Big Bang Nucleosynthesis A thesis for the degree Bachelor of Science Joakim Brorsson, Johan Jacobsson & Anton Johansson Supervisor: Christian Forssén Examiner: Gabriele Ferretti Department of Fundamental Physics Chalmers University of Technology Göteborg, Sweden 2010 Big Bang Nucleosynthesis Joakim Brorsson, Johan Jacobsson & Anton Johansson c©Joakim Brorsson, Johan Jacobsson & Anton Johansson, 2010 Supervisor: Christian Forssén Examiner: Gabriele Ferretti Department of Fundamental Physics Chalmers University of Technology Cover: The cosmic microwave temperature fluctuations from the 5-year WMAP data seen over the full sky. The average temperature is 2.725 Kelvin, and the colors represent the tiny temperature fluctuations, as in a weather map. Red regions are warmer and blue regions are colder by about 0.0002 degrees. Credit: NASA/WMAP Science Team Big Bang Nucleosynthesis Joakim Brorsson, Johan Jacobsson & Anton Johansson May 20, 2010 Supervisor: Christian Forssén Examiner: Gabriele Ferretti FUFX02 - Bachelor thesis at Fundamental Physics Department of Fundamental Physics Chalmers University of Technology Göteborg, Sweden 2010 Big Bang Nucleosynthesis Joakim Brorsson, Johan Jacobsson & Anton Johansson Supervisor: Christian Forssén Examiner: Gabriele Ferretti Department of Fundamental Physics Chalmers University of Technology SUMMARY The fundamental physical processes that govern the Big Bang nucleosynthesis (BBN) have been studied. BBN refers to the production of predominantly light nuclei in the early Universe, which occurs on the time scale of a few minutes after the bang. An initial intensive literature study was carried out, followed by computer simulations with the scientific code NUC123. The aim of the literature study was to build a theoretical basis from which observational support of BBN and key estimates of parameters could be understood, and in the case of the latter also reproduced. The emphasis has been placed on the time leading up to BBN, specifically the relation between time and temperature, the universal expansion and the baryon-to- photon ratio, in order to determine the onset of BBN. Additionally, different simulations, based on models with varying degrees of complexity, have been performed in order to verify the theoretical work and the estimates of key parameters. By mass the most important abundances were found to be 75.2 % 1H and 24.8 % 4He with help of the NUC123 software. These abundances were found to agree well with both observations and simulations referred to in literature. One important exception is 7Li for which the calculated abundance differs significantly from the observational values. Even though the over all good agreement is a strong evidence for the standard models for both BBN and the Big Bang, this discrepancy points to shortcomings in the theory. Simply put, neither of these models can be completely wrong, though they do not paint the whole picture either. Keywords: BBN, big bang nucleosynthesis, early Universe, nuc123, primor- dial nucleosynthesis. Contents 1 Introduction 1 1.1 Specific Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Standard Model of Particle Physics 4 2.1 Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 The Expansion 7 3.1 Hubble expansion . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Relativistic Model of the Expansion . . . . . . . . . . . . . . . 8 4 The Early Universe 9 4.1 The Very Early Universe . . . . . . . . . . . . . . . . . . . . . 9 4.2 The Early Universe . . . . . . . . . . . . . . . . . . . . . . . . 10 4.3 Freeze-out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Energy Density 19 5.1 The Baryon to Photon Ratio . . . . . . . . . . . . . . . . . . . 19 6 Relating Time and Temperature 23 6.1 Simple Model for Relating Time and Temperature . . . . . . . 23 7 Big Bang Nucleosynthesis 25 7.1 The Physical Process . . . . . . . . . . . . . . . . . . . . . . . 25 7.2 The Impact of η . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.3 Calculating the Fraction of High Energy Photons . . . . . . . 36 8 Simulations 38 8.1 Calculation of Tfreeze-out . . . . . . . . . . . . . . . . . . . . . . 38 8.2 Simple Predictions . . . . . . . . . . . . . . . . . . . . . . . . 38 8.3 Big Bang Nucleosynthesis Using NUC123 . . . . . . . . . . . . 40 8.4 Simulation of the Time Evolution of BBN . . . . . . . . . . . 42 8.5 BBN Calculations for a Range of Value on η . . . . . . . . . . 44 9 Discussion 49 Bibliography 52 A Glossary 56 v B List of Symbols 60 C Elaborate Deduction of t(T ) 62 C.1 t(T ) for Temperatures 1012 K > T > 5.5 · 109 K . . . . . . . . . 62 C.2 t(T ) for Temperatures 5.5 · 109 K > T > 109 K . . . . . . . . . 73 D Evaluation of Important Integrals 78 D.1 ∫∞ 0 x2e−x2 dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 D.2 ∫∞ 0 xm−1dx ex±1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 E Programs 81 E.1 bbn.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 E.2 manyruns.sh . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 E.3 analyzedata.m . . . . . . . . . . . . . . . . . . . . . . . . . . 81 E.4 analyzematdata.m . . . . . . . . . . . . . . . . . . . . . . . . 81 E.5 analyzeautodata.m . . . . . . . . . . . . . . . . . . . . . . . 81 E.6 analyzeautomatdata.m . . . . . . . . . . . . . . . . . . . . . 81 E.7 partphotons.m . . . . . . . . . . . . . . . . . . . . . . . . . . 81 E.8 freezeout.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 E.9 TempofTime.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 E.10 Blackbody.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 E.11 canon.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 List of Figures 1 Abundances in the solar system . . . . . . . . . . . . . . . . . 3 2 nn/np as a function of T . . . . . . . . . . . . . . . . . . . . . 17 3 Relation between time and temperature . . . . . . . . . . . . . 25 4 Small reaction network . . . . . . . . . . . . . . . . . . . . . . 27 5 Binding energies of nuclei . . . . . . . . . . . . . . . . . . . . 29 6 The evolution of nD/(nn · np) with temperature . . . . . . . . 33 7 Fraction of high energy photons . . . . . . . . . . . . . . . . . 39 8 Large reaction network . . . . . . . . . . . . . . . . . . . . . . 41 9 Abundances relative to hydrogen . . . . . . . . . . . . . . . . 43 10 Abundances in mass percentage . . . . . . . . . . . . . . . . . 43 11 Abundances as a function of η . . . . . . . . . . . . . . . . . . 45 12 Abundance of 7Li as a function of η . . . . . . . . . . . . . . . 46 13 Abundances as a function of η in mass percentage . . . . . . . 47 14 Onset of BBN as a function of η . . . . . . . . . . . . . . . . . 48 vi List of Tables 1 The four forces. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Properties of quarks. . . . . . . . . . . . . . . . . . . . . . . . 5 3 Properties of leptons. . . . . . . . . . . . . . . . . . . . . . . . 6 4 Prediction of relative abundances . . . . . . . . . . . . . . . . 36 5 Final abundances after decay . . . . . . . . . . . . . . . . . . 44 6 Simple estimates of light element abundances by mass. . . . . 44 7 Observed and calculated abundances. . . . . . . . . . . . . . . 