The Power Spectrum of Cosmological Number Densities

We study the cosmological power spectra (PS) of the differential and integral galaxy volume number densities $\gamma_i$ and $\gamma_i^{*}$, constructed with the cosmological distances $d_i$ $(i=A,G,L,Z)$, where $d_A$ is the angular diameter distance, $d_G$ is the galaxy area distance, $d_L$ is the luminosity distance and $d_z$ is the redshift distance. Theoretical and observational quantities were obtained in the FLRW spacetime with a non-vanishing $\Lambda$. The radial correlation $\Xi_i$, as defined in the context of these densities, is discussed in the wave number domain. All observational quantities were computed using luminosity function (LF) data obtained from the FORS Deep Field galaxy survey. The theoretical and observational PS of $\gamma_i$, $\gamma_i^{\ast}$, $\Xi_i$ and $\gamma_i / \gamma_i^\ast$ were calculated by performing Fourier transforms on these densities previously derived by Iribarrem et al. (2012) from the observed values $\gamma_{obs}$ and ${\gamma^\ast}_{obs}$ obtained using the galactic absolute magnitudes and galaxy LF Schechter's parameters presented in Gabasch et al. (2004, 2006) in the range $0.5 \le z \le5.0$. The results show similar behavior of the PS obtained from $\gamma$ and $\gamma^{\ast}$ using $d_L$, $d_z$ and $d_G$ as distance measures. The PS of the densities defined with $d_A$ have a different and inconclusive behavior, as this cosmological distance reaches a maximum at $z\approx 1.6$ in the adopted cosmology. For the other distances, our results suggest that the PS of ${\gamma_i}_{obs}$, ${\gamma^\ast_i}_{obs}$ and ${\gamma_i / \gamma^{\ast}_i}_{obs}$ have a general behavior approximately similar to the PS obtained with the galaxy two-point correlation function and, by being sample size independent, they may be considered as alternative analytical tools to study the galaxy distribution.