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Gennaro Miele » 7.Primordial Nucleosynthesis: theory and experimental data

Big Bang Nucleosynthesis as a probe for new physics

Big Bang Nucleosynthesis

Big Bang Nucleosynthesis (BBN) corresponds to the epoch of the early Universe when the primordial plasma behavied as a nuclear reactor synthetizing the abundances of light elements, mainly ²H, ³He, ⁴He and ⁷Li. For such elements the primeval abundances have been consequently modified by the galactic/stellar chemical evolution but at an extent which still allows for the inference of the primordial value. We will discuss this point later.

On the contrary, for heavier nuclei the amount synthesized in stars or as a consequence of stellar explosions is enormously larger than the primordial abundance thus making almost impossible to infere the intial value from present measurements.

The main framework of BBN, that emerged after the seminal paper by Alpher et al. (1948), has become one of the observational pillars of the hot Big Bang model. For sure it provides the most ancient and robust probe of the earliest phases of the Universe expansion nowdays testable.

We can simply describe BBN as the transition from the Universe at temperature of the order of 10 MeV (a hot and dense plasma of electrons, positrons, photons and nucleons in kinetic and chemical equilibrium), to an Universe at a temperature of few keV containing frozen abudances of light elements. For more details, we refer the reader to many excel- lent reviews on BBN such as (Sarkar, 1996), (Olive et al., 2000), (Steigman, 2007), (Iocco et al., 2009).

When the temperature was ≈ 10 MeV, Nuclear Statistical Equilibrium was at work, hence the Saha equation held. Fast nuclear and e.m. interactions kept nuclear species in chemical equilibrium. In that epoch since there were too many photons per baryons, η_B^-1≈10⁹ , baryon matter was all in the form of free neutrons and protons.

BBN in four steps

We can simply summarize the BBN process in terms of the following four steps

i) Initial conditions T > 1 MeV

ii) n/p ratio freeze out T ≈ 1 MeV

iii) D bottleneck T ≈ 0.1 MeV

iv) Nuclear chain 0.1 MeV > T > 0.01 MeV

Nuclei abundances at Nuclear Statistical Equilibrium (initial conditions)

Under Nuclear Statistical Equilibrium nuclei N(A,Z) with binding energy B(A,Z) are as abundant as

$\frac{n_{N(A,Z)}}{n_B} \sim A^{3/2} \eta_B^{\,A-1} \left( \frac{2 \zeta(3)}{\pi^2} \right)^{A-1} \left( \frac{2 \pi T}{m_N} \right)^{3(A-1)/2}$ $\exp\left( \frac{B(A,Z)}{T} \right)$

with the expanding of the Universe the temperature T decreased to such a value that CC weak reaction rates n ↔ p became too slow to guarantee n-p chemical equilibrium (soon after neutrinos decoupling). With the temperature below T_D≈ 0.7 MeV, the n/p density ratio departs from its equilibrium value and freezes out at the value n/p = exp(-Δm/T_D) ≈ 1/6, where Δm = 1.29 MeV is the neutron-proton mass difference. Such ratio is then reduced only by neutron decays.

At this stage, the photon temperature is already below the deuterium binding energy, but deuterium synthesis starts only when photodissociation process becomes ineffective. Such deuterium bottleneck is overcome at T_BBN ≈ 0.07 MeV for which the ratio n_2H/n_B ≈ 1.

Binding energy

Deuterium bottleneck

The Deuterium formation triggers the whole nuclear process network that produces heavier nuclei, until BBN eventually stops. As soon as deuterium forms, it is quite immediately burned into⁴He, which has the largest binding energy per nucleon among light nuclei. This nucleus represents the main BBN outcome. Deuterium is formed via proton neutron fusion reaction

$n + p \leftrightarrow \,^2\mbox{H} + \gamma$

The Saha equation for this process reads

$\frac{n_{2H}}{n_p n_n} = \frac34 \left(\frac{2 \pi \,(m_n+m_p-B_{2H})}{m_n m_p T} \right)^{3/2} \exp \left( \frac{B_{2 H}}{T}\right)$

B_2H≈ 2.2 MeV is the deuterium binding energy. When the temperature is of the order of 1 MeV it is easy to see that n_n = n_p = n_B= η_B n_γ and hence

