Cosmology for Alcock- Paczynski Test

On test of classical test of cosmology, derives from the assumption of isotropy of the universe. Thus objects at cosmological distances should not be oriented preferably with respect to our line-of-sight, and should thus be statistically as elongated along the line-of-sight as transverse to it. This provides a useful cosmological test since we measure transverse sizes of objects by the angle they subtend, and radial sizes by the redshift, , and the ratio of the constants of proportionality which related angle to distance, and increments in redshift to distance, , will depend on the comology. Thus be requiring that objects be statistically just as elongated in the radial and transverse direction will provide one with a measure of , and thus a measure of cosmology. This is the argument given by Alcock & Paczynski. There is, however, an important caveat to this argument, namely that the redshift is not really a measure of distance, but rather of radial velocity, and any peculiar velocities can effect the radial size of objects in redshift space but will have no effect on their transverse size. Such an effect is known as redshift-space distortion. Thus to implement this test in detail one must understand the redshift-space distortions. For the redshift range relevent to the Lyman- forest the difference in between cosmological models is small, and therefore a detailed knowledge of redshift-space distortions is required to make this test a good discriment between cosmological models.

The transverse comoving length corresponding to a give angle can be written where is the comoving distance to redshift , is the spatial curvature radius of the FRW spacetime, depending on whether the universe has zero, positive or negative spatial curvature (in the case one takes the limiting form : ). The Friedmann equatiion is

Normalizing the scale factor so that the redshift-comoving distance relationship is given by

so that
.
One defines then

For a cosmology with positive spatial curvature a more manifestly real expression is
.
In the limit of a flat universe both expressions approach
.
For very small redshifts all these formulae have the limiting form
.
We now proceed to apply these formulae to compute for a variety of cosmological models, i.e. different 's.