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

TL;DR

This work analyzes how astrophysical inputs—specifically the Earth’s full three-component velocity and the local WIMP velocity distribution—shape the WIMP annual modulation signal. By comparing accurate Earth-motion models with common approximations and by exploring physically motivated velocity distributions (notably the logarithmic ellipsoidal, triaxial halo), the authors show that phase and shape can shift by up to about $20$ days and tens of percent, respectively, though restricting parameters to observationally plausible ranges mitigates these changes. The study highlights the necessity of realistic Galactic dynamics when interpreting modulation data (e.g., DAMA) and suggests that such signals could, in principle, probe halo structure, including potential high-velocity streams. Overall, the results emphasize that careful astrophysical modeling is crucial for robust WIMP parameter inference and for leveraging modulation as a Galactic diagnostic; they also acknowledge an erratum noting total circular velocity $v_c(R_0) = 220 \, \pm \, 20$ km s$^{-1}$, with main conclusions remaining unchanged.

Abstract

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.

Figures (10)

• Figure 1: The components of the orbital velocity of the Earth in Galactic X,Y,Z co-ordinates ($v_{{\rm e, \, X}}$ top panel, $v_{{\rm e, \, Y}}$ middle panel and $v_{{\rm e, \, Z}}$ bottom panel) found using the assumptions discussed in the text: i) assuming that the Earth's orbit is circular and the ecliptic lies in the X-Y plane (eq. (\ref{['ve1']}), short dashed lines) ii) ignoring the ellipticity of the Earth's orbit and the non-uniform motion of the Sun in right ascension (eq. (\ref{['vegg']}) and Ref. gg, long dashed) iii) including the ellipticity of the Earth's orbit but not the non-uniform motion of the Sun (Ref. fs, dot-dashed) and iv) including the ellipticity of the Earth's orbit and the non-uniform motion of the Sun (eq. (\ref{['vels']}) and Ref. ls, solid). Time is measured in days from noon on Dec 31st 2002.
• Figure 2: The differences between the expressions for the velocity of the Earth in Galactic co-ordinates ($\Delta (v_{{\rm e, \, X}}), \, \Delta (v_{{\rm e, \, Y}}), \, \Delta (v_{{\rm e, \, Z}})$, top, middle and bottom panel respectively) relative to the full expression including the ellipticity of the Earth's orbit and the non-uniform motion of the Sun (eq. (\ref{['vels']}) and Ref. fs). Line types as in Fig. 1. Note the larger scale used for $\Delta (v_{{\rm e, \, Y}})$.
• Figure 3: Physically and observationally reasonable values of $\sigma_{{\rm r}}$ and $\sigma_{{\phi }}$ ($\sigma_{{\rm z }}=(v_{{\rm c, \, h}}^2- \sigma_{{\phi }}^2)^{1/2}$). Inside the dotted lines the ratio of any two of the velocity dispersions is no greater than 1:3 and inside the solid lines the anisotropy parameter $\beta$ is in the range $0 < \beta < 0.4$. Restricting the axial ratios of the density distribution would further rule out some sets of values inside the solid lines.
• Figure 4: The dependence of $T(v_{{\rm min}}, \, t)$ on the approximations used when calculating the Earth's orbit for, from top to bottom at t=0, $v_{{\rm min}}=100, 500, 400, 300$ and $200 \, {\rm km s^{-1}}$ with line types as in Fig. 1, for the standard halo model (upper panel) and for the fiducial triaxial model, see text for details, (bottom panel). Note the change in the phase of the annual modulation for $v_{{\rm min}}=100 \, {\rm km s^{-1}}$ (see text and Ref. revphase for further discussion of the change in phase which occurs for small $v_{{\rm min}}$). We have fixed $v_{{\rm c, \, h}} = 150 \, {\rm km \, s^{-1}}$ and $v_{\odot, \, {\rm pec}} = (10.0, 5.2, 7.2) \, {\rm km \, s^{-1}}$ here.
• Figure 5: As Fig. 4 for the dependence of $T(v_{{\rm min}}, \, t)$ on the values used for the Sun's velocity with respect to the LSR ($v_{\odot, \, {\rm pec}}$, in ${\rm km \, s^{-1}}$): (10.0, 5.2, 7.2) (Ref. hippbm, solid line), (0, 0, 0) (dotted line), (10, 15, 8) (Ref. stand, short dashed), (9, 12, 7) (Ref. ls, long dashed). The full expression for the Earth's orbit (Ref.ls and eq. \ref{['vels']}) is used here.