# Universe geometry article

1. May 30, 2013

### Mordred

Universe geometry article simpify?

article development for the Forum on geometry suggestions, as well as any errors etc are welcome
particularly on how to keep the FLRW metrics but simplify the explanation....

Universe geometry

The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a blackhole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.

The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.

In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 1800. A positive curvature will add up to greater than 1800, a negative curvature will add up to less than 1800

The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula

$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$

P=pressure
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega $\Omega$ so critical density is $\Omega crit$
total density is

$\Omega$total=$\Omega$dark matter+$\Omega$baryonic+$\Omega$radiation+$\Omega$relativistic radiation+${\Omega_ \Lambda}$

$\Lambda$ or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "0"for $\Omega$ shown in the image above denotes time in the present.

Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, and in many circumstances is an exact synonym.

$\Omega=\frac{P_{total}}{P_{crit}}$
or alternately
$\Omega=\frac{\Omega_{total}}{\Omega_{crit}}$

Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c2 = a2 + b2
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c2= a2 + b2 + c2
see image 2.0 below

On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
$\alpha$,$\beta$,$\gamma$ for a flat geometry this follows the relation

$\alpha$+$\beta$+$\gamma$=$\pi$.

image 1.0

image 2.0 reference (3)

On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
$d{s^2}=d{x^2}+d{y^2}$

If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation

$\alpha$+$\beta$+$\gamma$=$\pi+{AR^2}$

image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
$\alpha$+$\beta$+$\gamma$>$\pi$ are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and $\theta$ is the azimuthal. angle measured relative to the prime meridian,"(1) then the distance ds between a point (r; $\theta$) and another nearby point (r+dr+$\theta$+d$\theta$) is given by the relation

${ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2$

"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."(1) Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices $\alpha$
$\beta$,$\gamma$ obey the relation $\alpha$+$\beta$+$\gamma$=$\pi-{AR^2}$.

${ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2$

image 1.2

A negative curvature is an open topography

If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius

k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space

Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have
uniform negative curvature. If a three-dimensional space is flat (k = 0), it
has the metric

ds2 = dx2 + dy2 + dz2 ;

expressed in cartesian coordinates or

${ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2]$

If a three-dimensional space has uniform positive curvature (k = +1), its
metric is

${ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

A negative curvature in the uniform portion has the metric (k=-1)

${ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

Geometry in 4D

Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc link on the main page. Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form

$d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]$

references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu

Last edited: May 30, 2013
2. Jun 1, 2013

### Mordred

I could really use some advise on how to simplify or better explain the 3d and 4d sections of this article. The two D section is simple spacial geometry the images are probably enough (hopefully).
Any suggestions??

3. Jun 1, 2013

### yenchin

Note that the saddle surface as illustrated is not of constant curvature (thus a bad analogy). I know everyone does it, but perhaps this should be pointed out; I have seen people get confused by this in a cosmology course.

The fact that a complete two-dimensional surface with constant negative curvature cannot be isometrically immersed in 3-dimensional Euclidean space is proved by David Hilbert in 1901; so for example one could use a pseudosphere to illustrate a constant negative curvature space instead, but it is not complete. In Susskind's cosmology lecture, I remember he simply uses the hyperbolic disk, which evades all these problems, but that may be too technical for beginners. So perhaps a fine print pointing out the imperfect analogy would be enough.

4. Jun 1, 2013

### Mordred

Thats a good point I should include. Thanks for reminding me. I'll look at at Susskind's cosmology lecture if you have a link great if not I'll google it.
Your correct in needing to show Hilberts proof in the article.
A reference link might work best for beginners. I'll have to think about that aspect.

5. Jun 1, 2013

### yenchin

The lecture is on YouTube, unfortunately there are 10+ of those lectures [it's a lecture series] and I don't remember which one it is...

For a proof, see for example, p.203 of John McCleary’s "Geometry from a Differentiable Viewpoint".

6. Jun 1, 2013

### Mordred

No problem I'm sure I can find supportive material. Barbera Reiden briefly mentions Hibert in her "Introductory to Cosmology" referenced in the article. I may include that reference then go into further details. With one or two other supportive links. In order to keep the size down.

7. Jun 1, 2013

### Ibix

Your 3D Pythagoras image (2.0) has two x axes and a y axis, and the dz is parallel to the y while the dx and dy lie in the x-x plane.

I can have a crack at original images for this, if you'd like? Might take me a few days, though.

8. Jun 1, 2013

### Mordred

Absolutely and thanks for the help that portion of the image is a bit confusing lol so if you are willing to help find better images I would definetely appreciate it.

