From what I read Alpha Centauri formed 250 million years before the sun which was 4.85 billion years. So how far apart were they when the sun formed vs what they are now?
The expansion of the universe must be contributing to it's 4.4 light years. But because they are so close together I am wondering if it is significant on such short scales?
The Sun was very likely nowhere near Alpha Cen when it was born. Stars born in the Galactic disc have velocity dispersions of about 10-20 km/s, which grow over time. A km/s velocity difference amounts to a distance travelled of about 3 light years every million years.
That the Sun and Alpha Cen are close together now, is a fleeting moment in Galactic history. They were hundreds and thousands of light years apart in the past and will be again in the future
This is totally unrelated to the universal expansion, which has no noticeable effects on objects within a galaxy.
Cosmic expansion has no influence on how the distance between the Sun and Alpha Centauri evolves. Not even a negligibly small influence, but none whatsoever.
The idea that cosmic expansion creates a tendency for things to separate is a common misconception, driven by an overly literal interpretation of the idea of expanding space. As I've said in another answer, space is not expanding, at least not in the way that you are thinking. I won't repeat all the details, so if you are in doubt, check that link. The basic point is that cosmic expansion is fundamentally just that galaxies and other material are moving apart, in the large-scale average.
With that in mind, this question is essentially asking how the recession of distant galaxies affects the distance between the Sun and Alpha Centauri. There is no effect, because the gravitational influence of galaxies in opposite directions cancel. This is the shell theorem in Newtonian gravity or Birkhoff's theorem in general relativity; in principle they only apply to spherically symmetric systems, but corrections due to asphericity introduce tidal forces, not any net outward pulls.
With that being said, people often conflate dark energy with cosmic expansion. Dark energy's gravitational influence accelerates cosmic expansion, but they are different things. If you are wondering instead whether dark energy influences the Sun-Alpha Centauri separation, the answer is likely yes, but not in the obvious way.
There is probably dark energy pervading the Milky Way. That dark energy would tend to push the Sun and Alpha Centauri apart. However, we can't talk about the Sun and Alpha Centauri in isolation, because their dynamics are dominated by their orbits around the Milky Way galaxy. In the context of galactic dynamics, dark energy effectively reduces the Galaxy's gravitational attraction. This would slightly perturb the Sun's and Alpha Centauri's orbits; they would move slightly more slowly, precess slightly differently, etc. But that's all. There's no tendency to make the stars move apart.
But there would be secondary effects on the Sun-Alpha Centauri separation due to their altered orbits. The two stars are only nearby today by chance; they were not close in the past and will not be in the future. Any minor perturbation to their orbits, such as that supplied by dark energy, could drastically change their positions within the Galaxy today. So in some sense, dark energy is responsible for the Sun and Alpha Centauri being near each other today!
The Hubble constant tells us that the "expansion of the universe" is causing Sol and Alpha Centauri to move apart at about 8.5cm / 3.4in per second.
Velocity dispersion in the galactic disc is about 380m / 1200ft per second, our orbit around the galaxy is about 230km / 143mi per second and isn't quite parallel to Alpha Centauri's.
So Hubble expansion doesn't play much of part in the relative position of the two stars. And in a region as small as a galaxy its effect is so weak that it is overcome thoroughly by gravity, in effect having no effect.