$\frac{n_{2 H}}{n_B} \sim \eta_B \left(\frac{T}{m_N} \right)^{3/2} \exp \left( \frac{B_{2 H}}{T}\right)$

with m_N the average nucleon mass. From the previous expression one finds that for T ≈B_2Honly 10^-12baryons are in the form of deuterium. This means that in order to produce a sufficient amount of Deuterium to trigger the nuclear chain for the nuclides synthesis one has to wait till the photon temperature T reaches a value much smaller than B_2H . This is called the Deuterium bottleneck.

n/p ratio freeze out and Helium 4 production

As soon as deuterium forms (let us denote with T_BBNthe temperature at which the nuclear reaction network stars), it is quite immediately burned into ⁴He, which has the largest binding energy per nucleon among light nuclei. This nucleus represents the main BBN outcome, and its abundance can be quite accurately obtained by very simple arguments. Indeed, the final value n_4He , is very weakly sensitive to the details of the nuclear network, and a very good approximation is to assume that all neutrons which have not decayed at T_BBN are eventually bound into helium nuclei. This leads to the famous result for the helium mass fraction Y_p ≡ 4 n_4He/n_B

$Y_p \sim \frac{2}{1+ \exp(\Delta m/T_D)\exp(t(T_{\rm BBN})/\tau_n)} \sim 0.25$

with τ_n the neutron lifetime. To have a more accurate prediction of light elements yields one has to solve an involeved set of differential equations that will be presented in the following.

Neutron <–> Proton weak reaction rates

The weak reactions transforming n <–> p are the leading processes in fixing the neutron abundance at the onset of BBN and thus a key quantity in determining the ⁴He mass fraction. Inview of this, much effort has been devoted to refining the theoretical accuracy in evaluating these processes, which presently is at the order of 0.1%. The Born rates are the tree level estimates obtained with V-A theory and with infinite nucleon mass. As an example, for the neutron decay process (see (e) process on the figure) we have

$\omega_B ( n \rightarrow e^- + \overline{\nu}_e + p ) = \frac{G_F^2\left( C_V^2 + 3 C_A^2 \right)}{2 \pi^3} \,\int_0^\infty d {|\bvec{p}'| \,|\bvec{p}'|^2} \, q_0^2 \,\Theta(q_0)$

where p₀’= (p’²+m_e²)^1/2 and q₀= Δm – p₀’

List of weak rates transforming n <--> p and their rates as a function of e.m. temperature

Neutron <–> Proton weak reaction rates

${\times} \left[ 1 - f_{\bar{\nu}_e} (q_0)\right] \left[ 1 -f_e(p_0') \right]$

The rates for all other processes (a) – (f) can be simply obtained from the previous expression by changing: i) the statistical factors, ii) the expression for neutrino energy in terms of the electron energy and iii) electron energy integration limits. The accuracy of the results in the Born approximation is, at best, of the order of 9%. Such precision can be largerly improved by considering different types of corrections, which include electromagnetic radiative, finite nucleon mass and finite temperature radiative correction.

Nuclear reaction network

Nuclear processes during the BBN proceed in a hot e.m. plasma, but with a low nucleon density. The primordial medium has a significant population of free neutrons and expands and cools down very rapidly. These conditions lead to an out of equilibrium nuclear burning.

The low density suppresses the three–body reactions, and an enhanced effect of the Coulomb barrier inhibits any reaction with interacting nuclei charges Z₁ Z₂ ≥ 6. Due to the lack of tightly bound nuclides with A =5 – 8, a bridge for the synthesis of heavier and more stable isotopes such as ¹²C, BBN is producing low mass nuclei only, and mainly ⁴He. On the contrary, in stars due to a much higher density environment, ¹²C is copiously produced via the triple-α process, which is very suppressed during BBN.

In BBN The most efficient categories of reactions are proton, neutron and deuterium captures (p,γ), (n,γ), (d,γ), charge exchanges (p,n), and proton and neutron stripping (d, n), (d,p)$. Note that we are using the standard nuclear physicists notation. For example, p(n,γ)d denotes the neutron-proton fusion reaction n + p → ²H+γ. The leading nuclear reactions are graphically summarized in the Figure and listed in Table

Nuclear network

Toward Helium 4 – The main nuclear reaction channel for Helium 4 production during BBN

Nuclides considered for the nuclear chain

The set of differential equations ruling BBN

We consider N_nuc species of nuclides (the ones reported in the previous slide), whose number densities, n_i, are normalized with respect to the total number density of baryons, n_B, X_i=n_i/n_Bwhere i=n, p, ²H, ³He, ⁴He and ⁷Li

To quantify ²H, ³He, ⁴He and ⁷Li abundances, we also use in the following the parameters X_2H/X_p, X_3H/X_p, X_7Li/X_p, andY_p = 4 X_4He namely ²H, ³He and ⁷Li number densities normalized to hydrogen, and the ⁴He mass fraction respectively.

