How Does Inflation Solve the Flatness Problem in Cosmology?

[latentcorpse]

In the attached notes, there is a paragraph on p61 just under the eqn $\dot{R}^2=\dots$ that explains why inflation solves the flatness problem. Can someone explain to me how this works please? I cannot get my head around it.

http://www2.ph.ed.ac.uk/teaching/course-notes/documents/118/1446-AstrophysicalCosmology.pdf

Thanks.

 

[Dick]

It's really pretty simple. In the R dot equation if curvature where an important influence in the early universe after a period of inflation where R grows like exp(Ht), the R^2 and (dR/dt)^2 terms will grow enormously, while the k term will remain constant. So it naturally becomes small compared to the other terms.

 

[latentcorpse]

It's really pretty simple. In the R dot equation if curvature where an important influence in the early universe after a period of inflation where R grows like exp(Ht), the R^2 and (dR/dt)^2 terms will grow enormously, while the k term will remain constant. So it naturally becomes small compared to the other terms.

Thanks for the reply.

Why would the $R,\dot{R}$ terms grow so much?

Also, does this mean that inflation is in fact evidence for a flat universe?

Thanks.

 

[Dick]

Thanks for the reply.

Why would the $R,\dot{R}$ terms grow so much?

Also, does this mean that inflation is in fact evidence for a flat universe?

Thanks.

Because R=exp(Ht). If Ht increases by 100, R increases by a factor of e^100. That's a lot. And you've got the evidence backwards. A flat universe is evidence for inflation.

 

[latentcorpse]

Because R=exp(Ht). If Ht increases by 100, R increases by a factor of e^100. That's a lot. And you've got the evidence backwards. A flat universe is evidence for inflation.

In your original reply you wrote that if the curvature was significant then $R,\dot{R}$ would increase a lot. Surely we want them to increase a lot if curvature was insignificant (i.e. flat), no? I think I am getting confused about how the inflation kills off curvature...

 

[Dick]

In your original reply you wrote that if the curvature was significant then $R,\dot{R}$ would increase a lot. Surely we want them to increase a lot if curvature was insignificant (i.e. flat), no? I think I am getting confused about how the inflation kills off curvature...

Inflation isn't caused by the curvature. It's caused by another field that you add to the equations specifically to cause inflation, the simplest case is scalar field with a region of flat potential. And I don't see what's confusing. The curvature isn't 'killed off', it's still there. But the R terms become huge and the k term stays constant. After inflation the R and R' terms are so large that the k term is insignificant compared with them.

 

[latentcorpse]

Inflation isn't caused by the curvature. It's caused by another field that you add to the equations specifically to cause inflation, the simplest case is scalar field with a region of flat potential. And I don't see what's confusing. The curvature isn't 'killed off', it's still there. But the R terms become huge and the k term stays constant. After inflation the R and R' terms are so large that the k term is insignificant compared with them.

So why does the flatness problem motivate inflation? Is that just what you were saying earlier? That if we observe a k=0 universe (or close to it) then the universe must have been launched from a very precise set of initial conditions (a density close to the critical density) and the only way to explain this is with a finite inflationary period in the early universe where we go through a de Sitter like exponential phase in which R and R' grow enormously thus making k negligible. Is this about right?

In my lecture notes we have the Friedmann equation and then written that if $p=(\gamma-1)\rho$ then $\rho \tilde a^{-3 \gamma}$ and so $\rho$ falls slower than $\frac{k}{a^2}$ if $3 \gamma < 2$. Note that we can use the acceleration equation to show that this means $\rho + 3 p <0$ i.e. strong energy is violated which implies we have a vacuum energy fluid with antigravity properties right and the best candidate for this is a cosmological constant term being added in (as we can treat this as a fluid for which $\rho=-p$). Why though, do we want $\rho$ to fall slower than $\frac{k}{a^2}$?

What is the connection to dark energy? Is the fluid corresponding to the cosmological constant considered dark energy?


Thanks a lot!

