Distance measures (cosmology)

In a universe whose global development is described by the Friedmann equations, there is no unique measure of distance more. This contradicts the everyday human experience in the static Euclidean space, is dynamic and curved space-times as the universe but inevitable. Here, the propagation of light is significantly influenced by the space-time geometry and dynamics underlying.

Distance measurements

In shallow and static space-times, there are various methods of distance measurement, which all lead to exactly the same result, although the underlying measurement methods are very different. For example, can be determined from the transit time of a reflected signal, the distance of the targeted object at a known speed signal. This principle is used in radar measurements or the so-called " laser ranging". Other opportunities exist to derive from the apparent angular size or apparent brightness of an object whose distance. This purpose must be known the true size or the true brightness.

These three principles are also found in astrophysics, mostly, however, in a different context. They are used to determine actual luminosities or sizes of astronomical objects, or the time at which the observed object has emitted the light. For this purpose, use is made of astrophysics brightness distance, the angular diameter distance and the duration of removal. Further, there is also moved along the distance. As a common denominator acts the cosmological redshift, which allows the calculation of these distances as follows.

Duration Distance

The definition of the term distance (English: light travel time distance ) is based on the light propagation time between two events with the redshifts given by

Substituting the cosmological time as an integration variable with the observable redshift, we obtain

Here is the cosmological expansion factor, normalized to the value 1 to the present time. It applies (see the relativistic derivation of the cosmological redshift )

One then writes the Hubble function explicitly, one obtains the familiar expression for the term distance

A flat universe (), this integral can be solved analytically:

And this put the matter density and the vacuum energy density parameter ( cosmological constant ) dar. According to measurements with WMAP and this amount. The Hubble constant is km s - 1Mpc -1.

Comoving distance

In analogy to the term of distance gives the comoving distance (English: comoving distance). This is the distance between the source and the observer on a spacelike hypersurface, defined by events with a constant cosmological time ( today). Starting from the line element (see also Friedmann equations) arises

From which one derives

The big difference between running time and distance comoving distance is that the former is a distance over space and time. Term distance is the distance to the object as it appears to the observer, and this holds in a state of the past. Moved along that distance, however, is the distance, having the observer and the object at the same time to each other, i.e., a distance on a space-like hypersurface. In this state, the observer can not see the object, however, because the light has just been sent out from the object to him.

Angular diameter distance

The angular diameter distance (English: angular diameter distance) is defined in analogy to the Euclidean space-time, as the ratio between the source area and the solid angle under which the object appears to the observer:

Using the comoving distance, this results in

With

The function distinguishes between three-dimensional spacelike hypersurfaces of constant time with positive, vanishing or negative curvature.

Luminosity distance

Similarly, the luminosity distance ( engl: luminosity distance) results from the analogy to Euclidean geometry. Given the late arrival of photons at the observer by the intervening expansion of the universe, its redshift and the photon number conservation, we obtain

General characteristics of the different distance definitions

Through the pre-factors of and the non-linearity of, have neither the angular diameter distance nor the luminosity distance an additive property. Considering two objects 1 and 3, with an intervening object 2, the distance between 1 and 3 does not equal the sum of the distances between the object 1 and 2, and the object 2 and 3:

The term of distance and the comoving distance, however, are additive.

Numerical examples

For the following redshifts, the various distances (in billions of light-years ) to the observer ( ) result:

Here it is noticeable that the angular diameter distance is not a monotonic function of redshift, but has a maximum, to then again be smaller. This means that the same object for increasing redshifts appear ever smaller, reached at a minimum, and the observer appears again larger for larger distances.

The term distance is aiming for a constant value ( the value of the age of the universe in light years ) for infinitely large redshifts. The luminosity distance, however, strives to infinity, that is, the apparent brightness of an object decreases with increasing redshift from very strong. In fact, the surface brightness decreases.

Application Examples

A galaxy have the redshift 0.5. Thus, it appears that the light from it was five billion years traveling to the observer, and thus its running time distance to 5.0 billion light years. If you want to from the apparent magnitude of the galaxy close (eg Magnitude = 22) on their actual brightness, we may not use the term distance, but you have to use the luminosity distance. This is 9.1 billion light years. Analogous to this is the size determination: If the galaxy to the observer at an angle of 5 arc seconds, so you have to use the angular diameter distance of 4.1 billion light years to their actual size to be able to determine ( 99600 light years) on the tangent function.