50 vii 1 Introduction Big Bang nucleosynthesis, often abbreviated BBN, refers to the network of nuclear reactions governing the formation of light elements, most significantly 2H, 3He, 4He and 7Li, in the early Universe [1]. More precisely, BBN is thought to begin 0.01 seconds after the big bang before coming to an end about 30 minutes thereafter [1]. It is also estimated that the rapidly expand- ing Universe, filled with a dense gas of particles and radiation, cooled from about 1011 to 109K during this time [1]. Remarkably, the primordial nucleosynthesis is one of the most easily sim- ulated processes in the entire field of astrophysics [2]. As such, computational models of BBN yield results that are quite accurate compared to inherent errors in the observational and experimental data that are put into the equa- tions [1, 2]. Many of the physical constants of importance for this process can be accurately measured in laboratories, because the relevant energy ranges are obtainable in a laboratory environment [1]. Consequently, modern BBN calculations for determining the abundances of light elements are carried out with only a single parameter, the baryon density [1]. These easily achievable precision calculations, under the assumption that the standard model of the Big Bang holds true, has and hopefully will help to shed light on both the preceding and following history of the Universe [1]. Indeed, “there are presently three observational evidences for the Big-Bang model: the universal expansion, the Cosmic Microwave Background (CMB) radiation and Primordial or Big-Bang Nucleosynthesis (BBN)” [3]. In 1929 Edwin Hubble and Milton Humason discovered that the velocity at which galaxies travel away from the earth is proportional to the distance between the earth and the galaxy. This means that the Universe is expand- ing, and it confirmed what Georges Lemâıtre had proposed two years earlier in his “hypothesis of the primeval atom” which later was termed the Big Bang theory. At present the Universe is large and cold, but because of the expansion we can extrapolate backwards to when the Universe was very hot and dense. The idea of the primordial nucleosynthesis, that is the creation of nuclei before the galaxies were formed, first appeared in the 1940s in the work of Gamow and his collaborators [4]. Despite some errors with regards to the physics involved in the process, they were able to predict the existence of cosmic background radiation, which after it was discovered in 1965 gave essential evidence not only for BBN but the big bang model as a whole [4]. Since that time the subject has evolved significantly both with regard to the underlying theory and the computational models. During the last three decades BBN calculations has been able to determine the above mentioned 1 baryon density with an unprecedented accuracy [1]. Further evidence of BBN, as a theory, comes from the fact that the ratio of 1H and 4He, predicted abundances of the light elements, 2H, 3He, 4He and to a lesser extent 7Li, agrees very well with observational measurements[1, 2]. This despite of the fact that these values spans nine orders of magnitude, since the ratio of the mass density of 7Li to 4He is in the order of 10−9 [1, 2]. Such comparisons, however, have relied heavily upon the contemporary understanding of the chemical evolution, that is the constant change in the chemical composition of matter because of nuclear transformation in for ex- ample stars [1]. This predicament stems from the fact that the abundance measurements can only be compared with the output from standard BBN cal- culations once they have been extrapolated to primordial abundances [1, 2, 3]. The situation has since changed entirely in light of new precise measure- ments of the CMB, which have been used to fix the baryon density [1, 3]. Thus, the last unknown in BBN calculations has been deduced, which in turn determines the primeval abundances of the light elements. Therefore, it is now possible to use these exact calculations to research the chemical evolution that has since taken place [1, 2]. Even so, it should be noted that the discrepancy between 7Li abundances as calculated with BBN and the ob- served values remains quite large [2, 3]. While there exists many suggested explanations for this result, no clear solution to this problem has emerged as of yet [2, 3]. The success of the standard model for BBN has enabled it to be used as a tool for probing new physics, such as alternative theories of gravity or the existence of new light particle species [1, 2]. For instance, calculations on primordial nucleosynthesis used to give the best possible constraint on the number of neutrino flavors, before being overtaken by precise laboratory measurements in the late 1980’s [1]. However, now that the baryon density has been fixed it would be possible, if the uncertainties in determination of the 4He abundance can somehow be reduced, for BBN calculations to put a comparable limit on the number of neutrino flavors, thereby cooperating with laboratory experiments to put bounds on new physics [1]. This prospect serves to exemplify how the BBN theory will continue to nurture the bond that it had previously helped forge between cosmology and nuclear and par- ticle physics [1]. With regards to the amount of time and resources that is put into re- searching the big bang and its implications, it is apparent that the interest for these events within the scientific community is quite substantial. More- over, the diverse stories of creation that appears in scripture are a testimony to the fact that the origin of humanity has been an ever present subject within the minds of scholars and philosophers for thousands of years. With- 2 0 10 20 30 40 50 60 70 80 90 100 10 −4 10 −2 10 0 10 2 10 4 10 6 10 8 10 10 10 12 H He Li Be B C F Ne Na Si Ca Fe Ni Zn Ge Sr Mo Ag Sb Cs Ba Nd Yb Au Hg Pb Th U Atomic Number (Z) A bu nd an ce o n a S ca le w ith S i = 1 06 Abundances In the Solar System Figure 1: Abundances of elements in the solar system, data taken from [5]. out doubt, this will remain true at least for the foreseeable future and in so doing propel mankind to delve ever deeper into the story of the early Universe. 1.1 Specific Aims The project aim is to study BBN and the formation of the light atomic nuclei, and consists of two main parts. The first part consists of a literature study to find the main observations that support the Big Bang theory in general and BBN in particular. The goal is also to find, understand and be able to reproduce the main parameters and conditions that describe the Universe prior to BBN. Mainly because these properties are essential for any effort to determine the outcome and the duration of BBN. Important aspects of these quantitative estimates is the time frame of BBN and the production of key isotopes. The second part is to be based upon calculations with a computer model using the parameters and key estimates made from literature as input data. As the reaction networks that describe the BBN process are complex an avail- able scientific code will be used to calculate the abundances. Hopefully, these calculations will help to explain the measured abundances of the elements. For instance, these ought to yield some clues to why the elemental abun- dances in the solar system have been observed to be distributed according to figure 1 and [5]. 