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PArthENoPE: Public Algorithm Evaluating the Nucleosynthesis of Primordial Elements
O. Pisanti, A. Cirillo, S. Esposito, F. Iocco, G. Mangano, G. Miele, P. D. Serpico
TL;DR
BBN studies need accurate, public tools for predicting primordial nuclide abundances and examining cosmological scenarios beyond the standard framework. PArthENoPE solves the coupled BBN equations with updated reaction and weak-rate treatments, producing abundances for standard and extended models. Its predictions closely agree with corrected Wagoner/Kawano results for 4He in tested fiducial cases.
Problem
BBN research requires an updated public code for accurate abundance predictions and studies of physics beyond the Standard Model.
Method
PArthENoPE numerically solves the BBN equations using updated weak-rate treatments and a revised nuclear reaction network, with extensions for cosmological parameters and neutrino chemical potential.
Results
The code evaluates abundances for 26 nuclides and, in tested fiducial cases, its 4He predictions agree very closely with corrected Wagoner/Kawano results.
Takeaways & Limitations
PArthENoPE provides a public tool intended for accurate BBN-related studies and for constraining non-standard cosmological physics.
Abstract
from arXiv · showhide
We describe a program for computing the abundances of light elements produced during Big Bang Nucleosynthesis which is publicly available at http://parthenope.na.infn.it/. Starting from nuclear statistical equilibrium conditions the program solves the set of coupled ordinary differential equations, follows the departure from chemical equilibrium of nuclear species, and determines their asymptotic abundances as function of several input cosmological parameters as the baryon density, the number of effective neutrino, the value of cosmological constant and the neutrino chemical potential. The program requires commercial NAG library routines.
Program summary
PArthENoPE is a public numerical program for computing primordial light-element abundances by solving the coupled BBN equations across standard and extended cosmological scenarios.
- Implementation: The listed implementation uses Fortran 77, NAG libraries, and a BDF method for ODE integration, with a typical default run time of 90 seconds.The distributed program contains 4969 lines and 192 Kb of code, input card, and test data.
- Physical context: BBN produces light nuclides through a network of nuclear processes as the primordial plasma cools from a few MeV to approximately 10 keV.The resulting abundances can be compared with observations in different astrophysical environments.
- Purpose: PArthENoPE updates earlier public BBN codes and was released to support BBN studies and constraints on physics beyond the Standard Model.The program is publicly available and the paper describes its structure, use, extensions, and comparison with the Kawano code.
- Equation system: PArthENoPE follows N_nuc nuclide abundances together with the electron chemical potential, using thermodynamic and reaction-rate inputs.Nuclides are normalized to the total baryon number density, while reaction rates are thermally averaged cross section times relative velocity.
- Physical inputs: The program includes finite-temperature QED corrections and can follow neutrino-distribution distortions relevant to sub-percent BBN predictions.Its default scenario assumes vanishing neutrino chemical potential, while finite chemical potential can also be studied.
- Equation system: The model combines the Friedmann equation, baryon and entropy conservation, nuclide Boltzmann equations, and charge neutrality.The total energy density and pressure include photons, electrons, neutrinos, and baryons; reaction rates enter the nuclide evolution.
B. Numerical solution of the BBN set of equations
The numerical formulation recasts the BBN equations into an evolution problem in z = m_e/T and initializes the abundance variables at T = 10 MeV.
- Numerical formulation: PArthENoPE evolves the N_nuc + 1 unknowns (φ_e, X_j) as functions of z = m_e/T.In this formulation, charge neutrality supplies n_B as a function of φ_e.
- Initial conditions: The initial conditions are imposed at z_in = m_e/(10 MeV), corresponding to the high-temperature starting point of the numerical integration.The initial abundance construction uses the equilibrium formulation described in the paper.
- Initial conditions: The initial abundance expression is applied only when the resulting abundance exceeds the numerical zero YMIN, whose default value is 10^-30.The initial baryon-to-photon ratio is defined at T = 10 MeV.
- Reaction network: The nuclear reaction set includes a small network whose reactions are listed in Table II.The small network is the simpler PArthENoPE running option.
C. The Nuclear Chain
PArthENoPE provides selectable nuclear networks and parameterized extensions for non-standard physics, allowing users to vary reaction content and cosmological inputs.
- Non-standard physics: Beyond the standard scenario, PArthENoPE implements commonly used modifications involving neutrinos, gravity, and additional plasma degrees of freedom.In the standard case, the sole free BBN parameter is η or equivalently Ω_Bh^2.
- Network options: The intermediate and complete options add reactions to the small network, with their additions documented in Tables III and IV.The reaction tables correspond to progressively expanded network configurations.
- Non-standard physics: A cosmological constant contribution is parameterized by ρ_Λ and allowed over 0 ≤ ρ_Λ/MeV^4 ≤ 1.The parameter enters the equations through the energy-density treatment.
B. Extra degrees of freedom, ∆Neff
PArthENoPE parameterizes extra radiation and neutrino chemical-potential effects while exposing cosmological, nuclear-network, and output controls through interactive or card-based inputs.
