# How can $\Omega_m<1$ indicate an infinitely expanding universe?

I was reading up about the cosmological density parameters $$\Omega_m. \Omega_{rel}, \Omega_{\Lambda}$$ corresponding to total matter(baryonic+dark), relativistic particle matter and dark energy. I studied that if $$\Omega=\frac{\rho}{\rho_c}$$, which is the sum of the three is greater than one, the universe eventually collapses, if exactly one, its flat and if less than 1, it keeps expanding. This doesn't make sense to me because dark energy is what is driving the expansion in the first place, so I would normally expect the condition for infinite expansion to be something along the lines of $$\Omega_m+\Omega_{rel}<\Omega_{\Lambda}$$. Can anyone please explain if I'm missing something?

• "Dark energy is what is driving expansion" No. Dark Energy isn't a "pushing antigravity". Commented May 5 at 17:57
• eh, "dark energy is gravitationally repulsive" is actually the clearest way to understand its effects. The story is a bit more complicated if you are thinking about inhomogeneous dark energy models, but at least for homogeneous dark energy, the "gravitationally repulsive" mental model is exactly accurate mathematically.
– Sten
Commented May 5 at 19:21

I studied that if $$\Omega=\frac{\rho}{\rho_c}$$, which is the sum of the three is greater than one, the universe eventually collapses, if exactly one, its flat and if less than 1, it keeps expanding.

This is not exactly correct. It would be correct if there were no dark energy. In that case, comparison of the total density to the critical density indeed tells you whether the Universe will recollapse. However, it is not accurate if there is dark energy. In that case, it is even possible for a closed universe ($$\Omega>1$$) to expand forever (or even be static; Einstein's static universe has $$\Omega=\infty$$, although it is unstable).

This is essentially because dark energy is gravitationally repulsive and does not disperse as the Universe expands. If an expanding universe becomes sparse enough for dark energy's repulsion to overpower the attraction of the other components, then it will tend toward runaway exponential expansion. Indeed, dark energy drives $$\Omega$$ toward 1 over time, whereas in other cosmological phases, $$\Omega$$ tends away from 1 (such that any small deviation of $$\Omega$$ from 1 is amplified over time).

This doesn't make sense to me because dark energy is what is driving the expansion in the first place,

This is incorrect. Dark energy has been driving accelerated cosmic expansion for the last 6 billion years or so. However, it is not the reason the Universe is expanding. Dark energy was totally negligible for the vast majority of the history of the Universe. The Universe is expanding because -- for an unknown reason -- it was expanding in the initial conditions at a high enough rate that its self-gravity has only slowed but not stopped it.

Dark Energy isn't a type of pushing anti-gravity.

All matter and energy contribute to the curvature of spacetime and if there were "more" dark energy then the total density of the universe would be greater and the universe would have positive curvature, and could be a closed universe.

This may be a common misconception, based off ideas that "energy" can "cause things to expand" (perhaps from an image of hot gasses expanding), and so thinking that dark energy is pushing spacetime apart. This isn't the right mental image.

The difference between "dark energy" and "matter" (dark or light) is that dark energy seems to fill space and as space expands, there is more dark energy. It can be modelled as a single constant $$\Lambda$$ in the Equations that describe the large-scale curvature of space-time. It is constant across space and time. With matter (dark or light) the matter falls towards itself forming a network of clusters of galaxies, but the total amount of matter stays constant, so as the universe expands, it gets less dense.

Of course in a universe that is filled with dark energy, dark matter and regular matter, there is mass-energy all around you, the large scale distribution of mass-energy isn't pulling us in one particular direction.

The energy density of matter gets less as time continues, because the universe is expanding, but the energy density of dark-energy stays constant. If the universe contained no dark-energy, then as the matter moved apart it would be pulling on itself, tending to slow down expansion. But in fact, most of the energy-density of the universe is dark energy, and this doesn't get stretched out as the universe expands. And so, with time, the rate of expansion increases.

As the article "The Counterintuitive Reason Why Dark Energy Makes The Universe Accelerate" explains:

Dark energy doesn't make the Universe accelerate because of an outward-pushing pressure or an anti-gravitational force; it makes the Universe accelerate because of how its energy density changes (or, more accurately, doesn't change) as the Universe continues to expand.