Inflation & Ω: Dark Matter & Energy Impact

[Ranku]

Inflation leads to Ω=1. But we also have Ω_matter+Ω_{dark matter}+Ω_{dark energy} = 1. So if there were no dark energy and dark matter present in the universe, would Ω have eventually deviated away from 1?

 

[mathman]

Without dark matter and dark energy the universe would be completely different. $$\Omega$$ would have not started at 1 after inflation. Also, inflation might not have happened.

 

[Ranku]

Without dark matter and dark energy the universe would be completely different. $$\Omega$$ would have not started at 1 after inflation. Also, inflation might not have happened.

Could you please elaborate on why that is so?

 

[JMz]

The OP asks about something that is counter-factual. No problem, but in that case, what are you assuming is the same as reality? Are you still assuming general relativity; or if not, which theory of gravity? Are you assuming some specifics about the physics of inflation, like whether DE (or Λ) is a tiny residual of the inflation field? Which specifics?

 

[Ranku]

The OP asks about something that is counter-factual. No problem, but in that case, what are you assuming is the same as reality? Are you still assuming general relativity; or if not, which theory of gravity? Are you assuming some specifics about the physics of inflation, like whether DE (or Λ) is a tiny residual of the inflation field? Which specifics?

I am assuming the standard ∧CDM model. I am trying to clarify something I read in Alan Guth's book The Inflationary Universe, and I quote two passages, first about a universe without the cosmological constant (p.p.177) and the second with the cosmological constant (p.p. 178, footnote).

1. "The standard cosmological evolution would resume at the end of inflation, so any deviation from flatness would begin to grow. The universe, however, would be so nearly flat at the end of inflation that it would remain essentially flat until the present day. Thus, the inflationary theory leads to an important prediction that is in principle testable. The present value of omega should be very precisely equal to one"

2. "If the (cosmological constant) is non-zero, the effect of inflation is still to drive the universe to a state of geometric flatness.The flatness problem is solved in this case, also, since again the value of omega before inflation can be almost anything. Regardless of the initial value of omega, inflation will drive the universe to a state of nearly perfect flatness. Although the deviation from flatness will begin to grow once inflation ends, it will remain imperceptible to the present day"

In other words, the presence of cosmological constant alone makes no difference in the eventual deviation away from Ω = 1; it implies that it is only when dark matter is added that we permanently have Ω = 1 (although Guth for some reason does not mention this - even though the presence of dark matter was known at the time of the inflationary theory - and concludes that there will be eventual deviation away from Ω = 1.)

What I am trying to clarify, however, is does the presence of cosmological constant alone make any difference at all about eventual deviation away from Ω = 1, such as maybe the deviation would have happened somewhat later than it would have if the cosmological constant had not been present.

 

[timmdeeg]

What I am trying to clarify, however, is does the presence of cosmological constant alone make any difference at all about eventual deviation away from Ω = 1, such as maybe the deviation would have happened somewhat later than it would have if the cosmological constant had not been present.

It is assumed that the inflaton field (not the cosmological constant which is tiny) drives the universe to spatial flatness.

However it seems that some cosmologists consider this matter with a certain degree of skepticism.

https://ned.ipac.caltech.edu/level5/March03/Lineweaver/Lineweaver6.html
6. THE STATUS OF INFLATION
Down to Earth astronomers are not convinced that inflation is a useful model. For them, inflation is a cute idea that takes a geometric flatness problem and replaces it with an inflaton potential flatness problem. It moves the problem to earlier times, it does not solve it. Inflation doesn't solve the fine-tuning problem. It moves the problem from "Why is the Universe so flat?" to "Why is the inflaton potential so flat?". When asked, "Why is the Universe so flat?", Mr Inflation responds, "Because my inflaton potential is so flat." "But why is your inflaton potential so flat?" "I don't know. It's just an initial condition." This may or may not be progress. If we are content to believe that spatial flatness is less fundamental than inflaton potential flatness then we have made progress.

 

[JMz]

I will point out that positing that dark energy = 0, but using a lambda-anything model, is a bit of an oxymoron: Λ (the cc) is our best guess for DE -- a constant of the universe, always and everywhere. Neither that model nor that guess is the only game in town, but both are the current standards.

