# Chapter 8 Primordial Nucleosynthesis * From $t\sim1\text{ s}$, a number of nuclear reactions involving baryons took place - BBN * Results * Lock up free neutrons * Produce $\ce{^4He, D, ^3H, ^7Li, ^7B}$ ## The Neutron-to-Proton Ratio * Before neutrino decoupling at around 1 MeV, neutrons and protons are kept in mutual thermal equilibrium through charged-current weak interactions $$ \ce{n + e+ <-> p + \overline{\nu}*\text{e}}\\ \ce{p + e- <-> n + {\nu}*\text{e}}\\ \ce{n <-> p + e- + \overline{\nu}\_\text{e}} $$ The last reaction suggests that free neutrons are not stable ($m\_n>m\_p$). If protons were unstable, the world would look totally different from what it is today * The relative number densities of neutrons and protons is $$ \left(\frac{\mathrm{n}}{\mathrm{p}}\right)*{\mathrm{eq}}=\exp \left\[-\frac{\Delta m c^{2}}{k T}\right]=\exp \left\[-\frac{1.5}{T*{10}}\right] $$ where $$ \Delta m=\left(m\_{\mathrm{n}}-m\_{\mathrm{p}}\right)=1.29\ \mathrm{MeV}=1.5 \times 10^{10}\ \mathrm{K} $$ and $T\_{10}$ is the temperature in units of $10^{10}\text{ K}$ * The equilibrium will be maintained so long as the timescale for the weak interactions is short compared with the timescale of the cosmic expansion $$ \frac{\Gamma}{H} \simeq\left(\frac{T}{1.6 \times 10^{10} \mathrm{K}}\right)^{3} $$ That $\Gamma/H\propto t^3$ comes from * In radiation dominated era $$ H\approx H\_0\sqrt{\Omega\_{\text{rad},0}(1+z)^4}\propto\Omega\_{\text{rad}}^{1/2}\propto u^{1/2} $$ where $u$ is the radiation energy density and $$ u=\frac12 g\_\*aT^4 $$ thus $H\propto T^{2}$ * The rate per neutron of the reactions is proportional to * The number of electron neutrinos, $n\left(\nu*{\mathrm{e}}, \overline{\nu}*{\mathrm{e}}\right)\propto T^3$ * The weak interaction cross section, $\langle\sigma\rangle\propto T^2$ Thus $\Gamma\propto T^5$ * As $T$ increases, the temperature falls to $T\_\text{d}$ that $$ k T\_{\mathrm{d}} \approx 0.8\ \mathrm{MeV}=\left(m\_{\mathrm{n}}-m\_{\mathrm{p}}\right)-m\_{\mathrm{e}} $$ where weak interaction rate falls rather suddenly below the expansion rate and the ratio $\text{n/p}$ is frozen - **neutron freeze-out** $$ \frac{\mathrm{n}}{\mathrm{p}}=\exp \left\[-\frac{\Delta m c^{2}}{k T\_{\mathrm{d}}}\right]=\exp \left\[-\frac{1.3}{0.8}\right]=0.20=1 : 5 $$ * The higher $T\_\rm{d}$ is, the higher the $\rm{n/p}$ ratio and the abundance of $\ce{^4He}$ will be. The precise value of $T\_\text{d}$ $$ T\_{\mathrm{d}}^{3} \propto \tau\_{1 / 2}\left(\frac{11}{4}+\frac{7}{8} \mathcal{N}\_{\nu}\right)^{1 / 2} $$ where $\tau*{1/2}$ is the neutron half-life for free decay and $\mathcal{N}*\nu$ is the number of neutrino families * We currently realize that there are 3 flavors of neutrinos * We've detected the relative rest mass of each flavor but have no idea of the absolute rest mass for any of them ## Deuterium Formation * Before the formation of deuteron, the neutrons are 'naked' and will decay through $$ \ce{n <-> p + e- + \overline{\nu}\_\text{e}} $$ with a mean lifetime $\tau\_\text{n}=880.3\pm1.1$ * Deuteron formation $$ \ce{p + n <-> d + \gamma} $$ * Exothermic, release 2.225 MeV of energy * Strong interaction, proceeds efficiently (compared to weak interactions) * Deuterium bottleneck * At $t\sim$ a few seconds, the temperature is not much lower than 2.2 MeV and, given that photons are a billion times more numerous than baryons, there are sucient photons with energies $E*{\gamma}>E*{\mathrm{D}}$ to destroy newly formed $\ce{D}$ by photo-dissociation * Wait until the temperature is low enough for a substantial concentration of $\ce{D}$ * The formation rate of $\ce{D}$ exceeds its photo-dissociation rate at $T\_\ce{D}\simeq8\times10^8\text{ K}\sim 70\text{ keV}$, which is attained at $t\sim300\text{ s}$ * $\text{n/p}$ has fallen by a factor $\exp \[-300 /(880.3)]=0.71$ to $0.14$ or $1:7$ * This is not too much, because $t<\tau$ * If $\tau$ were much shorter, all the neutrons would have decayed before BBN and only hydrogen would remain * the primordial helium mass fraction $$ {Y}\_\text{p}=\frac{4 {\mathrm{n}}/{2}}{\mathrm{p}+\mathrm{n}}=\frac{2 \mathrm{n}}{\mathrm{p}+\mathrm{n}}=\frac{2 {\mathrm{n}}/{\mathrm{p}}}{1+{\mathrm{n}}/{\mathrm{p}}}=0.25 $$ because virtually all neutrons surviving after 300 s are later incorporated into $\ce{^4He}$ By number - $^{4} \mathrm{He} / \mathrm{H}=1 / 12$ ## Nuclear Reactions * BBN stops at $\ce{^7Li}$ * no stable nucleus of mass number 5 or 8 exists - no new nuclei can be formed in collisions of $\ce{He + He}$ or $\ce{He + p}$ * Collisions between three nuclei are far too rare to contribute * Parameter $\eta \equiv n*{\mathrm{b}} / n*{\gamma}$ * How far the BBN nuclear reactions proceed before they are frozen out by **low density and temperature** * Relate to the cosmic density of baryons in units of the critical density, $\Omega*{\mathrm{b}, 0} h^{2}$ via the temperature of the CMB today, $T*{\gamma, 0}$ $$ \eta\_{10}=273.3 \Omega\_{\mathrm{b}, 0} h^{2}\left(\frac{2.7255\ \mathrm{K}}{T\_{\gamma, 0}}\right)^{3} $$ ![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2Fb48df4678730c3db30771f5fb56ae4ccd5913c1a.png?generation=1608873193431661\&alt=media) * $Y\_\text{p}$ has a minor dependence on $\eta$ * Essentially all of the neutrons end up in $^4\ce{He}$ * For larger $\eta$, the balance between $\ce{D}$ formation and photo-dissociation only moves to slightly larger $T\_\text{D}$ - fewer neutrons decay * $\ce{D/H}$ and $^3\ce{He/H}$ depend inversely on $\eta$ * The higher $\eta$, the more sufficient is the conversion of $\ce{D}$ to $^4\ce{He}$ via two-body reactions with $\ce{^3He, p, n, \ce{D}}$ * The same with $\ce{^3He}$, but it declines more gently because this nucleus is more robust * $\ce{^7Li}$ has a bimodal behavior * Produced via two channels * Low baryon densities $\ce{T + \alpha -> {\gamma} + ^7Li}$ * High baryon densities $\ce{^3He + \alpha ->{ \gamma} + ^7Be}$, $\ce{^7Be + n -> p + ^7Li}$ ## Measures of Primordial Abundances of the Light Elements * The relative abundances of the light elements created in BBN remained unaltered for the first \~ 200 Myr * Once the first stars formed at $z\sim20$, the BBN abundances began to be changed by the process of galactic chemical evolution * Metallicity increased * Elements heavier than $\ce{B}$ were synthesised in the interior of stars * Identify astrophysical environments that have not been polluted by metals formed in stellar nucleosynthesis ### $\ce{^4He}$ * The abundance has increases during post-BBN era * Created by the fusion of 4 $\ce{H}$ nuclei in the hot cores of stars * In the local universe, **the least massive galaxies are also the ones with the lowest fraction of metals** * Cannot turn a significant fraction of gas into stars * Cannot retain products of stellar nucleosynthesis in their shallow potential wells * Observation * Measuring the $\ce{^4He/H}$ ratio from the emission lines of $\ce{H II}$ region in the lowest mass, most-metal-poor galaxies known * $Y*{\mathrm{p}}=0.245 \pm 0.004\Rightarrow 0.018 \leq \Omega*{\mathrm{b}, 0} \leq 0.059 \text { for } h=0.675$ * Not sensitive ### $\ce{D}$ * Sensitive *baryonmarker* - steep inverse dependence on $\eta$ * The abundance has decreases during post-BBN era * Burnt in the late stage of star formation * Observations * Analysis of absorption lines in pockets of interstellar gas at high redshift * Underwent minimum processing through stars * Low metallicity * $(\mathrm{D} / \mathrm{H})\_{\mathrm{p}}=(2.527 \pm 0.030) \times 10^{-5}\Rightarrow(4.91 \pm 0.11) \times 10^{-2}$ ### $\ce{^7Li}$ * Both sythesised and destroyed during the lifetime of stars/produced by the interaction of high energy cosmic rays with atoms in IGM * Observations * A pair of weak absorption lines in the atmospheres of cool stars (oldest stars in the halo of the Milky Way) * Stars with $-3 \leq\[\mathrm{Fe} / \mathrm{H}] \leq-1.5$ have a plateau $ ^{7} \mathrm{Li} / \mathrm{H}=(1.6 \pm 0.3) \times 10^{-10}$ (we *hope* that this is a good indication for the primordial value) * $\ce{Li}$ Problems * For stars with lower metallicity, $^{7} \mathrm{Li} / \mathrm{H}$ is lower * $^{7} \mathrm{Li} / \mathrm{H}$ is three times lower than the BBN prediction ## Dark Matter * Baryons contributes less than $5\\%$ of the critical density * Non-baryonic dark matter * Does not interact with photons * Interacts with ordinary matter only through gravity * Baryonic dark matter * Stars and gas in galaxies contribute $\Omega\_{\text { stars, } 0} \sim 0.003$ * The remaining baryons are in gas in the halos of galaxies and in between galaxies