9. Jun 4, 2013

### Ibix

OK - here are my versions of the two images. Comments welcome.

I can have a go at the rest if you want home-grown non-jpeggy versions. I assume that $z=x^2-y^2$ would be an acceptable negative curvature surface? Someone would have to help me with equations for the geodesics in that surface - my calculus of variations is too long in the past.

10. Jun 5, 2013

### Mordred

Looks good, I'll be doing some revisions to cover Hilberts findings. The pics also looks good. I'll incorperate both suggestions into the article probably be a few days though busy work week.
The simple relation for negative curvature you suggested will probably suffice.
We can see how it looks and adjust if need be. I'll probably add a few more geometry work ups to help correlate the affect on lightpaths.

11. Jun 5, 2013

### Mordred

I'm also thinking of covering the following k relations. (hopefully I can figure out the latex so bear with me lol)

$$S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}$$

whew looks like I got it lol. Do you think adding this is useful, or adds confusion?

12. Jun 5, 2013

### Mordred

Universe geometry

The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a blackhole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.

The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.

In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 1800. A positive curvature will add up to greater than 1800, a negative curvature will add up to less than 1800

The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula

$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$

P=pressure
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega $\Omega$ so critical density is $\Omega crit$
total density is

$\Omega$total=$\Omega$dark matter+$\Omega$baryonic+$\Omega$radiation+$\Omega$relativistic radiation+${\Omega_ \Lambda}$

$\Lambda$ or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "0"for $\Omega$ shown in the image above denotes time in the present.

Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, and in many circumstances is an exact synonym.

$\Omega=\frac{P_{total}}{P_{crit}}$
or alternately
$\Omega=\frac{\Omega_{total}}{\Omega_{crit}}$

Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c2 = a2 + b2
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c2= a2 + b2 + c2
see image 2.0 below

On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
$\alpha$,$\beta$,$\gamma$ for a flat geometry this follows the relation

$\alpha$+$\beta$+$\gamma$=$\pi$.

image 1.0

On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
$d{s^2}=d{x^2}+d{y^2}$

If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation

$\alpha$+$\beta$+$\gamma$=$\pi+{AR^2}$

image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
$\alpha$+$\beta$+$\gamma$>$\pi$ are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and $\theta$ is the azimuthal. angle measured relative to the prime meridian,"(1) then the distance ds between a point (r; $\theta$) and another nearby point (r+dr+$\theta$+d$\theta$) is given by the relation

${ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2$

"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."(1). David Hilbert proved that a constant negative curvature cannot be constructed in a Euclidean 3D space. Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices $\alpha$
$\beta$,$\gamma$ obey the relation $\alpha$+$\beta$+$\gamma$=$\pi-{AR^2}$.

${ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2$

image 1.2

A negative curvature is an open topography

If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius

k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space

Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have
uniform negative curvature.

The 3 possible metrics for homogeneous and isotropic 3D geometries can be represented in the form ds2=dr2=Sk(r)22

where
2=dθ2=sin2d$\phi$2

$$S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}$$

If a three-dimensional space is flat (k = 0), it
has the metric

ds2 = dx2 + dy2 + dz2 ;

expressed in cartesian coordinates or

${ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2]$

If a three-dimensional space has uniform positive curvature (k = +1), its
metric is

${ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

A negative curvature in the uniform portion has the metric (k=-1)

${ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

Geometry in 4D

Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc link on the main page. Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form

$d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]$

references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu

Last edited: Jun 5, 2013
13. Jun 7, 2013

### Ibix

Can I suggest a few more section headings and what I think should go under each? It's only the bit up to Geometry in 2D that I'm suggesting changing.

The visible universe - spherical volume of diameter 46Gly. Not (necessarily) the whole universe, which might be much bigger

The whole universe - could be finite or infinite, and flat or curved. Suggest the diagram of the three surfaces, but with a small circular patch on each - the visible universe (I can draw this if you like). Can differentiate the cases by studying light paths over long distances (parallel, converge, diverge). Has implications for the ultimate fate of the universe.

Critical Density, and why it's critical - $\Omega_\mathrm{crit}$, $\Lambda$ etc.

Geometry in 2D - describing curvature - plane (triangle angles, 2d-Pythagoras, open topography) - sphere (generalisation of triangle and its angles, geodesic, differential distance formula, closed topography) - hyperboloid (triangle angles, differential distance formula, open topography) - k and R

Geometry in 3D - general metric - metric form in k=0, k=+1 and k=-1 space

Geometry in 4D - FLRW metric

References

14. Jun 7, 2013

### Mordred

You've got some good suggestions above, I'll work on adding more differential geometry workup.
Also adding the observable definment, as well as fate of the universe in each k. The other suggestions are also good.
May take a bit as size is also an issue lol doesn't take long to grow.
if you have a visual in mind feel free to post.

