Cosmological Principle

[Mordred]

In my spare time I will be writing a series of useful articles to help answer common questions. As these are being designed for forum reference I feel strongly on cooperative review. Here is the first. Please look over and feel free to make suggestions. Any solid contributions will be accorded credit at the end of the final product.

 

(Key note all articles MUST comply with textbook descriptives, they are being designed as teaching aids)

 

[latex]\textbf{The Cosmological principle}[/latex]

 

is defined as "at sufficiently large scales, the universe appears as homogeneous and isotropic."

 

[latex]\underline{Homogenous}[/latex]

 

is oft defined as " no preferred location"

 

[latex]\underline{Isotropic}[/latex]

 

is oft defined as "no preferred direction"

 

Obviously at smaller localized scales we can see numerous examples of systems that are inhomogeneous and anisotropic (planets, stars galaxies and large scale structures). However if you increase the radius of measurements sufficient enough those non uniform regions essentially become negligible or more accurately averages out.

A good analogy is look at the surface of a lake. At small scales you can discern waves and ripples.

As you increase in height or distance from the lake those non uniform regions become a uniform appearing surface.

The cosmological principle works the same way. The scale commonly used is 100 Mpc mega parsecs.

 

Speed of light in a vacuum: [latex]c\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}[/latex]

 

The parsec (symbol: pc) is a unit of length used in astronomy, equal to about 30.9 trillion kilometers (19.2 trillion miles). In astronomical terms, it is equal to 3.26 light-years, and in scientific terms it is equal to 3.09×1013 kilometers

 

The cosmological principle has an added reward in that complex systems can be modelled as good approximations with far less complicated mathematics.

 

However it should be noted that if measurements and observations disagree with the cosmological principle those metrics become invalid.

We're lucky though as the body of evidence fully support the cosmological principle. A commonly referred to example being the CMB.

 

Cosmic microwave background. Although the temperature images look chaotic, the difference in temperature of the blue regions and red regions are roughly 1/1000 of a degree.

 

Certainly supports the cosmological principle.

 

The cosmological principle is of importance in telling us that the Universe did not have an origin point nor is the result of an explosion.

 

This is of primary importance in regards to expansion and inflation.

Lets detail this a bit further.

 

Take any number of points, three or more. As the volume of space increases, the same ratio of change will occur between any two points and the angles between those points also do not change.

 

This mathematically is only possible via a uniform change regardless of location.

 

A good analogy is the balloon analogy or the raisin bread analogy.

 

 

http://www.phinds.com/balloonanalogy/: A thorough write up on the balloon analogy used to describe expansion

http://tangentspace.info/docs/horizon.pdf:Inflation and the Cosmological Horizon by Brian Powell

 

The other consequence of the cosmological principle is that the universe cannot have a rotation. All rotating bodies have a center of rotation and rotation imparts a preferred direction.

Edited May 30, 2015 by Mordred

[ajb]

The cosmological principal looks good when you get to the scale of clusters of galaxies; as does the notion of Hubble expansion. This must be your 100 Mpc scale or there abouts.

[Nicholas Kang]

[latex]\underline{Isotropic}[/latex]

 

is oft defined as "no preferred direction"

Compare with

The other consequence of the cosmological principle is that the universe cannot have a rotation. All rotating bodies have a center of rotation and rotation imparts a preferred direction.

 

Hello, Mordred. Can I ask you one question?

 

By comparing the 2 sections in the above quotes, what if the universe is rotating and we haven`t know it yet. Then shouldn`t we (cosmologists) come up with a revised or maybe totally new principle to explain that, instead of keeping on agreeing with the "old" cosmological principle?

Edited May 30, 2015 by Nicholas Kang

[michel123456]

In my spare time I will be writing a series of useful articles to help answer common questions. As these are being designed for forum reference I feel strongly on cooperative review. Here is the first. Please look over and feel free to make suggestions. Any solid contributions will be accorded credit at the end of the final product.

 

(Key note all articles MUST comply with textbook descriptives, they are being designed as teaching aids)

 

[latex]\textbf{The Cosmological principle}[/latex]

 

is defined as "at sufficiently large scales, the universe appears as homogeneous and isotropic."

