dark matter isn’t connected to the expansion of space

To be double-precise (😜) both dark matter and dark energy are related to the expansion of space. What you probably meant to say is that dark energy is responsible for the accelerating rate of space expansion. Dark matter slows the rate down.

Even if there was no dark energy, the universe would still have been expanding... at first. If there was enough matter+dark matter+light, the expansion rate would slow down to a zero and reverse, resulting in a Big Crunch. But if the amount was below a critical threshold (Ω~M~ = 1 in this picture: https://en.wikipedia.org/wiki/File:Friedmann_universes.svg) then the rate would never slow down to exactly zero (same as a rocket on escape velocity always slows down but never returns back to Earth), and expansion would continue forever.

When you add dark energy back in, the expansion rate starts accelerating, but only when the density of dark energy is greater than the density of matter+dark matter+light. The expansion of our universe was actually decelerating until 5 billion years ago, because matter+dark matter dominated. With the right ratios, we could even have had a Big Crunch despite having dark energy.

So matter, dark matter, and dark energy don't cause the expansion, they only gradually increase/decrease the rate it happens at. But then why did expansion happen in the first place? That's the initial boundary condition of the universe. The Big Bang theory doesn't explain why Big Bang happened, only how.