tools you need to redo them would be lost, but others would get intensive interest and so development.
At the very least, at our technological level of development, research into infertility treatments could continue and would be urgent, both to preserve genetic diversity and to simply increase the population.
Not only possible, but pretty much probable even without an extinction event.
The original "dark age" happened in the first place because the life was "good enough" (at least, good enough for those in a position to do research or make inventions). All you need to repeat the feat is to kill the social elevators (that's why deep inequality is considered bad).
Contrary, a great repopulation effort will require (and reward!) new technologies (or even reinventing old ones). It will also make possible a lot of different social experiments. Hardly a dark age recipe.
The big problem with most extinction level events is that in order to kill enough people to make technology stagnate you have to minimize the population to a point that makes all sorts of existing technology unachievable. Not just forward progress. A few hundred people won't run around for more than a few years using computers and cell phones because they will be unable to maintain the space programs required to keep the satellites going. They will be unable to maintain the multi-billion dollar factories that manufacture them, they will be unable to gather all the exotic materials from across the globe needed in thier construction. They won't even be able to devote the manpower to keep a modern steel mill running. Since you bring up 100 as an example, your society is pretty much guaranteed to crumble back to the iron age as it did for the surface people with so few people left. It would then take centuries or even millenia for people to repopulate enough to pick back up where we started.
Modern technology is so heavily reliant on economy of scale and global trade that you need to maintain a world population in the millions to keep it from all falling apart, but millions of people thrust into a harsh new survival scenario will tend to innovate just as fast as a larger more comfortable population because they will have more needs to meet.
These factors together mean that you need an event that does less to cause a population driven dark age, but a cultural one. One really good way to do this is a robot uprising. The robots could kill off a large portion of the human population without completely destroying our ability to maintain a modern infrastructure, but the resulting fear of AI would paralyze our ability to technologically progress. Modern tech is at a point now where nearly every new invention hinges on highly developed computer systems to make happen, but if people fear AI too much to use it, then moving forward past our current levels will slow down to a snail's pace.
To achieve your modern micro population scenario, you could make the AI uprising's devastation asymmetric. Have some nations that were completely wiped out while others survived mostly intact. The the surviving nations would need to recolonize the wild zones by first sending small expeditions that may only be in the hundreds.
Modern technology requires a long supply chain -- not just physical supply chain but also a chain of specialized knowledge. Those suppliers don't exist everywhere in the world, and it would take a long time to build up. For example, over the past 20-30 years China has been building up the supply chain for electronics. They have not only the device assemblers but also the chip makers and the suppliers of the various raw materials and components. It has taken decades for them to build the network of companies to dominate that field. If we decide to try to move those industries back to America, it will also be a project of a few decades.
This can also be seen in other complex industries such as auto manufacturing, pharmaceuticals, etc. There are many countries without an auto manufacturer. And you have some countries, like China, that don't yet have an internationally successful car company, but they have lots of the auto parts supply chain.
Simply put, you could create a "dark age" with regards to certain industries, simply by knocking out certain countries in your armageddon scenario. Without the USA, Japan, Germany, and South Korea, you wouldn't have much of a modern car industry. There are some car makers in China, India, etc., but you might think of their vehicles as a technological step backwards (to be fair: I haven't taken a test drive of any of these lately) and it might take them several years to catch up to present day expectations or exceed them.
In your "repopulating the earth" story, people could have all the written-down knowledge of the present, but still not be able to duplicate what we can accomplish. This is the actual case today in many countries. Vietnam or Cuba or Ireland could check all the engineering books out of any major library but that wouldn't give them an auto industry. Canadians could learn online how to make decent bacon, but their agriculture just isn't organized for it.
On the other hand, your settlers might find ways to innovate with less infrastructure. Maybe instead of having a specialized computer chip for every device, they could learn to reproduce one chip (i.e. an Arduino) and utilize it for all kind of different purposes. So their factories and farms might not look like modern ones or like the ones of our past, but more of a DIY hybrid.
If you place the story 20-30 years forward from now, you could use nano-technology aka robots to fill in the gaps. The humans have the knowledge and your robot army does the field work.
We need coal! -> robot mine. We need to build x -> robot builds.
On a side note; many stories imagine that this will be how future colonization of worlds will work. You can't turn a human off (yet), you can turn off a robot. Besides that their needs are very shallow; power and maintenance. Where as a human has a pletora of needs that need to be met during transit and after planetfall.