The quadrupole in the local Hubble parameter: first constraints using Type Ia supernova data and forecasts for future surveys

The cosmological principle asserts that the Universe looks spatially homogeneous and isotropic on sufficiently large scales. Given the fundamental implications of the cosmological principle, it is important to empirically test its validity on various scales. In this paper, we use the Type Ia supernova (SN~Ia) magnitude-redshift relation, from both the Pantheon and JLA compilations, to constrain theoretically motivated anisotropies in the Hubble flow. In particular, we constrain the quadrupole moment in the effective Hubble parameter and the dipole moment in the effective deceleration parameter. We find no significant quadrupole term regardless of the redshift frame we use. Our results are consistent with the theoretical expectation of a quadrupole moment of a few percent at scales of $\sim 100 h^{-1}$ Mpc. We place an upper limit of a $\sim 10\%$ quadrupole amplitude relative to the monopole, $H_0$, at these scales. We find that we can detect a $\sim 7\%$ quadrupole moment at the 5$\sigma$ level, for a forecast low-$z$ sample of 1055 SNe~Ia. We find an exponentially decaying dipole moment of the deceleration parameter varies in significance depending on the redshift frame we use. In the heliocentric frame, as expected, it is detected at $\sim 3 \sigma$ significance. In the rest-frame of the cosmic microwave background (CMB), we find a marginal $\sim 2 \sigma$ dipole, however, after applying peculiar velocity corrections, the dipole is insignificant. Finally, we find the best-fit frame of rest relative to the supernovae to differ from that of the CMB.