Tilted physics: A cosmologically dipole-modulated sky

Physical constants and cosmological parameters could vary with position. On the largest scales such variations would manifest themselves as gradients across our Hubble volume, leading to dipole modulations of the cosmic microwave background anisotropies. Here we derive the cosmic microwave background covariance matrix for models with such gradients, and show that they generically lead to a correlation between adjacent multipoles in the spherical harmonic expansion of the sky. Our results generalize previous studies of anisotropic primordial spectra: essentially any quantity which can be considered as a parameter of the cosmological model could in principle be modulated in this way, yielding a distinctive signal which should be searched for in future data sets. For the case of the fine structure constant ${\ensuremath{\alpha}}_{e}$, we find that the Planck satellite could detect variations as low as $\ensuremath{\Delta}{\ensuremath{\alpha}}_{e}/{\ensuremath{\alpha}}_{e}\ensuremath{\approx}{10}^{\ensuremath{-}4}$ across our observable volume.