Polypolar spherical harmonic decomposition of galaxy correlators in
  redshift space: Toward testing cosmic rotational symmetry
  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. Fisher matrix forecasts of their constraints
indicate that the $Planck$-level sensitivity could be achieved by the SDSS or
BOSS-CMASS data, and an order-of-magnitude improvement is expected in a near
future survey as PFS or Euclid by virtue of an increase in accessible Fourier
mode. Our methodology is model-independent and hence applicable to the searches
for various types of statistically anisotropic fluctuations.