3 2 The Standard Model of Particle Physics As explained in [6] the theory came out of advances made in physics in the 20th century. Dirac combined quantum mechanics, electromagnetism and special relativity in his famous equation forming the first step towards quantum field theory. The first interaction to be successfully described within a field theory was that between the electron and the electromagnetic field. According to the standard model there are four fundamental forces, or interactions, in nature [7]. These are gravity, the weak nuclear force, the elec- tromagnetic force and the strong nuclear force. Each type of interaction has its own associated particles, called bosons, as outlined in table 1. Particles in a quantized interaction field will, in other words, interact by exchanging bosons. The members of this group of particles are characterized by having integer spins and that they obey Bose-Einstein statistics. Particles that have half integer spins are instead called fermions and obey Fermi-Dirac statistics. Table 1: The four forces. Force Boson Spin gravity gravitons (hypothetical) 2 weak nuclear force W+,W−,Z 1 electromagnetic force photons 1 strong nuclear force gluons 1 Particles are divided into groups depending on which force that they can interact with. On the scale that concerns particles, gravity plays a minor role and it will not be dealt with any further. Charged particles, such as electrons, interact with the electromagnetic force, while The weak force interact with all particles. The strong force however, only interact with at particular set of different species. Specifically, particles that can interact with the strong force are called hadrons and those that do not are called leptons. 2.1 Hadrons Hadrons are particles formed from quarks that interact with the strong force. The quarks are in turn elementary particles that can not exist freely and hence have to be combined. These are termed elementary since they can not be divided into smaller particles. There are six different types of quarks that all have corresponding anti particles, the properties of which are shown in table 2, as can be read in [7]. 4 Table 2: Properties of quarks. Quark Symbol Mass [MeV/c2] Charge [e] B Anti particle up u 5 +2/3 +1/3 ū down d 10 −1/3 +1/3 d̄ charm c 1500 +2/3 +1/3 c̄ strange s 200 −1/3 +1/3 s̄ top t 1.7·105 +2/3 +1/3 t̄ bottom b 4300 −1/3 +1/3 b̄ Quarks can be combined in two ways, either three quarks taken together or one quark and one anti quark. The former combination forms a group called baryons and the latter forms mesons. The most familial baryons, that is the proton and the neutron, both consists of up and down quarks, with the proton having (uud) and neutron (udd) [6]. Since a certain anti quark have the same mass as the corresponding quark but negative baryon number and charge, the baryon numbers of baryons and mesons are 1 and 0 respectively. This follows since, to our current knowledge, all reactions conserve the baryon number. 2.2 Leptons The first elementary particles to be discovered was the electrons, which are part of the group of particles named leptons, as described in [6]. There are three families of leptons, each of which consists of a particle and an accom- panying neutrino as well as the corresponding anti-particles. The properties of each member of the above mentioned families are shown in table 3. It shall also be noted that both particles in an particle-antiparticle pair have the same mass and spin, yet opposite charge. As in the case of baryons, there exists a so called lepton number, which equals 1 for leptons, -1 for their corresponding antiparticles and 0 for non- leptons. Like the baryon number, both lepton number and electric charge are conserved in any reaction. 2.3 Bosons As was mentioned earlier, each type of fundamental interaction in nature can be described as an exchange of bosons. The weak force is carried by W+,W− and Z bosons, the first two are charged and forms a particle-antiparticle 5 Table 3: Properties of leptons. Particle Symbol Mass [MeV/c2] Charge [e] Anti particle Electron e− 0.511 −1 e+ Electron neutrino νe < 1 · 10−7 0 ν̄e Muon µ− 105.7 −1 µ+ Muon neutrino νµ < 1 · 10−7 0 ν̄µ Tau τ 1777 −1 τ̄ Tau neutrino ντ < 1 · 10−7 0 ν̄τ pair while Z is uncharged [7]. As a result of the uncertainty principle, these particles, with masses between 80-90 GeV/c2, are very short ranged [6]. On the other hand photons, like gluons, are massless and thus expected to have infinte range. The latter species are carriers of the strong force and are therefore responsible for making the quarks stick together as well as getting protons and neutrons to combine to form nuclei. 6 3 The Expansion 3.1 Hubble expansion In 1929 Edwin Hubble noted that all distant galaxies in all directions seemed to be moving away from us [8], and even more remarkably, that their velocities were directly proportional to the intermediate distance. In short, the velocity was found to be described by Hubbles law (3.1.1) [8]: v = HR (3.1.1) where H is the Hubble parameter and R is the relative distance between the two objects. Furthermore, the current value of H is often referred to as the Hubble constant, H0, which in turn is sometimes expressed in terms of the dimensionless Hubble parameter, h, in accordance with (3.1.2) [9, 10]. H0 = h · 100km/(sMpc) ≈ 100h 3.0857 · 1019 m/(sm) ≈ 3.241 · 10−18 · h m/(sm) (3.1.2) as derived from the latest WMAP measurements, since 1 pc = 3.0857 ·1016 m [11, 12]. If distant objects seem to be moving away from Earth in all possible directions it might be assumed that the earth would be in the very center of the visible Universe[8]. Although this would undoubtedly be remarkable, the truth is even more so. Even though it may appear as if distant objects move away relative to the earth, it is in fact space itself that stretches between the earth and the objects [8]. This means that neither of them actually moves [8]. To illustrate this effect it is possible to paint spots on a half inflated balloon and watch how the spots appear to move away from each other as the balloon is filled with air. Alternatively, one can drink a shrinking potion, like Alice did in wonderland. As one shrinks together with Alice it may appear as if she is moving away, when in actuality both are standing still. With this new view on the expansion it is now possible to regard R from (3.1.1) as a cosmic scale function [8]. Since (3.1.1) is linear, there is no reason, if neglecting gravitational effects, to think that the Hubble constant, and thus the expansion rate, has changed from the time of the early Universe[8]. If this assumption holds true it would be possible to find an upper limit on the age of the Universe(3.1.3)[8]. t = R/v = R/H0R = 1/H0 (3.1.3) 7 With H0 = 71.0±2.5 km/(sMpc) ≈ 71.0 km/s/Mpc ≈ 71.0/3.0857·1019 m/s/m ≈ 2.3009 · 10−18 m/s/m, since 1 pc = 3.0857 · 1016 m, one finds that t ≈ 13.78 billion years [12]. 