### 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)`

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| |

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(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