An idea for calculating the probability that the Universe is flat

[Buzz Bloom]

TL;DR Summary

From other thread discussions (see the body for examples) , a consensus seems to be that it is not possible to know whether or not the universe is flat. This thread is seeking some expert understanding regarding the reasonableness of calculating a plausible probability that the universe is flat.

References

(1) https://www.physicsforums.com/threa...-that-the-universe-is-absolutely-flat.971984/​

(2) https://www.physicsforums.com/threa...bability-that-the-universe-is-finite.1011826/​

Suppose the Friedmann equation is used to analyze two models.

(1) H₀ and all four Ωs can in principle have various values.​

(2) H₀ and three of the four Ωs can in principle have various values,​

but Ω_k = 0.​

For both of these two models, a best fit to the same database of values is calculated., say F₁ and F₂. I am not sure what the proper method would be for calculating a fit value, but the best fit is the smallest value, and the best bit includes the specific variable values that produced this best fit. For the purpose of this discussion, I will assume the method is the following for the fit value.

F = Σ_i,j (V_db,i,j-V_m,i,j)²​

i is an index over the database sets of variable values.​

j is an index for the variables within each set of variables​

V_db is a variable in the database​

V_m is a corresponding variable whose values are calculated using the​

Friedmann equation.​

Since F₁ includes the results of one more Friedmann equation variable than F₂, I would expect that

F₁ < F₂.​

If F₂-F₁ is small compared with F₁, then I think it would be likely that F₂ should be considered to be a better fit than F₁. I do not know the math for quantifying this, but this does seem to me to be reasonable. There should be a mathematical way to compare F₁ and F₂ so that an adjustment can be made for the difference in the number of variables.

The following is just a example of a concept about making such a comparison. I do not think it is a correct method. It is intended only as an illustration of a result of a method allowing for a reasonable comparison between F₁ and F₂.

Suppose we define G as follows.

G = F^1/2 / N,​

where N is the number of model variables. Therefore

G₁ = F₁^1/2 / 5, and​

G₂ = F₂^1/2 / 4.​

Since the only difference is that F₁ has a range of values for Ω_k, but F₂ has Ω_k=0. I suggest that P₁ is the probability that G₁ is correct, and P₂ is he probability that G₂ is correct. I calculate as follows:

P₁ = G₁/(G₁+G₂)​

P₂ = G₂/(G₁+G₂)​

P₁ + P₂ = 1​

The result means that P₂ is the probability that the universe is flat. P₁ is the probability that the universe is not flat.

Reference 2 makes a specific assumption that the universe is not flat. Based on this assumption, approximately the probability the universe is finite is 70%, and the probability the universe is infinite is 30%. Base on the concept that P₁ and P₂ can be calculated, using these probabilities gives the following results.

The probability that the universe is infinite and flat is P₂.​

The probability that the universe if a finite 3D hyper-sphere is P₁ x 0.7.​

The probability that the universe is is an infinite 3D hyperbolic shape is P₁ x 0.3.​

​

 

[anorlunda]

I did not try to follow your arguments. But just contemplate the difficulty of distinguishing the difference between an approximately flat universe and an exactly flat universe. Is any kind of heuristic equation likely to do that for us?

Even if your logic is correct, how would we ever gather the evidence to verify it?

 

[Buzz Bloom]

Is any kind of heuristic equation likely to do that for us?

Hi @anorlunda:

If you would read the details. there is a presentation of a conceptually possible (but not necessarily practical) process for assigning plausible probabilities to the two models. If you want verifying evidence, and you and I agree on what verify means, I think this is not part of physics. Nothing in physics is certain, although some ideas may be very close to being certain. Evidence can (and frequently does) support the likelihood of a conjecture being correct. I am pretty certain that it is not possible to be certain that the universe is flat. When two similar (not identical) models are used and the results are very close to each other, then I think it is reasonable to conclude that the two choices are about equally likely to be correct.

 

[timmdeeg]

@Buzz Bloom We have no clue to calculate the probability that the universe is flat.
In case the universe wasn't flat before inflation then the prediction - in case we trust the inflationary theories - is that the universe is almost flat which excludes exact flatness. If the universe was flat before inflation then we can calculate no probability because we lack any understanding for reasons causing flatness.

 

[Buzz Bloom]

We have no clue to calculate the probability that the universe is flat.