$\frac{\dot{R}}{R} = H = \sqrt{\frac{8\, \pi G_N}{3}~ \rho}$ (the Hubble parameter, H; definition)

$\frac{\dot{n}_B}{n_B} = -\, 3\, H$ (the rate of dilution due to Universe expansion of baryonic number density)

$\dot{\rho} = -\, 3 \, H~ (\rho + {\rm p})$ (entropy conservation)

$L \left(\frac{m_e}{T}, \varphi_e\right) \equiv n_{e^-} - n_{e^+} = \frac{n_B}{T^3}~ \sum_j Z_j\, X_j$ (Universe neutrality)

$\dot{X}_i = \sum_{j,k,l}\, N_i \left( \Gamma_{kl \rt ij}\,\frac{X_l^{N_l}\, X_k^{N_k}}{N_l!\, N_k !} \; - \; \Gamma_{ij \rt kl}\,\frac{X_i^{N_i}\, X_j^{N_j}}{N_i !\, N_j !} \right) \equiv \Gamma_i(X_j)$

where ρ and p denote the total energy density and pressure, respectively.

Neutrality of primordial plasma

$\rho = \rho_\gamma + \rho_e + \rho_\nu + \rho_B = \rho_{NB} + \rho_B$

${\rm p} &=&{\rm p}_\gamma + {\rm p}_e + {\rm p}_\nu + {\rm p}_B = {\rm p}_{NB} + {\rm p}_B$

where i,j,k,l denote nuclides, ρ_B and ρ_NB are the baryon and non baryonic energy density respectively, Z_i is the charge number of the i-th nuclide, and the function L(ξ,ω) is defined as

$L(\xi,\omega) \equiv \frac{1}{\pi^2}\int_\xi^\infty d\zeta~\zeta\, \sqrt{\zeta^2-\xi^2} \left(\frac{1}{e^{\zeta-\omega}+1} - \frac{1}{e^{\zeta+\omega}+1}\right)$

The set of N_nuc Boltzmann equations of the previous slide describes the density evolution of each nuclide specie, with Γ_klij the rate per incoming particles averaged over kinetic equilibrium distribution functions. While in fact chemical equilibrium among nuclides cannot be assumed, as BBN strongly violates Nuclear Statistical Equilibrium, it is perfectly justified to assume kinetic equilibrium, which is maintained by fast strong and electromagnetic processes. The equation of the previous slide involving the function L states the Universe charge neutrality in terms of the electron chemical potential, with φ_e≡μ_e/T and T the temperature of e^±, γ plasma.

PArthENoPE a public code for BBN

Primordial abundances as function of photon temperature

Astrophysical Observations

To compare the BBN predictions with the astrophysical observations one has to be able to infer the primordial abundances from the present data. This can be done for the light elements abundaces, since they have been modified by stellar/galactic evolution but at an extent which still allows for extracting the primordial value.

The main strategies to overcome this problem are the following:

1) To restrict the analysis to systems that due to their nature result only sligthly contaminated by stellar evolution

2) To develope models which are able to take into account the stellar/galactic evolution and hence to trace back, from the present observation, the initial and primordial value.

We will consider both these approaches by studing the possibility to infer the primordial abundances of ²H, ⁴He, ³He, ⁷Li and ⁶Li rispectively.

Deuterium

The astrophysical environments which seem the most appropriate are the hydrogen-rich clouds absorbing the light of background QSO’s at high redshifts.

To apply the method one must require that:

(i) neutral hydrogen column density is in the range 17 < log[N(HI)/cm-2] < 21;

(HI regions are interstellar cloud made of neutral atomic hydrogen)

(ii) the metallicity [M/H] is low in order to reduce the chances of deuterium astration;

(iii) the internal velocity dispersion of the atoms of the clouds is low, allowing the isotope shift of only 81.6 km/s to be resolved.

