We revisit Wagoner et al., a classic contribution in the development of Big Bang Nucleosynthesis. We demonstrate that it presents an incorrect expression for the temperature of the early universe as a function of time in the high temperature limit, Т>˜ 1010 K. As this incorrect expression has been reproduced elsewhere, we present a corrected form for the initial conditions required for calculating

the formation of the primordial elements in the Big Bang

Barnes, Luke A. (R19196)

Lewis, Geraint F.

Sharpe, Charlie

English

Western Sydney ResearchDirect

OPEN ACCESSBig Bang Nucleosynthesis Initial Conditions: RevisitingWagoner et al. (1967)Charlie Sharpe , Geraint F. Lewis , and Luke A. BarnesPublished May 2021 • © 2021. The Author(s). Published by the American Astronomical Society. Research Notes of the AAS, Volume 5, Number 5Citation Charlie Sharpe et al 2021 Res. Notes AAS 5 106DOI 10.3847/2515-5172/abf9ac306 Total downloads Turn on MathJaxShare this article1.  2.  3  4  AbstractWe revisit Wagoner et al., a classic contribution in the development of Big Bang Nucleosynthesis. Wedemonstrate that it presents an incorrect expression for the temperature of the early universe as afunction of time in the high temperature limit,  K. As this incorrect expression has beenreproduced elsewhere, we present a corrected form for the initial conditions required for calculatingthe formation of the primordial elements in the Big Bang.Export citation and abstract Previous article in issue Next article in issueOriginal content from this work may be used under the terms of the Creative CommonsAttribution 4.0 licence. Any further distribution of this work must maintain attribution tothe author(s) and the title of the work, journal citation and DOI.1. IntroductionBig Bang Nucleosynthesis (BBN) is considered an outstanding success and pillar of moderncosmology, explaining the abundance of primordial elements within the universe (Tytler et al. 2000;Barnes & Lewis 2020). Calculating the output of BBN requires integrating a series of coupleddifferential equations representing the sources and sinks of various elements, encompassing cross-sections that are temperature dependent. Such integrals, of course, depend upon the initialconditions.The structure of this research note is as follows; in Section 2, we discuss the process Wagoner et al.(1967) take in deriving the initial conditions for the integration of BBN in the early universe. We thendiscuss the issue with their results in Section 3 before concluding in Section 4.2. BBN Initial ConditionsIn their classic paper, Wagoner et al. (1967) derive the dependence of the temperature of the earlyuniverse on time, in the high temperature limit  K. To do this, they first derive the work-energyequation for the Friedmann–Lemaıtre–Robertson–Walker metric. The baryon mass density  scalesas  and the neutrino energy density  scales as , where R is the scale factor of the universe.After some algebra, they arrive at the following first-order differential equation for the expansion ,where  is the sum of the energy densities of electrons and radiation, and  is thesum of the pressures of electrons and radiation. They then combine this equation with theFriedmann equation to arrive at a differential equation for ,Now, at such high temperatures (  K), electrons are highly relativistic with energy density ,the same as neutrinos. The coefficient of  comes from the number of species and allowed spinstates of each respective particle. This gives  and . Hence, assuming an equation ofstate of  for highly relativistic species, they finally arrive at a relationship between the temperatureof the universe and time (their Equation (A15)),Here, T  is the temperature, G is Newton's gravitational constant,  is the radiation densityconstant,  is the Stefan-Boltzmann constant, and c is the speed of light. The right-hand expression gives the temperature of the universe in units of 10 K when the time, t, is measuredin seconds from t = 0.There are some interesting observations to be made about Equation (3), but perhaps the mostimportant is that while the second equality, , is correct, the first equality is incorrect. Mostobviously, the dimensions are incorrect. Working in cgs units, the central expression in Equation (3)has units of s  K , which is clearly inconsistent with T  having units of kelvin.3. Correcting the Initial ConditionsWagoner et al. (1967) state that details that are relevant to their derivation of Equation (3) areprovided by Alpher et al. (1953). However, this earlier paper finds that temperature follows, which has a different numerical factor from Equation (3). This discrepancy arises fromthe fact that Alpher et al. (1953) do not consider the presence of electrons or neutrinos at earlytimes, and hence use .Directly solving Equation (2) we find that, for time in seconds,This expression is dimensionally correct, with the central equation having units K, and yields thesame numerical factor as presented in Wagoner et al. (1967).4. ConclusionsNormally, such a minor typographical error in a paper has no significant consequences and so canpass unnoted. However, the incorrect expression presented in Wagoner et al. (1967) has beenpropagated into other key publications, and numerical codes for calculating BBN. We highlight twoof these below;Kawano (1992) presents a fortran implementation of the integration of the coupled BBNequations named NUC123. This directly reproduces the expression from Wagoner et al.(1967) as the initial condition for the integration (their Equation (D.6)). However, an examinationof the fortran source code reveals that the initial time is set to be t = 1/(const1∗t9)∗∗2!Initial time (Ref 1) where PARAMETER (const1  =  0.09615) !Relationbetween time and temperature, so only the numerical aspect of Equation (3) isemployed.Arbey et al. (2020) presents a C implementation of BBN known as AlterBBN. They too citeEquation (3) in determining the initial conditions of their integration. Exploring the source code,the initial time is set to double t = sqrt(12.∗pi∗G∗sigma_SB)/pow(Ti,2.); and sodirectly implements the incorrect form of the initial time. We note that in the natural (energy)units employed in AlterBBN results in a very early starting time, well into the high temperaturelimit, and so does not impact the resulting integration. We have discussed this with theAlterBBN author(s) and the initial conditions will be corrected in a forthcoming update (Arbey2021, private communication).This work was undertaken as part of Charlie Sharpe's honors project at the University of Sydney'sSchool of Physics.Footnoteshttps://github.com/ckald/KAWANO-sterilehttps://alterbbn.hepforge.org 1 1 21Article and author informationBibTeX  RISRelated links99−1 −19 3  4 Show ReferencesJournals Books Publishing Support Login Abstract1. Introduction2. BBN Initial Conditions3. Correcting the InitialConditions4. ConclusionsFootnotesReferences Back to topJournals Books About IOPscience Contact us Developing countries access IOP Publishing open access policy© Copyright 2023 IOP Publishing Terms & conditions Disclaimer Privacy & cookie policy This site uses cookies. By continuing to use this site you agree to our use of cookies.ReferencesThis site uses cookies. By continuing to use this site you agree to our use of cookies. 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Big Bang Nucleosynthesis initial conditions : revisiting Wagoner et al. (1967)

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Big Bang Nucleosynthesis initial conditions : revisiting Wagoner et al. (1967)

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