- ΔNeff represents additional radiation energy density from non-electromagnetically interacting particles entering the Hubble expansion rate.
- The program accepts ΔNeff values from −3.0 to 15.0.
- Neutrino chemical potentials are represented by one shared degeneracy parameter ξ because neutrino oscillations imply approximate chemical equilibrium before BBN.
- For small |ξ|, the dominant BBN effect is a shift in proton–neutron beta equilibrium, while radiation-density and neutrino-reheating effects are subleading or negligible.
- Users can select physical parameters, one of three nuclear networks, reaction-rate variations, tracked nuclides, and output files through interactive or input-card modes.
B. PARTHENOPE
PArthENoPE resolves the BBN differential equations through initialized numerical routines, with carefully selected solver settings and a linearized reaction system.
- The resolution driver initializes the calculation, calls the NAG solver, and then prints the final output.
- NAG resolution parameters control the integration method and numerical accuracy and were chosen to optimize performance.
- The independent variable z runs from values corresponding to Ti = 10 MeV to Tf = 1/130 MeV, and these settings should be varied cautiously.
- Initialization computes nuclear parameters, nuclide abundances, and the electron chemical potential by inverting an implicit equation with a NAG routine.
- FCN evaluates thermodynamics and reaction rates, linearizes the equations into an Nnuc × Nnuc system, and solves it by Gaussian elimination.
E. OUTEVOL, OUTEND
PArthENoPE combines updated physical inputs, reaction networks, thermodynamics, and numerical solvers with a customizable interface and output system. Comparisons with Kawano show component updates can offset one another, while the code achieves high numerical accuracy and closely reproduces selected helium results.
- Physical updates: PArthENoPE implements improved n-p reaction fits, finite-nucleon-mass effects, non-thermal neutrino distortions, and neutrino asymmetry effects.
- Physical updates: The code improves electromagnetic and neutrino thermodynamics by accounting for entropy transfer during e+−e− annihilation rather than imposing simple entropy conservation.
- Numerical resolution: PArthENoPE replaces several numerical routines with more efficient algorithms, including a multistep BDF method implemented through NAG routines.The default accuracy parameters target a compromise between accuracy and running time.
- Numerical resolution: 10^-4 relative accuracy is reached for the 4He mass fraction, keeping numerical error below theoretical uncertainties.
- Physical updates: Improved weak-rate, plasma, neutrino, and nuclear-network treatments have effects with different signs that almost accidentally cancel.New and updated reactions nevertheless produce a large difference in the predicted BBN metallicity, probably insufficient to change primordial-cloud chemistry.
- Conclusions: PArthENoPE computes abundances for 26 nuclides in standard and extended BBN models, and its 4He results closely agree with appropriately corrected Kawano results.The public code incorporates updated weak-rate and nuclear-network studies for BBN-related applications.
APPENDIX A: DERIVATION OF THE PARTHENOPE SET OF EQUATIONS
The appendix derives the differential system used by PArthENoPE by combining cosmological evolution, plasma and neutrino thermodynamics, charge neutrality, and nuclide Boltzmann equations. The formulation evolves electron chemical potential and nuclide abundances using z = me/T, while modeling neutrino entropy transfer through N(z).
- Evolution variables: The numerical formulation evolves Nnuc + 1 unknowns, (φe, Xj), as functions of z = me/T, with charge neutrality supplying nB as a function of φe.
- Cosmological evolution: The cosmological equations combine total energy density and pressure from electromagnetic, baryonic, and neutrino components.The appendix writes ρ = ρeγB + ρν and p = peγB + pν.
- Neutrino treatment: The temperature-evolution equation includes N(z), a neutrino auxiliary function associated with entropy transfer during e+−e− annihilation.
- Nuclide evolution: The abundance equations describe nuclide-density evolution through thermally averaged reaction rates and charge numbers.Reaction-rate symbols represent cross-section times relative-velocity averages for incoming particles.
APPENDIX B: COMMON VARIABLES USED IN PARTHENOPE
Appendix B lists common physical, numerical, reaction-network, and input/output variables used by PArthENoPE. These variables configure the selected nuclides and reactions, reaction-rate changes, solver controls, cosmological quantities, and card-mode operation.
- Physical and nuclear variables: The common variables include nuclide atomic numbers and charges, spin degrees of freedom, nuclear partition-function coefficients, and weak-reaction fit parameters.
- Network variables: Reaction-network variables specify the number of nuclides and reactions, tracked nuclides, reaction changes, incoming-nuclide types, and multiplicative rate factors.
- Interface variables: Input and output variables include electron chemical potential, running mode, screen-evolution and overwrite options, output filenames, and card-reading controls.
- Numerical variables: Solver variables cover the independent-variable stepsize, previous stepsize, convergence and matrix-inversion flags, numerical abundance zero, and nucleosynthesis equations.
- Cosmological variables: Cosmological and thermodynamic variables include adimensional baryon density and pressure, the Hubble function, the neutrino-to-photon temperature ratio, and neutrino auxiliary function N(z).