 

[latex]\underline{Homogenous}[/latex]

 

is oft defined as " no preferred location"

 

[latex]\underline{Isotropic}[/latex]

 

is oft defined as "no preferred direction"

 

Obviously at smaller localized scales we can see numerous examples of systems that are inhomogeneous and anisotropic (planets, stars galaxies and large scale structures). However if you increase the radius of measurements sufficient enough those non uniform regions essentially become negligible or more accurately averages out.

A good analogy is look at the surface of a lake. At small scales you can discern waves and ripples.

As you increase in height or distance from the lake those non uniform regions become a uniform appearing surface.

The cosmological principle works the same way. The scale commonly used is 100 Mpc mega parsecs.

 

Speed of light in a vacuum: [latex]c\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}[/latex]

 

The parsec (symbol: pc) is a unit of length used in astronomy, equal to about 30.9 trillion kilometers (19.2 trillion miles). In astronomical terms, it is equal to 3.26 light-years, and in scientific terms it is equal to 3.09×1013 kilometers

 

The cosmological principle has an added reward in that complex systems can be modelled as good approximations with far less complicated mathematics.

 

However it should be noted that if measurements and observations disagree with the cosmological principle those metrics become invalid.

We're lucky though as the body of evidence fully support the cosmological principle. A commonly referred to example being the CMB.

 

Cosmic microwave background. Although the temperature images look chaotic, the difference in temperature of the blue regions and red regions are roughly 1/1000 of a degree.

 

Certainly supports the cosmological principle.

 

The cosmological principle is of importance in telling us that the Universe did not have an origin point nor is the result of an explosion.

 

This is of primary importance in regards to expansion and inflation.

Lets detail this a bit further.

 

Take any number of points, three or more. As the volume of space increases, the same ratio of change will occur between any two points and the angles between those points also do not change.

 

This mathematically is only possible via a uniform change regardless of location.

 

A good analogy is the balloon analogy or the raisin bread analogy.

 

 

http://www.phinds.com/balloonanalogy/: A thorough write up on the balloon analogy used to describe expansion

http://tangentspace.info/docs/horizon.pdf:Inflation and the Cosmological Horizon by Brian Powell

 

The other consequence of the cosmological principle is that the universe cannot have a rotation. All rotating bodies have a center of rotation and rotation imparts a preferred direction.

I still am worried with the cosmological principle as commonly presented.

Let me try to explain.

Distance is related to time. When you reach cosmological scales, the sphere of 100 Mpc mega parsecs is a sphere that spreads in space AND in time.

That is to say that observers that are physically related inside or on the surface of this sphere are not in the same time frame. But that makes no importance at all for the cosmological principle to work.

IOW the cosmological principal teaches us that the universe should look roughly "the same" independently of position and direction AND time.

 

The "and time" part added by me.

 

And thus, if I understand correctly, the cosmological principle is not in agreement with the BB Theory, because following the BB Theory, the Universe should not look the same through time, but should change (because it expands).

 

Or the other way round should be to say that NO, the Cosmological principle states that the Universe looks roughly "the same" but in a very specific time frame, and then how do you make a "sphere" of 100 Mpc? It is not a sphere any more.

Edited May 30, 2015 by michel123456

[Strange]

The "and time" part added by me.

 

And there is the flaw in your argument.

[Mordred]

Compare with

 

Hello, Mordred. Can I ask you one question?

 

By comparing the 2 sections in the above quotes, what if the universe is rotating and we haven`t know it yet. Then shouldn`t we (cosmologists) come up with a revised or maybe totally new principle to explain that, instead of keeping on agreeing with the "old" cosmological principle?

If measurement and observations are found that disagree with the Cosmological principle then yes it would need to be dropped.

 

Thus far all the best datasets and measurements still find the principle accurate. These datasets are not limitted to WMAP and Planck, those two are merely the more popularly known.

The cosmological principal looks good when you get to the scale of clusters of galaxies; as does the notion of Hubble expansion. This must be your 100 Mpc scale or there abouts.

Correct. This is the value also provided in numerous textbooks as well. Afaik cosmology papers are still currently using this value.

teaches us that the universe should look roughly "the same" independently of position and direction AND time.

 

The "and time" part added by me.

 

 

The metrics of the BB model accounts for the aspects of time vs distance.

There is several different types of time used in cosmology. Cosmic, conformal and proper time. There is also different categories of distance measure. Proper conformal and commoving distance.

Think of it this way.

