Chapter 8 Primordial Nucleosynthesis

PreviousChapter 10 Recombination and CMB NextChapter 7 Thermal History of the Universe

Last updated 4 years ago

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
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, $T{\gamma, 0}$

$Y_\text{p}$ 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

PreviousChapter 10 Recombination and CMB NextChapter 7 Thermal History of the Universe

Last updated 4 years ago

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, $T{\gamma, 0}$
  $\eta_{10}=273.3 \Omega_{\mathrm{b}, 0} h^{2}\left(\frac{2.7255\ \mathrm{K}}{T_{\gamma, 0}}\right)^{3}$

$Y_\text{p}$ 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

{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

\eta_{10}=273.3 \Omega_{\mathrm{b}, 0} h^{2}\left(\frac{2.7255\ \mathrm{K}}{T_{\gamma, 0}}\right)^{3}