The flux power spectrum (or power spectrum of transmitted flux, or Ly$\alpha$ forest power spectrum) is a measure of the variance in the amplitude of the Fourier transform coefficients of the transmitted flux in Ly$\alpha$ forest in spectra of distant quasars. This function is used for statistical analysis of density fluctuations of neutral intergalactic hydrogen. The first such work was made by Croft et al. [1].
The data (quasar spectrum) is given in the form of pixels with wavelength label $\lambda _{i}$ and the flux value $F_{i}$ and the distance between pixels in units of the local velocity scale is usually defined as (see e. g. [2]) where the wavelength at the mean redshift $\bar {z}$ is $\bar {\lambda } = \lambda _{\alpha }(1+\bar {z})$, and $\Delta r_{i}$ is the comoving distance between pixel $i$ and a pixel at the mean redshift, where one needs to assume some cosmological model.
The Fourier transform of the variance (or fluctuations) of transmitted flux over the velocity interval $\Delta v$ is and the flux power spectrum is where $\langle F\rangle$ is the mean transmitted flux.
The dimensionless form $kP_{F}(k)/\pi$ is often more convenient; it forms a Fourier transform pair with the flux autocorrelation function, $\xi _{F}(\Delta v)$, through Several different conventions for $P_{F}(k)$ exist in the literature, some without a mean normalization, some differing by a factor of $2\pi$.
The observed flux power spectrum is one-dimensional (along the line-of-sight). The problem of recovering the three-dimensional flux power spectrum on one-dimensional is complicated and requires taking into account redshift-space distortion effects.