 

At any specific moment in time. The average universe density and thus rate of expansion is uniform. So at any moment of time. The Cosmological principle applies.

 

The FLRW metrics accounts for this via the scale factor.

 

It is a good point to raise in regards to time.

One of the articles provided by Brian Powell that I included goes into this detail.

Edited May 30, 2015 by Mordred

[michel123456]

If measurement and observations are found that disagree with the Cosmological principle then yes it would need to be dropped.

 

Thus far all the best datasets and measurements still find the principle accurate. These datasets are not limitted to WMAP and Planck, those two are merely the more popularly known.

Correct. This is the value also provided in numerous textbooks as well. Afaik cosmology papers are still currently using this value.

The metrics of the BB model accounts for the aspects of time vs distance.

There is several different types of time used in cosmology. Cosmic, conformal and proper time. There is also different categories of distance measure. Proper conformal and commoving distance.

Think of it this way.

 

At any specific moment in time. The average universe density and thus rate of expansion is uniform. So at any moment of time. The Cosmological principle applies.

 

The FLRW metrics accounts for this via the scale factor.

 

It is a good point to raise in regards to time.

One of the articles provided by Brian Powell that I included goes into this detail.

Oh. In this case, we are not talking about a sphere of 100 Mpc, but a circle of 100Mpc. A circle including all objects that have exactly the same cosmological age with us. Because with a sphere, all objects are cosmologically closer to the BB than we are. Is that correct?

[Mordred]

Not quite, light takes roughly 326 light years to travel 100 Mpc. The changes in that time period is negligible compared to the age of the universe. So one can say roughly the same age. However when you take your measurements. You calculate the proper distance between two or more measurement points. Any time you take any measurements you must account for observer influences.

 

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

 

Here is the 4d Freidmann equation for distance measures. K is the curvature constant. a(t^2) is the scale factor which takes expansion at a point in time into account.

 

http://en.m.wikipedia.org/wiki/Scale_factor_(cosmology)

 

Note the scale factor also accounts for cosmological redshift.

 

Any specific point in time will have the thermodynamic properties. The universe will be roughly the same temperature throughout. (Though the temperature variations isn't significant in 326 years, except in the early universe) You cannot directly see the same point in time throughout the universe. So you must calculate where objects will be at that point in time.

 

I really wish I could post the charts from the lightcone calculator in my signature. However one can use that tool to see the changes in 326 years.

 

For some reason this site doesn't like the latex the calc uses. You can refine the time period being calculated via the S_upper and S_lower values.

Edited June 1, 2015 by Mordred

[imatfaal]

Not quite, light takes roughly 326 light years to travel 100 Mpc. ....

 

Light years as a measure of time!?

 

 

I know it's a typo but - Snigger Snigger

 

And it would take 326 MILLION years to travel that distance - you lost a mega somewhere.

[Mordred]

 

Light years as a measure of time!?

 

 

I know it's a typo but - Snigger Snigger

 

And it would take 326 MILLION years to travel that distance - you lost a mega somewhere.

Lol oops, evidently I hadn't completely woken up lol Edited June 1, 2015 by Mordred

[michel123456]

........[resuming from speechless mode]

........

 

Are 326 million years negligible to the 13 billion years?

it is about one in 39.

[Mordred]

Not really that negligable. Try to understand a key detail.

 

If you look further and further away, your naturally looking further back in time. In this context one can state there is a preferred direction but this isn't entirely accurate. The further you look back in time the denser the universe will appear.

 

This isn't what the Cosmological Principle is really stating.

 

One way to think of it is, " At any specific moment in time, the universe is homogeneous and isotropic." This includes key dynamics such as those involved in thermodynamic processes, or the ideal gas laws. Pressure, temperature, energy density and expansion.

 

The CMB is one such moment. However any point in time will have a uniform density throughout the universe.

 

Today that critical density is roughly 10^29 grams/cubic metre with average blackbody temperature of 2.73 Kelvin.

 

Using the equations of the FLRW metric one can use density as our clock. Fundamentally cosmic time does just that.

Edited June 3, 2015 by Mordred

[michel123456]

Not really that negligable. Try to understand a key detail.

 

If you look further and further away, your naturally looking further back in time. In this context one can state there is a preferred direction but this isn't entirely accurate. The further you look back in time the denser the universe will appear.

 

This isn't what the Cosmological Principle is really stating.

