I Spatial flatness and cosmological constant

[Ranku]

Was the universe always spatially flat due to the presence of the cosmological constant, or has it become flatter in the late-time universe?

 

[kimbyd]

The two are rather different.

The cosmological constant makes the universe more spatially-flat at late times. But before a few billion years ago, the impact of spatial curvature would have been increasing. Which means that the curvature, when compared against the matter density, had to be incredibly tiny in the early universe.

To see this, the effect of the curvature scales as $(z+1)^2$, while matter density scales as $(z+1)^3$. Right now, the measured spatial curvature is less than a few percent of the matter density. Go back to the time the CMB was emitted ($z=1090$), and the spatial curvature would have been a few thousandths of a percent of the matter density.

So something else must have caused the very, very small spatial curvature in the early universe, which was around long before the current cosmological constant was relevant to the expansion. This fact has long been one of the primary motivations for cosmic inflation, which drives the universe towards flatness very rapidly in a manner similar to the cosmological constant.

 

[Ranku]

The two are rather different.

The cosmological constant makes the universe more spatially-flat at late times. But before a few billion years ago, the impact of spatial curvature would have been increasing. Which means that the curvature, when compared against the matter density, had to be incredibly tiny in the early universe.

To see this, the effect of the curvature scales as $(z+1)^2$, while matter density scales as $(z+1)^3$. Right now, the measured spatial curvature is less than a few percent of the matter density. Go back to the time the CMB was emitted ($z=1090$), and the spatial curvature would have been a few thousandths of a percent of the matter density.

So something else must have caused the very, very small spatial curvature in the early universe, which was around long before the current cosmological constant was relevant to the expansion. This fact has long been one of the primary motivations for cosmic inflation, which drives the universe towards flatness very rapidly in a manner similar to the cosmological constant.

So when the universe was expanding deceleratingly due to gravitation, before it began to expand acceleratingly due to cosmological constant, the universe was less spatially flat?

 

[timmdeeg]

So something else must have caused the very, very small spatial curvature in the early universe, which was around long before the current cosmological constant was relevant to the expansion. This fact has long been one of the primary motivations for cosmic inflation, which drives the universe towards flatness very rapidly in a manner similar to the cosmological constant.

" drives the universe towards flatness " seems to suggest that before inflation the curvature constant wasn't $k=0$. In this case this is true till today and our universe could e.g. be a very very large sphere. Is this reasoning correct?

If however the curvature constant was $k=0$ before inflation which means euclidean flatness then this holds till today and our universe would be spatially infinite (if we disregard a non-trivial topology e.g. torus) . - Should we neglect this case because we have some reason to think that it is extremely unlikely?

 

[kimbyd]

So when the universe was expanding deceleratingly due to gravitation, before it began to expand acceleratingly due to cosmological constant, the universe was less spatially flat?

Nobody knows. But usually physicists expect to see numbers that aren't ridiculously small or large when comparing things in a particular way.

Take the dimensionless electromagnetic coupling constant (aka the fine structure constant) $\alpha$. This number is approximately 1/137. Which isn't terribly big or small.

Similarly, one might expect that the spatial curvature when our observable universe was started would have been a medium number. Like 2 or 0.1.

Instead, if the current curvature is 0.01, then the curvature near the big bang would have been something ridiculous like $10^{-30}$ (note: this number is for illustration only and it's not precise in any sense). Theorists generally feel such a tiny number is something which needs explaining.

Cosmic inflation is one attempt to solve this problem.

 

[Catastrophe]

kimbyd, you stated:
"So something else must have caused the very, very small spatial curvature in the early universe, which was around long before the current cosmological constant was relevant to the expansion"

Please excuse a question from a very basic learner who does not know enough. Does your above quoted statement in any way relate to decreasing curvature towards the Big Bang, which might suggest a nexus, a thinning passage, rather than a singularity.

Catastrophe :)