= Solution
For a fixed comoving cell, take physical volume $V=R^3$ and energy density $u=\rho c^2$. With negligible chemical potentials and fixed relativistic species, $u=AT^4$ and $P=u/3$. The <first law of thermodynamics> gives
$$
T\,dS=d(uR^3)+\frac u3d(R^3),\qquad dS=d\!\left(\frac{4AR^3T^3}{3}\right).
$$
Choosing zero entropy at zero temperature yields \b[$S=4R^3u/(3T)$]. Reversible expansion conserves this entropy; when species disappear, the effective entropy degrees of freedom must be changed rather than keeping $A$ fixed.
Before electron-positron annihilation, photons contribute two states, electrons plus positrons four, and three neutrino/antineutrino species six. Thus
$$
g_*=2+\frac78(4+6)=\frac{43}{4},\qquad\boxed{u=\frac a2g_*T^4=\frac{43}{8}aT^4.}
$$
After neutrinos decouple, the electromagnetic entropy alone is conserved through annihilation: $(2+7\times4/8)T_i^3R_i^3=2T_\gamma^3R_f^3$, while $T_\nu R_f=T_iR_i$. Consequently
$$
\boxed{\frac{T_\nu}{T_\gamma}=\left(\frac4{11}\right)^{1/3}\simeq0.714.}
$$
This assumes instantaneous decoupling, negligible neutrino masses/chemical potentials and complete electromagnetic thermalization. Small noninstantaneous corrections are outside this approximation.
In the <Pati-Salam model>, right-handed quarks and leptons fill a $(4,1,2)$ multiplet. The neutrino component has no colour, $T_L=0$ and hypercharge $Y=T_{3R}+(B-L)/2=1/2-1/2=0$. Hence $\nu_R$ is a <Standard Model gauge singlet neutrino>, with no ordinary weak or electromagnetic gauge coupling. It is \b[sterile under the low-energy Standard Model], not a singlet of the full unbroken Pati-Salam group. A heavy Majorana mass can arise after symmetry breaking. The representation alone fixes neither its mass nor the stability, mixing or relic abundance needed for dark matter.<Lepton number as the fourth color by Jogesh Pati and Abdus Salam>[1974 gauge embedding and right-handed neutrino]{ref}
\b[Space example: <AMS-02>], a magnetic spectrometer on the International Space Station. Tracker curvature determines rigidity and charge sign; transition-radiation detection and an electromagnetic calorimeter distinguish electrons/positrons from the much larger proton background. It measures charged cosmic-ray spectra, including positrons that could arise from halo-particle annihilation. This is an \b[indirect dark-matter search], despite directly recording the charged products: it does not measure scattering of neutral halo WIMPs, and astrophysical sources can mimic a positron excess. Calling it direct WIMP detection would misstate the technique.<First Result from the Alpha Magnetic Spectrometer on the International Space Station by AMS Collaboration>[2013 positron fraction and detector discrimination]{ref}
\b[Underground example: <XENON100>], a <dual-phase xenon detector>. A candidate WIMP scatters from a xenon nucleus, producing prompt liquid scintillation S1 and ionization electrons. Drift and extraction into gas produce delayed proportional scintillation S2. Their delay and light patterns reconstruct the interaction position; S2/S1 discriminates nuclear recoils from electron-recoil backgrounds. Underground shielding, fiducial cuts and calibration reduce backgrounds, though neutrons can imitate nuclear recoils. This is direct recoil detection; a null excess constrains cross sections for an assumed halo and interaction model.<The XENON100 Dark Matter Experiment by the XENON100 Collaboration>[Dual-phase time-projection chamber, scintillation and ionization]{ref}<Dark Matter Results from 225 Live Days of XENON100 Data by the XENON100 Collaboration>[2012 search found no significant dark-matter excess]{ref}
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2014/paper-1-pa2014-search-techniques.png]
{title=Underground scattering and space-based annihilation-product searches test different dark-matter signals}
{source=paper-1-pa2014-search-techniques.py}
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