Exactly. To me, for the Cosmoligical principle to apply, the further you look back in time, the Universe should remain roughly the same.

 

One way to think of it is, " At any specific moment in time, the universe is homogeneous and isotropic." This includes key dynamics such as those involved in thermodynamic processes, or the ideal gas laws. Pressure, temperature, energy density and expansion.

 

The CMB is one such moment. However any point in time will have a uniform density throughout the universe.

 

Today that critical density is roughly 10^29 grams/cubic metre with average blackbody temperature of 2.73 Kelvin.

 

Using the equations of the FLRW metric one can use density as our clock. Fundamentally cosmic time does just that.

Does this comes from a measurement? Or is this a supposition? Edited June 3, 2015 by michel123456

[Mordred]

Why do I think your still misunderstanding?

 

The cosmological principle has nothing to do with how the universe evolves over time. It is specifically describing the distribution of thermodynamic processes and distribution of matter at specific moments in time.

 

We can directly measure today's temperature. It's the temperature of our local group. We can also confirm the universe expands by taking measurements further and further back in time. Just as we can measure the change in distance measurements and redshift.

[michel123456]

Why do I think your still misunderstanding?

 

 

 

Because you still are talking about a sphere of 100Mpc, and if I understand correctly, this sphere is not observable. It is the collection of objects that are 50Mpc around us (for example, since we are the observers) and that have the same cosmological age with us.

However all observable objects that we are able to measure have a different age with us. The further they are, the youngest. Since everything that we observe lie into the past, we are the oldest observable objects in the Universe.

Edited June 3, 2015 by michel123456

[Mordred]

Didn't I also state you account for age? I did mention the scale factor and post the distance formula in 4d.

 

All datasets must account for observer influences. This includes time

"You calculate the proper distance between two or more measurement points. Any time you take any measurements you must account for observer influences.

 

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

This was posted in a previous post this thread.

[Mordred]

Here is something to consider.

 

Take a telescope pick a star, now you have the problem. How far away is that star. So to determine that one technique is redshift. However for redshift you need an original frequency. So you look at known elements. Hydrogen spectral lines are handy. Already you have to use calculated values. The cosmological redshift formula. One cannot use visual data to measure the universe directly. You always have to find ways to calculate and determine distance.

[Sorcerer]

The cosmological principle is of importance in telling us that the Universe did not have an origin point nor is the result of an explosion.

 

This is of primary importance in regards to expansion and inflation.

Lets detail this a bit further.

 

Take any number of points, three or more. As the volume of space increases, the same ratio of change will occur between any two points and the angles between those points also do not change.

 

This mathematically is only possible via a uniform change regardless of location.

 

A good analogy is the balloon analogy or the raisin bread analogy.

 

I'm sorry, I don't how the cosmological principle prevents an origin point. An origin point is perfectly isotropic and homogenous.

 

If you take the balloon analogy and interpolate, you end up with all 3 points converging, the same ratio of change still occurs between points at all times even at the first (or last) instant, when it is undefined for all of them, only the angles drop to 0 at 0 distance between points. And I don't see how this violates the principle.

 

Am I missing something?

 

By origin point, I assume you mean as in a mathematical point, one of 0 size, what about an origin point of non zero size?

Edited June 8, 2015 by Sorcerer

[Mordred]

By origin point, I assume you mean as in a mathematical point, one of 0 size, what about an origin point of non zero size?

Excellent question. To answer this detail we need to clarify one key aspect.

 

"What is the size of the entire universe"

 

Well pop media shows and programs will almost always show this explosion like origin, starting from a volume less than an atom in size. This is entirely wrong.

 

In point of detail we do not know the size of the entire universe....

We only know the size of our observable universe.

 

http://en.m.wikipedia.org/wiki/Observable_universe

 

The big bang model starts at [latex]10^{-49}[/latex] forward in time. Prior to that we cannot accurately describe due to the conditions prior to that point. We call this a singularity conditions, but not a point like singularity such as a BH.

 

Our entire universe could be either finite or infinite in size. The big bang model only describes how our Observable universe evolves in volume since [latex]10^{-49}[/latex] forward. This is the detail missed in pop media literature.

 

The other detail is that at all times the universe surrounds us. No matter what direction you look in, the further you look, the further back in time you look. However you will see not see a particular direction look any different from any other direction. So assuming you could directly see 10^-49 seconds forward you will still see the observable universe in every direction and the same thermodynamic conditions at a given distance in every observation direction.

