The (2D) flux correlation 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 are related to density fluctuations of neutral intergalactic hydrogen. As introduced in [1, 2] the (unnormalized) flux correlation function is where $F$ is normalized (on continuum) flux along two lines of sight with separation $\theta$ and $\bar {F}$ is the mean transmitted flux.
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. [3]) 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. And the transverse distance corresponding to angular separation $\theta$ can be written in velocity units as Hence the formula for $\xi _{F}(\theta ,\Delta \lambda )$ can be rewritten in another notations with $\Delta v_{\parallel }$ and $\Delta v_{\perp }$ instead of $\Delta \lambda$ and $\theta$.
A bit different definition of $\xi _{F}(\theta ,\Delta \lambda )$ can be found in [4]: i.e. the mean transmitted fluxes in points defined by $\{0,\lambda \}$ and $\{\theta ,\lambda +\Delta \lambda \}$ are considered to be different.
The special cases of the (2D) flux correlation function are its (1D) projections:

• onto the line of sight — flux autocorrelation function, $\xi _{F}(\Delta v_{\parallel })$
• onto the plane perpendicular to the line of sight — flux cross-correlation function, $\xi _{F}(\theta )$