We say that the universe has a particular shape (flat, sphere or saddle) depending on what the ratio between the average energy density and the critical energy density is, something we call omega.

But as dark energy starts to dominate going towards the future, will this also affect the shape of the universe? Suppose that "right now" we observe a flat universe with omega = 1. If that changes even slightly because dark energy starts to dominate, would the Universe's shape as we observe it suddenly "snap" to a saddle or sphere shape?

In other words, as the ratio between the dark energy, dark matter and normal matter contributions change (with dark matter starting to dominate its contribution to the energy density and the other two becoming irrelevant), shouldn't this also change the overall energy density, thus changing the value of omega?

From what I understand, the proposed solution is that yes, the ratios will change but the shape will remain the same as omega itself remains the same. So, instead of our currently observed 70/25/5 dark energy/dark matter/normal matter contributions to the energy density we will have something like 95/4/1 in the future, but omega will remain 1.

What I don't understand is why would omega remain 1? Aren't the contributions different depending on who is contributing? I am imagining that the dark energy contribution "weight" is different (it is of a different nature, it's not structured like normal matter, for example) than dark matter or normal matter.

Can anyone chip in on what the mistake in my thinking is?

We say that the universe has a particular shape (flat, sphere or saddle) depending on what the ratio between the average energy density and the critical energy density is, something we call $\Omega$.

But as dark energy starts to dominate going towards the future, will this also affect the shape of the universe? Suppose that "right now" we observe a flat universe with $\Omega=1$. If that changes even slightly because dark energy starts to dominate, would the Universe's shape as we observe it suddenly "snap" to a saddle or sphere shape?

In other words, as the ratio between the dark energy, dark matter and normal matter contributions change (with dark matter starting to dominate its contribution to the energy density and the other two becoming irrelevant), shouldn't this also change the overall energy density, thus changing the value of $\Omega$?

From what I understand, the proposed solution is that yes, the ratios will change but the shape will remain the same as $\Omega$ itself remains the same. So, instead of our currently observed 70/25/5 dark energy/dark matter/normal matter contributions to the energy density we will have something like 95/4/1 in the future, but $\Omega$ will remain 1.

What I don't understand is why would $\Omega$ remain 1? Aren't the contributions different depending on who is contributing? I am imagining that the dark energy contribution "weight" is different (it is of a different nature, it's not structured like normal matter, for example) than dark matter or normal matter.

What's the mistake in my thinking?