  We show that cosmological quantum relaxation predicts an anisotropic
primordial power spectrum with a specific dependence on wavenumber k. We
explore some of the consequences for precision measurements of the cosmic
microwave background (CMB). Quantum relaxation is a feature of the de
Broglie-Bohm pilot-wave formulation of quantum theory, which allows the
existence of more general physical states that violate the Born probability
rule. Recent work has shown that relaxation to the Born rule is suppressed for
long-wavelength field modes on expanding space, resulting in a large-scale
power deficit with a characteristic inverse-tangent dependence on k. Because
the quantum relaxation dynamics is independent of the direction of the wave
vector for the relaxing field mode, in the limit of weak anisotropy we are able
to derive an expression for the anisotropic power spectrum that is determined
by the power deficit function. As a result, the off-diagonal terms in the CMB
covariance matrix are also determined by the power deficit. We show that the
lowest-order l-(l+1) inter-multipole correlations have a characteristic scaling
with multipole moment l. Our derived spectrum also predicts a residual
statistical anisotropy at small scales, with an approximate consistency
relation between the scaling of the l-(l+1) correlations and the scaling of the
angular power spectrum at high l. We also predict a relationship between the
l-(l+1) correlations at large and small scales. Cosmological quantum relaxation
appears to provide a single physical mechanism that predicts both a large-scale
power deficit and a range of statistical anisotropies, together with
potentially testable relationships between them.
