= Solution
\b[Particle candidate.] A stable, electrically neutral <weakly interacting massive particle> can become nonrelativistic before <cosmological particle freeze-out>. Its <annihilation> rate per particle $\Gamma=n\langle\sigma v\rangle$ eventually falls below the <Hubble parameter>; the surviving comoving number then remains approximately fixed. A weak-scale <annihilation> <cross section> can give a cosmologically significant <thermal relic>. This is a mechanism and a parameter-scale motivation, not a guarantee that every weakly interacting particle has the right abundance.
<supersymmetry> supplies <neutralinos>, mixtures of neutral electroweak <gauginos> and <higgsinos>. If a <neutralino> is the <lightest supersymmetric particle> and <R-parity> is conserved, it cannot decay solely into <Standard Model> particles and is stable. Its <electroweak interactions> can make it a <WIMP>. Conservation of <R-parity> and a neutral lightest state are additional assumptions; <R-parity violation>, <coannihilation>, composition and resonant <annihilation> can substantially change the conclusion.
\b[An underground test.] <XENON100> at Gran Sasso used a liquid-xenon target in a two-phase <time projection chamber>. A <WIMP> scattering elastically from a <xenon> nucleus could produce a low-energy <nuclear recoil>. The interaction makes prompt <particle scintillation>, S1, and <ionization> electrons. An <electric field> drifts the electrons upward; their extraction into <xenon> gas produces delayed <electroluminescence>, S2. The delay measures depth, while the S2 light pattern determines the transverse position. An inner <fiducial volume> suppresses surface backgrounds through <xenon> self-shielding; underground overburden reduces cosmic-ray <muons>, and shielding and material selection reduce <radioactivity>.<The XENON100 Dark Matter Experiment by the XENON100 Collaboration>[Detector design, signal formation and background-rejection principles]{ref}
The S2/S1 ratio separates most <electron recoils> from <nuclear recoils>. Gamma-source <calibration> maps the electron-recoil leakage; an americium–beryllium <neutron> source calibrates nuclear-recoil response and cut acceptance. Purity and position corrections must be measured, and selection fixed before unblinding the search region. <neutrons> can themselves produce single <nuclear recoils>, so discrimination is not perfect; material assays and transport simulations constrain this background. In the 224.6-live-day, 34-kg analysis, two candidate events were compatible with the expected $1.0\pm0.2$ background events. The result constrained scattering <cross sections> rather than establishing a discovery.<Dark Matter Results from 225 Live Days of XENON100 Data by the XENON100 Collaboration>[Pages 2–4, calibration, blinded selection, neutron background and candidate counts]{ref}
A predicted count also depends on the local <dark matter halo> density, distribution of <velocities>, recoil threshold and nuclear response. Thus a null result constrains a particle model under stated astrophysical assumptions; it does not exclude all <dark matter>. A collider missing-momentum signal would likewise need independent evidence for a cosmological lifetime.
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-xenon-signals.png]
{title=Original functional schematic of prompt light and delayed charge readout in a two-phase xenon detector}
\b[Relic <energy density> in the stipulated freeze-out approximation.] Let $M=mc^2=1900\ \mathrm{GeV}$ and $\theta=k_BT$. Writing the given distribution in <energy> units gives
$$
n_f=6\left(\frac{M\theta_f}{(\hbar c)^2}\right)^{3/2}e^{-P},
\qquad
\theta_f=\frac{M}{P}=52.78\ \mathrm{GeV}.
$$
By <sharp-freeze-out relic dilution>, after <cosmological particle freeze-out> conserve <particle number>, $n a^3=\mathrm{constant}$. In the approximation $aT=\mathrm{constant}$ for the background bath,
$$
n_0=n_f\left(\frac{\theta_0}{\theta_f}\right)^3
=6P^{3/2}e^{-P}\left(\frac{\theta_0}{\hbar c}\right)^3.
$$
Inserting the printed inputs yields
$$
n_f=5.5242\times10^{39}\ \mathrm{m^{-3}},\qquad
n_0=4.8766\times10^{-4}\ \mathrm{m^{-3}},
$$
and hence
$$
\boxed{\rho_0c^2=M n_0=0.927\ \mathrm{GeV\,m^{-3}},
\qquad\Omega_\chi=\frac{0.927}{5}=0.185.}
$$
This is approximately $93\%$ of the stipulated dark-matter <energy density>, $0.2\times5=1\ \mathrm{GeV\,m^{-3}}$, so the toy calculation gives the right order and is close to the target. At fixed $P$ the cancellation of $M$ in $n_0$ is exact for the given prefactor, while the present <energy density> remains proportional to $M$.
The equilibrium exponential must be evaluated at freeze-out, \b[not at today's <temperature>]: equilibrium no longer maintains the <particle number> afterward. More accurately, <comoving entropy conservation> gives $a^3g_{*s}T^3=\mathrm{constant}$ and multiplies the displayed $n_0$ by $g_{*s,0}/g_{*s,f}$, when both counts use the same bath-temperature convention. Thus $0.185$ is the constant-$g_{*s}$ estimate supported by the supplied toy model. A refined abundance would require additional <entropy degrees of freedom> and <annihilation> history: realistic <entropy> release, gradual freeze-out and the model's <annihilation> channels matter.
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-relic-sensitivity.png]
{title=Original calculation of the relic-density sensitivity to freeze-out parameter, with the printed mass and constant-entropy-degrees approximation}
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