# Flat universe and accelerating expansion

As current data suggests that the universe is flat, a flat universe would imply a slowing expansion rather than an accelerating one which we observe. Neither in an open nor a closed universe there is such a thing as accelerating expansion. Dark energy is thought to be responsible for that, but wouldn't that drive $\Omega$ away from 1, although we still measure it to be 1?

## 2 Answers

No. It is perfectly possible to have a flat universe that expands forever and is accelerating. Dark energy is what makes this possible. Whilst the curvature of the universe is defined by the sum of all the energy densities in it, the effects of matter (baryonic or dark) and dark energy are quite different on its dynamics.

It is in fact quite possible to have a closed universe with an accelerating expansion.

Open and closed are not synonymous with acceleration and deceleration.

• This happens because the geometry is defined by the total mass/energy density, but the time evolution of the scale factor is defined by the equation of state of the stuff making up that density. Dark energy has a different equation of state to matter. Specifically the matter density scales as $1/a^3$ while the dark energy density is constant. You might be interested to read How does the Hubble parameter change with the age of the universe? and the links within it. – John Rennie Apr 12 '15 at 17:17

Einstein made an error in concluding from the universe being static that it has a cosmological constant. Being static is perfectly well explained by matter being more disperse at larger distances. It turns out he was right anyway. Observations have shown that the expansion of the universe is accelerating.

The only way for the cosmological constant to be observed as the same from any point in space-time going at any velocity and have the tendency for any object in free fall to follow a geodesic of space-time is if space-time has the same geometry as a set of point equidistant from a given point in a flat Mikowski space with 1 time like dimension. Our universe can be described as having spherical geometry with its size varying with time as c(sqrt(t^2 + 1/Λ^2) where Λ is the cosmological constant. I think that in reality only the regions space time in the future light cone of the big bang exist but for simplicity, I will describe it as though regions outside of it exist. Because simultaneity is relative, for any frame of reference accelerating only by the acceleration of the universe and not by gravity at any time going at any velocity, which I will call an inertal frame of reference, there is a coordinate system where the universe behaves in that way where the object is considered to be at time t=0 with zero velocity at that time. All inertial frames of reference have an antihorizon very far away where once an object enters comes in through it, it can never leave back through it and an event horizon where once an object leaves through it, it can never reenter through it. Once two objects get close enough that there exists an interital frame of reference whose antihorizon contains both of them, there can never stop being one and once they get far enough away from each other that there's no interial frame of reference whose event horizon contains both of them, there can never start being one again. The region of space-time outside of the event horizon of any inertial frame of reference appears to be an infinitely big black hole that that surrounds all of space except for a finite spherical region we're in. If someone goes beyond that event horizon, they'll observe themself pass it in a finite amount of time but you'll never observe them reach the event horizon and will see them keep getting more redshifted without bound unless you cross it yourself. Unlike a real black hole where space-time beyond its antihorizon doesn't exist, matter has sometimes entered through its antihorizon and you can see it from a time before it entered. Different interial frames of reference have a different event horizon and what's beyond the event horizon of one inertial frame of reference might not be beyond the event horizon of another inertial frame of reference. For instance, if someone has passed the event horizon of one intertial frame of reference and you're another inertial frame of reference that has not yet passed the event horizon, they might not be beyond your event horizon because it's not too late for them to make it back to you because you're going to fall past the event horizon.