#### Buzz Bloom

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- What is the probability/certainty-level that the universe is absolutely flat?

My questions are based on material from the following source.

pg 26 Eqs 27 & 28

P 27 eq 30

P 39 eq 50

Q`1: I am not sure I understand what the "95%" means in equation 30. Which of the following does it mean?

Q2: Does the final statement referring to "a 1σ accuracy of 0.25%" mean that the following probability statements are all true:

Assuming both the above guesses are correct, I have a few additional questions motivated by the following goals:

I have been trying to think of an approach to come up with a way of justifying a statement of something similar to the following form:

Q3: Does anyone know of any scientifically reliable source that includes a statement similar to something similar to the above indented statement?

The best idea I have been able to come up with for what may be a possible scientifically reliable related statement is:

Assuming that no PF participant has a handy "YES" answer to Q3, I offer a fourth question.

Q4: Does the thought experiment I describe below seem reasonable as a theoretical approach that (given sufficiently precision instruments) could lead to a reliable statement of the form S?

I assume that there is some limit to the accuracy with which an astronomer can measure that position in the sky of a distant object. In particular I am thinking of the identifiable patterns in the CBR temperatures. If we arbitrarily define the North Pole, P

gives the following value for the radius of the OU.

Given equation 50, we can derive from probabilty statements of Q2 a,b, and c, the following regarding the radius of curvature R:

If it is currently technically impossible to distinguish two angles which are different by 5 trillionths of an arcsec, is it then reasonably to conclude that the following S-like sentence is true?

pg 26 Eqs 27 & 28

In the base CDM model, the Planck data constrain the Hubble constant H

_{0}and matter density Ωm to high precision:H

_{0}=67.3±1.0 km s^{−1}Mpc^{−1}Ω

_{m}=0.315±0.013 PlanckTT+lowP. (27)With the addition of the BAO measurements, these constraints are strengthened significantly to

H0=67.6±0.6 km s−1Mpc−1

Ω

_{m}=0.310±0.008 PlanckTT+lowP+BAO. (28)P 27 eq 30

The R11 Cepheid data have been reanalysed by Efstathiou (2014, hereafter E14) using the revised geometric maser distance to NGC 4258 of Humphreys et al. (2013). Using NGC 4258 as a distance anchor, E14 finds

H

_{0}=70.6±3.3 km s−1 Mpc−1, NGC 4258, (30)which is within 1σ of the Planck TT estimate given in Eq. (27). In this paper we use Eq. (30) as a “conservative” H

_{0}prior.P 39 eq 50

The constraint can be sharpened further by adding external data that break the main geometric degeneracy. Combining the Planck data with BAO, we find

Ω

_{m}=0.000±0.005 (95%,PlanckTT+lowP+lensing+BAO). (50)This constraint is unchanged at the quoted precision if we add the JLA supernovae data and the H_0 prior of Eq. (30). . . . Our Universe appears to be spatially flat to a 1σ accuracy of 0.25%.

Q`1: I am not sure I understand what the "95%" means in equation 30. Which of the following does it mean?

a. The mean value 0.000 of the distribution in Equation 30 is accurate with a confidence of 95%.

b. The error range of ±0.005 of the distribution in Equation 30 is accurate with a confidence of 95%.

c. Both the mean value the error range of the distribution in Equation 30 is accurate with a confidence of 95%.

d. It means something else.

My guess is that the answer is (c).Q2: Does the final statement referring to "a 1σ accuracy of 0.25%" mean that the following probability statements are all true:

a. PROB{Ω

_{m}>+0.005} = 0.25%, andb. PROB{Ω

_{m}<-0.005} = 0.25%, andc. PROB{Ω

My guess is that the answer "YES"._{m}>-0.005} AND PROB{Ω_{m}<+0.005} = 99.5%?Assuming both the above guesses are correct, I have a few additional questions motivated by the following goals:

I have been trying to think of an approach to come up with a way of justifying a statement of something similar to the following form:

The probability that the universe is perfectly flat is x (where x is a value close to 1.)

It is clear that a distribution statement like Equation 50 fails to support such a statement.Q3: Does anyone know of any scientifically reliable source that includes a statement similar to something similar to the above indented statement?

The best idea I have been able to come up with for what may be a possible scientifically reliable related statement is:

S: The probability that our universe is definitely not perfectly flat is less than x (where x is a small fraction).

Assuming that no PF participant has a handy "YES" answer to Q3, I offer a fourth question.

Q4: Does the thought experiment I describe below seem reasonable as a theoretical approach that (given sufficiently precision instruments) could lead to a reliable statement of the form S?

I assume that there is some limit to the accuracy with which an astronomer can measure that position in the sky of a distant object. In particular I am thinking of the identifiable patterns in the CBR temperatures. If we arbitrarily define the North Pole, P

_{N}, of the sky as the statistically determined point towards which the various CBR telescopes are moving, giving them (by Doppler effects) the center of the area with the largest temperature. Then presumably it can be determined what the angular distance is between the North Pole and a specific spot on some identifiable temperature pattern, say P. It should be possible to choose a point P such that the position of the telescope, O, P_{N}and P form an equilateral triangle. The distance D between O and P is the same as the distance between O and P_{N}. It is approximately the radius of the observable universe (OU). (I understand that the source of the CBR is approximately 400,000 years younger than the age of the universe, which make the distance to the CBR smaller than the radius of the OU, but this is an insignificant reduction with respect to the thought experiment.) The choice of P should be such that the distance (as light travels) between P and P_{N}is D. If the universe is flat, then this is straight forward since the angular distance will be exactly 60 degrees. If the universe is not flat, then the angle with be a very small amount less or more than 60 degrees. This amount depends on the absolute value of the radius of the curvature.### Observable universe - Wikipedia

en.wikipedia.org

r = Radius of observable universe = 93 Gly = 9.3 x 10

^{10}lyGiven equation 50, we can derive from probabilty statements of Q2 a,b, and c, the following regarding the radius of curvature R:

ρ

_{crit}= 3h_{0}^{2}/8πGρ

_{k}= c_{2}/πG|R^{2}|I am unable to find the source where I got this formula. The URL I had for this source has become lost, and I cannot find it again. Does any reader know a source for this formula can be found?

Ω

_{k}= ρ_{k}/ρ_{crit}|R

^{2}| = (8/3) c^{2}/ (Ω_{k}h_{0}^{2})PROB{|R|< 1.254 x 10

The area A of a flat equilateral traingle with sides of length r is^{19}ly} = 0.5%A = (1/4) √π r

^{2}α = sum of angles - π

A = α x R

^{2}α= (1/4) √π (r/R)

^{2}= (2.44 10^{-17}) radians1 radian = 57.296 degrees = 206,265 arcsecs = 2.06265x10

^{5}arcsecsα= 5.03 x 10

^{-12}arcsecsIf it is currently technically impossible to distinguish two angles which are different by 5 trillionths of an arcsec, is it then reasonably to conclude that the following S-like sentence is true?

The probability that our universe (using current technology) is distinguishable from a perfectly flat universe is less than 0.5%.

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