Reconstruction of a direction-dependent primordial power spectrum from
  Planck CMB data
  We consider the possibility that the primordial curvature perturbation is
direction-dependent. To first order this is parameterised by a quadrupolar
modulation of the power spectrum and results in statistical anisotropy of the
CMB, which can be quantified using `bipolar spherical harmonics'. We compute
these for the Planck DR2-2015 SMICA map and estimate the noise covariance from
Planck Full Focal Plane 9 simulations. A constant quadrupolar modulation is
detected with 2.2 sigma significance, dropping to 2 sigma when the primordial
power is assumed to scale with wave number k as a power law. Going beyond
previous work we now allow the spectrum to have arbitrary scale-dependence. Our
non-parametric reconstruction then suggests several spectral features, the most
prominent at k ~ 0.006/Mpc. When a constant quadrupolar modulation is fitted to
data in the range 0.005 < k Mpc < 0.008, its preferred directions are found to
be related to the cosmic hemispherical asymmetry and the CMB dipole. To
determine the significance we apply two test statistics to our reconstructions
of the quadrupolar modulation from data, against reconstructions of
realisations of noise only. With a test statistic sensitive only to the
amplitude of the modulation, the reconstructions from the multipole range 30 <
l < 1200 are unusual with 2.1 sigma significance. With the second test
statistic, sensitive also to the direction, the significance rises to 6.9
sigma. Our approach is easily generalised to include other data sets such as
polarisation, large-scale structure and forthcoming 21-cm line observations
which will enable these anomalies to be investigated further.