3.2 Relativistic Model of the Expansion As it turns out, the Universe does not behave as linearly as one would as- sume, for which reason it is necessary to involve general relativity[8]. In the following reasoning, taken from D.E. Neuenschwander[8], the main ideas of this approach are discussed. Most importantly, the model as defined must be able to predict the behaviour and the end of the Universe. In relativity one must thus define an invariant distance between points in space time, so that there exists a proper time between nearby events in space time. This distance, dtp, is given by: dt2p = dt2 − (dx2 + dy2 + dz2) (3.2.1) where the speed of light, c, is set to unity. Furthermore, equation (3.2.1) can be written in spherical coordinates as: dt2p = dt2 − (dr2 + r2dω2) (3.2.2) where dω2 ≡ dθ2 + sin2 θdφ2. By mixing the equations above with the scale function R(r) and by al- lowing space to be non-Euclidian one arrives at: dt2p = dt2 −R2(r) [( dr2 1− kr2 ) + r2dω2 ] (3.2.3) Here k is the curvature parameter, which has three possible values. • Case 1: k = −1 Space is hyperbolic. • Case 2: k = 0 Space is Euclidean. • Case 3: k = 1 Space is elliptic. Case 1: Space will continue to expand forever with a non-vanishing ve- locity. This leads to what is called an “open Universe” Case 2: The expansion velocity of space will decrease towards zero until equilibrium is reached with regards to the gravitational potential, at which point the Universe will have reached a fixed size. Case 3: The gravitational potential is larger than the kinetic energy and will hence pull the Universe together again, resulting in what is usually called the “big crunch” 8 4 The Early Universe 4.1 The Very Early Universe The time period lasting from the beginning of time, t = 0, until approxi- mately one second after the bang, often referred to as the very early Universe, can roughly be broken down into the following epochs [13, 14, 15]: • Planck epoch, 0 s < t < 10−43 s. • Grand Unification, 10−43 s < t < 10−35 s. • Inflation & Baryon genesis, 10−35 s < t < 10−33 s. • Separation of the weak and electromagnetic forces, 10−33 s < t < 10−5 s. • Protons and neutrons are created, 10−5 s < t < 1 s. The first of these eras, the Planck epoch, is framed by two fundamental points in time, specifically the birth of the Universe in the form of a singu- larity at t = 0 s and the Planck instant at t ≈ 10−43 s. The latter marks the moment after which quantum effects no longer dictates all physical processes, which follows from the fact that the general theory of relativity breaks down during the Planck epoch. The physics of this era is largely unknown, partly because of the high temperature, T ≈ 1032 K. Although equally difficult to imagine, the physics of the following era, The Grand Unification, is more in line with classical theory [14]. Yet, the temperature was still high enough, that is ∼ 1029 K, for all fundamental forces apart from gravity to be indistinguishable, which therefore is true for a number of particle species as well [14]. The Universe continues to grow and cool however, and eventually reaches a temperature just below 1028 K. As this occurs, the strong force begins to dominate over the other interactions, which in turn influences strongly on the nature of matter [14]. More to the point, the separation of forces shifts the equilibrium for the composition of matter, thereby provoking what can be described as a phase transition [14]. During this period 10−36 s < t < 10−33 s called the inflation the universal expansion takes place at an exponential rate [14]. Remarkably, by the end of this time period the Universe has expanded by a factor of approximately 1025 [14]. At t ≈ 10−9 s the temperature in the Universe has dropped to about 1015 K and the electromagnetic and weak forces start to separate, while si- multaneously becoming significantly decoupled from the strong nuclear force 9 [14]. Though less potent than the inflation, this later shift of the fundamen- tal forces results in a perturbation of the matter content by introducing a small, yet significant, asymmetry in the number of particles as compared to antiparticles [14]. The ratio of the number of baryons and leptons is conserved during the later stages of the Universe, as these amounts are thought to have been, al- most, fixed during the baryon genesis, at t ≈ 10−34 s, and the electro-weak transition, at t ≈ 10−10 s, respectively [14]. Additionally, the number of lep- tons per baryon is related to the number of photons per baryon since photons were created as a result of the annihilation of leptons at high temperatures [14]. The latter quotient is in turn a measure of the entropy per particle [14]. As was mentioned in the previous section, hadrons are composed of quarks, held together by gluons associated with the strong nuclear force [14]. These two particles species did not begin to form into hadrons until t ≈ 10−5 s, though [15]. Previously, that is from the baryon genesis and forwards, the Universe is filled with a quark-gluon plasma that also contains electron-positron pairs, neutrinos and photons [14, 15]. During the phase transition that follows, bubbles of hadron gas forms and grows in what is best described as a sort of nucleation process. At t ≈ 10−4 s small droplets of gluons and quarks remain in the, at this point, dominating gas of hadrons and leptons [15]. When this period comes to an end, the protons and neutrons contained within the hadron gas are in thermal equilibrium [14]. 4.2 The Early Universe Before delving into the details of the final era of the very early Universe, that is the time period 0.01 s < t < 1.9 s, it is helpful to present, as a reference, a list of events to be discussed, together with the approximate times at which they are thought to have begun [15]. • Neutrino oscillations are initiated, t ≈ 0.1 s. • The neutrinos decouple, t ≈ 1 s. • Simultaneously, the neutrons freeze-out, t ≈ 1 s. With reference to the first of these happenings, it is important to keep in mind that its occurrence is not predicted by the standard model for cos- mology, since it includes the assumption that all neutrinos are massless [10]. In the present day model for particle physics however, no conflict exists[10]. Specifically, there are no theoretical restraints that compels the neutrino masses to be either zero or non-zero [10]. Given that the latter holds true, 10 it would be possible for the weak eigenstates of the neutrinos to be formed from linear combinations of mass eigenstates, thereby providing a route for transitions between different neutrino flavors, often referred to as neutrino oscillations [10]. Neutrinos, after photons, are the most abundant particle species in the Universe [10]. Therefore it is not far fetched to assume that a non-zero neu- trino mass, together with oscillations, would severely effect the cosmological evolution [10]. Indeed, the first of these deviations from the standard model would alone result in a profound contribution to the total energy density of the Universe [10]. Furthermore, neutrino oscillations is bound to have af- fected the universal expansion rate, neutrino densities and energy spectrum together with the asymmetry between neutrinos and anti-neutrinos as well as the neutrino dependent