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We revise the spin-dependent neutralino-nucleus elastic scattering comparingthe formalisms and approximations found in literature for the momentum transferdependent structure functions. We argue that one of the normalized structurefunctions of Divari, Kosmas, Vergados and Skouras is all that one needs tocorrectly take into account the detailed nuclear physics information providedby shell-model calculations. The factorization of the particle physics degreesof freedom from the nuclear physics momentum dependent structure functionsimplied by this formalism allows for a better understanding of the so-calledmodel independent method for setting upper limits. We further discuss thepossibility of experiments with spin-dependent sensitivity like COUPP to testor set limits on the proton spin-dependent cross section in the framework ofthe stau co-annihilation region of the constrained minimal supersymmetricstandard model. For this model with $A_0 =0$, we provide a fitting formula bywhich it is possible to convert an upper limit on the spin-independent crosssection as a function of the neutralino mass directly into an exclusion plot inthe ($m_{1/2}$, $\tan\beta$) plane.

On the formalism and upper limits for spin-dependent cross sections in dark matter elastic scattering with nuclei