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Redshifts and Distance in an Expanding Universe

Posted: 26.06.2006, 18:50
Redshifts and Distance in an Expanding Universe

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In P&A, someone asked (correctly so), how we can interpret the observed red-shifts of spectral lines in galaxies in terms of an expanding Universe.
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Since I am lurking here in Purgatory, far away from the "seriousness" of the P&A board, ... on the "party mile" so to speak , let me summarize some respective thoughts...

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First thing to remember: measuring distance in an expanding Universe is a tricky business!
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To see this clearly, let's best return to the good old 2d balloon model of the Universe: in this intuitive 2d illustration, the Universe's expansion corresponds to an increasing balloon size with time. The galaxies and other objects are distributed over the balloon surface with their physical mutual distances increasing in ALL directions with increasing time/balloon size!

We can immediately contemplate two basically different definitions of distance.

1) Let us inscribe on the balloon surface a square with coordinates

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`(0,0)---------(1,0)   |            |   |            |   |            |   |            |(0,1)---------(1,1)`

You may e.g. imagine some galaxies sitting in the 4 corners. This frame of reference is called a "comoving frame" in cosmology: since despite the expansion, in this frame the distances between the neighboring galaxies will always remain to be 1 (in some suitable units!).

2) The physical distance however, increases with time, since it's proportional to the comoving distance times the cosmic scale factor a(t). The scale factor at our present time is conveniently set to one, implying that a(t) < 1 at earlier times. In our simple balloon picture, the scale factor accounts for the increasing balloon size!

Besides the scale factor and its evolution with time, our Universe is characterized by another crucial piece of input: it's geometry! As you meanwhile know, there are 3 basic possibilities: flat, open and closed Universes! If the Universe is not flat we must distinguish positive (spherical!) and negative curvature (saddle shaped, doghnuts,...) of the geometry.

The geometry of the Universe also affects the definition of distance, of course.

In a flat expanding Universe, Cosmologists use 3 popular measures of distance:

a) the comoving distance
b) the luminosity distance
c) the angular diameter distance

The comoving distance I have intuitively explained already.

Conventionally, the luminosity distance (b) )is defined as follows:

We measure the flux F from an object of known luminosity L. It depends on the "luminosity" distance d_L as

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`F = L/(4 Pi* d_L^2),`

as most of you know. Let me just tell you here (without derivation) that this definition may be precisely generalized to the case of an expanding Universe! The modification entering here involves one additional, observed quantity, the redshift z of the object under consideration.

Finally a word about the angular diameter distance c) :

It goes back to the classic way to determine distances in astronomy by measuring the angle ?? subtended by an object of known physical size r. The distance of that object is then (conventionally):

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`d_A = r/??   `

Again this distance definition can easily be extended
to an expanding Universe, using the comoving distance as a helping concept. Once more the only new experimental quantity entering is the redshift z
of the object.

In summary:
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These three distance definitions are different in an expanding Universe, but by using them carefully along with the observed redshift z, there is good experimental evidence that indeed the Universe is expanding.

Bye Fridger

Posted: 28.06.2006, 11:47
Hello Fridger
Have you ever thought about the pubblication of these intersting articles as Wikimedia/Wikipedia/Wikibook contents?

Perhaps it would help in reducing "noise", it is possible to discuss about the articles, and would have a better visibility than purgatory. Perhaps a wiki hosted in shatters.net or in celestial matters instead that in wikibooks should be an effective soution.

Kind regards

Posted: 28.06.2006, 12:26
Paolo wrote:Hello Fridger
Have you ever thought about the pubblication of these intersting articles as Wikimedia/Wikipedia/Wikibook contents?

Perhaps it would help in reducing "noise", it is possible to discuss about the articles, and would have a better visibility than purgatory. Perhaps a wiki hosted in shatters.net or in celestial matters instead that in wikibooks should be an effective soution.

Kind regards

Paulo,

thanks for the proposal. Indeed, after opening of CelestialMatters, there will be a much more adequate environment for such things that what is available to me here in Purgatory . In the P&A board of shatters.net, there was obviously (and unfortunately) more interest for "lighter diet" ...

Cheers,
Fridger

Um...Ok

Posted: 28.06.2006, 13:37
Ok... so, what if what we're observing is really a change in the shape of the Local group or a gravitational bubble of sorts (possibly the expansion or c ontraction of the Solar Wind.. like a lens)? How do we KNOW that we're seeing the motion of these galaxies if the light is "billions of years" old? How do we know it hasn't already begun to contract 2.5 billions of years ago? How do we know that the Milky way didn't shink and cause a wierd lighting effect? I'm not saying we shouldn't measure.. just wondering if crazy things like this aren't being taken into account. If there's a supermassive black hole at the center of our galaxy... how does that affect our visual perspective? I know, i'm crazy.. but, when i read what scientists have to say, i usually, think they consider what the others say... Or what has been observered or what is in current theory. I like the explanation but, is it really like a balloon? Is this what we've observed? Or is our observation tainted by the methodology? One observataion point per study? (I don't count the Earth studies, so much as, well, the earth is in the way.) Am i a whacko? are these questions dismissable?