Hi @timmdeeg:

NOTE: I edited this post below, based on a suggestion from post #7. This edit added text in italics.

At the present time, the particular solution of the four Ωs Friedmann model with a calculated value of Ω_k=0.0007 is very close to zero. Suppose, hypothetically, that instead of the current reality, this model calculated a best fit value like Ω_k=0.3. Note that along with the 0.3 value, the standard deviation of the Ω_k distribution of values would also be useful in order to calculate the number of standard deviation values between 0.3 and 0.0. Would you then think that the universe was likely to be flat? I am guessing that you would not, but might you think it was still possible it might be flat?

This thread is suggesting that it might be possible to calculate a plausible probability by comparing the best fit of two models: (1) the best fit for the four Ω model which calculates the particular solution including Ω_k=0.0007+/- 0.0019, and (2) the best fit particular solution for the model which assumes Ω_k=0.

What in particular in my description of a hypothetical procedure for calculating a probability for Ω_k=0 do you find to be illogical?

Regards,
Buzz

 

[PeterDonis]

Suppose the Friedmann equation is used to analyze two models.

Then any "probability" you claim to get from any such analysis (and I haven't gone through your proposal in detail to see whether such a claim is warranted, although based on our discussion in the previous thread you referenced I think I have a pretty good idea what you're proposing, and as far as calculating distributions for free variables in particular models there is no problem with it) can only be a probability based on the claim that the two models you are analyzing are the only possible models that can explain the data.

But of course we can never know that for sure, so your general method here is open to the same objection that I made to calculating the probability in the context of a single model in the previous thread. You're still not answering the question that the title of this thread raises: you're not calculating "the probability that the universe is flat", period. You're only calculating the probability that the universe is flat in the context of a particular set of models.

 

[PeterDonis]

At the present time, the particular solution of the four Ωs Friedmann model with a calculated value of Ω_k=0.0007 is very close to zero. Suppose, hypothetically, that instead of the current reality, this model calculated a best fit value like Ω_k=0.3. Would you then think that the universe was likely to be flat?

First, note that just the best fit value itself would not be enough; you would also need to know the error bars around it, so that you would know how many standard deviations away from the best-fit value $\Omega_k = 0$ was, for the "standard" model that you describe. And you would need to know the same thing for any other specific models under consideration.

Second, most cosmologists would probably think that the universe was unlikely to be flat in such a scenario, but you need to be clear about the reason why. It would not be simply because $\Omega_k = 0$ was some large number of standard deviations away from the "standard" model's best fit value. It would be because, if the data were such that the "standard" model's best fit value was $\Omega_k = 0.3$, cosmologists would have a hard time imagining any model that could provide a reasonable fit to the data while enforcing $\Omega_k = 0$. In other words, it would be based on some form of reasoning about the set of all possible or reasonable models, not just about the single model for which the best fit value, in your scenario, is $\Omega_k = 0.3$.

 

[timmdeeg]

Hi @timmdeeg:

Suppose, hypothetically, that instead of the current reality, this model calculated a best fit value like Ω_k=0.3. Would you then think that the universe was likely to be flat?

No, I wouldn't. And I wouldn't trust the inflationary theories anymore.

What in particular in my description of a hypothetical procedure for calculating a probability for Ω_k=0 do you find to be illogical?

I haven't gone through it, because I'm convinced you can't calculate said probability (and lack of time )

How can you calculate the probability for a physical phenomenon in case you don't know the cause for this phenomenon?

 

[PAllen]

The probability that the universe is consistent with what we measure is 1

 

[Buzz Bloom]

How can you calculate the probability for a physical phenomenon in case you don't know the cause for this phenomenon?

Supposed I observed a phenomenon which occurred some days always at noon, but not every day, in a pattern of days that seemed random, and after 100 days counting the first occurrence as day 1, it had occurred
29 days the first 50 days and 35 days the second 50 days. Would you find it to be unreasonable for me to estimate the probability of the occurrences of the phenomenon as 64% +/- 3%, even though I had absolutely no idea about what was causing the phenomenon?

 

[PAllen]

Supposed I observed a phenomenon which occurred some days always at noon, but not every day, in a pattern of days that seemed random, and after 100 days counting the first occurrence as day 1, it had occurred
29 days the first 50 days and 35 days the second 50 days. Would you find it to be unreasonable for me to estimate the probability of the occurrences of the phenomenon as 64% +/- 3%, even though I had absolutely no idea about what was causing the phenomenon?

