B What is the probability that the Universe is absolutely flat?

[metastable]

Is it necessary or only convenient to reference a 3-space?

I define an arc with n=arbitrarily high # turns. I make a straight line from point t=0 to point t=1. Next I bisect line t=0, t=1 creating a point A. The vector is created first by a line from point A, through the referenced spiral point B to point C, a point in space. Points in space are referenced as Vector: t=0.14563... Distance: 686,739,974.97969...km

If we're talking about the universe, then yes, since spatial slices of the universe are 3-spaces.

I don't know what you're trying to get at with the arcs.

wolframalpha.com:

ParametricPlot3D [ { [//math:cos(2*pi*10^27*t)*sin(pi*t)//] , [//math:sin(2*pi*10^27*t)*sin(pi*t)//] , [//math:cos(pi*t)//] } , { t , [//number:0//] , [//number:1//] } ]

if on this scale 1 = Earth radius, loops = 10^27

unless I've made a mistake in my calculation we have:

~0.000020037... femtometers position specification accuracy & separation distance between loops at Earth radius

 

[Ibix]

...so you are trying to use distance along your spiral to specify angular position and radius as in regular spherical polars? How do you specify a point that does not lie on a line through your spiral?

 

[metastable]

How do you specify a point that does not lie on a line through your spiral?

I make a straight line from point t=0 to point t=1. Next I bisect line t=0, t=1 creating a point A. The vector is created first by a line from point A, through the referenced spiral point B to point C, a point in space.

^With a point A at the bisection of a straight line from point t=0 to t=1

 

[Buzz Bloom]

the ratio of the actual density to the critical density

Hi timmdeeg:

The above quote is the definition of Ω_M.
Please see

https://en.wikipedia.org/wiki/Friedmann_equations​

in the section Density parameter.

Regards,
Buzz

 

[Buzz Bloom]

And thus to be consistent with known data, models of inflation must work in universes that have negative spatial curvature.

However, that appears to be a moot point for inflation models, since the reference you give notes that they do in fact work with either positive or negative spatial curvature.

Hi George and Peter:

I apologize for my error. My bad. My previous careless exposure to discussions of inflation have involved explanatory pictures of finite shapes getting bigger. I failed to realize that the metaphor in the pictures intended the inclusion of both kinds of non-flat universes.

https://en.wikipedia.org/wiki/Inflation_(cosmology)#/media/File:History_of_the_Universe.svg​

http://www.ctc.cam.ac.uk/outreach/origins/inflation_zero.php
Regards,
Buzz

 

[Ibix]

^With a point A at the bisection of a straight line from point t=0 to t=1

And the position of A along that line is your third number.

 

[timmdeeg]

The above quote is the definition of Ω_M.
Please see

https://en.wikipedia.org/wiki/Friedmann_equations​

in the section Density parameter.

I'm not sure how you come to this conclusion. This article states "The density parameter (useful for comparing different cosmological models) is then defined as:" ... and describes thereafter $\Omega$ as the ratio of the actual density to the critical density.

If $\Omega$ in the expression in brackets (post #60) were $\Omega_m$ then how could this expression be identical zero in case of exact flatness?

 

[metastable]

And the position of A along that line is your third number.

Are you saying if I want to specify 3 chosen subsequent particle positions it would take me more than 6 numbers? (3 spiral points and 3 distances from point A)

 

[Ibix]

Are you saying if I want to specify 3 chosen subsequent particle positions it would take me more than 6 numbers? (3 spiral points and 3 distances from point A)

Yes. Because you cannot specify a position that does not lie on a line through your spiral.

 

[metastable]

Because you cannot specify a position that does not lie on a line through your spiral.

In a 3 space can I specify any irrational position?

 

[Buzz Bloom]

k=0 requires Ω=1

If Ω in the expression in brackets (post #60) were Ω_m then how could this expression be identical zero in case of exact flatness?

