Research suggests that the number of children people have is strongly affected by the economics of children. When having many children is economically advantageous, birthrates go up, and when it's not, birthrates go down.

During most of history, the cost of raising children to successful adulthood was relatively low and the benefits of having a large family fairly high. Today, in all of the advanced societies, it is very expensive to raise a child to successful adulthood -- it takes 30-50% longer, and the education needed to be successful is very expensive. So families today are appreciably smaller than they were in medieval times.

(Two other causes -- the availability of birth control, and the dramatically lower rate of child mortality -- are also important. But I don't think you propose to reverse either of these, so they are off the table in this discussion. That would be pretty dystopian, anyway.)

So basically, you need to change the economics of having children. Having a large family today requires one parent to stay at home to raise the kids -- a big hit to income -- and has huge schooling and housing costs. Widespread automation will eventually mean many fewer people working while at the same time raising the general standard of living. This should pretty much eliminate the one-parent-stays-home issue. So find a way to bring down the cost of education, and you have a plausible scenario for renewed population growth.

Start a war that requires the draft

The single biggest reason for the massive population boom of the 1950's in the U.S. was World War II. At the beginning of the war it required nearly everybody — especially young men who were no longer in a situation to recognize that having children bore consequences. They were all marrying quickly and siring quickly (possibly because they didn't expect to return... at least that would be part of it).

At the end of World War II came one of the biggest economic upturns in U.S. history. Technology had rising dramatically during the war. Enough people had died to ensure a manpower shortage. Perhaps feeling their mortality, young men returned home with the desire to start families.

Yes, these observations are massive oversimplifications....

But it's not enough to start a war. You also want a Fifth Great Awakening

Nothing says "build a big, happy family!" like religion. Most religions (especially the Abrahamic religions: Judeo-Christian & Islamic) are very big on very big families. A man's worth was dictated by the size of his family. This is especially useful since many religions, being pro-family, are anti-contraception.

Make it cheap to have a big family

Other answers have proposed things like tax incentives to build big families. That isn't worth much because the cost of raising a child is substantially larger than any tax incentive that could possibly be passed.

On the other hand, passing some laws with unintended consequences would do the trick nicely. Here are some fictional laws that might help things along:

• The National Free College Education Act (Necessary college savings: $0)


• The Textile Manufacturers Subsidy and Revitalization Act (Cheap clothes during those critical first 15 years)


• The Workforce Equalization Act: Section I - Employer-required child daycare


• The Workforce Equalization Act: Section II - Employer-required child health care


• The Workforce Equalization Act: Section III - Teen Apprenticeship Program

That last one is important

IMO, one of the biggest reasons for the shift to declining family size is the shift from agricultural to city-based living. You need big families on farms as a source of cheap labor (and character-building. Oh I wish we had character-building today. It's such a pain to hire today's teens. Soapbox-mode:off.). On the other hand, when all you're doing is sending one parent to the office everyday, children are simply a luxury. And if both parents must or want to work at their careers, then children are a problem. This really drives down the "do you want another child?" statistics.

Give the parents a right to a percentage of the income tax paid by their children and (a smaller one) of their grand children. This would encourage them to raise successful tax paying children, and would go a long way to solving the pension problem a the same time. It is really just institutionalizing the traditional human practice of children taking care of their parents in old age.

Getting the percentage right would be tricky because your feedback loop is measured in many decades.

I have heard that there has been some discussion of this approach in a Scandinavian country, but not sure which one.

Factory babies.

Making babies used to be a mom and pop enterprise, so to speak - a cottage industry. No longer. Absent a need for humans to gestate the fetus, the government can make all the babies it wants. And it wants a lot - the space colonies are hungry for pioneers. The baby boom is a factory phenomenon.

This premise - humans grown to fulfill societal needs - dates back to Brave New World.

https://archive.org/stream/ost-english-brave_new_world_aldous_huxley/Brave_New_World_Aldous_Huxley_djvu.txt

One egg, one embryo, one adult-normality. But a bokanovskified egg
will bud, will proliferate, will divide. From eight to ninety-six
buds, and every bud will grow into a perfectly formed embryo, and
every embryo into a full-sized adult. Making ninety-six human beings
grow where only one grew before. Progress.

You just need to meet two conditions

The first, is that people need to feel like they can support children. Some people will still have kids no matter how desperate there situation, but many will want to wait till they feel stable. Stable income, stable (and preferably roomy) housing, stable relationships, stable (bright) future. Sounds like you have the first two points there solved, relationships happen mostly on their own in society, but you could throw something in to boost that too if you want, and if you have avoided any dystopian/societal break down elements, that last is not hard.

