Flux autocorrelation function

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The flux autocorrelation function is a two-point statistic used for description of fluctuations of the transmitted flux in Ly$\alpha$ forest in spectra of distant quasars, which is used for statistical analysis of density fluctuations of neutral intergalactic gas and can be defined as a line-of-sight (or longitudinal) projection of more general case — the (2D) flux correlation function. That is why sometimes flux autocorrelation function is called longitudinal flux correlation function (see e. g. [1]).
The flux autocorrelation function was introduced by Zuo & Bond [2]. For velocity separation $\Delta v_{\parallel }=\Delta v = v_{2}-v_{1}$ it was defined by them as
where $F(v)$ denotes the pixel flux at velocity $v$ and $\bar {F}=\left \langle F\right \rangle$ is the measured mean transmitted flux at the relevant redshift. Note, that the data (quasar spectrum) is given in the form of pixels with wavelength label $\lambda _{i}$ and the flux value $F_{i}$. Thus the line-of-sight distance between pixels $\lambda _{}$ and $\lambda +\Delta \lambda$ in units of the local velocity scale is usually defined as (see e. g. [6])
where the wavelength at the redshift $z$ is $\lambda =\lambda _{\alpha }(1+z)$, and $\Delta r_{\parallel }$ is the line-of-sight comoving distance between two pixels. In most of the literature the mean flux changes over the velocity separations are assumed to be negligible. Thus more common definition of $\xi _{F}(\Delta {v})$ (see e.g. [3],[4],[5]) is
where $\delta _{F}$ is the variance (or fluctuations) of transmitted flux.
The flux autocorrelation function is related to (1D) flux power spectrum$P_{F}(k)$ through the following formula:

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