Polypolar spherical harmonic decomposition of galaxy correlators in redshift space: Toward testing cosmic rotational symmetry

Understand

We propose an efficient way to test rotational invariance in the cosmological perturbations by use of galaxy correlation functions.

• In symmetry-breaking cases, the galaxy power spectrum can have extra angular dependence in addition to the usual one due to the redshift-space distortion, $ \hat{k} \cdot \hat{n}$.
• We confirm that, via the decomposition into not the usual Legendre basis ${\cal L}_\ell(\hat{k} \cdot \hat{n})$ but the bipolar spherical harmonic one $\{Y_{\ell}(\hat{k}) \otimes Y_{\ell'}(\hat{n})\}_{LM}$, the symmetry-breaking signal can be completely distinguished from the usual isotropic one since the former yields nonvanishing $L \geq 1$ modes but the latter is confined to the $L = 0$ one.
• As a demonstration, we analyze the signatures due to primordial-origin symmetry breakings such as the well-known quadrupolar-type and dipolar-type power asymmetries and find nonzero $L = 2$ and $1$ modes, respectively.