Testing the statistical isotropy of large scale structure with multipole vectors

A fundamental assumption in cosmology is that of statistical isotropy—that the Universe, on average, looks the same in every direction in the sky. Statistical isotropy has recently been tested stringently using cosmic microwave background data, leading to intriguing results on large angular scales. Here we apply some of the same techniques used in the cosmic microwave background to the distribution of galaxies on the sky. Using the multipole vector approach, where each multipole in the harmonic decomposition of galaxy density field is described by unit vectors and an amplitude, we lay out the basic formalism of how to reconstruct the multipole vectors and their statistics out of galaxy survey catalogs. We apply the algorithm to synthetic galaxy maps, and study the sensitivity of the multipole vector reconstruction accuracy to the density, depth, sky coverage, and pixelization of galaxy catalog maps.

Figure 1

Multipole vectors of our sky, with the corresponding multipoles $\ell =2–8$ computed from WMAP’s 3-year Internal Linear Combination map [52]. The lobes represent the CMB temperature pattern seen at each multipole, where the observer is at the center and the observed sky anisotropy can be projected to a sphere of a fixed radius. The sticks are the multipole vectors, each pointing in a fixed direction (or its opposite) on this sphere. Figure kindly provided by Craig Copi.

Figure 2

Illustration of the efficacy of our reconstruction scheme for a mock galaxy survey with $N_g={10}^6$. The top panel shows our starting map. The middle panels show the map made up from the cut-sky coefficients (i.e., using Eq. (19)), while the bottom row shows the full-sky reconstruction that we adopted. The left columns show the full-sky case, while the right columns show the case where $\pm 4.5°$ galactic cut (removing $\simeq 8\%$ of pixels) have been applied.

Figure 3

The theoretical angular power spectra calculated using the radial number density function $n(z)$ from the SDSS for different redshifts at which the radial number density of objects peaks. See Appendix  for details of the calculation.

Figure 4

Reconstruction of the coefficients $a_{\ell m}$ for $\ell =2–4$ for 300 realizations with $N_g={10}^4$ (top row), $N_g={10}^5$ (middle row) and ${10}^6$ (bottom row). We show results for three HEALPix map resolutions: pixelizations of $N_{\mathrm{side}}=4$ (blue), 8 (black) and 16 (red). The total number of pixels on the full sky is $N_{\mathrm{pix}}=12\times N_{\mathrm{side}}{}^2$. The true underlying $a_{\ell m}$ are shown by the dotted line. An increase in resolution (i.e., higher $N_{\mathrm{side}}$) improves the accuracy of the reconstruction only for mock catalogs of size $N_g={10}^6$ and higher.

Figure 5

Effects of the number density of LSS tracers. Histograms of the dot product of the true and reconstructed MVs, $\cos ({\Theta }^{(\ell ,i)})=v^{(\ell ,i)}·v_{\mathrm{true}}^{(\ell ,i)}$, from 300 realizations for surveys with $N_g={10}^4$ (top row), $N_g={10}^6$ (middle row), and $N_g={10}^8$ (bottom row). We assume a fixed pixelization level of $N_{\mathrm{side}}=8$, and the radial distribution of objects $z_{\mathrm{peak}}=0.2$. An improvement in accuracy is indicated by a closer proximity to 1, at which the MVs are reconstructed perfectly. The narrowing of the histograms suggests a considerably better recovery of the MVs as the survey size is increased.

Figure 6

Effects of the sky cut. Histogram of the dot products of the true and reconstructed MVs $\cos ({\Theta }^{(\ell ,i)})=v_{\mathrm{true}}^{(\ell ,i)}·v^{(\ell ,i)}$, from 300 realizations when the following areas of the sky are removed: 0 (top row), $\pm 4.5°$ (middle row), and $\pm 9°$ (bottom row). The second and third case correspond to $f_{\mathrm{sky}}\simeq 0.92$ and 0.84, respectively. The pixelization level is fixed at $N_{\mathrm{side}}=8$ and we assume a survey with $N_g={10}^6$ objects which radial distribution of tracers that peaks at $z_{\mathrm{peak}}=0.2$.

Figure 7

Effects of the survey depth. Histogram of the dot products of the true and reconstructed MVs $\cos ({\Theta }^{(\ell ,i)})=v_{\mathrm{true}}^{(\ell ,i)}·v^{(\ell ,i)}$ from 300 realizations of a full sky for a surveys with $z_{\mathrm{peak}}=0.1$, 0.2 and 0.4 (top to bottom rows). The adopted pixelization is $N_{\mathrm{side}}=8$ and the total number of sources is $N_g={10}^6$.

Figure 8

Summary of all effects. Plot of the average angle between the reconstructed and input MVs, ${\Theta }^{(\ell ,i)}=\arccos (v_{\mathrm{true}}^{(\ell ,i)}·v^{(\ell ,i)})$ as a function of $N_g$, with error bars indicating the 16–84 percentile ranges for different choices of $z_{\mathrm{peak}}$; $z_{\mathrm{peak}}=0.1$ (left column), $z_{\mathrm{peak}}=0.2$ (middle column) and $z_{\mathrm{peak}}=0.4$ (right column) for $\ell =2$ (top row), $\ell =3$ (middle row) and $\ell =4$ (bottom row). The different colors indicate different sky masks: 0° (black), $\pm 4.5°$ (blue) and $\pm 9°$ (red).

Figure 9

Detectability of perfectly aligned quadrupole and octopole in a mock survey using the $B_{\mathrm{signal}}$ statistic (see Eq. (29)). Each histogram is based on 50 000 Monte Carlo realizations. Solid lines shows survey with $N_g={10}^6$ objects and the radial distribution that peaks at $z_{\mathrm{peak}}=0.1$, while the dashed lines show a survey with $N_g={10}^9$ and $z_{\mathrm{peak}}=0.4$ (the “unaligned” case, shown with the red solid line, is independent of the presence of the cut and the values of $z_{\mathrm{peak}}$ and $N_g$). The grey region covers values of $B_{\mathrm{signal}}$ which correspond to the bottom 5% of the isotropic (i.e., unaligned) sky cases. We find that 98–99% of the aligned skies without the galactic cut (and for either of the two $z_{\mathrm{peak}}$ cases) lie below this value—in other words, it is roughly at the $20:1$ odds that the value of $B_{\mathrm{signal}}$ found below this value favors the aligned model.