cosmological processes [10]. Before discussing further the implications of neutrino oscillations, one would benefit from having a rough estimate of the upper bound for the neu- trino masses. This is possible, thanks to the requirement that the total mass density for all neutrino species should be less than equal to that of matter, ρm, as stated in (4.2.1). ∑ ρνf ≤ ρm. (4.2.1) For these calculations it will be assumed, in agreement with present day observations, that the non relativistic matter density in the Universe, ρm, is less than 30% of the so called critical mass density, ρc, defined by (4.2.3) [1, 3, 10]. ρc = 3H2 0 8πG (4.2.2) where G is Newtons gravitational constant and H0 is the Hubble constant. Before continuing with this discussion it is convenient to introduce the prop- erty Ωi, which represents the contribution of species i, by fraction, to the critical mass density [1, 3]. Thus, it is related to ρc by equation (4.2.4), where ρi is the mass density for species i [1, 3]. Ωi = ρi ρc (4.2.3) By combining (4.2.2) and (4.2.3) one can easily derive the expression (4.2.4) for ρi [3]. 11 Ωi = ρi ρc = ρi 8πG 3H2 0 ⇔ ρi = 3H2 0Ωi 8πG (4.2.4) By substituting ρm in (4.2.1) for (4.2.4) one thus arrives at the inequality in (4.2.5). ∑ ρνf ≤ Ωm · 3H2 0 8πG (4.2.5) The procedure necessary to arrive at a precise limit for the sum of the neutrino masses, ∑ mνf , is a bit to involved to be attempted here, as such only the result (4.2.6) shall be stated [10].∑ mvf . 94 eV/c2 · Ωmh2 (4.2.6) As was stated above, it has been inferred that ρm < 0.3 · ρc or equally that Ωm < 0.3. Additionally, it will be assumed that the Hubble parameter h = 0.7, in agreement with section 3. Upon inserting the above values into (4.2.6) one finally arrives at the sought limit, (4.2.7) [10].∑ mvf ≤ 15 eV/c2 (4.2.7) Even so, there exists much more precise limits on the neutrino masses as obtained from observations, experiments and BBN calculations [10]. For example, some measurements indicate that massive neutrinos could be can- didates for hot dark matter if mvf ∼ 5 eV, which suggests that the usefulness of the estimate presented above is perhaps limited [10]. The kinetic decoupling of the neutrinos can be described as a decrease in thermal contact between these particles and the rest of the plasma. The process begins when t ≈ 0.12 s at a temperature of T ≈ 3 · 1010 K and then comes to a close ∼ 1.1 s after the bang [13]. Specifically, this means that the rates of the weak interactions, such as e++e− � ν+ν̄, whereby the neutrinos are kept in thermal equilibrium with the plasma drops below the expansion rate of the Universe [16, 17]. Afterward, the neutrinos only influence the cosmological evolution by their addition to the total mass-energy density of the Universe [13]. Lastly, it shall be noted that during the entirety of the time period that has been discussed the Universe is filled predominantly with photons, neu- trinos and antineutrinos together with electron-positron pairs [13]. The neu- trons, protons and electrons meanwhile are mixed into the primordial gas only 12 in trace amounts [13]. Furthermore, the temperature, ranging from 1011 K at t ≈ 0.01 s to T . 1010 K once t . 1.9 s, is sufficiently high for e± pairs to be produced. As such the particles within the gas mixture are relativistic and the total behaviour of the fluid resembles more that of radiation than matter [13]. 4.3 Freeze-out Since the neutron number greatly influences the outcome of BBN, it is impor- tant to be able to calculate, at least approximately, the time for the neutron- to-proton freeze-out. Similarly to the neutrino decoupling, this freezeout is assumed to have occurred when the overall interconversion rate of protons and neutrons λn,p fell below the universal expansion rate, due to decreasing temperature [3]. Specifically, it would seem likely that as the average time between collisions, that is reciprocal of the conversion rate, grows compared to the time scale in the Universe, as measured by 1/H, these events will oc- cur ever more seldom. One would thus expect the n-to-p interconversion to become more ineffective to sustain the equilibrium that had existed between the two species before the freeze-out [4]. It is therefore not unreasonable to assume an estimate temperature at the neutron freeze-out would correspond to the time at which the equality λn,p = H was satisfied. Before continuing with this discussion however, it is of the essence to describe the relationship between neutrons and protons at times when these were still kept in chemical equilibrium through the reactions in (4.3.1), (4.3.2) and (4.3.3), n + e+ p + ν̄e (4.3.1) n + νe p + e− (4.3.2) n → p + e− + ν̄e (4.3.3) The fact that the mass difference between the species, Q = mnc 2−mpc 2 ≈ 1.293 MeV, is greater than zero implies that there were fewer neutrons than protons in the early Universe, or equally that nn/np < 1 [2]. Yet, the ratio of the neutron to proton number densities, is predicted to approach unity as the temperature goes to infinity, at least according to (4.3.4) [13]. nn np = exp ( −Q kBT ) (4.3.4) In deriving equation (4.3.4), one proceed by first finding suitable expres- sion for the neutron and proton number densities. As is shown in appendix 13 C, the number of particles of species i per unit volume and with momentum in the interval [q,q + dq] is given by equation (4.3.5). ni(q)dq = 4πgi h3 q2dq exp ( Ei(p,q)−µi kBT ) ± 1 (4.3.5) Both protons and neutrons have half integer spins and are thus fermions, for which reason the variant of (4.3.5) with a plus sign on the right hand side applies [11]. Furthermore, it can be assumed that both protons and neutrons were non-relativistic at the time of the n-to-p freeze-out, which shall later be shown to have occurred when T ≈ 1010 K [13, 18, 19]. This assumption is justified by the fact that the electrons and positrons, with masses three orders of magnitude less than the nucleons, seized to be relativistic at similar temperatures [13, 19]. It follows that the energy of these particles, Ei(p,q) in (4.3.5), can be written on the form (4.3.6) [11, 18]. Ej(q) = mjc 2 + q2 2mj (4.3.6) where the subscript j has been included to distinguish the nucleons from the relativistic particles discussed in section C, with j = n for neutrons and j = p for protons. Most importantly, the chemical potentials, compared to those of positrons and electrons, do not vanish in this case. By proceeding in a manner identical to when deriving equation (C.1.29), µ−+µ+ = 0, in section C one ought to be able to prove that µp = µn = µ. For example, one could substitute N− for Nn and N+ for Np and then use the fact that the chemical potentials of each of the species e−, e+ and ν, which appear in reactions (4.3.1) to (4.3.3), are zero. In other words, what would be shown is that the chemical potentials are additively conserved in each of the named reactions, which in fact generally holds true [13]. Lastly, nucleons, being fermions with spin 1/2, have two spin degrees of freedom, gj = 2. With the above statements taken into account the expression (4.3.7), for the total number of particles j per unit volume, results when integrating (4.3.5) . nj = ∫ ∞ 0 nj(q)dq = ∫ ∞ 0 4π · 2 h3 q2dq exp ( mjc2+q2/(2mj)−µ kBT ) + 1 ⇔ nj = 8π h3 ∫ ∞ 0 q2dq exp ( mjc2−µ kBT ) exp( q2 2mjkBT ) + 1 (4.3.7) 14 In order to ascertain an analytical solution to (4.3.7) it can be assumed that the exponential term in the denominator is much larger than unity, effectively inferring that the nucleons follow Maxwell-Boltzmann statistics. The assumption is the most critical for particles with small momentum q, for which exp[q2/(2mjkBT )] ≈ 1. Therefore, ensurance that the magni- tude of the factor exp[(mjc 2 − µj)/(kBT )] is high enough for an appropriate temperature is sufficient evidence to validate this approximation. To show that this factor is indeed large compared to 1, without taking the chem- ical potential into account, one can determine the magnitude of the term mjc 2/(kBT ). For this purpose, the temperature can be taken to be 1010 K. With very simple estimates of the physical constants, one thus finds that mjc 2/(kBT ) ≈ 10−27 · 1017/(10−23 · 1010) = 10−27+17+23−10 = 103 [11]. As exp[mjc 2/(kBT )] ≈ e1000 � 1 unarguably, the stated assumption ought to be justified for all q, under which (4.3.7) will now be shown to reduce to the form (4.3.8). Note that the integral on the right hand side of (4.3.7) was evaluated with help of formula (D.1.3), derived in appendix D.1. nj ≈ 8π h3 ∫ ∞ 0 q2dq exp ( mjc2−µ kBT ) exp ( q2 2mjkBT ) = 8π h3 exp ( µ−mjc 2 kBT )∫ ∞ 0 q2 exp ( − q2 2mjkBT ) dq = { x = q√ 2mjkBT ⇔ √ 2mjkBTx = q ⇒ dq = √ 2mjkBTdx } = 8π h3 exp ( µ−mjc 2 kBT )∫ ∞ 0 (2mjkBT )x2e−x2√ 2mjkBTdx = 8π h3 exp ( µ−mjc 2 kBT ) (2mjkBT )3/2 ∫ ∞ 0 x2e−x2 dx ⇔ nj ≈ 8π h3 exp ( µ−mjc 2 kBT )(2mjkBT )3/2 √ π 2 ⇔ nj ≈ 4(2πkBT )3/2 h3 exp ( µ kBT ) m 3/2 j exp ( −mjc 2 kBT ) (4.3.8) The expression (4.3.9) is obtained by forming the ratio nn/np and then introducing (4.3.8). Finally, (4.3.4) stated earlier follows from (4.3.9), by inferring that the smallness of the neutron to proton mass difference means that mn/mp ≈ 1. 15 nn np ≈ ( mn mp )3/2 exp ( −mnc2 kBT ) exp ( −mpc2 kBT ) ⇒ nn np ≈ ( mn mp )3/2 exp ( −Q kBT ) (4.3.9) As mentioned though, equation (4.3.4) is only applicable at thermal equi- librium, times preceding the neutron to proton freeze-out. Thereafter, the number of neutrons decreases because of beta decay, according to the reaction in(4.3.10) [2]. n → p + e− + ν̄e (4.3.10) Given that the neutrons in a particular system is neither consumed nor created by any reaction except (4.3.10) one can calculate the number of neutrons at any time t, later than t0, from (4.3.11). Nn = Nn,0 exp ( −(t− t0) τn ) , (4.3.11) where Nn,0 = Nn(t0) is the, known, number of neutrons at a particular time t0, for example the time at the neutron to proton freeze-out, and τn ≈ 885.7± 0.8 s is the mean neutron life time [11, 19]. Figure 2 shows the preditcted evolution of the neutron-to-proton ratio for a decreasing temperature based on the previous discussion. Specifically, the temperature dependence of this ratio is governed by (4.3.9) for T < Tfreeze-out and by (4.3.11) for T > Tfreeze-out . From the statement above the simple relation (4.3.11) ceases to hold when the Universe has become cool enough for the nucleosynthesis to begin [2]. During this era most neutrons are fused into different nuclei, primarily 4He [2]. Once the BBN process has come to an end however, the conditions for the neutrons return to those that persisted just before the onset and the remaining neutrons are thus comparably slowly converted into protons, through reaction (4.3.10), as time progresses [2]. With the above discussion in close mind, it is convenient to return the problem of calculating the temperature at the nucleon freeze-out. As was sug- gested earlier one ought to be able to estimate this temperature by solving the equation obtained by setting the Hubble parameter equal to the neutron to proton conversion rate. In order to achieve this however, one must first find an expression for the conversion rate and the Hubble parameter H as 16 10 9 10 10 10 11 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ratio of neutrons per protons as a function of temperature Temperature [K] N eu tr on s pe r pr ot on s Freeze−out Figure 2: The evolution the neutron to proton ratio as a function of temperature before the Big Bang Nucleosynthesis, specifically (4.3.9) for T < Tfreeze-out and (4.3.11) for T > Tfreeze-out . The asterisk, ∗, marks the point that corresponds to the n-to-p freeze-out, as calculated from equation (4.3.14). Before the freeze-out the ratio is just a function of the canonical ensemble and thereafter only of neutron decay. 17 functions of time. Because time and temperature of the early Universe are tightly linked, an almost equivalent approach would be to determine the tem- perature below which the protons and neutrons are no longer in equilibrium. Deducing equation (4.3.12), that shall be used for this comparison, is far beyond the scope of this text though, and as such it will be stated without proof [18]. λn,p = 255 τnx5 ( 12 + 6x + x2 ) , x = Q kBT (4.3.12) Moving on to the Hubble parameter, one have by definition H = Ṙ/R. With the help of expressions (C.1.6) and (C.1.38) derived in C, it is therefor possible to arrive at the formula (4.3.13) for H(T ). 1 Ṙ R = √ 8πG 3c2 ε (C.1.6) ε ≈ 43 8 aT 4 (C.1.38) ⇒H = Ṙ R = √ 8πG 3c2 ε ≈ ( 8πG 3c2 43 8 aT 4 )1/2 ⇔H(T ) ≈ ( 43πaG 3c2 )1/2 T 2 (4.3.13) An estimate of the temperature at the nucleon freeze-out can be calcu- lated solving the equation,(4.3.14) , by setting the Hubble parameter equal to the neutron to proton conversion rate. H(T ) = λn,p ⇔ ( 43πaG 3c2 )1/2 T 2 = 255 τnx5 ( 12 + 6x + x2 ) , x = Q kBT (4.3.14) With numerical values for the physical constants appearing in (4.3.14), the temperature below which the neutrons and protons were no longer in equilibrium is calculated to be Tfreeze-out ≈ 7.8965 · 109 K , as explained in section 8.1. 1As was mentioned previously, (C.1.38) differs from equation 8.62 deduced by Islam since the contribution of the τ neutrino and its corresponding antiparticle has not been taken into account in the latter case [13]. Also, in deriving the same expression Islam has set the speed of light equal to unity, c = 1 [13] 18 5 Energy Density 5.1 The Baryon to Photon Ratio Over the years accurate and independent experimental measurements have successively improved the estimates of the original input parameters to Big Bang Nucleosynthesis simulations. Eventually, these were pinned to within ranges that essentially promoted BBN to a model with a sole parameter, namely the baryon to photon ratio η [3]. Thanks to the Wilkinson Microwave Anisotropy Probe satellite, WMAP, this situation has recently changed rather dramatically [2]. After its launch by NASA in 2001, WMAP has mapped the cosmic microwave background, CMB, over the entire sky in great detail [20]. Specifically, multi-parameter expressions have been fitted to the observed anisotropy of the background radiation [21, 22]. The errors in the predicted values on these parameters has been further refined through comparison with other observational data [21, 22]. The baryon density was one of those chosen parameters and has, as such, been determined with an unprecedented accuracy [22]. As will be shown it is possible to deduce the photon number density given the black-body temperature that correspondence to the cosmic background radiation, 2.743 K [11]. This deduction will be based upon the assumption that radiation energy density of CBR follows Planck’s radiation law, both in terms of frequencies (5.1.1) and wavelengths (5.1.2) [11]. Indeed, this is also what has been observed, mind the small fluctuations mentioned above [23]. du = 8πh c3 ν3dν exp ( hν kBT ) − 1 (5.1.1) du = 8πhc λ5 dλ exp ( hc kBTλ ) − 1 (5.1.2) The formulas (5.1.1) and (5.1.2) give the energy content per unit volume of black body radiation in the intervals [ν, ν +dν] and [λ, λ+dλ] respectively. Thus, the total energy density of the radiation emitted by a black body of temperature T can be deduced by integrating (5.1.1) over all frequencies ν according to (5.1.3). 