Ranku said:
In the sentence "WIMP-nucleon cross sections of 1.2x10-47cm2 at 1 TeV/c2 WIMPs", there is a relationship between cross section and mass. Is there a general formula that relates the two quantities, in that if there is a certain cross section that means it will be associated with a certain mass?
How Are Maximum Possible Cross-Sections Of Interaction Calculated For Different Possible Dark Matter Particle Masses In Direct Dark Matter Detection Experiments?
Their Assumptions About Dark Matter Near Earth
Direct Dark Matter Detection experiment scientists have modeled how many dark matter particles should have passed through their detector's area in the time period during which they gathered data.
This is based upon the estimated dark matter density (mass per volume) in the vicinity of the Earth, divided by different possible dark matter particle masses, and adjusted for plausible velocities of the dark matter particles relative to the detector.
The assumptions about the local dark matter mass density are based upon models of mass density of the inferred dark matter halo of the Milky Way in the vicinity of Earth, which in turn is based upon the dynamics of the Milky Way galaxy relative to a Newtonian, no dark matter assumption.
The assumptions about the velocity of dark matter particles are based upon galaxy dynamics and the amount of observed structure in the universe (which support a conclusion of warm or cold dark matter, that are defined by the mean velocity of the dark matter particles -- lower velocities would produce way too many satellite galaxies and subhalos, while higher velocities would prevent galaxies from forming at the rates that they are observed to form).
One paper from the LUX dark matter detection experiment for example, modeled what a signal ought to look by making the following assumption (on page four of
this paper) about the velocities of the dark matter particles and the Earth, in plain text, and about the estimated local dark matter density in the vicinity of Earth, which I have put in bold:
Nuclear-recoil energy spectra for the WIMP signal are derived from a standard Maxwellian velocity distribution with v0= 220 km/s, vesc= 544 km/s, ρ0= 0.3 GeV/cm3, average Earth velocity during data taking of 245 km/s, and a Helm form factor
Using Their Assumptions To Make Calculations
Then, the scientists determine based upon the actual data and the uncertainty in their data, the maximum number of events in excess of the predicted number of background events that they expect to see over the course of the experiment at different strengths of nuclear recoil in the protons and neutrons in the detector.
The amount of nuclear recoil in a signal event of a collision between a proton or neutron and a dark matter particle of a particular mass and assumed velocity can be related to the mass of the dark matter particle in the rare cases where it does interact with a nucleon.
More massive dark matter particles should give rise to larger average nucleon recoils but should happen less frequently (once backgrounds are subtracted out), while less massive dark matter particles should give rise to smaller average nucleon recoils but should happen more frequently (once backgrounds are subtracted out).
The more precise the nuclear recoil event detections are and the more precisely and accurately the background expectation nuclear recoil events can be modeled, the smaller the potential signal that can be due to experimental uncertainty alone will be, and the smaller the amount of the maximum dark matter cross-section which isn't just due to experimental uncertainty will be.
The maximum possible cross-section of interaction at a given hypothetical dark matter particle mass is then calculated based upon the maximum possible dark matter signal divided by the number of dark matter particles that passed through the area of their detector during the course of the experiment if the particles had that mass and produced the appropriate nuclear recoil in the cases where they did interact with the protons and neutrons in the detector.
The data are then analyzed for a wide range of dark matter particle masses hypotheses, doing a statistical hypothesis test at each possible dark matter particle mass.
This is then converted to appropriate units and displayed in a chart like this one:
From this
2015 paper (chosen at random).
Real Direct Detector Exclusion Curves Analyzed
The constraints on the cross-section of interaction at low masses gets much weaker because the backgrounds get bigger when you approach the masses of nucleons themselves making it hard to distinguish nucleon-nucleon collisions giving rise to nuclear recoils in a given mass range from dark matter-nucleon collisions for dark matter particles of similar mass.
The uncertainties at high masses arises, in part, from the fact that for a high enough dark matter particle mass, the predicted number of dark matter particles to pass through the detector during the experiment is so low that even with a fairly high cross-section of interaction it becomes more and more plausible that there would be no signal events at all during your data taking period as a matter of simple random chance.
The sweet spot in between where the cross-section is constrained to be smallest experimentally is where the backgrounds are small and well controlled, but the number of dark matter particles that are expected to pass through the detector during the course of data taking is still fairly large.
Real Collider Exclusions Curves Analyzed
The constraints from CMS (one of the LHC collider experiments) has a different shape because it is making assumptions about the energy scales at which dark matter particles could be produced at different masses in its collisions compared to missing energy at different energy scales that could be due to dark matter particles, after subtracting out the expected neutrino background (and other sources of missing energy like charged particle detector inefficiency).
All missing energy can be detected at CMS, even though the source of the missing energy has to be inferred in light of the experiment's models of the missing energy producing backgrounds.
So, the cross-section is estimated based upon what cross-section would be necessary to produce enough dark matter particles of a specific mass to account for all potentially non-background missing energy, rather than based upon the number of dark matter particles that could have interacted with a detector.
This approach turns out to be much less sensitive to the mass of the hypothetical dark matter particle.
Calibration
You can also test your direct dark matter detector to see that you haven't missed something important in your design by exposing it to a stream of ordinary particles with a known cross-section of interaction and known energies from a source that emits those particles.
For example, you could put an exactly measured amount of a very pure substance that experiences beta decay at a known rate with a known range of energies to see if the electrons and neutrinos that the beta decay from the source produces and sends through your detector matches what your model says it should in addition to your estimated backgrounds.
If it does, you can be confident that your modeling is experimentally confirmed and that dark matter particles with the properties you are testing for should behave the way that you expect them to if dark matter particles with those properties exist.
What Scientists Have Learned So Far
The state of the art direct dark matter detection experiments, like LUX, have determined for dark matter particle mass ranges from below 1 GeV to up to about 1,000 GeV that if dark matter particles in this mass range exist and have any interactions with ordinary protons and neutrons at all, that
the strength of this interaction described by the cross-section of interaction, is on the order of millions to billions of times weaker than the cross-section of interaction of a neutrino (which is the most feebly interacting kind of ordinary matter definitively determined to exist in the Standard Model so far).
In and of itself, that isn't a huge blow to the general idea of a dark matter particle hypothesis. One important class of proposed dark matter particles (which are called "sterile") has a zero cross-section of interaction with ordinary matter, and the LambdaCDM "standard model of cosmology" assumes that dark matter is, at a minimum, "nearly collisionless".
But the state of the art results are a big problem for theories like supersymmetric WIMP dark matter particle proposals, where the model you are using assumes that your dark matter particle candidates will have a particular, calculable, non-zero cross-section of interaction.
For example, if your dark matter particle candidate has a mass of 1 GeV to 1000 GeV and interacts with other particles only via the weak force and gravity, and has the same weak force cross-section of interaction as a neutrino does (which was precisely what scientists were expecting at first in the 1980s when both dark matter and supersymmetry were brand new promising ideas), your ship has sailed and you get a participation medal, but no Nobel prize, because that possibility was ruled out long ago.
