# I Does Euclidean geometry require initial fine-tuning?

#### timmdeeg

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Summary
The question

doesn't Ω=1 require k=0 at the big bang (before inflation started)?"

So to clarify this issue, does Ω=1 require k=0 at the big bang?
If yes would this mean that we have a fine-tuning problem regarding the initial density of the universe here (comparable to the *old* Flatness-Problem)?
The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe.

The fine-tuning problem of the last century was solved by introducing the theory of inflation which flattens out any initial curvature so that our actual universe looks almost spatially flat. The inflation doesn't create euclidean flatness though.

Would the claim that our universe has euclidean geometry raise the old fine-tuning problem regarding the initial energy densities again?

#### Orodruin

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Exact flatness requires that $\Omega_k = 0$. However, inflation would drive it unobservably close to zero even if it was not to start with.

#### timmdeeg

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Exact flatness requires that $\Omega_k = 0$.
Ok, thanks for confirming that.
Would you agree that this requires initial fine tuning?

However, inflation would drive it unobservably close to zero even if it was not to start with.
Yes. I think from our observation we can't distinguish (now and probably also in future) between $\Omega_k$ is "close to zero" (but not zero, due to inflation) and $\Omega_k = 0$ (due to initial fine-tuning).

If this is correct so far, what would we assume to be more likely, the former or the latter?

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Would you agree that this requires initial fine tuning?
However, inflation would drive it unobservably close to zero even if it was not to start with.
Why is this not an acceptable answer?

#### timmdeeg

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Why is this not an acceptable answer?
This answer concerns one aspect of my question, which involves 2 initial possible states of the universe:

A The initial spatial curvature is non-zero, implying $\Omega_k\neq 0$ (due to inflation "unobservably close to zero though") and thus excluding euclidean geometry.

B The initial spatial curvature is zero, implying $\Omega_k = 0$ (raising the fine-tuning problem though) und thus euclidean geometry.

A and B are not distinguishable by observation. But I think it is legitimate to ask which possibility seems more likely?
Also please correct wrong reasoning.

#### Orodruin

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But I think it is legitimate to ask which possibility seems more likely?
It is not legitimate. The answer is utterly dependent on your prior.

#### timmdeeg

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It is not legitimate. The answer is utterly dependent on your prior.
Oh, then I was mistaken, thanks for clarifying.

Remembering the flatness problem my personal view is that $\Omega_k = 0$ is very unlikely.

May I ask what’s the personal view of others around here?

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Staff Emeritus
May I ask what’s the personal view of others around here?
Science does not progress by vote, nor by personal views. However, my personal view is worrying about whether a number is x or immeasurably close to x is effort best expended elsewhere.

#### timmdeeg

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Science does not progress by vote, nor by personal views.
I fully agree, but this doesn't preclude that scientists don't hesitate to talk about personal views, e.g. about the interpretations of Quantum Mechanics.

Another question in this context, hopefully more on scientific grounds and less on personal views:

Can we ever ascertain from our observation whether or not our universe has euclidean geometry?

#### Orodruin

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I fully agree, but this doesn't preclude that scientists don't hesitate to talk about personal views, e.g. about the interpretations of Quantum Mechanics.
That some scientists like to discuss something does not make it scientific.

Edit: Also, many scientists do prefer to leave personal opinion out of the discussion as much as possible.

Can we ever ascertain from our observation whether or not our universe has euclidean geometry?
No. On two levels. First of all, being Euclidean is not something that can be experimentally established beyond experimental accuracy. Second, you are talking about an approximation that is only valid on very large scales and in a very particular set of coordinates. It is clear that this approximation is not applicable on small scales and the Universe clearly is not flat on small scales so the question in itself becomes rather moot.

#### timmdeeg

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That some scientists like to discuss something does not make it scientific.

Edit: Also, many scientists do prefer to leave personal opinion out of the discussion as much as possible.

No. On two levels. First of all, being Euclidean is not something that can be experimentally established beyond experimental accuracy. Second, you are talking about an approximation that is only valid on very large scales and in a very particular set of coordinates. It is clear that this approximation is not applicable on small scales and the Universe clearly is not flat on small scales so the question in itself becomes rather moot.
Yes, I'm talking about the universe in the context of the Friedmann Equations. Our observable universe could have overdensity or underdensity (as some people are presently discussing regarding the Hubble tension). So I don't think it makes any sense to consider a value for $\Omega_k$ locally.

Wikipedia writes If the initial density of the universe could take any value, it would seem extremely surprising to find it so 'finely tuned' to the critical value $\rho_c$..

From this and the fact that Euclidean geometry of the universe can't "be experimentally established" it seems that it is not personal view to argue that Euclidean geometry is rather unlikely. Not sure if you agree.
If correct this assumption would not support any reasoning that the universe is infinite (but rather the opposite).

#### PeterDonis

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From this and the fact that Euclidean geometry of the universe can't "be experimentally established" it seems that it is not personal view to argue that Euclidean geometry is rather unlikely.
As @Orodruin has already pointed out, "unlikely" requires a prior. Different priors will lead to different answers.

