The Cosmic Teardrop

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The Cosmic Teardrop

Postby BurtJordaan on January 4th, 2017, 4:05 am 

The 'cosmic teardrop' is essentially the past lightcone of a comoving observer (at rest w.r.t. the CMB) in an Einstein-de Sitter expanding universe, shown in proper-distance, proper-time coordinates.

Image

The teardrop shape is caused by the way that light propagates to us from two directions, starting from the CMB, through the (variable rate) expansion of the intervening space. In the beginning, the expansion rate was so fast that the two wave fronts (propagating locally at c) moved away from us. Around time 4 Gy, the expansion rate slowed down enough to enable the wave fronts to make headway in our direction.

A more enlightening equivalent can be depicted in circular coordinates, shown below for values compatible with the latest cosmological data.[1]

Image

This is essentially the 'cosmic teardrop' drawn inside the cosmological balloon analogy, connecting the wave front positions at progressive circular slices of constant cosmic time. But note that this does not make the radial direction a time coordinate. It is a fictitious extra spatial coordinate, which is a rather complex function of cosmic time and total energy density. It can also be viewed as representing the cosmic scale factor (a) multiplied by the Hubble constant.

The "2 Gy rings" show that the expansion rate was very fast in the beginning, gradually decreasing during the 'middle ages' and then increased again lately. The 46 Gly "pizza slices" represent the observable universe's present spatial dimensions and the total circumference the minimum size of the total universe (200 pi = 627 Gly).

-J

[1] Note that it assumes that the spatial curvature is positive, just 0.1% off perfectly flat. This is within the limits of present observational errors, which tends towards favoring zero curvature, but still allows a small positive curvature. One can obviously not represent a perfectly flat space as circular, unless you imagine an infinitely large circle.

PS: The above was written very compact and terse, in the hope that the diagrams might convey more than the words, so please ask questions if it is not clear what is going on.
Last edited by BurtJordaan on January 4th, 2017, 1:28 pm, edited 1 time in total.
Reason: Minor clarification and PS.
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Re: The Cosmic Teardrop

Postby zetreque on January 4th, 2017, 4:38 am 

Hi BurtJordaan,

This sounds cool. Could you possibly sum it up in laymen terms? :)
It might help if I knew what Gy and Gly are.
Something to do with absorbed radiation it seems.
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Re: The Cosmic Teardrop

Postby BadgerJelly on January 4th, 2017, 5:07 am 

I am pretty sure that is layman terms? Haha

Afterall he did say "speed" and "expansion". Our puny human brians try and make sense by drawing diagrams and pretty shapes to represent streams of numbers which don't physically exist! Haha

Sorry, coudln't help myself ;)
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Re: The Cosmic Teardrop

Postby BurtJordaan on January 4th, 2017, 5:35 am 

Hi Zet, sorry for being too brief. ;)

Giga years and Giga lightyears, i.e. American billions...
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Re: The Cosmic Teardrop

Postby zetreque on January 4th, 2017, 4:27 pm 

Image

I'm still at a loss. Now I wonder what negative giga light years are.
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Re: The Cosmic Teardrop

Postby BurtJordaan on January 4th, 2017, 11:50 pm 

Hi Zet, both sides of the origin are positive, so it should be taken as radial distance from the origin. Yea, I know from a pure math point of view, since the referenced diagram is not in circular coordinates (but flat), I should have signed the left-side numbers negative. But that looked 'funny' to me at the time!

Here is an annotated chart showing the symmetrical nature better. Imagine rotating the spacetime parabola horizontally around the origin and you get a 3-D spacetime parabolic cup. Add one more spatial dimension (not possible to visualize) and you have a 4-D parabolic shape.

Cosmic Teardrop.png


Einstein and de Sitter found this simplified solution to Einstein's field equations around 1932. It was for a spatially flat universe with matter and radiation only, fitting Hubble's discovery of the expanding universe and Hubble's law. They set the cosmological constant to zero and this model prevailed until in the late 1990's. As you can see, there is no sign of accelerating expansion in the parabola.
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Re: The Cosmic Teardrop

Postby BurtJordaan on January 5th, 2017, 8:49 am 

The Einstein-de Sitter parabola that I have pictured above may have caused some confusion, because it does not represent our observed accelerating expansion universe correctly. Here is a more representative curve with a cosmological constant causing later accelerated expansion, i.e. the modern Lambda-cold-Dark-Matter (LCDM) model.

Image

The blue teardrop pictures the spacetime paths of light from two galaxies in opposite directions from us, each today observed at a redshift of z=8.6. The purple and the red curves are the spacetime paths of the two galaxies respectively, from the emission up to today (observation). One can just-just see the accelerating expansion as the curves flex to the outside, in contrast to the parabola before, where the flexing was always to the inside.

BTW, at the time when I made this graph (2010), z=8.6 was the highest confirmed galaxy redhsift (translating to a present distance of ~30 Gly). It has lately (2016) been increased by the HST to z=11 (~32 Gly).
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Re: Circular Reasoning

Postby Faradave on January 9th, 2017, 7:04 pm 

Image
BurtJordaan wrote:A more enlightening equivalent can be depicted in circular coordinates, shown below for values compatible with the latest cosmological data.

This is essentially the 'cosmic teardrop' drawn inside the cosmological balloon analogy, connecting the wave front positions at progressive circular slices of constant cosmic time. But note that this does not make the radial direction a time coordinate. It is a fictitious extra spatial coordinate, which is a rather complex function of cosmic time and total energy density. It can also be viewed as representing the cosmic scale factor (a) multiplied by the Hubble constant.


Nonetheless, I consider this depiction a refreshing step in the direction of (3RC)!

The circular coordinates above suggest that a single, convenient spatial dimension is represented by any given circumference. If we let the circles instead represent 3-spheres, the radial coordinate simplifies by replacing the energy density term with a constant, representing the total mass-energy of the universe, a conserved quantity. Either way, I agree that the expansion is not linear (as with an ordinary circumference) but rather "S" shaped, as with this cubic function. Common ground?

P.S. I'm hoping the recent debate over Type 1a supernovae is settled. It makes modeling a tricky business.
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