Redshift space

The redshift space is an alternative coordinate system in cosmology , for which the observable instead of the real distance is used.

On cosmological scales, distances can only be measured (e.g. to quasars or distant galaxy clusters ) by measuring the redshift . Therefore, instead of the “natural” coordinate system consisting of two angular coordinates and the real distance between observer and object, one is forced to switch to the redshift space consisting of the two angular coordinates and the redshift distance.

Distance after redshift

Would create the observed redshift  z only by the cosmic expansion , the Rotverschiebungsentfernung would  s equal to the real distance  D . But if the object has a peculiar velocity  v , for example a gravitationally bound cluster galaxy, then the following applies  approximately up to a redshift of z = 0.2 (see diagram):

${\displaystyle s=D+{\frac {v}{H_{0}}}\approx {\frac {c\ z}{H_{0}}}}$

Here designated

• ${\displaystyle c}$the speed of light
• ${\displaystyle H_{0}}$the Hubble constant .

At higher redshifts, the change in the rate of expansion over time must also be taken into account. Since the universe continues to expand during the light travel time, a distinction is made between the distance when the light was sent out and the distance when it was received:

${\displaystyle D_{damals}=\int _{1}^{\text{a}}-{\frac {{\text{a}}\ c}{a^{2}\ H(a)}}\,{\text{d}}a}$

and

${\displaystyle D_{heute}={\frac {D_{damals}}{\text{a}}}}$

where scale factor and redshift in relation to each other ${\displaystyle {\text{a}}}$ ${\displaystyle {\text{z}}}$

${\displaystyle 1/{\text{a}}=z+1}$

stand. The Hubble parameter dependent on the scale factor is

${\displaystyle H({\text{a}})=H_{0}{\sqrt {\Omega _{\text{R}}\ {\text{a}}^{-4}+\Omega _{\text{M}}\ {\text{a}}^{-3}+\Omega _{\text{K}}\ {\text{a}}^{-2}+\Omega _{\Lambda }}}}$

with for the expansion rate, for the radiation density and for the mass density. Since the curvature of the universe is practically flat according to current measurements, the following also applies to the dark energy density and to the cosmic curvature parameter . ${\displaystyle H_{0}=67150{\frac {\text{m}}{\text{Mpc s}}}}$ ${\displaystyle \Omega _{\text{R}}=5\cdot 10^{-5}}$ ${\displaystyle \Omega _{\text{M}}=0.315}$ ${\displaystyle \Omega _{\Lambda }=1-\Omega _{\text{R}}-\Omega _{\text{M}}}$ ${\displaystyle \Omega _{\text{K}}=0}$

Example: since the cosmic background radiation received today has a redshift of , i.e. was emitted at a time when the universe was times smaller than today, the surface of the last scattering from where it was emitted was at the time of the emission (around 400,000 Years after the Big Bang) , and is now at a distance of around , i.e. just in front of the particle horizon . ${\displaystyle z=1089}$ ${\displaystyle 1/{\text{a}}=z+1=1090}$ ${\displaystyle D_{damals}=40{\text{ Mio. Lj.}}}$ ${\displaystyle D_{heute}=45{\text{ Mrd. Lj.}}}$

left: normal, right: redshift space

In general, the observed structures can be distorted when displayed in redshift space. For example, a roughly spherical cluster of galaxies in a virialized state can be so distorted by the peculiar velocities that it becomes an elongated ellipsoid along the direction of observation . These phenomena give astronomers finger of god - finger of God .