The second, is that children have to seem desirable. If you have a culture that says don't loose your independence, whether it is a macho kids tie you down attitude, or a feminist don't give up your career for kids attitude, birth rates will go down. You need having kids to be seen a good thing. This could be because of anything as simple as a couple of celebrities deciding to bring their kids on to the red carpet with them, or a surge in births from the royal family.

Meet those two conditions, and the likelihood of births goes up.

A new entrant into the supply chain for the active chemical ingredients in contraceptives appears on the scene, ten years later it's discovered the company has been supplying chalk dust or substandard or other cheap substitutes as the real mccoy to the contraceptive manufacturing industry.

Think of the unfit for human consumption chicken meat being sold in supermarkets scandal of not so long ago in the UK : that guy was supplying the supermarket chains for a fair old while before he was caught : if you imagine his activities occurring during a period of greater cut backs in government spending on health & safety inspectors he might have got away with it for longer.

The further up the supply chain this is the more contraceptive manufacturers are likely to be effected.

Nice & simple, slightly funny & gets you your baby boom.

A reversal in the policies that discourage development of bonding social skills, break up young couples, and advocate against early childbirth.

Part of the reasons may be found in The Atlantic: Why Are Young People Having So Little Sex?

But the more complete list of problems in my view are:

• We coddle our children, being afraid to leave them alone to play with other kids. The younger they start to independently socialize, the quicker they learn to form meaningful friendships (which is the basis of romantic relationships). Facing threats together like bullies or danger teaches them that friendship has meaning beyond superficial fun.


• We rip apart early romances by sending kids off to college in different parts of the country, and then recalling them back.


• We expect young people to raise their own children, without a lot of help from grandparents.


• We teach young people that they must have an education and a career, and that kids get in the way of education and career, and that grandparents can't take off the workload and help them to achieve these things despite having kids.

Basically, my view is that the key to a baby boom is young people. You won't get very far convincing older (late 20s+) people to get married and have kids more than they already do. Not without simply paying them, I guess.

People like sex (citation needed). Sex produces babies. The reason population growth slowed down is obviously an easy access to contraception (and to a lesser degree - abortion). There are two ways of causing baby boom - a) contraception stops working, b) people choose not to use it.

1. No contraception available

For some reason contraception became less available. It can be religious, political, technological or any other. Religious fundamentalist politics is done to death in novels and tv these days, techno is more interesting.

1a Technoloy

When the oil runs out many technologies will become very expensive, almost everything we make is based on oil products, like plastics. That includes condoms, but also a lot of medicines. The problem with this story is that you have bigger problem than lack of condoms when you run out of oil.

1b Biology

Evolution decided to save us from going extinct and people developed severe allergic reaction to condoms or pills or both. Or even better, pills stop working. This will take longer than 50 years though, you would have to move your story 200 years or more in the future. If a gene of resistance to contraception appears it will obviously spread quite quickly, in couple of generations it will dominate the population because people without it have 0-1 child and people with it have 6-10.

2. Contraception is dangerous

More interesting option is some disease that affects women on pill only. If taking a pill makes you likely to die you will have no option but to abstain from sex unless you are ready for children, most likely in a stable marriage.

3. No need for contraception

People try to prevent pregnancies, even in stable marriage, because children are very costly. They cost the mother a year or more of her career. They need a spare room, new clothes every few months, toys, school fees, etc. If your civilization somehow becomes post-scarcity many of these problems go away.

Conclusion

There will obviously always be women who just don't want children and do everything to prevent or terminate pregnancy, and if all else fails even resort to infanticide ("fourth trimester abortion"). But even if only 25% women have 5 children each, and another 50% have 3 each, that's already population explosion.

Hmmm. Baby boom is the result of a widespread behavioral change (likely). So perhaps a fast-spreading, ridiculously popular gospel of lotsa kids embedded in a religion "gone viral".

Or, baby boom is the result of a change in the way procreation works, but only for some people. It could be an emergent property expressed only in certain haplogroups causing ludicrous fertility, greatly increased twin/triplet birth percentages, etc. You don't have to explain WHY some haplogroups would be affected (although people in your universe would certainly seek this information), and you only have to throw a bone as to HOW.

One of my favorite books is "Darwin's Radio" by Greg Bear. It treats a fascinating mechanism for emergent radical changes in human development. If you're looking for a "scientific enough" mechanism, this is a great source from which to fashion a jumping-off point for your own treatment.

You alredy have the most important condition - Cheap Energy. Humans are not that different from all other living beings in that they expand their numbers as fast as the free energy in the environment allows. The long population boom that brought mankind from 1 billion around 1750 to 8 billions today was due the cheap energy of the fossil fuels. If your fusion power is at least as good as the fossil fuels were (which is not something to take for granted due to EROEI) they you will have your boom and that boom will go on until there is so many people consuming so much energy that the energy per capita starts falling, like it started in the 1970's in relation to fossil fuels.

