Distance measure

In a universe whose global development is described by the Friedmann equations , there is no longer any clear measure of distance . This contradicts the everyday human experience in static Euclidean space , but is unavoidable in dynamic and curved spacetime like the universe. There the light propagation is significantly influenced by the underlying spatiotemporal geometry and dynamics.

Distance measures

Different methods of distance measurement exist in flat and static spacetime, all of which lead to exactly the same result, although the underlying measurement methods are very different. For example, if the signal speed is known, the distance to the targeted object can be determined from the transit time of a reflected signal. This principle is used for radar measurements or so-called "laser ranging". Other possibilities are to derive its distance from the apparent angular size or the apparent brightness of an object. For this, the true size or the true brightness must be known.

These three principles can also be found in astrophysics, but mostly in a different context. They are used to determine the actual brightness or sizes of astronomical objects, or the time at which the observed object emitted the light. For this purpose, astrophysics uses the brightness distance , the angular diameter distance and the travel time distance . There is also the moving distance . The cosmological redshift acts as a common denominator, which allows these distances to be calculated as follows.

Run-time distance

The definition of the term distance (ger .: light travel time distance ) based on the light propagation time between two events with the redshifts given by ${\displaystyle z_{2}>z_{1}}$

${\displaystyle {\mathrm {d} }D_{\mathrm {prop} }(z_{1},z_{2})=-c\,{\mathrm {d} }t}$

If one substitutes the cosmological time as an integration variable by the observable redshift, the result is

${\displaystyle {\mathrm {d} }D_{\mathrm {prop} }(z_{1},z_{2})=-c\,a/(a\,{\dot {a}})\,{\mathrm {d} }a=-c/(aH)\,{\mathrm {d} }a\,.}$

Here is the cosmological expansion factor, normalized to the value 1 at the present time. The following applies (see the relativistic derivation of the cosmological redshift ) ${\displaystyle a(t)}$

${\displaystyle a={\frac {1}{1+z}}\;.}$

If you then write out the Hubble function explicitly, you get the common expression for the runtime distance ${\displaystyle H}$

${\displaystyle D_{\mathrm {prop} }(z_{1},z_{2})={\frac {c}{H_{0}}}\int _{a(z_{2})}^{a(z_{1})}\left[{\frac {\Omega _{0}}{a}}+(1-\Omega _{0}-\Omega _{\Lambda })+a^{2}\,\Omega _{\Lambda }\right]^{-1/2}\,{\mathrm {d} }a\;.}$

For a flat universe ( ) this integral can be solved analytically: ${\displaystyle 1-\Omega _{0}-\Omega _{\Lambda }=0}$

${\displaystyle {\begin{aligned}D_{\mathrm {prop} }(z_{1},z_{2})&={\frac {2\,c}{3H_{0}{\sqrt {\Omega _{\Lambda }}}}}\;&&\left[\ln \left(a^{3/2}\Omega _{\Lambda }+{\sqrt {\Omega _{0}\Omega _{\Lambda }+a^{3}\Omega _{\Lambda }^{2}}}\right)\right]_{a(z_{2})}^{a(z_{1})}\\&={\frac {2}{\sqrt {3\Lambda }}}\;&&\left[\ln \left(a^{3/2}\Omega _{\Lambda }+{\sqrt {\Omega _{0}\Omega _{\Lambda }+a^{3}\Omega _{\Lambda }^{2}}}\right)\right]_{a(z_{2})}^{a(z_{1})}\,.\end{aligned}}}$

${\displaystyle \Omega _{0}}$and represent the matter density and vacuum energy density parameters ( cosmological constant ). According to measurements with WMAP , these are and . The Hubble constant is km s ⁻¹ Mpc ⁻¹ . ${\displaystyle \Omega _{\Lambda }}$ ${\displaystyle \Omega _{0}=0{,}27}$ ${\displaystyle \Omega _{\Lambda }=0{,}73}$ ${\displaystyle H_{0}=71}$

Moving distance

The evolution of the universe and its horizons in moving coordinates

The comoving distance is obtained in analogy to the travel-time distance . This is the distance between the source and the observer on a space-like hypersurface, defined by events with constant cosmological time (today). Starting from the line element (see also Friedmann equations ) the result is ${\displaystyle t=t_{0}}$

${\displaystyle {\mathrm {d} }D_{\mathrm {com} }(z_{1},z_{2})={\mathrm {d} }w=-c/a\,{\mathrm {d} }t=-c/(a^{2}H)\,{\mathrm {d} }a\,,}$

from which one deduces

${\displaystyle D_{\mathrm {com} }(z_{1},z_{2})={\frac {c}{H_{0}}}\int _{a(z_{2})}^{a(z_{1})}\left[a\,\Omega _{0}+a^{2}\,(1-\Omega _{0}-\Omega _{\Lambda })+a^{4}\,\Omega _{\Lambda }\right]^{-1/2}\,{\mathrm {d} }a=w(z_{1},z_{2})\,.}$

The big difference between travel-time distance and moving distance is that the former is a distance across space and time. Time-of-flight distance is the distance to the object as the observer sees it, and he sees it in a state of the past. The moving distance, on the other hand, is the distance between the observer and the object at the same point in time, i.e. a distance on a space-like hypersurface. In this state, however, the observer cannot see the object because the light was just sent out from the object to him.

