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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, ηB-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≈ 0.7 MeV, the n/p density ratio departs from its equilibrium value and freezes out at the value n/p = exp(-Δm/TD) ≈ 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 TBBN ≈ 0.07 MeV for which the ratio n2H/nB ≈ 1.

Binding energy

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)

B2H ≈ 2.2 MeV is  the deuterium binding energy. When the temperature is of the order of 1 MeV it is easy to see that nn = np = nB = ηB nγ 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 mN the average nucleon mass. From the previous expression one finds that for T ≈B2H only  10-12 baryons 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 B2H . This is called the Deuterium bottleneck.

n/p ratio freeze out and Helium 4 production

As soon as deuterium forms (let us denote with TBBN the 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 n4He , 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 TBBN are eventually bound into helium nuclei. This leads to the famous result for the helium mass fraction Yp ≡ 4 n4He/nB

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

with τn 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+me2)1/2 and q0= Δm – p0

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

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

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 Nnuc species of nuclides (the ones reported in the previous slide), whose number densities, ni, are normalized with respect to the total number density of baryons, nB, Xi=ni/nwhere i=n, p, 2H,  3He, 4He and 7Li

To quantify 2H, 3He, 4He and 7Li abundances, we also use in the following the parameters X2H/X, X3H/Xp, X7Li/Xp , and Yp = 4 X4He  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, ρB and ρNB are the baryon and non baryonic energy density respectively, Zi 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 Nnuc Boltzmann equations of the previous slide describes the density evolution of each nuclide specie, with Γklij 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 φe≡μe/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

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 Yp can be derived by extrapolating the Yp-O/H and Yp - 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) YP in different galaxies plotted as function of O and N abundances.

iii) Regression to “zero metallicity”

Regression to zero metallicity for different data sets

Regression to zero metallicity for different data sets


Different analyses for primordial 4He

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

ii) Olive et al 04 quoted the value  Yp=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 Yp=0.250 ± 0.004

iv) Peimbert et al 07, present a new 4He mass fraction determination, yielding Yp=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 Yp 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

Yp = 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 Yp, 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.

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

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 ΩB h2 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 Ωh2, 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 - Nν) coming from CMB, SNIa, HST, etc. (G. Steigman ArXiv:0807.3004 [astro-ph]).


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