= Solution
Gravitational evidence requires much more nonbaryonic matter than can be supplied by ordinary luminous material or by the baryon abundance permitted by <nucleosynthesis>. A stable, electrically neutral massive particle is a plausible cold component. Thermal production followed by <cosmological particle freeze-out> gives an additional motivation: an <annihilation> rate near a weak-interaction scale can leave an appreciable relic abundance. This is a mechanism and search hypothesis, not proof that the <dark matter> must be a <WIMP> or have any particular elastic-scattering <cross section>.
The plotted regions are conditional <WIMP> interpretations of different measurements. The experiments use different targets and distinguish signal from background differently:
* <DAMA experiment> uses NaI(Tl) <particle scintillation> and seeks an annual variation of low-energy single-hit counts as the Earth's velocity through the halo changes. Multiple-hit events and other <energy> ranges provide controls, but <particle scintillation> alone does not identify every event as an <electron> or <nuclear recoil>. Converting the modulation into <mass> and <cross section> requires a <halo model> and <nuclear-recoil quenching factor> assumptions.<New results from DAMA LIBRA by R. Bernabei and others>[abstract and single-hit annual-modulation analysis]{ref}
* <CoGeNT experiment> uses a low-capacitance p-type point-contact <germanium> diode to reach low <ionization> thresholds. Low-energy event <radiation spectrum> and a possible annual modulation can suggest light <dark matter>, but surface charge collection, internal x-ray lines and threshold efficiency must be distinguished from bulk recoil events. Its 2011 modulation paper described a modulated component of unknown origin rather than an unambiguous particle identification.<Search for an Annual Modulation in a P-type Point Contact Germanium Dark Matter Detector by C. E. Aalseth and others>[abstract and pulse/energy-event analysis]{ref}
* <CDMS experiment> measures <phonons> and <ionization> in cryogenic <germanium> or <silicon>. The smaller <ionization> yield of <nuclear recoils> and <phonon> timing help reject <electron> and surface events. The lowered-threshold <germanium> analysis improves low-mass sensitivity, at the cost of more difficult background discrimination near threshold; this explains why distinct 2010 and 2011 curves appear.<Results from a Low-Energy Analysis of the CDMS II Germanium Data by CDMS Collaboration>[abstract; low-threshold analysis and background discussion]{ref}
* <XENON100> uses a <dual-phase xenon detector>. A recoil produces prompt liquid <particle scintillation>, S1, and <ionization> <electrons>. A drift field transports the charge to the surface; extracted <electrons> produce gas <electroluminescence>, S2. Drift delay and the S2 light pattern reconstruct position, while S2/S1 statistically separates <electron> and <nuclear recoils>. An inner <fiducial volume> suppresses external radiation. Neutron and gamma <calibrations> establish recoil response and leakage, and low-light-yield <calibration> is particularly important for light <WIMPS>. Neutrons can themselves mimic <nuclear recoils>.<The XENON100 Dark Matter Experiment by the XENON100 Collaboration>[detector principle, signal reconstruction and calibration sections]{ref}
The plot also includes <CRESST experiment>, whose cryogenic calcium-tungstate modules combine <phonon> and scintillation-light signals. Different nuclei contribute to its recoil interpretation. Its excess region is another conditional candidate region, not an extra confirmed detection.<Results from 730 kg days of the CRESST-II Dark Matter Search by G. Angloher and others>[abstract and detector/recoil-population analysis]{ref}
Around ten GeV, the shaded regions favour <cross sections> roughly $10^{-41}$–$10^{-40}\,\mathrm{cm^2}$, while the <XENON100> solid curve and the improved low-mass CDMS curve cut into or exclude much of those regions under the common elastic, <spin-independent WIMP-nucleus scattering> interpretation expressed per <nucleon>. \b[There is tension within that shared model; the plot does not show mutually confirmed discoveries.] <electron> backgrounds, threshold response, <nuclear-recoil quenching factors>, channeling assumptions and the halo velocity distribution can shift interpretations. Changing the interaction model can also alter target comparisons, but cannot establish a <WIMP> by assertion. The 100-live-day <XENON100> result itself reported candidates compatible with estimated background.<Dark Matter Results from 100 Live Days of XENON100 Data by XENON100 Collaboration>[abstract; background-compatible candidates and elastic spin-independent limit]{ref}
For a rough dimensional event estimate, $1\,\mathrm{cm^2}=10^{-4}\,\mathrm{m^2}$ gives
$$
10^{-41}\text{--}10^{-40}\,\mathrm{cm^2}=0.01\text{--}0.1\,\mathrm{fb},
\qquad \mathcal L\sigma\sim20(0.01\text{--}0.1)=\boxed{0.2\text{--}2\ \text{events}}.
$$
If a collider production <cross section> had this value, that small yield, before acceptance and backgrounds, would not support a likely discovery. \b[The plotted nuclear-scattering <cross section> is not the LHC production <cross section>], however. Converting between them needs a mediator, couplings, parton distributions and production kinematics; a light stable <WIMP> would normally be inferred from <missing transverse momentum> with a visible recoil object, not a visible <WIMP> track. The <mass> makes <pair production> kinematically accessible, but the plot and <integrated collider luminosity> alone do not decide discovery probability.
For the relic estimate, define $M=mc^2$ and $\theta=k_BT$. The <sharp-freeze-out relic dilution> model gives
$$
\theta_f=\frac{M}{P}=\frac{10}{30}\,\mathrm{GeV},\qquad
n_f=6\left[\frac{M\theta_f}{(\hbar c)^2}\right]^{3/2}e^{-P}
=4.27\times10^{35}\,\mathrm{m^{-3}}.
$$
Assume stable particles, no subsequent <annihilation>, and $RT=\mathrm{constant}$. Conserved comoving particle number then gives
$$
n_0=n_f\left(\frac{T_0}{T_f}\right)^3
=6P^{3/2}\left(\frac{k_BT_0}{\hbar c}\right)^3e^{-P}
=\boxed{0.1497\,\mathrm{m^{-3}}}.
$$
The present rest-energy density and its critical fraction are
$$
\boxed{\rho_{\chi,0}=Mn_0\simeq1.50\,\mathrm{GeV\,m^{-3}},\qquad
\Omega_\chi\simeq\frac{1.50}{5}=0.30}.
$$
Thus the supplied toy model yields a cosmologically substantial density, of the right broad scale to motivate a dark-matter candidate. It is not an independent precision prediction: $P$ is imposed and the abundance depends exponentially on it. If <entropy degrees of freedom> change, <entropy> conservation instead gives an extra factor $g_{*s,0}/g_{*s,f}$ in $n_0$ when using the <photon> <temperature>. The species content needed to evaluate that correction is not supplied. A realistic calculation also follows residual <annihilations> with the abundance <Boltzmann equation> rather than stopping exactly at equilibrium freeze-out.
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2012/paper-1-recoil-signals.png]
{title=Complementary nuclear-recoil signals}
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