Effect of realistic astrophysical inputs on the phase and shape of the WIMP annual modulation signal

The orbit of the Earth about the Sun produces an annual modulation in the WIMP direct detection rate. If the local WIMP velocity distribution is isotropic then the modulation is roughly sinusoidal with maximum in June, however if the velocity distribution is anisotropic the phase and shape of the signal can change. Motivated by conflicting claims about the effect of uncertainties in the local velocity distribution on the interpretation of the DAMA annual modulation signal (and the possibility that the form of the modulation could be used to probe the structure of the Milky Way halo), we study the dependence of the annual modulation on various astrophysical inputs. We first examine the approximations used for the Earth's motion about the Sun and the Sun's velocity with respect to the Galactic rest frame. We find that overly simplistic assumptions lead to errors of up to ten days in the phase and up to tens of per-cent in the shape of the signal, even if the velocity distribution is isotropic. Crucially, if the components of the Earth's velocity perpendicular to the motion of the Sun are neglected, then the change in the phase which occurs for anisotropic velocity distributions is missed. We then examine how the annual modulation signal varies for physically and observationally well-motivated velocity distributions. We find that the phase of the signal changes by up to 20 days and the mean value and amplitude change by up to tens of per-cent