Angular diameter distance

The evolution of the universe and its horizons in physical coordinates

The angular diameter distance is defined in analogy to Euclidean space-time, as the ratio between the source area and the solid angle at which the object appears to the observer: ${\displaystyle \delta A}$ ${\displaystyle \delta \Omega }$

${\displaystyle D_{\mathrm {ang} }(z_{1},z_{2})=(\delta A/\delta \Omega )^{1/2}=a(z_{2})f_{K}\left(w(z_{1},z_{2})\right)\,.}$

Using the moving distance results from this

${\displaystyle D_{\mathrm {ang} }(z_{1},z_{2})=a(z_{2})f_{K}\left[D_{\mathrm {com} }(z_{1},z_{2})\right]\;,}$

With

${\displaystyle f_{K}(w)={\begin{cases}{\frac {1}{\sqrt {K}}}\;\sin({\sqrt {K}}w)&K>0\\w&K=0\;.\\{\frac {1}{\sqrt {-K}}}\;\sinh({\sqrt {-K}}w)&K<0\end{cases}}}$

The function differentiates between three-dimensional space-like hypersurfaces of constant time with positive, vanishing or negative curvature . ${\displaystyle f_{K}(w)}$ ${\displaystyle t}$ ${\displaystyle K}$

Luminosity distance

Likewise, the luminosity distance results from the analogy to Euclidean geometry. If one takes into account the delayed arrival of the photons at the observer due to the intermediate expansion of the universe, their redshift and the conservation of the number of photons, one obtains

${\displaystyle D_{\mathrm {lum} }(z_{1},z_{2})={\frac {a(z_{1})^{2}}{a(z_{2})}}\,f_{K}\left[D_{\mathrm {com} }(z_{1},z_{2})\right]\;.}$

General properties of the various distance definitions

Due to the prefactors of and the non-linearity of , neither the angular diameter distance nor the luminosity distance have an additive property. If you look at two objects 1 and 3 with an object 2 in between, then the distance between 1 and 3 is not equal to the sum of the distances between objects 1 and 2, and objects 2 and 3: ${\displaystyle a}$ ${\displaystyle f_{\mathrm {K} }}$

${\displaystyle D(z_{1},z_{3})\neq D(z_{1},z_{2})+D(z_{2},z_{3})}$

The transit time distance and the moving distance, however, are additive.

Numerical examples

For the following redshifts there are the different distances (in billions of light years) to the observer ( ): ${\displaystyle z=0}$

${\displaystyle z}$	0.1	0.5	01.0	03.0	006.0
Run-time distance ${\displaystyle D_{\mathrm {prop} }}$	1.280	4,970	07,600	11,190	012,370
Moving distance ${\displaystyle D_{\mathrm {com} }}$	1,340	6.070	10.620	20,430	026,510
Angular diameter distance ${\displaystyle D_{\mathrm {ang} }}$	1.220	4.050	05.310	05.110	003.790
Luminosity distance ${\displaystyle D_{\mathrm {lum} }}$	1,480	9.110	21,240	81.710	185.540

It is noticeable here that the angular diameter distance is not a monotonous function of the redshift, but rather shows a maximum in order to then become smaller again. This means that the same object appears smaller and smaller for increasing redshifts, when it reaches a minimum, and again appears larger to the observer for greater distances. ${\displaystyle z=1{,}6}$ ${\displaystyle z=1{,}6}$

The transit time distance strives for a constant value for infinitely large redshifts (the numerical value of the age of the universe, in light years). The luminosity distance, on the other hand, tends towards infinity, which means that the apparent brightness of an object decreases very sharply with increasing redshift. In fact, the surface brightness decreases with it . ${\displaystyle \propto (1+z)^{-4}}$

Application examples

A galaxy has a redshift 0.5. This means that the light traveled from her to the observer 5.0 billion years, and thus her travel time was 5.0 billion light years. If one would like to infer its actual brightness from the apparent brightness of the galaxy (e.g. magnitude = 22), one must not use the transit time distance, but one must use the luminosity distance. This is 9.1 billion light years. The size determination is analogous to this: If the galaxy appears to the observer at an angle of 5 arc seconds, the angular diameter distance of 4.1 billion light years must be used in order to be able to determine its actual size (99,600 light years) using the tangent function.

See also

• Observable Universe

Web links

• Cosmology Calculator
• Richard Powell: The Distance Scale of the Universe

literature

• C. Misner, KS Thorne, JA Wheeler: Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0 .
• JA Peacock: Cosmological Physics . Cambridge University Press, 2001, ISBN 0-521-42270-1 .