# Lambda CDM Model and Spatial Flatness

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• timmdeeg
In summary: No. If ##\Omega = 1## exactly, then ##k = 0## for any value of ##a##. That's what the equation says. So this equation by itself cannot tell you whether the universe is spatially finite or spatially infinite.Yes.
timmdeeg
Gold Member
[Moderator's note: Spun off from previous thread due to topic/level change.]

PeterDonis said:
Since, according to our best current model, the universe is and always has been spatially infinite, there were plenty of such places at the early times you mention.
Which current model? Doesn't "spatially infinite" imply Euclidean flatness which as far as I'm aware of requires ##\Omega = 1##? The theories of inflation don't predict that. They predict a value for ##\Omega## which is very close to ##1## but not a value which equals ##1##, because during inflation the growth of ##a## is finite.
Isn't is more likely from this point of view that the universe isn't spatially infinite?
Shouldn't we say at least we don't know?

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timmdeeg said:
Which current model?

The Lambda CDM model.

timmdeeg said:
Doesn't "spatially infinite" imply Euclidean flatness

For spatial slices, yes.

timmdeeg said:
which as far as I'm aware of requires ##\Omega = 1##?

It requires ##\Omega = 1## to be within the current error bars of the data, yes. Right now that is the case. There are some cosmologists who believe it will cease to be the case as our data becomes more precise and the error bars narrow. But others don't think so. If that ever does happen, obviously our best current model would need to be re-evaluated.

timmdeeg said:
The theories of inflation don't predict that. They predict a value for ##\Omega## which is very close to ##1## but not a value which equals ##1##, because during inflation the growth of ##a## is finite.

I don't know that this is true for all inflation models, though it is true for some of them, yes.

timmdeeg said:
Isn't is more likely from this point of view that the universe isn't spatially infinite?

For purposes of this thread, it doesn't really matter, because in inflationary models where ##\Omega## is not precisely equal to ##1##, the spatial extent of the universe is still so much larger than the extent of our observable universe that it is effectively infinite when you're thinking about things like whether there are events at early times, like the emission of the CMBR or light emitted by early galaxies, from which light signals would not yet have reached us.

Of course you could turn this around and argue that the fact that we are still seeing light signals from such early events--that not all such light signals have passed us by now--does not establish that the universe must be spatially infinite, only that its spatial extent is much, much larger than the spatial extent of our observable universe.

PeterDonis said:
I don't know that this is true for all inflation models, though it is true for some of them, yes.
I should have shown more precisely to what I take reference.

##(\Omega^{-1}-1)\rho{a}^2=-\frac{3c^2}{8\pi{G}}k##

It doesn't really make a difference which inflation model we are talking about. During inflation the expansion, the growth of ##a## is large but finite and hence ##(\Omega^{-1}-1)## can't be zero as it should in order to yield ##\Omega = 1##. So this equation seems to predict that the universe is spatially finite. Please clarify where I'm misled.

However if correct, don't we have the good old flatness problem again then?(*) Regarding the above euclidean flatness seems to require ##\Omega = 1## before inflation started. I haven't seen this reasoning somewhere else which makes me suspicious. If true the inflation still explains a lot about the CMB.

EDIT (*) No, the flatness problem had to do with the observation that the universe is almost flat.

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timmdeeg said:
During inflation the expansion, the growth of ##a## is large but finite

Yes.

timmdeeg said:
and hence ##(\Omega^{-1}-1)## can't be zero

No. If ##\Omega = 1## exactly, then ##k = 0## for any value of ##a##. That's what the equation says. So this equation by itself cannot tell you whether the universe is spatially finite or spatially infinite.

Also, you seem to be mistakenly assuming that if the universe is spatially infinite, then the growth of ##a## must be infinite. That's not correct. A spatially infinite universe still has a finite scale factor and scale factor growth.

PeterDonis said:
No. If ##\Omega = 1## exactly, then ##k = 0## for any value of ##a##. That's what the equation says. So this equation by itself cannot tell you whether the universe is spatially finite or spatially infinite.
Agreed. But it tells me that the universe can't be infinite due to inflation thereby neglecting the case ##k = 0##. So if you say "to our best current model, the universe is and always has been spatially infinite" you seem to mean the case ##k = 0##. My point is that I don't see that the Lambda CDM model predicts this case.

timmdeeg said:
it tells me that the universe can't be infinite due to inflation thereby neglecting the case ##k = 0##.

Inflation is perfectly compatible with ##k = 0##. Maybe you need to read this again:

PeterDonis said:
you seem to be mistakenly assuming that if the universe is spatially infinite, then the growth of ##a## must be infinite. That's not correct. A spatially infinite universe still has a finite scale factor and scale factor growth.

PeterDonis said:
Inflation is perfectly compatible with ##k = 0##.
Yes, but I didn't say otherwise. I only said that if we have the case that ##k## isn't ##0## then the inflation can't produce a spatially infinite universe.