Hi timmdeeg:

It is Ω_k = k = 0 then

Ω_m = 1, and the universe is flat.​

When Ω_m > 1,

k=+1 and Ω_k < 0.​

The universe is then finite and hyperspherical.​

When Ω_m < 1,

k=-1 and Ω_k > 0.​

The universe is then infinite and hyperbolic.​

Regards,
Buzz

 

[George Jones]

No, $\Omega$ in $(\Omega^{-1}-1)\rho{a}^2=-\frac{3c^2}{8\pi{G}}k$ (post #13) is the ratio of the actual density to the critical density.

The above quote is the definition of Ω_M.
Please see

https://en.wikipedia.org/wiki/Friedmann_equations​

in the section Density parameter.

I'm not sure how you come to this conclusion. This article states "The density parameter (useful for comparing different cosmological models) is then defined as:" ... and describes thereafter $\Omega$ as the ratio of the actual density to the critical density.

I agree with @timmdeeg

In this section, density is given by
$$\rho = \rho_r + \rho_m + \rho_\Lambda$$

and

$$\begin{align} \Omega &= \frac{\rho}{\rho_c} \nonumber \\
&= \frac{\rho_r + \rho_m + \rho_\Lambda}{\rho_c} \nonumber \\
&= \frac{\rho_r}{\rho_c} + \frac{\rho_m}{\rho_c} + \frac{\rho_\Lambda}{\rho_c} \nonumber \\
\Omega &= \Omega_r + \Omega_m + \Omega_\Lambda . \nonumber
\end{align}$$

 

[Ibix]

In a 3 space can I specify any irrational position?

If you can specify the irrational number you mean, then sure. $(x,y,z)=(\pi,e,\sqrt{2})$, for example.

 

[metastable]

Can I write a complete list of the x,y,z positions an electron had as it traveled a meter, even in principle?

 

[Buzz Bloom]

In this section, density is given by

ρ=ρ_r+ρ_m+ρ_Λ​

and

Ω=ρ_r+ρ_m+ρ_Λ/ρ_c​

=ρ_r/ρ_c+ρ_m/ρ_c+ρ_Λ/ρ_c​

Ω=Ω_r+Ω_m+Ω_Λ.​

Hi George:

I edited the format of quote so that I could read it more clearly.

I confess I was not previously familiar with this usage of Ω, but I now see at the bottom of the Wiki section:

Ω_0,k - 1-Ω₀.​

So I yield, and I am now convinced I was previously mistaken. Thank you for pointing out my error.

Regards,
Buzz

 

[Buzz Bloom]

Hi @metastable:

I am puzzled by the figure with a spiral on a sphere in your post #35. Please explain in words what this is intended to represent.

Regards,
Buzz

 

[metastable]

Hi @metastable:

I am puzzled by the figure with a spiral on a sphere in your post #35. Please explain in words what this is intended to represent.

Regards,
Buzz

In rough terms: suppose you take a standard desk globe of the earth, and you attach a motorized gantry to the arch which supports it at the poles, and you spin the globe at a constant velocity, and you move the gantry from the north pole to the south pole at constant angular speed, with a pen attached to it, the pen will trace out this spiral on the globe depending on the relative speeds of the gantry and the globe.

 

[PeterDonis]

suppose you take a standard desk globe of the earth, and you attach a motorized gantry to the arch which supports it at the poles, and you spin the globe at a constant velocity, and you move the gantry from the north pole to the south pole at constant angular speed, with a pen attached to it, the pen will trace out this spiral on the globe depending on the relative speeds of the gantry and the globe.

What does this have to do with the thread topic?

 

[metastable]

What does this have to do with the thread topic?

I was asking if 3 spaces are necessary or only convenient, because I thought, via the illustration, I could specify an infinite subset points in space, not all along the same plane, with only 2 dimensions: 1) distance along spiral from pole and distance along straight line from point A through 1)

 

[PeterDonis]

I thought, via the illustration, I could specify an infinite subset points in space, not all along the same plane, with only 2 dimensions

Since the motion in your scenario is restricted to the surface of a 2-sphere, obviously it is only describing points in a space with 2 dimensions, since a 2-sphere is a 2-dimensional manifold. The fact that you are describing the 2-sphere using its embedding in 3-dimensional space does not change that.

