70 Equations ius rN that maximizes the radial number density nN ( r ) r2 of target nucleus alues, the contact interaction limit fails for mA 0 . 3 MeV. Rather than mentum term altogether, a slightly more sophisticated approach would be tution p2 (1 cos ✓CM ) ! µ2 N w2 . In this work, however, we keep the full p propagator and evaluate the capture rate numerically so that our results ut parameter space. We have confirmed that our results reproduce those n the corners of parameter space where simplifying assumptions are valid. match Ref. [48] in the large- mA 0 , point-like cross section limit. apture rates, it is convenient to re-express the di↵erential cross section ecoil energy ER = µ2 N w2 (1 cos ✓CM ) /mN in the lab frame. In the non- he expression simplifies to [49] d N dER ⇡ 8 ⇡"2 ↵X↵Z2 N mN w2 (2 mN ER + m2 A 0 )2 | FN |2 . (9) the Helm form factor [50], | FN ( ER )|2 = exp [ ER/EN ] , (10) y Gould [13, 14, 16] where v ( r ) is the escape velocity at radius r and u is the dar elocity asymptotically far from the Earth. The total capture rate is obtained by integrating Eq. (11) over the region of p pace where the final state dark matter particle has energy less than mXv2 ( r ) / hus gravitationally captured. The escape velocity v ( r ) and number densities n etermined straightforwardly from the density data enumerated in the Preliminary R arth Model [51]. Following Edsj¨ o and Lundberg [20], the target number den modeled by dividing the Earth into two layers, the core and the mantle, with ensities and elemental compositions given in Table I. The capture rate is then P N CN cap , where the rate on target N is CN cap = nX Z R 0 dr 4 ⇡r2 nN ( r ) Z 1 0 dw 4 ⇡w3 f ( w, r ) Z Emax Emin dER d N dER ⇥( E ) . ere ⇥( E ) = ⇥( Emax Emin ) imposes the constraint that capture is kinematically y enforcing that the minimum energy transfer, Emin , to gravitationally capture 6 radius r , which is distorted from the free-space Maxwell–Boltzmann distribution, f ( u ), by the Earth’s motion and gravitational potential. We follow the velocity notation introduced by Gould [13, 14, 16] where v ( r ) is the escape velocity at radius r and u is the dark matter velocity asymptotically far from the Earth. The total capture rate is obtained by integrating Eq. (11) over the region of parameter space where the final state dark matter particle has energy less than mXv2 ( r ) / 2 and is thus gravitationally captured. The escape velocity v ( r ) and number densities nN ( r ) are determined straightforwardly from the density data enumerated in the Preliminary Reference Earth Model [51]. Following Edsj¨ o and Lundberg [20], the target number densities are modeled by dividing the Earth into two layers, the core and the mantle, with constant densities and elemental compositions given in Table I. The capture rate is then Ccap = P N CN cap , where the rate on target N is CN cap = nX Z R 0 dr 4 ⇡r2 nN ( r ) Z 1 0 dw 4 ⇡w3 f ( w, r ) Z Emax Emin dER d N dER ⇥( E ) . (12) Here ⇥( E ) = ⇥( Emax Emin ) imposes the constraint that capture is kinematically possible by enforcing that the minimum energy transfer, Emin , to gravitationally capture the dark 6 This is hard to read! The real problem is that the audience has to read it at all.
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