So ,if you can create 100 universes ….

 

[Vanadium 50]

Buzz, if yu didn't follow the argument the first time, will asking essentially the same question again again be likely to help?

 

[Buzz Bloom]

First, note that just the best fit value itself would not be enough; you would also need to know the error bars around it,

In other words, it would be based on some form of reasoning about the set of all possible or reasonable models, not just about the single model for which the best fit value, in your scenario, is .

Hi @PeterDonis:

It was an oversight for me to forget the need for error bars. I will edit to correct that.

I am confused about "all possible models". I feel you are referring to some non-Friedmann equation based models, and I guess that they are not related to the same five variables which are in the Friedmann equation. I cannot think of any Friedmann based models related to the Ω_k value other than the two we have discussed. I am guessing they would relate to non-Friedmann related models which do not relate to the five Friedmann variables, but they would relate to entirely different variables. I would much appreciate it you would help me understand this issue in some detail.

Regards,
Buzz

 

[Buzz Bloom]

The probability that the universe is consistent with what we measure is 1

Hi @PAllen:

This certainly seems to be non-refutable. The problem with this view is that what is measured can (and frequently does) change over time. For example, the value of H₀ has been measured many times with the measured value changing frequently.

However, the interpretation of the quote is a bit unclear. I understand there is no measurement of the specific value of Ω_k. There is a calculated mean of a distribution of values of Ω_k together with an error range which can be recalculated as a standard deviation. I understand that this distribution of Ω_k values may only be useful as a probability distribution if specific assumptions are made which limit the interpretation.

Regards,
Buzz

 

[Buzz Bloom]

Buzz, if yu didn't follow the argument the first time, will asking essentially the same question again again be likely to help?

Hi @Vanadium 50:

I would like to understand the quote, but I need some help. I have been posting a lot in two threads about various relations regarding Ω_k. I do not get what "argument" you are referring to. I also do not get what the "same question" is which has been asked two times.

Regards,
Buzz

 

[Buzz Bloom]

So ,if you can crate 100 universes ….

Hi @PAllen:

I apologize for my ignorance, which seems to grow with my age. I do not get the meaning of the quote at all, but it does seem to have some humor.

Regards,
Buzz

 

[PeterDonis]

I am confused about "all possible models". I feel you are referring to some non-Friedmann equation based models

I have no idea why you are confused or why you would feel that I am referring to non Friedmann equation based models. I proposed one other possible model in the previous thread, which is also based on the Friedmann equation. One could concoct a number of Freidmann equation based models with different choices for which of the five parameters you listed are free variables and which ones are fixed by some constraint. Indeed, as I mentioned in the previous thread, up until the 1990s, the "best fit" model had a fixed value of $\Omega_\Lambda = 0$.

I would much appreciate it you would help me understand this issue in some detail.

We went into considerable detail in the previous thread. I see no point in rehashing that again.

he value of H0 has been measured many times with the measured value changing frequently.

However, the interpretation of the quote is a bit unclear. I understand there is no measurement of the specific value of Ωk. There is a calculated mean of a distribution of values of Ωk together with an error range which can be recalculated as a standard deviation.

Your understanding of the "standard" model you are using appears to be flawed. None of the five parameters you listed, including $H_0$, are directly measured. All of them are estimated from other measurements, and those estimates take the form of a distribution of values, which is standardly given as a "best fit" value and an error range. Directly measured quantities include redshifts, luminosities, and angular sizes of distant objects such as galaxies and supernovas, as well as light curves for supernovas and rotation curves for galaxies. There are also direct measurements of the properties of the CMB, mainly its average temperature and its angular variation of temperature over the sky. There are others, but I think the ones I've listed are the most important ones for estimating the five parameters you are interested in.

Indeed, you also appear to have a flawed understanding of direct measurements themselves. Those also consist of a distribution of values, which is standardly given as a "best" value and an error range. There is no such thing as a directly measured quantity that is known for certain to have a single specific value.

Given that your understanding of these very basic points appears to be flawed, you might want to reconsider whether you are ready to tackle the more ambitious task of trying to interpret what the "best fit" distribution for $\Omega_k$ in the current "standard" model means with regard to the title questions of this thread and the previous one.

