An Unbiased Method of Modeling the Local Peculiar Velocity Field with Type Ia Supernovae

We apply statistically rigorous methods of nonparametric risk estimation to the problem of inferring the local peculiar velocity field from nearby supernovae (SNIa). We use two nonparametric methods - Weighted Least Squares (WLS) and Coefficient Unbiased (CU) - both of which employ spherical harmonics to model the field and use the estimated risk to determine at which multipole to truncate the series. We show that if the data are not drawn from a uniform distribution or if there is power beyond the maximum multipole in the regression, a bias is introduced on the coefficients using WLS. CU estimates the coefficients without this bias by including the sampling density making the coefficients more accurate but not necessarily modeling the velocity field more accurately. After applying nonparametric risk estimation to SNIa data, we find that there are not enough data at this time to measure power beyond the dipole. The WLS Local Group bulk flow is moving at 538 +- 86 km/s towards (l,b) = (258 +- 10 deg, 36 +- 11 deg) and the CU bulk flow is moving at 446 +- 101 km/s towards (l,b) = (273 +- 11 deg, 46 +- 8 deg). We find that the magnitude and direction of these measurements are in agreement with each other and previous results in the literature.