Power's out... NO INTERNET!!!

Imagine there's a massive power outage for days, or weeks even, and there's no computers or tablets or tv's or phones left working, no electronic devices at all to keep people entertained, but most importantly there's NO INTERNET!!! Many jobs are even unworkable, leaving lots of people everywhere bored stiff, with nothing else to do...

Maybe a few times a year these kinds of power & internet outages occur, if it's during colder weather then people sometimes even huddle together for warmth, "sleeping" overnight...

Perhaps your world's birth control is dependent on electricity somehow, or needs refrigeration, or shipping, and becomes ineffective or unobtainable after a while without electricity.

Baby booms after blackouts appear to be a real occurrence, for plain power outages in general, or after hurricanes, especially power outages in winter:

Forty-four percent more [babies]... villages lost power for 50 hours in December

The community is battling a declining birth rate, like the rest of the Netherlands -- which ranks among the lowest in the world. And while the power cut method worked well, Maasdriel doesn't plan on a deliberate repeat.

Or after ice storms:

“We just tried to stay warm,” said Hay-Mendoza, 20. One deed in particular was more effective than the rest.

“We were pretty active,” she said with a laugh. The couple got extra cozy a few times a day, she said. “There was nothing else to do, really… It was just cold.”

Why would your power become so unreliable? Maybe you've recently switched to a new clean energy source, but it's unstable. Maybe solar flares knock out the power grid or power plants, maybe there's fuel shortages, or terrorists, or thieves stealing vital equipment, or just unusually strong & widespread ice storms.

At least the idea is very conceivable...

A major war comprising most of Western Europe and the Americas, e.g., between the EU and the US. Historically, there is always a baby boom after a war ends.

provide basic income only for families with more than 2 children, based on the number of children, and enough that with three children the basic income covers enough of the family's needs so that only one parent needs to work. 6 children should allow both parents to stay home. essentially, pay families to have children.

that's likely not enough, so also structure the cost of living in such a way that one or even two salaries are not enough to live (like extremely high taxes on childless couples or singles), and only the basic income from having children allows you to reach a comfortable living level.

add some exceptions for those who can't have children for no fault of their own (but encourage them to adopt)

What could cause a baby boom across the Western world:

Economic

As other posters have mentioned, change economic conditions to make raising a family affordable.

Eliminate inheritance taxes and estate taxes, allowing families to accumulate wealth and retain property over generations. Drastically lower or eliminate property taxes.

Develop a robust varied economy with sectors for all skill levels. Implement economic policies to encourage skilled labor jobs, blue collar and white collar. Lower taxes on businesses, enabling them to hire more workers. Implement balanced tariffs on foreign goods to prevent product dumping and destruction of industries. Drastically limit immigration so the value of labor will increase and the cost of housing will decrease. The goal of these reforms is to make it possible for a family with multiple children to prosper with one working parent, as in decades past.

Educational

Raise educational standards and have schools focus on nothing but academics instead of indoctrination. Have different quality level of schools, so motivated children can go to schools where the students are serious about what they're doing. Develop trade education programs for students who want to enter a career field which doesn't require university education. Remove disruptive, violent and delinquent individuals from schools so kids who want to study can be safe.

Promote personal responsibility in schools, since school is like a kid's job.

Social

Eliminate anti-family media bias. Respect women's choices even if that choice is to get married and have children.

Eliminate media promotion of promiscuity, which damages marriage satisfaction.
Source
Source

Reform public assistance law to incentivize marriage.
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Reform family court to decrease the devastating financial consequences of divorce, thereby encouraging marriage. Eliminate no fault divorce and grant custody of the children to the parent best able to care for them, rather than court systems habitually awarding custody to the mother. Require paternity to be proven with a DNA test before granting child support. Honor pre-nuptial agreements. Require the same standards of evidence in family court as in civil and court.

Reform domestic violence law and apply enforcement of the law equally to both men and women. Enforce perjury laws against individuals who make false statements to law enforcement.

Promote general values based character education with a focus on consideration for others, personal responsibility, and honesty.

Maybe too dystopian:

In 2050, gene manipulation has progressed enough to make designer babies, however the exact combination of gene expressions that give a certain trait is not 100% accurate. To get around this limitation, people create multiple embryos with slightly different variations, and grow them for a few weeks in artificial wombs. The baby with the best expression of the desired characteristics is chosen and the rest are terminated.

Opponents of designer babies finally get their way though, and the practice is banned. However, it is also ruled that the terminated babies have a right to life, and each unwanted variation is recreated, resulting in a population boom of sets of almost identical people.