**PHYS291 project**

For my PHYS291 project, I have been looking at recoil spectra of collisions of dark matter particles with nuclei in detectors on earth.

**Theory**

The aim is to work out a differential event rate dR/dE as a function of the recoil energy the nucleus receives. These spectra can be compared both between different masses for the dark matter particle and for different detector elements. The calculations will not rely on other properties of the dark matter particle than its mass, and an assumed cross section for scattering on nucleon which has been set to 4*10^{-43} cm^{2} for my calculations (no specific model assumed, just that it has to be very low based on experiments that have been done and not found anything).

I assumed that a dark matter particle collides elastically with a detector nucleus. Further it is assumed that the galactic dark matter distribution is an isothermal sphere, where the particles have a Maxwellian velocity distribution.

These assumptions give rise to the following simplified formula:

Where R0 is the total rate per unit mass per day, E0 is the most probable kinetic energy of incident DM particle (with the assumed velocity distribution this is around 30 keV) and r is a kinematic factor calculated from the particle masses.

F(q) is a nuclear form factor. This can have both spin-indepedendent and spin-dependent contributions. In this project I have only taken into account the spin-independent contributions, since the spin-dependent are in general much smaller and only relevant for some nuclei.

I have chosen to use the Helm form factor as an approximation for F(q), which is a common approximation for these types of calculation.

For more info on calculation of dark matter recoil spectra, see: https://iktp.tu-dresden.de/bndschool/lecture2_baudis.pdf

**Method**

The first script, PlotSpectra.C, asks the user for how many different recoil spectra they want to compare and whether the y-axis should be logarithmic or not. Then it asks the user to input the assumed mass of the dark matter particle and the mass number and element for the nucleus, for each graph. It adds all the spectra to a multigraph, draws the graph and saves the multigraph to a ROOT file.

The second script, CompareRates.C, takes the ROOT file made by the first script and compares the total event rates for the different graphs. The script asks the user to input a minimum threshold energy that the detector can measure. It then integrates the different graphs, from this minimum energy to the maximum energy in the plot, using the trapezoid method. In the end it outputs the various total rates to the user, and redraws the spectra with a vertical line indicating the threshold energy.

Scripts:

**Results**

Comparison between different dark matter masses for Xe 131:

Comparison between different detector elements for 100 GeV dark matter particle:

Comparison of total rates for Ge and Xe detectors for 100 Gev and 200 GeV dark matter particles with 25 keV threshold energy for detector:

Output from CompareRates.C:

Total rate for Ge 73 with 100 GeV DM: 1.76017e-05 events per day per kilogram

Total rate for Xe 131 with 100 GeV DM: 3.2073e-05 events per day per kilogram

Total rate for Ge 73 with 200 GeV DM: 3.61613e-06 events per day per kilogram

Total rate for Xe 131 with 200 GeV DM: 4.50087e-06 events per day per kilogram

**Conclusion**

The calculations show the dependence of the recoil spectra on both nucleus mass and mass of dark matter particle. Heavier nuclei get higher rates for low recoil energies than lighter nuclei. Another effect to consider is that heavier nuclei get "dips" in event rates from form factors at much lower energies than lighter nuclei. These types of calculation must be taken into consideration when designing experiments to detect dark matter.

**Further work**

The calculations in this project are estimates that show general features of dark matter spectra. A number of steps could be taken to improve accuracy. Some of these include:

- Using a more accurate dark matter velocity distribution.

- Using better approximation for form factors.

- Including contributions from spin-dependent scattering.

In addition, an interesting thing to include in the model would be the motion of the Earth relative to the dark matter halo. My calculations assume a stationary earth, but small annual modulations of the spectra due to the Earth's motion around the sun are an expected signature of dark matter for future experiments to find.