 

However you will never be able to see the entire universe. You will only ever be able to measure how our Observable portion evolves.

Edited June 8, 2015 by Mordred

[Sorcerer]

I understand this. But I was just wondering what part of the cosmological principle tells us "that the Universe did not have an origin point nor is the result of an explosion."

The Big Bang model, if it only describes time from 10^-49 seconds onwards doesn't tell us that the universe did not have an orgin point. It doesn't tell us anything about time before then. The logical reasoning which led to the big bang theory, can still be taken to its conclusion and lead to a singularity. Just because current science, (or any science ever), can't explain it doesn't invalidate extrapolation of the expansion of the universe backwards in time using general relativity to yield an infinite density and temperature at a finite time in the past.

[Mordred]

An explosion regardless of when you measure it has a point of origin. As well as a preferred direction. It is inhomogeneos and isotropic. So when you measure how an explosion expands. It will be nothing like the balloon analogy. You will measure an increase in volume change with a preferred direction and location. All objects are moving from the center outward.

 

However if you measure expansion you will not measure a preferred direction and location. That is precisely the point of the cosmological principle. The distance between all objects are increasing in all directions not an outward direction.

 

Here Ned Wright's tutorial has some good visuals.

 

http://www.astro.ucla.edu/~wright/balloon0.html

 

Beaz has a decent write-up.

 

http://math.ucr.edu/home/baez/physics/Relativity/GR/centre.html

 

Metrics expansion of space shows one decent image.

 

http://en.m.wikipedia.org/wiki/Metric_expansion_of_space

 

http://oneminuteastronomer.com/6949/where-is-the-center-of-the-universe/

 

http://csep10.phys.utk.edu/ojta/c2c/largescale/cosmology/geometry_tl.html

 

 

Granted the balloon analogy is 2d, the raisin bread is 3d.

 

http://fundamentalweirdness.blogspot.ca/2010/03/raisin-bread-universe.html?m=1

 

Here is a decent YouTube video 11 minutes long.

 

https://www.khanacademy.org/science/cosmology-and-astronomy/universe-scale-topic/big-bang-expansion-topic/v/hubble-s-law

[Sorcerer]

Yes, yes, but I had objection to there not being an explosion, it is not necessary for a point or singularity to be viewed as an explosion, when it moves from 0 spatial dimension to 3.

Why would you consider 0 dimension changing to 3 dimensions an explosion? A point in the middle of a sphere is equidistant from the edge and thus the dimensional change is isotropic and homogenous, but there is no increase in volume, just the dimensional shift to it, its illogical to consider no volume becoming volume an increase. Just as it would be illogical to consider a point a shorter line, than a line segment, because it isn't a line. It is also illogical to consider there being a change in direction, a point has 2 cooridinates but this one has no reference so they are meaningless and there is no time, there is no direction and there can't be change when time begins after the point.

Maybe if you have to use calculus, you will find that your argument holds, but calculus is only an approximation. It can only describe limits as they approach 0, never actually 0 itself.

Edited June 8, 2015 by Sorcerer

[Strange]

Note that the idea of a singularity of zero volume is not part of the mainstream big bang theory; this just describes the evolution of the universe from an early hot dense state.

[Sorcerer]

Note that the idea of a singularity of zero volume is not part of the mainstream big bang theory; this just describes the evolution of the universe from an early hot dense state.

That was pointed out the theory only describes the universe from a time of 10^-49 seconds onwards (according to what Mordred posted before). But the reasoning that lead to the creation of the idea, inevitably leads to the crossing of spacetime at the origin, where it has no volume and is infinitely dense.

 

This logical consequence was abandoned because we lack the tools to describe it with our current knowledge. But that doesn't necessarily mean it isn't what happened before that. It doen't necessarily mean it is either, however that would be the conclusion from the reasoning which first created the theory.

 

The fact that you need to give a time of 10^-49 seconds, implies that time before it.

Edited June 8, 2015 by Sorcerer

[ajb]

But the reasoning that lead to the creation of the idea, inevitably leads to the crossing of spacetime at the origin, where it has no volume and is infinitely dense.

Right, but this is according to the classical theory of gravity. The presence of such a singularity signals that the classical theory breaks down as we approach this singularity.