Could you elaborate how the Lambda CDM model predicts a spatially infinite universe?

timmdeeg said:
I didn't say otherwise

It sure seemed like you were saying that ##k = 0## is not possible in any inflation model, which is saying otherwise.

timmdeeg said:
I only said that if we have the case that ##k## isn't ##0## then the inflation can't produce a spatially infinite universe.

Well, of course not, since ##k > 0## by definition means a spatially finite universe. (I assume you actually meant ##k > 0##, not ##k \neq 0##, since a ##k < 0## universe, like ##k = 0##, is spatially infinite.) But that's not at all the same as saying that inflation requires ##k > 0##. It doesn't.

timmdeeg said:
Could you elaborate how the Lambda CDM model predicts a spatially infinite universe?

Um, by having ##k = 0##? (More precisely, by having ##\Omega = 1##, which implies ##k = 0##.)

https://en.wikipedia.org/wiki/Lambda-CDM_model

PeterDonis said:
It sure seemed like you were saying that ##k = 0## is not possible in any inflation model, which is saying otherwise.
I said "neglecting the case ##k = 0##", "not possible" wasn't meant. Perhaps I've a problem with correct wording. Of course inflation doesn't exclude a certain sign of ##k##.
PeterDonis said:
I assume you actually meant ##k > 0##
Yes, thanks for correcting.
PeterDonis said:
https://en.wikipedia.org/wiki/Lambda-CDM_model
The author discusses the measurement

 1.0023

and mentions "flat or almost flat". I still can't see that the model predicts that our universe is spatially infinite. In my opinion we will never be able to establish ##\Omega = 1## experimentally.

timmdeeg said:
I still can't see that the model predicts that our universe is spatially infinite.

The model explicitly says that ##\Omega = 1## exactly in the model. It's a fixed parameter. That's what I pointed you at in the Wikipedia article. So the model by definition predicts ##k = 0## and a spatially infinite universe.

timmdeeg said:
In my opinion we will never be able to establish ##\Omega = 1## experimentally.

Of course not, since we can never make measurements to infinite accuracy. But that doesn't mean a model can't adopt ##\Omega = 1## as a fixed parameter. That's what the Lambda CDM model does.

If we ever establish that ##\Omega## cannot equal ##1## exactly (because the error bars from observation exclude that value), then our current Lambda CDM model will be falsified and we will need to re-evaluate what our best current model of the universe is. But that hasn't happened, so the Lambda CDM model is our best current model.

timmdeeg said:
The author discusses the measurement

The measurement is not the model. The model is a mathematical model that makes predictions. The measurement data is what you compare the predictions with.

PeterDonis said:
The model explicitly says that ##\Omega = 1## exactly in the model. It's a fixed parameter. That's what I pointed you at in the Wikipedia article. So the model by definition predicts ##k = 0## and a spatially infinite universe.
Yes the model predicts that. But didn't you say that our our universe is spatially infinite in #2? Doesn't this claim require the knowledge that regarding our universe ##\Omega = 1## ?

timmdeeg said:
didn't you say that our our universe is spatially infinite in #2?

No. Go back and read the post again, carefully.

PeterDonis said:
No. Go back and read the post again, carefully.
I understood "Since, according to our best current model, the universe is and always has been spatially infinite, there were plenty of such places at the early times you mention." such that you mean our universe.

timmdeeg said:
I understood "Since, according to our best current model, the universe is and always has been spatially infinite, there were plenty of such places at the early times you mention." such that you mean our universe.

See the bolded text above.

PeterDonis said:
See the bolded text above.
My reasoning is that the model can't predict anything unless we have a definitive value of ##k##.

A definite value of ##k## can only be fixed empirically though and only in principle, e.g. if the size of the universe isn't too large.

To my knowledge ##k = 1## can't be excluded, so we have to leave the question open as to whether or not the universe is spatially infinite.

Kindly clarify where I am wrong.

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timmdeeg said:
My reasoning is that the model can't predict anything unless we have a definitive value of ##k##.

The model does have a definite value of ##k##, ##k = 0##.

timmdeeg said:
A definite value of ##k## can only be fixed empirically though

Wrong. A model can assume any definite value of ##k## that it wants. You are confusing experimental data with model assumptions and parameters. They're not the same. If we could never make models without having empirical measurements to fix all the parameters, we could not do science.

Also, you have already said (and I agreed) that we can't ever fix an exact value of any quantity empirically, because our measurements always have some finite error. But that would mean we could never fix empirically a definite value of ##k##, or any parameter, which, by your reasoning, would mean we could never make models at all.

timmdeeg said:
Kindly clarify where I am wrong.

You seem to have a flawed understanding of how scientific models work. See above.

timmdeeg said:
To my knowledge ##k = 1## can't be excluded, so we have to leave the question open as to whether or not the universe is spatially infinite.