 

[Dick]

So why does the flatness problem motivate inflation? Is that just what you were saying earlier? That if we observe a k=0 universe (or close to it) then the universe must have been launched from a very precise set of initial conditions (a density close to the critical density) and the only way to explain this is with a finite inflationary period in the early universe where we go through a de Sitter like exponential phase in which R and R' grow enormously thus making k negligible. Is this about right?

In my lecture notes we have the Friedmann equation and then written that if $p=(\gamma-1)\rho$ then $\rho \tilde a^{-3 \gamma}$ and so $\rho$ falls slower than $\frac{k}{a^2}$ if $3 \gamma < 2$. Note that we can use the acceleration equation to show that this means $\rho + 3 p <0$ i.e. strong energy is violated which implies we have a vacuum energy fluid with antigravity properties right and the best candidate for this is a cosmological constant term being added in (as we can treat this as a fluid for which $\rho=-p$). Why though, do we want $\rho$ to fall slower than $\frac{k}{a^2}$?

What is the connection to dark energy? Is the fluid corresponding to the cosmological constant considered dark energy?


Thanks a lot!

That's about right. You want rho to fall slower the k/a^2 because they are the competing terms in the Friedmann equation. The smaller k/a^2 is compared with rho the closer the universe is to critical density. Dark energy has some features in common with inflation, but inflation had to happen early in the universe at very high energy and curvature scales. And it also had to end, or we wouldn't be here. Dark energy, on the other hand, is at about the same energy scale as the ordinary forms of energy and matter in the universe. And it doesn't seem to be ending. That's a puzzle. If dark energy could have chosen any energy scale to be at, then if it were much higher it would have made the universe a bad place for us. If it were much lower, it wouldn't become important until later in the universes history and we'd never notice it. That's the 'why now' puzzle.

 

[latentcorpse]

That's about right. You want rho to fall slower the k/a^2 because they are the competing terms in the Friedmann equation. The smaller k/a^2 is compared with rho the closer the universe is to critical density. Dark energy has some features in common with inflation, but inflation had to happen early in the universe at very high energy and curvature scales. And it also had to end, or we wouldn't be here. Dark energy, on the other hand, is at about the same energy scale as the ordinary forms of energy and matter in the universe. And it doesn't seem to be ending. That's a puzzle. If dark energy could have chosen any energy scale to be at, then if it were much higher it would have made the universe a bad place for us. If it were much lower, it wouldn't become important until later in the universes history and we'd never notice it. That's the 'why now' puzzle.

So vacuum energy and dark energy are completely different, yes?

When we are talking about inflation we should only really be talking about vacuum energy as it has antigravity properties. What is the exact relation between inflation and vacuum energy though?

Thanks.

 

[Dick]

The biggest thing they have in common is that no one understands much about either one in a fundamental physics sense. Strictly speaking pure vacuum energy is the energy zero point energy of the quantum fields. A naive estimate says it ought to be huge, near the Planck scale. Clearly it's not. Something is cancelling that effect. Vacuum energy also doesn't decay. What caused inflation operated for a while and then decayed. There are ad hoc models for the inflaton, but none of them, as far as I know are connected with other known physical fields. They are just added on to do inflation. The questions you asking are ones nobody has good answers for.

 

[latentcorpse]

The biggest thing they have in common is that no one understands much about either one in a fundamental physics sense. Strictly speaking pure vacuum energy is the energy zero point energy of the quantum fields. A naive estimate says it ought to be huge, near the Planck scale. Clearly it's not. Something is cancelling that effect. Vacuum energy also doesn't decay. What caused inflation operated for a while and then decayed. There are ad hoc models for the inflaton, but none of them, as far as I know are connected with other known physical fields. They are just added on to do inflation. The questions you asking are ones nobody has good answers for.

Thanks, I was wondering though...if we end up with a flat universe and we begin with de Sitter (which i think is hyperbolic geometry), then does the curvature of spacetime change during evolution? Is this allowed?I do have another cosmology question though. What is the definition of the comoving Hubble radius? And why is it given by $\mathcal{H}^{-1}$? Surely the inverse of the conformal Hubble parameter should give the conformal time by analogy with the inverse of the Hubble parameter giving the Hubble time, no?

Thanks.