19 u = ∫ ∞ 0 8πh c3 ν3dν exp ( hν kBT ) − 1 ⇔ u = 8πh c3 ∫ ∞ 0 ν3dν exp ( hν kBT ) − 1 (5.1.3) The relation (5.1.3) can be rearranged into the form (5.1.4), where x = hν kBT , that is more easily solvable. u = 8πh c3 ∫ ∞ 0 ( kBT h )3( hν kBT )3 1 exp ( hν kBT ) − 1 kBT h hdν kBT ⇔ u = 8π (kBT )4 (hc)3 ∫ ∞ 0 x3 ex − 1 dx (5.1.4) The integral on the right hand side of (5.1.4) has, as shown in appendix D.2, the solution (5.1.5). This result can be inserted into (5.1.4) to yield the formula (5.1.6) for the CMB energy density [24]. (m− 1)! ∞∑ n=1 1 nm ∣∣∣∣ m=4 = 6 · π4 90 = π4 15 ⇒ ∫ ∞ 0 x3dx ex − 1 = π4 15 (5.1.5) ⇒ u = 8π5 15 (kBT )4 (hc)3 (5.1.6) Equally, (5.1.2) can be rewritten in terms of the number density of pho- tons, Nγ, thus yielding the equation (5.1.7) since the photon energy equals hν = hc/λ and u = hν ·Nγ. dNγ = du hν = λ hc du = λ hc 8πhc λ5 dλ exp ( hc kBTλ ) − 1 ⇔ dNγ = 8π λ4 dλ exp ( hc kBTλ ) − 1 (5.1.7) Expression (5.1.8) for the total number density of photons follows from (5.1.7) by integrating both sides of the equation over all wavelengths. 20 Nγ = ∫ ∞ 0 8π λ4 dλ exp ( hc kBTλ ) − 1 ⇔ Nγ = 8π ∫ ∞ 0 λ−4dλ exp ( hc kBTλ ) − 1 (5.1.8) By the same token as (5.1.4), (5.1.9) represents a form of (5.1.8) that is more readily solvable, where y = (hc)/(kBTλ) ⇒ dy = −(hc)/(kBT )λ−2dλ. Nγ = 8π ∫ ∞ 0 ( kBT hc )2( hc kBTλ )2 1 exp ( hc kBTλ ) − 1 kBT hc hc kBT λ−2dλ ⇔ Nγ = 8π ( kBT hc )3 ∫ ∞ 0 y2 ey − 1 dy (5.1.9) Though the integral on the right hand side of (5.1.9), compared to that in (5.1.4), cannot be obtained as an precise number, it can still be evaluated in the same manner as before. This results in the estimate (5.1.10), with which the final relation (5.1.11) is obtained [24]. (m− 1)! ∞∑ n=1 1 nm ∣∣∣∣ m=3 ≈ 2 · 1.202 ≈ 2.404 ⇒ ∫ ∞ 0 x2dx ex − 1 ≈ 2.404 (5.1.10) ⇒ Nγ ≈ 2.404 · 8π ( kBT hc )3 ⇒ Nγ ≈ 60.42 · ( kBT hc )3 (5.1.11) Hence, the photon number density in the cosmic background radiation is found, by evaluating (5.1.11) for T = 2.743 K [11]. Combined with the WMAP data this yields the following result. Nγ ≈ 60.42 · ( 1.381 · 10−23 · 2.743 6.626 · 10−34 · 2.998 · 108 )3 ⇒ Nγ ≈ 4.190 · 108 m−3 21 Yet to find the sought baryon to photon ratio, one must first deduce the baryon number density nb. Given the baryon mass density density ρb, the number density is most easily calculated by assuming the mass per baryon to be equal to that of a proton, mb ≈ mp ≈ 1.6726216 · 10−27 kg [11, 13]. The main problem is therefore determining ρb. Fortunately, the baryon mass density can be calculated from the dimensionless number Ωb that has been accurately fitted to the WMAP observations, as mentioned above. As was discussed in section 4.2 Ωb is by definition the baryonic contribution, by fraction, to the so called critical mass density ρc, defined by (4.2.2) [1, 3]. Furthermore, this property conveniently appears in the expression (5.1.13) for the baryon mass density density, obtained simply by substituting the index i for b in (4.2.4) [3]. Ωb = ρb ρc (5.1.12) ⇒ ρb = 3H2 0Ωb 8πG (5.1.13) The gravitational constant will be taken as G = 6.6726 · 10−11 Nm2/kg2 while the most up to date WMAP measurements give Ωbh 2 = 0.02258+0.057 −0.056 [11, 12]. With H0, given by (3.1.2), one can thus calculate the baryon number density from (5.1.14). nb = ρb mb = 3H2 0Ωb 8πGmb (5.1.14) ⇒ nb = 3 · (3.241 · 10−18 · h)2 · 0.02258/h2 8π · 6.6726 · 10−11 · 1.6726216 · 10−27 ⇒ nb = 3 · (3.241 · 10−18)2 · 0.02258 8π · 6.6726 · 10−11 · 1.6726216 · 10−27 ≈ 0.2536 m−3 The sought baryon to photon ratio, η = nb/nγ, is thus found to be η = 0.2536 4.190 · 108 ≈ 6.1 · 10−10. 22 6 Relating Time and Temperature 6.1 Simple Model for Relating Time and Temperature The Hubble parameter H0 does not remain constant on large time scales and it is therefore necessary to find how H(t), now without the subscript, varies with the expansion. A much more thorough derivation than what is below is found in appendix C. Intuitively a relation to a radius would be practical, but since the Universe lacks one the scale factor R will be used instead [25]. This parameter only depends on time and thus the distance between two point in space can be predicted given an expression for R(t) and the magnitude of this distance at some arbitrarily time t0 [26]. The relation between H and R is [25] H = 1 R dR dt . (6.1.1) To obtain the relation for the time evolution of H the tensor equation of the theory of general relativity needs to be solved. The result is stated below [25]. H2 = (dR/dt)2 R2 = 8πG 3 ρ(t)− kc2 R2 + Λ 3 , (6.1.2) where G is the gravitational constant, ρ(t) is the sum of the mean mass and the energy density of the Universe, k is the curvature parameter the value of which depends on whether the Universe is open, closed or flat, as was discussed in section 3. Lastly, Λ is the cosmological constant which will be ignored from this point onwards. Furthermore, the Universe will be assumed to be flat, which simplifies the calculations since the geometrical factor k is zero in this case. In order to integrate (6.1.2) the dependence of ρ on R is needed. The early Universe was dominated by radiation as well as particles moving at relativistic speeds, for which reason it can be assumed that the radiation-like relationship E = hc/λ was obeyed. Hence, one can use the radiant energy density ρR, which represents the energy content of the radiation per unit volume. In turn ρR is equal to the cross product of the energy per quantum and the number of quanta per unit volume [25] ρR = energy volume = energy per quantum× quanta per volume. The energy per quantum is proportional to 1/R and the quanta per unit volume is proportional to 1/R3 [25]. ρ in (6.1.2) will therefore be assumed to have the form ρR = C/R4, with C a constant that will be shown to disappear 23 in the calculations below. An approximate form for (6.1.2) is thus H2 = ( 1 R dR dt )2 = 8πG 3 C R4 ⇔ H = 1 R dR dt = √ 8πGC 3 1 R2 , (6.1.3) which can be integrated to yield t = √ 3C 32πGR4 = √ 3 32πGρR . (6.1.4) To arrive at the desired relation between time and temperature the tem- perature dependence of ρR is required. As was previously mentioned, the early Universe was dominated by radiation and relativistic particles. There- fore the energy density ρR can be taken as that of black body radiation for a radiating system, u(T ), at temperature T [25] u(T ) = σT 4, (6.1.5) where σ is the Stefan-Boltzmann constant. The relation between tempera- ture, in Kelvins, and the elapsed time since the big bang , in seconds, is T = ( 3 32πGσ )1/4 · 1 t1/2 , (6.1.6) which reduces to (6.1.7) upon inserting numerical values for the physical constants. T ≈ 1.5 · 1010 t1/2 K · s1/2. (6.1.7) The relation in (6.1.7) is shown in figure 3. 