general relativity - Is the acceleration of the expansion of the universe caused by expansion of spacetime itself?

When considering the accelerating expansion of the universe, it is often stated that dark energy is required to cause acceleration of expansion in order to overcome the pull of gravity.

However, it seems that if the rate of expansion is great enough, this would overcome the effect of gravity that is causing deceleration.

In other words, the distance gain of the expanding universe is happening at a rate greater than the force pull of gravity. So although gravity does work to reduce the expansion velocity, the additional distance gained causes an increasing amount of space to expand. I.e. things are accelerating faster apart because at each step, there is more space between them to expand (which is increasingly greater than the gravity pull).

Let me try to define this better:

Given two bodies in the universe very far apart a,b.

vel_expand - the rate at which the distance between a,b is increasing due to the expansion of spacetime between them

acc_from_expand - the increase of vel_expand caused because of the greater distance between a,b

So as a,b move apart, the space between them increases (which is an increase in the amount of expanding space time between them), which causes a greater increase due to expansion of space time - i.e. an acceleration of expansion due to the expansion itself.

Now on the opposite side of the equation is the pull of gravity (the curvature of space time) caused by the mass of a,b. However, gravity decreases at an inverse square to the distance so it quickly becomes smaller as distance changes.

So as distances become large between a,b there is a point where the acceleration due to expansion of spacetime between a,b is greater than the deceleration due to gravity between them.

So at that point they will continue to expand apart at an accelerating rate.

(This concept is similar to the edge of the observable universe. The distance from which speed of light can overcome the expansion of the universe.)

Maybe this is all completely wrong. But I would like any feedback on whether this makes any sense.

Also in my description, the acceleration of expansion would be a function of the distance between objects - so the further apart 2 bodies in the universe, the faster they would be accelerating apart. I don’t know if that matches measurements - but I have read that there are discrepancies between the rate of expansion measured in the background radiation versus white dwarves vs supernovas.

Does my concept make sense?

If so, is there an article that discusses the acceleration of the expansion of the universe due to the expansion of the universe itself?

If not, where am I wrong in my concept?