Dark Matter Direct Detection - 2D Recoil Spectrum and Annual Modulation

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How to use

Purple dots are WIMPs streaming through an underground detector; a bright flash is a nuclear recoil whose energy is drawn from the spectrum below. Try: (1) Xenon, 100 GeV and watch the steep exponential spectrum; (2) Light WIMP, silicon - low mass needs a light target and a low threshold; (3) Argon, heavy WIMP - heavier WIMPs reach higher recoil energies; (4) Annual modulation - raise the time lapse and detector mass and follow the blue curve over a year; (5) switch the nuclear form factor off to see how coherence is lost at high momentum transfer. Rates are textbook values with 100% efficiency; the time lapse and the flash size are for display. Space pauses, R clears the event counts.

The science

Kinematics. In an elastic collision of a WIMP (mass m, speed v) with a nucleus (mass m_N) the nuclear recoil energy is E_R = (mu^2 v^2 / m_N)(1 - cos theta*), at most 2 mu^2 v^2 / m_N, with mu = m m_N/(m + m_N). A 100 GeV WIMP at 300 km/s gives a xenon recoil of only a few tens of keV.

Rate. For spin-independent coupling the cross-section on a nucleus is enhanced coherently, sigma_N = sigma_p A^2 (mu_N/mu_p)^2. The differential rate per nucleus is

dR/dE = (rho / m) sigma_N F^2(E) (m_N / 2 mu^2) c * integral_{v_min}^{v_esc} f(v)/v dv, v_min = sqrt(m_N E / 2 mu^2)

F^2 is the Helm form factor (loss of coherence at large momentum transfer). f(v) is the halo Maxwellian of width v0 seen from the moving Earth, truncated at the escape speed. Dividing by the nuclear mass gives events per kg per day per keV.

Annual modulation. Earth's speed through the halo is v_E = v_sun + v_orb cos(2 pi (t - 152.5 d)/365.25), highest in early June, so the rate for a heavy WIMP peaks then. For light WIMPs and thresholds near the kinematic limit the phase can invert.

Checks. (1) head-on collision conserves momentum and energy and gives 2 mu^2 v^2/m_N; (2) numerical velocity integral against 2 exp(-v_min^2/v0^2)/(sqrt(pi) v0) for a Maxwellian at rest; (3) a Monte Carlo of random WIMP velocities with isotropic scattering (flux weighted) reproduces the analytic recoil spectrum.

Simplifications. Spin-independent only, standard halo model with a sharp cut in the Earth frame, no detector efficiency or resolution, no backgrounds, no gravitational focusing, no Earth shadowing. Cross-sections here are illustrative; no claim about whether dark matter has these properties.