Only a small bunch of QAS’s pass the exam!

Observation of Lyman absorption lines in gas clouds in QAS’s at high redshift (z ≈ 2 – 3) with low metallicity C/H ≈ 0.01 – 0.001 (C/H)solar

Primordial Deuterium in QAS’s

Deuterium primordial abundance

Helium 4

⁴He evolution can be simply understood in terms of nuclear stellar processes which through successive generations of stars have burned hydrogen into⁴He and heavier elements, hence increasing the ⁴He abundance above its primordial value. Since the history of stellar processing can be tagged by measuring the metallicity (Z) of the particular astrophysical environment, the primordial value of ⁴He mass fraction Y_p can be derived by extrapolating the Y_p-O/H and Y_p - N/H correlations to O/H and N/H → 0. Hence the measurement strategy consists of the following steps:

i) Observation of ionized gas (He II - HeI recombination lines in HII regions) in Blue Compact Galaxies (BCGs) which are are the least chemically evolved known galaxies

ii) Y_P in different galaxies plotted as function of O and N abundances.

iii) Regression to “zero metallicity”

Regression to zero metallicity for different data sets

Different analyses for primordial 4He

i) Izotov et al. 04 reported the estimate Y_p=0.2421 ± 0.0021

ii) Olive et al 04 quoted the value Y_p=0.249 ± 0.009. A small sample size used and large uncertainties affecting analysis are responsible in this case for the very large error

iii) Fukugita et al. 06, based on a reanalysis of a sample of 33 HII regions from i) determined a value of Y_p=0.250 ± 0.004

iv) Peimbert et al 07, present a new 4He mass fraction determination, yielding Y_p=0.2477 ± 0.0029. This result is based on new atomic physics computations together with observations and photoionization models of metal-poor extragalactic HII regions.

All recent estimates are dominated by systematics. In Iocco et al. 09 we take as central value of Y_p the average (without weights) of the four determinations, while the systematic error is estimated as the semi-width of the distribution of the four best values

Y_p = 0.247 ± 0.002(stat) ± 0.004 (syst)

Finally, CMB anisotropies are sensitive to the reionization history, and thus to fraction of baryons in the form of ⁴He. Present data only allow a marginal detection of a non-zero Y_p, and even with PLANCK the error bars from CMB will be larger than the present systematic spread of the astrophysical determinations.

Helium 3

In stellar interior it can be either produced by ²H-burning or destroyed in the hotter regions.

All ³He nuclides surviving the stellar evolution phase contribute to the chemical composition of the InterStellar Medium (ISM). Stellar and galactic evolution models are necessary to trace back the primordial ³He abundance from the post-BBN data, at least in the regions where stellar matter is present.

Terrestrial Mesurement

From balloon measurements ³He/⁴He ≈ 10^-6
From continental rock ³He/⁴He ≈ 10^-8

Large spread of values, confirm the idea that the terrestrial He has no cosmological nature. Most of it is ⁴He produced by the radioactive decay of elements such as uranium and thorium. No natural radioactive decay produces ³He, hence its observed terrestrial traces can be ascribed to unusual processes such as the testing of nuclear weapons or the infusion of extraterrestrial material.

Helium 3 – Solar System and Local ISM

In the Solar System

The most accurate value was measured in Jupiter’s atmosphere by the Galileo Probe. The measurements support the idea of a conversion of D initially present in the outer parts of the Sun into ³He via nuclear reactions.

ProtoSolar Material (PSM) ³He/⁴He =(1.66 ± 0.05) 10^-4
Meteoritic gases ³He/⁴He =(1.5 ± 0.3) 10^-4

In the Local ISM
By counting the helium ions in the solar wind, the Ulysses spacecraft has measured

LISM ³He/⁴He =(2.48 ^+0.68_-0.62) 10^-4

note that ⁴He/H ≈ 0.1 . The result is not inconsistent with the idea that ³He at our galaxy location might have grown in the last 4.6 billion years since the birth of the Sun.

Helium 3 – Beyond Local ISM

The spectral spin-flip transition at 3.46 cm, the analog of the widely used 21-cm line of hydrogen is a powerful tool for the isotope identification, as there is no corresponding transition in ⁴He+.

The emission is quite weak, hence ³He has been observed outside the solar system only in a few HII regions and Planetary Nebula (glowing shell of gas and plasma formed by certain types of stars when they die) in the Galaxy.

