What force causes the universe to expand?
The universe only expands at large distances. It doesn't expand locally. You are not getting bigger, nor the city you live in.
Distant galaxy clusters only are observed to recede from us with a velocity proportional to their distance. They are sufficiently far and free, that is not bound by gravitational forces like our solar system is, nor by electromagnetic forces like we are, to have the Friedman Robertson Walker (FRW) metric as solution of Einstein's equations. That metric has a scale factor which is growing, i.e. it describes an expanding space. But ONLY at very large distances.
In addition, because the big bang model is not enough to explain isotropy of the cosmic radiation background, the universe is assumed to have inflated, and that inflation is caused by dark energy, that is negative energy (not to be confused with dark matter which is matter we cannot see).
This is the answer to your question. You notice I'm not using the word "Force" when describing the expansion of the universe. Force is a Newtonian concept. It will get you into trouble when trying to understand General Relativity. The force concept is replaced by the metric.