For example: suppose there is some constraint that requires zero spatial curvature. Then Euclidean geometry is imposed by this constraint. We don't currently know of any such constraint, but we don't currently have a complete understanding of all the factors involved either. So if I assign enough prior probability to there being such a constraint, Euclidean geometry is no longer "unlikely".

this assumption would not support any reasoning that the universe is infinite (but rather the opposite)
This is not correct, because it is perfectly possible to have an inflation model in which the curvature starts out negative and gets flattened out by inflation. A universe with negative curvature would be spatially infinite. So the question of whether there is spatial curvature or not is separate from the question of whether the universe is spatially infinite or not.

#### timmdeeg

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suppose there is some constraint that requires zero spatial curvature. Then Euclidean geometry is imposed by this constraint. We don't currently know of any such constraint, but we don't currently have a complete understanding of all the factors involved either. So if I assign enough prior probability to there being such a constraint, Euclidean geometry is no longer "unlikely".
Perhaps I misunderstand the meaning of "prior probability" (my native language is german). Why can't I understand the Wikipedia quote in #11 such that the prior probability for Euclidean geometry is "unlikely"?
Doesn't at least result a week constraint from this quote?
This is not correct, because it is perfectly possible to have an inflation model in which the curvature starts out negative and gets flattened out by inflation. A universe with negative curvature would be spatially infinite. So the question of whether there is spatial curvature or not is separate from the question of whether the universe is spatially infinite or not.
Yes, I've overlooked this aspect, thanks.

#### PeterDonis

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Why can't I understand the Wikipedia quote in #11 such that the prior probability for Euclidean geometry is "unlikely"?
You can, but that just means the author of that quote has a particular opinion on the prior probability for Euclidean geometry. It does not mean it's the only possible opinion or that that particular opinion has any special weight. Assignment of prior probabilities--i.e., the probabilities you assign to various alternatives prior to looking at any data at all--is always subjective. You can't use one person's subjective judgment as an argument against another person who has a different subjective judgment.

#### Orodruin

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Yes, I'm talking about the universe in the context of the Friedmann Equations. Our observable universe could have overdensity or underdensity (as some people are presently discussing regarding the Hubble tension). So I don't think it makes any sense to consider a value for ΩkΩk\Omega_k locally.
You are missing the point. If you are interested in a single value for $\Omega_k$, then even small local deviations from flatness on your Euclidean space is going to spoil everything for you. It simply is no longer a good idea to apply the approximation and talk about a value of $\Omega_k$ with that kind of precision given what we already know about the Universe's non-flatness.

#### timmdeeg

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You can, but that just means the author of that quote has a particular opinion on the prior probability for Euclidean geometry. It does not mean it's the only possible opinion or that that particular opinion has any special weight.
Perhaps I'm wrong. As I understood it "that quote" describes the origin of the flatness problem und hence was the general view among cosmologists at that time.

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Is the universe flat everywhere? (Picking up pen and dropping it). Clearly not.

Therefore the only way to tell if it is globally universally flat is some sort of average. Can I find an average that is zero? Always.

The statement Ωk is close to zero. even immeasurably close to zero is a scientific one. The statement that it is exactly zero is a statement about your choice of averaging procedure.

#### timmdeeg

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You are missing the point. If you are interested in a single value for ΩkΩk, then even small local deviations from flatness on your Euclidean space is going to spoil everything for you. It simply is no longer a good idea to apply the approximation and talk about a value of ΩkΩk with that kind of precision given what we already know about the Universe's non-flatness.
Sorry, I don't understand the message.

The inflation predicts $\Omega_k$ extremely close to zero regardless any initial curvature. In an arbitrary observable universe people will confirm this prediction by measurement. Nowhere they will confirm $\Omega_k = 0$ because of lack of measurement accuracy. So the initial state of the universe (curved or flat remains unknown)

As to "a single value for $\Omega_k$ " (a value valid for the universe as a whole) assuming the Cosmological Principle the inflation predicts $\Omega_k$ extremely close to zero regardless any initial curvature. If the initial state is $\Omega_k = 0$ then on small enough scales slight deviations from this seem possible (*) but not on very large scales and not regarding the universe as a whole.

I might still be missing something and hope the above allows you to identify what.

(*) Whereby I'm not sure if local values of $\Omega_k$ make sense at all. But strictly a local overdensity would require $\Omega_k = 1$.

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#### PeterDonis

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"that quote" describes the origin of the flatness problem
Not quite. The origin of the flatness problem is described a little later in that same article:

"The current density of the universe is observed to be very close to this critical value. Since the total density departs rapidly from the critical value over cosmic time,[1] the early universe must have had a density even closer to the critical density"

The key statement is in bold above; it is only true for models that include ordinary matter and radiation but nothing else, and which do not start out exactly flat spatially. It is not true for inflationary models; it is also not true for models that are dominated by a positive cosmological constant (as our current universe is). And it is also not true for a model that starts out exactly spatially flat.

So any one of those three possibilities--inflation, a positive cosmological constant, or exact spatial flatness, for example imposed by some constraint as I mentioned in a previous post--would solve the flatness problem. In the absence of evidence that rules out one or more of the three possibilities, which one you prefer is a matter of your choice of prior probabilities--i.e., which one you subjectively consider to be the least unlikely given your other subjective beliefs.

#### timmdeeg

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In the absence of evidence that rules out one or more of the three possibilities, which one you prefer is a matter of your choice of prior probabilities--i.e., which one you subjectively consider to be the least unlikely given your other subjective beliefs.
Thanks for this excellent explanation. I haven't seen these interrelations.