Re: Um...Ok

Posted: 28.06.2006, 18:41
Startyger wrote:Ok... so, what if what we're observing is really a change in the shape of the Local group or a gravitational bubble of sorts (possibly the expansion or c ontraction of the Solar Wind.. like a lens)? How do we KNOW that we're seeing the motion of these galaxies if the light is "billions of years" old? How do we know it hasn't already begun to contract 2.5 billions of years ago? How do we know that the Milky way didn't shink and cause a wierd lighting effect? I'm not saying we shouldn't measure.. just wondering if crazy things like this aren't being taken into account. If there's a supermassive black hole at the center of our galaxy... how does that affect our visual perspective? I know, i'm crazy.. but, when i read what scientists have to say, i usually, think they consider what the others say... Or what has been observered or what is in current theory. I like the explanation but, is it really like a balloon? Is this what we've observed? Or is our observation tainted by the methodology? One observataion point per study? (I don't count the Earth studies, so much as, well, the earth is in the way.) Am i a whacko? are these questions dismissable?

You are asking far too many questions at once (13! ), which makes me doubt whether you are really interested in their answer...

The whole art in science is to reorder and simplify the large amount of questions we ALL have to Nature into a systematic tree such as to ask the less important ones at a later stage of understanding.

Of course the universe is not really like a balloon, yet the simple balloon model illustrates what expansion is all about: it's NOT that individual galaxies a flying apart from each other in space, but rather it's the expanding geometry of space (like in an expanding balloon with galaxies distributed over its surface) that effectively leads to an increased physical separation with time. As I pointed out in other (explanatory) posts, the universe may well have a much more complex topology than that of an expanding sphere.

Earlier you were asking about determination of the velocities of galaxies. This is a sophisticated industry, since quite a while.

It is important to note that the total velocity of a galaxy
may be separated into the so-called Hubble velocity v_H and a direction-dependent perturbation, the so-called peculiar velocity describing its movement relative to the co-moving frame.

The Hubble velocity of a galaxy,

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`v_H = d_cm * da(t)/dt`

is of course the component we are really interested in!

d_cm is the co-moving distance from us, which I explained in my post about measuring cosmic distances. The galaxy's physical distance from us therefore would be a(t) *d_cm, where a(t) is the cosmic scale factor (<=> the increasing balloon size!). da(t)/dt denotes the change of the scale factor with time (i.e. it's gradient).

Altogether, the galaxy's Hubble velocity v_H only depends in a known manner on the observed redshift z for that galaxy! The redshift is just measured by spectral analysis of the emitted light.

I don't know whether you have ever seen the most impressive universal plot with 10000nds of distant galaxies fitting beautifully the universal Hubble
law, expressing the expansion of the Universe...
(after the peculiar velocities have been separated off!)

It's usually the peculiar velocity correction that is accounted for by a tedious process of applying corrections. But that's just known technology....

Bye Fridger

Posted: 29.06.2006, 13:06
Since all galaxies are measured to be moving away from us due to the Big Bang, is it possible that the 'universe' is like a globular cluster? I wonder if stars there appear to move away from each other in the same respect due to its original 'Big Bang' effect... Im not sure on those objects weither or not they have a starting point similar to this...

With that said it might also be safe to say the Big Bang is in essence similar to a supernova... An occurance that repeats itself over and over to recreate new matter from old matter... Einstien did say everything is somewhat relative...

But from what you state and have stated before Fridger... You believe the universe is sort of shaped by time and not dimention? Am I following you here?

Posted: 29.06.2006, 18:15
No, i wasn't asking the questions to evade an answer. Living on Earth, there are many variables that affect our existence. One or two might be singled out.. but, if all of them factor in our existence, is it logical to leave the other things out?

My question is when we determine the motion of all things, do we consider the motion of all things? If i say, "this rabbits velocity can be easily determined", how do we determine the velocity of ALL rabbits? Don't we need to set parameters and consider at least the variables of all the rabbits (observed/observable) speeds and directions to be able to determine their velocities?)

retrograde motion is the observed aparrent motion of mars(or other planet), not the action of its true orbit. I'm not really looking for answers to every question. But, my question is how do we determine the actual motion without getting confused by the rest of the motion of the things for which we are trying to determine a greater motion?

yeesh.. did that simplify my question?

Posted: 29.06.2006, 20:08
Rassilon wrote:Since all galaxies are measured to be moving away from us due to the Big Bang, is it possible that the 'universe' is like a globular cluster? I wonder if stars there appear to move away from each other in the same respect due to its original 'Big Bang' effect... Im not sure on those objects weither or not they have a starting point similar to this...

Ras,

certainly a reasonable question to ask: if the increasing distance of celestial objects is due to an increasing cosmic scale factor fired by the BigBang, then also stars in clusters should be subject to this.

However, in case of clusters, it appears much harder to separate off "peculiar velocity" components that are very typical for objects of our local environment.

But from what you state and have stated before Fridger... You believe the universe is sort of shaped by time and not dimention? Am I following you here?

Well it's more complex: Let's assume Einstein's General Relativity + Inflation as the "general framework" for a homogeneous and isotropic BigBang Universe. Then it's evolution and the feedback of mass on geometry as function of time is entirely predictable! So "time" effects are under control.

What is a totally different issue, is the Topology of the Universe including questions of multiple connectedness etc. (e.g. doghnut shaped vs. spherical...) This cannot be predicted from within GR! We need a stronger framework here...

Bye Fridger