I would like to understand the quote, but need some help.

What he is saying is that the discussion in this thread is turning out to be very similar to the discussion in the previous thread, rehashing the same points with nothing new of any real substance being added. And he is wondering whether it's worth continuing the discussion given that that is the case.

 

[PeterDonis]

I do not get the meaning of the quote at all

He means that on the standard frequentist interpretation of "probability", the only way to find out the probability that the universe is flat is to create a large ensemble of universes using the same process that created ours and count how many of them are flat.

He was, of course, being facetious, but the underlying point about what "probability" actually means is a serious one and you should consider it carefully.

 

[PeterDonis]

So ,if you can create 100 universes ….

I'm reminded of somebody's spoof of a final exam where the biology question is something like: "Create life, and write an essay commenting on the implications." The corresponding cosmology question (which was, alas, not on that spoof exam) is of course obvious.

 

[Buzz Bloom]

I have no idea why you are confused or why you would feel that I am referring to non Friedmann equation based models.

Hi @PererDonis:

My confusion was due to my inability to imagine that any of the other Friedmann models than the two we had discussed could be relevant to a meaningful calculation of a probability that the universe is flat. However, after further thought I think I have found a third model that might be relevant, but I do not have much confidence about this.

The third model is one in which Ω_r=0, and the other variables can take any value. Comparing this third model with model #1, the best fit values for these two models might be very close to each other, just as the best fit values for models #1 and #2. Models #1 and #3 having close best fit values would imply that radiation is not very likely to have any influence on the Friedmann calculations. From this conclusion, it seems likely that comparing the best fit between models #1 and #2, both having the same number of non-specified variable values (exactly one variable set to zero) makes it easier to compare the best fir values for the purpose of estimating a value for the likelihood that Ω_k=0, that is, that the universe is flat.

I would certainly like to know your views about this. Also, if you have a specific idea for choosing a model other than the three I discuss here, and which has relevance to the issue regarding the likelihood that the universe is flat, I would be very interested in understanding how this other model has relevance.

Your understanding of the "standard" model you are using appears to be flawed. None of the five parameters you listed, including , are directly measured. All of them are estimated from other measurements, and those estimates take the form of a distribution of values, which is standardly given as a "best fit" value and an error range.

My understanding is exactly what you describe in this quote. I must have expressed this understanding poorly.

Regards,
Buzz

 

[PeterDonis]

My confusion was due to my inability to imagine that any of the other Friedmann models than the two we had discussed could be relevant to a meaningful calculation of a probability that the universe is flat.

Since the data has now ruled out ("now" meaning "since the late 1990s") a model with $\Omega_\Lambda$ fixed at $0$, and since we know the universe has a significant density of matter, the only other of the $\Omega$s that could potentially be fixed at $0$ to simplify the model would be $\Omega_r$, as you suggest. In our current universe we know that is a very good approximation in any case, because the only significant radiation present is the CMB and we know that its energy density is several orders of magnitude smaller than any of the others. But of course that was not always the case; in the early universe the radiation density was higher than the matter density. So a model that fixes $\Omega_r = 0$ for all time is not viable.

However, none of that makes any of the models with one or more of the $\Omega$s fixed instead of being free parameters impossible. It just makes them unviable in the light of our best current understanding of the data. Which is just another way of saying that cosmologists do not look at only one model when trying to answer questions like the title question of this thread. They look at a family of models. And their judgment of which model of a given family is the best fit can change over time as new data comes in: the fact that before the 1990s the "best fit" model was one in which $\Omega_\Lambda$ is fixed at $0$ instead of being a free parameter is an example of that. If cosmologists had only been willing to consider one model, they would never have been able to switch to the current "best fit" model from the previous one.

All that said, it is also possible that at some point cosmologists will need to consider models that are not solely based on the Friedmann equations as well. For example, there is already published literature on models that are not fully homogeneous or isotropic. So far such models have not gained significant traction among cosmologists in general, but that could change if we get further data that makes such models seem more viable.

In short, we can never know for sure that our current "best" model, or even our current understanding of which family of models we should be considering in order to pick our "best" model, will stay the same in the future. And because of that, we can never fully answer questions like the title question of this thread in an unqualified manner. Any answer we give will always have to include qualifiers like "assuming our best current model is correct" that are open to correction in the future.

My understanding is exactly what you describe in this quote.

Ok, good.