Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 3

The new neutrino species undergoes thermal decoupling in cosmology when its interaction rate becomes comparable to the Hubble parameter. Equating the given rates givesWith $g_{\ast }\simeq 100$, $G_A=10^{-12}{\mathrm{GeV}}^{-2}$ and $m_{\mathrm{Pl}}\simeq 10^{19}\mathrm{GeV}$,This is an order-of-magnitude electroweak-scale estimate under the given approximation. The mass in this formula is the unreduced Planck mass, not the reduced Planck mass.

Count one neutrino species as the usual two thermally populated particle-antiparticle helicity states, with negligible chemical potential. Its entropy weight is $g_{\ast s,X}=(7/8)2=7/4$. In the ideal high-temperature Standard Model, the bosonic count is $16+8+4=28$ from gluons, electroweak gauge bosons and the Higgs field. The fermionic count is $72+12+6=90$ from quarks, charged leptons and the three ordinary neutrino species. Hence the relativistic entropy content of the Standard Model is $g_{\ast s,D}=28+(7/8)90=106.75$ for the remaining bath. Before the new species leaves, the total count is $108.5$; its separately cooling states are excluded from the subsequent bath count.

Near ordinary neutrino decoupling, the still-coupled bath hasfrom photons, electron-positron pairs and the three ordinary neutrinos. After those neutrinos decouple, the electromagnetic bath falls from $2+(7/8)4=11/2$ to two when the pairs annihilate. Cosmological entropy conservation in that bath gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$.

The new species has cooled as $T_X\propto a^{-1}$ since its earlier decoupling, whereas annihilation of Standard Model species heated the remaining bath. The temperature ratio of an early-decoupled neutrino is thusThese are ideal-gas estimates with the high-temperature count implicit in the question; precise electroweak thresholds change them slightly.

Entropy degrees of freedom weight temperature ratios with their cube, while energy degrees of freedom use their fourth power. The remaining Standard Model entropy today, in photon-temperature units, is $g_{\ast s,0}^{\mathrm{SM}}=2+(7/8)6(4/11)=43/11$. The new contribution is $(7/4)(T_X/T_{\gamma }{)}^3$. Therefore the fractional entropy of a relativistically decoupled species isAbout 1.6% of the total entropy is in the new species. Separate comoving entropy conservation also derives this fraction directly at decoupling. A model populating four helicity states instead would require the corresponding doubled species count.

Assume the new neutrino is stable and retains the abundance established by relativistic thermal decoupling in cosmology. While it remains relativistic, its contribution to the effective number of neutrino species isThus its effect on the radiation density and expansion during Big Bang nucleosynthesis is reduced relative to a fully heated extra neutrino species.

After it becomes nonrelativistic, its energy density is approximately $m_X{n}_X$. A two-state relativistic fermion relic has $n_X/n_{\gamma }=(3/4)(T_X/T_{\gamma }{)}^3\simeq 0.0275$. The density bound on a relativistically decoupled massive relic then givesHere ${\Omega }_{X,\max }$ is the allowed present matter-density fraction and $⟨E_{\gamma }⟩={\pi }^4{T}_{\gamma ,0}/(30\zeta (3))$. Choosing the total available matter fraction yields the basic overclosure bound. Its mass cannot be raised freely at a fixed relic number density.

There is also a perturbation constraint. Neutrino free streaming suppresses clustering below the distance traveled before the species becomes nonrelativistic. The ratio $T_X/m_X$ controls this distance, and the relic matter fraction controls the strength of the suppression. A significant hot or warm relic component must therefore satisfy the observed structure-formation constraints in addition to the abundance bound. Its allowed dark-matter contribution is limited jointly by these two effects. Numerical observational bounds depend on the chosen matter budget and perturbation data; the temperature and abundance dilution derived here are the factors needed to apply them.