 

[metastable]

Is the spiral not 1 dimensional since no area no volume?

 

[Ibix]

Can I write a complete list of the x,y,z positions an electron had as it traveled a meter, even in principle?

$\vec r(t)=\int_0^t\vec v(t')dt'$

And a critical problem with your approach is that you cannot use calculus to describe motion across the grain of your spiral because you don't have a smooth map from space to your coordinate system. So you've gone a long way towards disabling the only tool you can use to describe things moving in rigorous mathematical terms. And you haven't achieved your goal of specifying points in 3-space using two numbers because it's impossible to do so.

 

[metastable]

And a critical problem with your approach is that you cannot use calculus to describe motion across the grain of your spiral because you don't have a smooth map from space to your coordinate system.

After specifying a position, I was toying with the idea in my head of describing its heading with a second spiral at the specified point and its instantaneous motion as a curve which would be represented by the circumference of a circle with a specified radius and orientation, but I am far from claiming any of this represents reality, but I mention it because you mentioned not being able to use calculus with the approach and I wondered if specifying instantaneous motions as such curves could work with calculus?

 

[Buzz Bloom]

In rough terms: suppose you take a standard desk globe of the earth, and you attach a motorized gantry to the arch which supports it at the poles, and you spin the globe at a constant velocity, and you move the gantry from the north pole to the south pole at constant angular speed, with a pen attached to it, the pen will trace out this spiral on the globe depending on the relative speeds of the gantry and the globe.

Hi metastable:

I am wondering why you associate this concept with the topic of this thread?

Regards,
Buzz

 

[Ibix]

After specifying a position, I was toying with the idea in my head of describing its heading with a second spiral at the specified point and its instantaneous motion as a curve which would be represented by the circumference of a circle with a specified radius and orientation, but I am far from claiming any of this represents reality, but I mention it because you mentioned not being able to use calculus with the approach and I wondered if specifying instantaneous motions as such curves could work with calculus?

Then you have three numbers to specify positions in 3d space. Inevitably. I suspect there's more than one way to identify a location using this system, so this has nasty mathematical properties. But knock yourself out.

 

[Ibix]

Hi metastable:

I am wondering why you associate this concept with the topic of this thread?

Regards,
Buzz

That's a good question. I'll stop contributing to this hijack now.

 

[metastable]

Hi metastable:

I am wondering why you associate this concept with the topic of this thread?

Regards,
Buzz

asking if 3 spaces are necessary or only convenient

^I was trying to ascertain whether "3 spaces" are a "requirement" & "necessary" to describe the universe or merely "mathematically convenient."

 

[George Jones]

^I was trying to ascertain whether "3 spaces" are a "requirement" & "necessary" to describe the universe or merely "mathematically convenient."

These "3 spaces" can be shown mathematically to be the only cosmological spaces that are both spatially isotropic and spatially homogeneous, but the mathematics (of Killing vectors) is appropriate for a thread that has an "A" label.

More spaces result if one/both of these conditions is/are relaxed.

 

[PeterDonis]

I was trying to ascertain whether "3 spaces" are a "requirement" & "necessary" to describe the universe or merely "mathematically convenient."

And I already answered that way back in post #36. Please do not discuss this question, or your questions about whether it takes 3 numbers to specify a point in 3-space (it does), any further in this thread.

 

[timmdeeg]

And thus to be consistent with known data, models of inflation must work in universes that have negative spatial curvature. The Lyth and Liddle reference that I gave above explicitly notes that inflation smooths inhomogeneities in both $\Omega > 1$ and $\Omega < 1$ universes.

So regardless the global curvature locally the universe can have small areas like our observable universe which are positively or negatively curved depending on local inhomogeneities?

If this is correct so far would some kind of averaging over local curvatures in principle yield the global curvature? It would also mean that even exact knowledge about local curvature provides no clue to global curvature.