The values found in PN result one order of magnitude larger than in PSM and LISM ³He/H =(2 – 5) 10^-4 confirming a net stellar production of ³He in at least some stars.

Expected a gradient in ³He abundance versus metallicity and/or distance.

In this framework applying a conservative approach one finds:

³He/H < (1.1 ± 0.2) 10^-5

No 3He dependence on environment metallicity, as predicted by chemical evolution models of Galaxy. Known as 3He problem.

Lithium 7

Lithium’s two stable isotopes, ⁶Li and ⁷Li, continue to puzzle astrophysicists and cosmologists. Spite & Spite (1982) showed that the lithium abundance in the warmest metal-poor dwarfs is independent of metallicity for [Fe/H]< -1.5. This is commonly called the Spite plateau and may be the lithium abundance in pre-Galactic gas provided by the BBN. The very metal-poor stars in the halo of the Galaxy or in similarly metal poor galactic globular cluster (GGC) thus represent ideal targets for probing the primordial abundance of lithium. Several technical and conceptual difficulties have been responsible for quite a long tale of ⁷Li determinations: [⁷Li/H ] ≡ 12 + log₁₀(⁷Li/H )

(Bonifacio et al. 97) [⁷Li/H ] = 2.24 ± 0.01
(Ryan et al. 99, 00) [⁷Li/H ] = 2.09^+0.19_-0.13
(Bonifacio et al. 02) [⁷Li/H ] = 2.34 ± 0.06
(Melendez et al. 04) [⁷Li/H ] = 2.37 ± 0.05
(Charbonnel et al. 05) [⁷Li/H ] = 2.21 ± 0.09
(Asplund et al. 06) [⁷Li/H ] = 2.095 ± 0.055
(Korn et al. 06) [⁷Li/H ] = 2.54 ± 0.10

Spite & Spite plateau

Lithium 7 – primordial abundance

It is unclear how to combine the different determinations in a single estimate, or if the value measured is truly indicative of a primordial yield. A conservative approach (similar to the one used for ⁴He) is to quote the simple (un-weighted) average and half-width of the above distribution of data as best estimate of the average and “systematic” error on ⁷Li/H, obtaining

⁷Li/H = (1.86 ^+1.30_-1.10) 10^-10

We have a substantial disagreement, as a factor 1.5 – 2, between BBN prediction and observations. What could be the reason?

a value of Ω_Bh² lower than a factor 1.5 – 2 at the BBN epoch with respect to the best fit deduced from CMB data is excluded by the agreement between D observations and CMB value for Ω_Bh², and also by the inferred upper limit of the primordial ³He abundance.
underestimated errors in the adopted nuclear reaction rates are now excluded: the laboratory measurements of the crucial ³He(α,γ)⁷Be cross section, its inferred rate from solar neutrino data and the measurement of the proposed alternative channel for ⁷Be destruction ⁷Be(d; p)2α
Systematic errors in the analysis, although in principle still possible, seem very unlikely. 3D model atmospheres did not result in a significant upward revision of results obtained from more primitive 1D atmospheres.

Perhaps, and more likely, the lithium abundance of very metal-poor stars is not the one of the primordial gas

Lithium 6

Even assuming some diffusion and turbulent mixing mechanism to explain the ⁷Li problem, still an issue remains with ⁶Li.

The presence of the fragile ⁶Li isotope, which is produced during BBN at the level of ⁶Li/H ~ 10^-15 – 10^-14 , has been recently confirmed in a few metal-poor halo stars, with some hint of a plateau vs. metallicity with abundance as high as ⁶Li/H ~ 6 x 10^-16

A great help in solving this issue might come from detecting lithium in a different environment!

Likelihood BBN Analysis

Bound on Baryon fraction and Effective number of neutrinos from BBN

As one can see from the upper figure, the Deuterium is much more sentitive to baryonic fraction than Helium 4. For this reason the Deuterium is the real responsible for fixing such cosmological paramater.

On the othe hand, as can be seen from the lower figure, ⁴He is much more sensitive to the amount of radiation (namely the effective number of neutrinos). In the same figure are overimposed the bounds on the same parameters (η₁₀ - N_ν) coming from CMB, SNIa, HST, etc. (G. Steigman ArXiv:0807.3004 [astro-ph]).