You are still failing to address what I actually said. What I actually said (in the post that originally spawned this discussion) is that according to our best current model, the universe is spatially infinite. That is a true statement. It is still true even when we recognize that our best current model, like all scientific models, is not necessarily final and might have to be changed in the future when we have more data and more accurate data then we have now.

You seem to be confusing two different statements: the statement that our best current model, which says ##k = 0##, might have to be changed in the future, and the statement that our best current model doesn't actually say ##k = 0##. The first statement is true. But all of your posts are trying to assert the second statement, which is false.

PeterDonis said:
The model does have a definite value of ##k##, ##k = 0##.
The model does have also a definitive value of ##k##, ##k = 1##.
Why do you have a preference for ##k = 0##?

timmdeeg said:
The model does have also a definitive value of ##k##, ##k = 1##.
Why do you have a preference for ##k = 0##?
No, the model doesn't directly specify a value for k, because k=0 exactly follows from the fixed parameters (i.e. those assumed, not empirically determined). The validity of the assumptions is then tested by ongoing observations.

timmdeeg said:
The model does have also a definitive value of ##k##, ##k = 1##.

No, ##k = 0##, ##\Omega = 1##.

timmdeeg said:
Why do you have a preference for ##k = 0##?

I have said nothing at all about any preference of mine. I have merely said, repeatedly, what the Lambda CDM model says.

PAllen said:
No, the model doesn't directly specify a value for k, because k=0 exactly follows from the fixed parameters (i.e. those assumed, not empirically determined). The validity of the assumptions is then tested by ongoing observations.
Do you mean the assumption ##\Omega = 1##?

If yes, we then assume euclidean flatness. But we don't know for sure, so we can't claim it. It's just an assumption, how can we ever be sure? It is also possible that we live in a large sphere. This was my main point.

Another question, doesn't ##\Omega = 1## require ##k = 0## at the big bang (before inflation started)?

PeterDonis said:
I have said nothing at all about any preference of mine. I have merely said, repeatedly, what the Lambda CDM model says.
I understand, thank you for your patience.

timmdeeg said:
If yes, we then assume euclidean flatness. But we don't know for sure, so we can't claim it.

Which is why I said, according to our best current model, the universe is spatially infinite. That was the whole point of my putting in that qualifier.

timmdeeg said:
doesn't ##\Omega = 1## require ##k = 0## at the big bang (before inflation started)?

Yes.

timmdeeg
Thanks and sorry for some misunderstandings, my fault.

PeterDonis said:
timmdeeg said:
doesn't ##\Omega = 1## require ##k = 0## at the big bang (before inflation started)?
Yes.
Hm, after rethinking, ##k = 0##, seems to require extreme fine tuning which isn't too much liked by physicists. I wonder if from this point of view assuming euclidean geometry of the universe is not really realistic.

timmdeeg said:
after rethinking, ##k = 0##, seems to require extreme fine tuning which isn't too much liked by physicists. I wonder if from this point of view assuming euclidean geometry of the universe is not really realistic.

That is a completely separate question from the topic of this thread, which is what the Lambda CDM model says. If you want to discuss whether the Lambda CDM model, which says ##k = 0##, involves fine-tuning, and you can find references where that topic is discussed, you are welcome to start a new thread on it based on those references.

## 1. What is the Lambda CDM model?

The Lambda CDM model is a cosmological model that describes the evolution of the universe. It is based on the theory of general relativity and incorporates the presence of dark energy (represented by Lambda) and cold dark matter (CDM) to explain the observed expansion of the universe.

## 2. What is the significance of dark energy in the Lambda CDM model?

Dark energy is a hypothetical form of energy that is thought to make up about 70% of the total energy in the universe. In the Lambda CDM model, it is used to explain the observed acceleration of the expansion of the universe.

## 3. How does the Lambda CDM model explain the flatness of the universe?

The Lambda CDM model assumes that the universe is spatially flat, meaning that the curvature of space is zero. This is consistent with observations of the cosmic microwave background radiation. The presence of dark energy and dark matter in the model helps to balance out the overall mass-energy density of the universe, resulting in a flat geometry.

## 4. What evidence supports the Lambda CDM model?

The Lambda CDM model is supported by a variety of observational data, including the cosmic microwave background radiation, the large-scale structure of the universe, and the observed acceleration of the expansion of the universe. These observations are consistent with the predictions of the model.

## 5. Are there any challenges or limitations to the Lambda CDM model?

While the Lambda CDM model has been successful in explaining many observations, there are still some challenges and limitations. For example, the model does not fully explain the observed distribution of galaxies on small scales, and the nature of dark energy and dark matter remains a mystery. Additionally, there are alternative cosmological models that can also explain the observed data. Further research and observations are needed to fully understand the universe and its evolution.

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