 

[Chronos]

I'm inclined to agree with Lineweaver: inflation does not eliminate the need for initial conditions assumptions. It remains debatable if, and how much progress that constitutes.

 

[kimbyd]

Inflation leads to Ω=1. But we also have Ω_matter+Ω_{dark matter}+Ω_{dark energy} = 1. So if there were no dark energy and dark matter present in the universe, would Ω have eventually deviated away from 1?

The $\Omega$ parameters are defined as density fractions. Their sum is always equal to one. The fact that $\Omega = 1$ has no physical significance, it's just a statement about how the numbers are defined.

 

[mathman]

The $\Omega$ parameters are defined as density fractions. Their sum is always equal to one. The fact that $\Omega = 1$ has no physical significance, it's just a statement about how the numbers are defined.

Omega is the curvature of space and is not necessarily equal to one.

https://en.wikipedia.org/wiki/Shape_of_the_universe

 

[kimbyd]

Omega is the curvature of space and is not necessarily equal to one.

https://en.wikipedia.org/wiki/Shape_of_the_universe

Eh, you're right. I mixed up my notations. I was thinking of it as including $\Omega_k$, which isn't the case.

Answering the OP again:
Unless there was some symmetry that fixed $k=0$ exactly, then yes, eventually it would deviate away from one. This is easiest to see by examining how spatial curvature changes with the expansion. From the first Friedmann equation (where I'm wrapping $\Lambda$ into the density term for simplicity):

$$H^2 = {8\pi G \over 3} \rho - {k c^2 \over a^2}$$

As long as $\rho$ dilutes faster than $1/a^2$, then the curvature term will eventually dominate. If there is any component of $\rho$ that dilutes more slowly (e.g. dark energy), then the $kc^2/a^2$ term will continue to remain negligible.

 

[JMz]

Unless there was some symmetry that fixed k=0 exactly...

FWIW, I seem to recall that -- before inflation became accepted -- either Bekenstein or Shimon Malin (or maybe both?) posited that there were in fact several cosmological symmetries that are laws of nature, not merely accidents of nature. They were on exactly the same footing as the equivalence principle, but in addition to them: isotropy, homogeneity, flatness. This lent itself to a group-theory formulation of gravity, one that is somewhat different from GR. I don't know the status of those ideas now, though it's my impression that inflation is simpler.

 

[kimbyd]

FWIW, I seem to recall that -- before inflation became accepted -- either Bekenstein or Shimon Malin (or maybe both?) posited that there were in fact several cosmological symmetries that are laws of nature, not merely accidents of nature. They were on exactly the same footing as the equivalence principle, but in addition to them: isotropy, homogeneity, flatness. This lent itself to a group-theory formulation of gravity, one that is somewhat different from GR. I don't know the status of those ideas now, though it's my impression that inflation is simpler.

I would believe it. One of the interesting discussions I had with a theorist some years ago involved him flatly and loudly proclaiming that it was just not possible for the cosmological constant to be non-zero, asserting that the numerical value of the constant was just too small to be possible.

That kind of strong belief can be useful for a theorist, because it can give them the motivation and drive they need to spend months or (sometimes) years working out the mathematical consequences of their idea.

I really don't think that there really can be a symmetry that sets these things to zero, but it's not a bad thing that there are (or, at least, have been) working physicists who have believed those things. We all have biases, and science works best when different people with different biases work out the consequences of their beliefs and communicate the results.

 

[JMz]

That strikes me as a particularly enlightened post.

 

[JMz]

a symmetry that sets these things to zero

Apropos of that, I vaguely recall that someone showed that such a group-theoretic model obeyed Mach's principle (as Einstein called it): An empty universe would leave particles with no inertial at all. Something that GR does not, even though Einstein thought it should, and supposedly expected it to as he was developing the theory.

strong belief can be useful for a theorist

I suppose I have to admit that, even if inflation seems simpler (now, to me), the quantum folks have shown how valuable group theory can be, when it does apply. So it's easy to see why someone would look in that direction and put some work into it.