CMB statistical anisotropy from noncommutative gravitational waves
  Primordial statistical anisotropy is a key indicator to investigate early
Universe models and has been probed by the cosmic microwave background (CMB)
anisotropies. In this paper, we examine tensor-mode CMB fluctuations generated
from anisotropic gravitational waves, parametrised by $P_h({\bf k}) =
P_h^{(0)}(k) [ 1 + \sum_{LM} f_L(k) g_{LM} Y_{LM} (\hat{\bf k}) ]$, where
$P_h^{(0)}(k)$ is the usual scale-invariant power spectrum. Such anisotropic
tensor fluctuations may arise from an inflationary model with noncommutativity
of fields. It is verified that in this model, an isotropic component and a
quadrupole asymmetry with $f_0(k) = f_2(k) \propto k^{-2}$ are created and
hence highly red-tilted off-diagonal components arise in the CMB power spectra,
namely $\ell_2 = \ell_1 \pm 2$ in $TT$, $TE$, $EE$ and $BB$, and $\ell_2 =
\ell_1 \pm 1$ in $TB$ and $EB$. We find that B-mode polarisation is more
sensitive to such signals than temperature and E-mode polarisation due to the
smallness of large-scale cosmic variance and we can potentially measure $g_{00}
= 30$ and $g_{2M} = 58$ at 68% CL in a cosmic-variance-limited experiment. Such
a level of signal may be measured in a PRISM like experiment, while the
instrumental noise contaminates it in the $Planck$ experiment. These results
imply that it is impossible to measure the noncommutative parameter if it is
small enough for the perturbative treatment to be valid. Our formalism and
methodology for dealing with the CMB tensor statistical anisotropy are general
and straightforwardly applicable to other early Universe models.
