## Geometric flattening

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The dependences of longitudinal (along the line of sight) and transverse (along the direction perpendicular to the line of sight) distances on cosmological parameters (like $\Omega _{M}$) are different. Hence if one takes an intrinsically spherically symmetric object and calculates its sizes in different directions, this object can appear distorted if one assumed false cosmological model. This effect, called geometrical flattening, was considered by Alcock and Paczynski [1] and proposed by them as a powerful cosmological test (known as Alcock-Paczynski text) namely to determine $\Omega _{M}$.
The best'' spherically-symmetric object for this test (assuming the isotropy of the Universe) is real-space (2D) power spectrum (or two-point correlation function). Following [1] Ballinger et al. [2] showed that the relation between the true galaxy power spectrum and the power spectrum in assumed (wrong) cosmology is:
where $n$ is the local spectral index of the power spectrum, $\mu _{a}$ is a cosine of the angle between the wavevector $\vec {k}$ and the line of sight, $F=\frac {f_{\parallel }}{f_{\perp }}$ is the geometric flattening factor, $f_{\parallel }=A_{t}/A_{a}$, $f_{\perp }=B_{t}/B_{a}$, and for spatially flat $\Lambda$CDM cosmology:

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