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- 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

Suppose the Friedmann equation is used to analyze two models.

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 F

Suppose we define G as follows.

The result means that P

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 P

Suppose the Friedmann equation is used to analyze two models.

(1) H

_{0}and all four Ωs can in principle have various values.(2) H

_{0}and three of the four Ωs can in principle have various values,but Ω

For both of these two models, a best fit to the same database of values is calculated., say F_{k}= 0._{1}and F_{2}. 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 databaseV

_{m}is a corresponding variable whose values are calculated using theFriedmann equation.

Since F_{1}includes the results of one more Friedmann equation variable than F_{2}, I would expect thatF

If F_{1}< F_{2}._{2}-F_{1}is small compared with F_{1}, then I think it would be likely that F_{2}should be considered to be a better fit than F_{1}. 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 F_{1}and F_{2}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 F

_{1}and F_{2}.Suppose we define G as follows.

G = F

where N is the number of model variables. Therefore^{1/2}/ N,G

_{1}= F_{1}^{1/2}/ 5, andG

Since the only difference is that F_{2}= F_{2}^{1/2}/ 4._{1}has a range of values for Ω_{k}, but F_{2}has Ω_{k}=0. I suggest that P_{1}is the probability that G_{1}is correct, and P_{2}is he probability that G_{2}is correct. I calculate as follows:P

_{1}= G_{1}/(G_{1}+G_{2})P

_{2}= G_{2}/(G_{1}+G_{2})P

_{1}+ P_{2}= 1The result means that P

_{2}is the probability that the universe is flat. P_{1}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 P

_{1}and P_{2}can be calculated, using these probabilities gives the following results.The probability that the universe is infinite and flat is P

_{2}.The probability that the universe if a finite 3D hyper-sphere is P

_{1}x 0.7.The probability that the universe is is an infinite 3D hyperbolic shape is P

_{1}x 0.3.

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