15. Jun 21, 2013

### Mordred

Last edited: Jun 21, 2013
16. Jul 3, 2013

### Naty1

Hi Mordred:

The 3D geometry section is a topic that seemed a bit obscure to me for quite a while while trying to learn this stuff on my own.

Consider this as a possible simple introduction if accompanied by a brief explanation:

Chronos posted this simple one which could easily be used to explain how the scale factor relates to the distance [metric] in cosmology.....It took me quite a while to figure that out, but I had no text source which probably would have helped...

If you still care enough after all your work, send me a PM with your phone number [if you are i the US] or I can send you mine and I'd be happy to set up a convenient time to offer verbal comments and ask some questions about both your documents. It would be fun to discuss such topics since my wife, daughter and Yorkshire terriers are not all that interested!!!!

Closely related to the metric Chronos posted is the Leonard Susskind introduction video which I linked to in the above thread. That could be useful link for your article if you like it as I did.

17. Jul 3, 2013

### Mordred

Yeah Chalnoth posted a good single dimension example on that thread, by removing spatial coordinates Dy and Dz. He showed a clear example of the a(t) relation with the Ds coordinate. The section that this would work well in is in the 4d section.

I live in Canada so phone calls could be expensive, I can set up a skype account with a headset conference style on my new comp. Give me a few days to set that up for my new comp and I will email you the skype info.

18. Aug 5, 2015

### Buzz Bloom

Hi Mordred:

You seemed to have left out Ωk. If the universe is not flat, isn't this term needed?

BTW, in The Astrophysical Journal, February 10, 2012, (see http://iopscience.iop.org/0004-637X/746/1/85/pdf/apj_746_1_85.pdf ) on pg 17, Table 7, in the bottom line under the sub-table headed "oΛCDM", there is a calculation for the value of Ωk = 0.002 +/- 0.005. I calculate that the 0.002 value corresponds to a radius of curvature for the universe of (1/ sqrt(0.002) × c / H0 = 308 Gly. This implies a half circumference of 967 Gly compared with a observable universe radius of 46.5 Gly (see https://en.wikipedia.org/wiki/Observable_universe ). The +/- 0.005 is specified as 95%, that is 2 standard deviations. Assuming a Gaussian distributinoferrors, this would also mean that the probabilitiy that the universe is finite is approx 0.79.

I just located another more recent article:
Astronomy & Astrophysics, manuascript: Planck 2015 results. XIII. Cosmological parameters, February 9, 2015 (http://xxx.lanl.gov/pdf/1502.01589v2.pdf ). On pg 31, table 5 gives Ωk = 0.0008 +0.0040 / -0.0039. The 0.0008 corresponds to a curvature radius of 487 Gly. The +0.0040 / -0.0039. is again specified as 95%, and if Gausian errors, then the probabilitiy that the universe is finite is as of this paper is approx. 0.66.

Regards,
Buzz

Last edited: Aug 5, 2015
19. Aug 5, 2015

### Mordred

The equation your questioning is meant to show the involvement of the different particle species. Without going into to great of detail on their equations of state relations.

Yes $$\Omega_k$$ is the spatial curvature constant. However most textbooks seldom derive the specific value. In those textbooks its usually just represented as the curvature constant via the 4d FLRW equation in that article. ( last equation) the value at the introduction level is set at 1,0,-1. The value you posted is a more exact figure. However going into how that figure was beyond the introduction level of the article. I kept the article as close to the metrics included in Barbers Rydens Introductory to Cosmology as at that time she had one of the easiest methodology to introduce the FLRW metric.

It would essentially work out from
$$\rho_k=\frac{3kc^2}{3\pi G a^2}$$. The use of $$\Omega_k$$ is a derived value (not an actual density, though it is convenient to treat it as such) with which one can derive specific influences from that calculated value. However I hadn't covered the scale factor in the article.
In simpler form.
$$\Omega_k=\frac{\rho_k}{\rho_{crit}}$$

Last edited: Aug 5, 2015
20. Aug 5, 2015

### Buzz Bloom

Hi Mordred:

Thanks for explaining the absence of Ωk.

I wasn't suggesting that you add material about the values of Ωk and implications, I just thought you might find this information of interest.

Regards,
Buzz