24 10 −2 10 −1 10 0 10 1 10 2 10 3 10 9 10 10 10 11 Time [s] T em pe ra tu re [K ] Temperature as a function of time Figure 3: .Relation between time and temperature in the radiation dominated era. 7 Big Bang Nucleosynthesis 7.1 The Physical Process The Big Bang Nucleosynthesis represents an era in the history of the Universe that is said to have lasted from about a second until thirty minutes after the Big Bang [1, 4]. During this process protons and neutrons were combined, as governed by a complex reaction network, to form a multitude of light nuclei [1, 4]. Before moving on to discuss the details of the primordial nucleosynthesis, it is convenient to discuss this and similar events in the early Universe in more general terms. Most importantly, the processes that have been or shall be described neither ceases nor are initialized at specific times, or temper- atures. Indeed, regarding any changes as momentaneous, though helpful as a simplification for calculation and modeling purposes, is inherently flawed. Therefore, the specific times or temperatures related to these events, in many cases, represent points marking a shift in dominance of one physical property over another. For instance, the time for the neutron freeze-out is calculated by comparing the rate of the neutron to proton conversion and the expansion of the Universe respectively. This could falsely lead to the conclusion that 25 no protons were converted into neutrons once the point of intersection had been reached. In reality though this process did occur, be it at an ever slower pace. The value of the neutron to proton ratio at the beginning of BBN is one of the key factors that determines its outcome [9]. It is therefore convenient, given the previous example, to return to the era at hand. Nontheless, in order to fully appreciate the impact of the most crucial parameters, such as the n-to-p ratio, on the primordial nucleosynthesis a qualitative understanding of the underlying physics is required. During the period at hand the Universe was still quite dense, with regards to the total energy content per unit volume, and the temperature correspond- ingly high [4, 13]. Furthermore, the Universe was predominantly inhabited by photons and the recently decoupled neutrinos, while the protons, neu- trons and electrons were present only in trace amounts [4, 13]. It thus seems reasonable that the only nuclear reactions of interest would have involved interactions between, at most, two particles [4]. By the same token, it can be assumed that the more complex nuclei than D, could only have formed through a chain of two-body collisions [4]. This suggests that the formation of the lightest nuclei, the deuteron, could be regarded as the beginning of BBN. Indeed, the starting point for the primordial nucleosynthesis is often referred to as the time when the rate of deuterium nuclei formation, through (7.1.2), exceeded that of the reverse reaction [2]. Also, the right hand side of (7.1.2) suggest that the destruction of d was, at this stage, primarily due to photodissociation. The vast number of photons per baryon meant that this process was highly effective in hindering the deuterium nuclei to survive long enough for it to take part in other nuclear reactions [2]. This deadlock prevails long after the temperature, or rather kB · T , has dropped below the binding energy of the deuteron, Bd ≈ 2.23 MeV. Specifically, the Big Bang Nucleosynthesis is estimated to have begun in earnest when kB · T ≈ 0.080 MeV [2]. The nucleosynthesis then proceeded through a intricate network of reac- tions, of which (7.1.1) to (7.1.11), shown in figure 4, represented the twelve of these with highest significance [2, 3]. 26 p d t n 3He 4He 7Li 7Be 1 2 3 4 5 6 7 8 9 10 11 12 Figure 4: The twelve most important reactions in the reaction network that governs the BBN process. The reactions represented by numbers are repro- duced in equations (7.1.1) – (7.1.11). 1. n → p + e− + ν̄e (7.1.1) 2. p + n → d + γ (7.1.2) 3. d + p →3 He + γ (7.1.3) 4. d + d →3 He + n (7.1.4) 5. d + d → t + p (7.1.5) 6. t + d →4 He + n (7.1.6) 7. t + 4He →7 Li + γ (7.1.7) 8. 3He + n → t + p (7.1.8) 9. 3He + d →4 He + p (7.1.9) 10. 3He + 4He →7 Be + γ (7.1.10) 11. 7Li + p →4 He + 4He (7.1.11) 12. 7Be + n →7 Li + p (7.1.12) In discussing nuclear reactions, it is sometimes useful to examine the nuclear binding energies, defined as the difference in mass between a nucleus and the sum of its individual nucleons. Specifically, the comparison of this 27 property, evaluated for different nuclei, yields an important estimate of their relative stabilities. Given values for the binding energy per nucleon for the lightest elements it is thus possible to make qualitative predictions on BBN with regards to both the physical process and its outcome. Most importantly, in going from d through 3He and T to 4He the degree of binding increases, even though the tritium nuclei is in actuality unstable and later decays to 3He [3, 11, 18]. By the same logic used to assess the starting point for BBN, it could be inferred that each nuclear species with a binding energy higher than the deuteron would in fact have been stable at temperatures higher than 0.080 MeV [13]. More precisely, in the sense that not enough energetic pho- tons would have been available to photo fission them efficiently [13]. It is therefore helpful to think of the formation of d as a sort of bottleneck that delayed the nucleosynthesis. In addition, the tighter binding of the, slightly, more massive nuclides meant that these would be expected to have readily formed once this barrier had been breached [13]. It seems likely that most neutrons, which were outnumbered by the pro- tons by a factor of at least 6, would have been incorporated into the most stable nuclei [1, 13]. Because of the mass gaps at A = 5 and A = 8, the most tightly bound nuclide produced during the primordial nucleosynthesis was 4He [13]. Specifically, 4He constitutes a local maximum for the binding energy per nucleon as a function of the nucleon number, A, as is seen in figure 5 [18, 27]. Comparison of reactions (7.1.10) and (7.1.7) with those higher up in the list suggests that particles with higher positive charge, that is at least one with charge +2e instead of +e, must have collided in order for the heavier nuclei, A = 7, to have formed. Since the radius of these light particles are in the order of a few fm, the interactions through which 7Li and 7Be were created would most definitely have involved significantly higher coloumb barriers [11, 13]. Additionally, the continuous expansion of the Universe rapidly drained away the energy that was needed to surmount these barriers. Therefore reactions (7.1.10) and (7.1.7) ought to have given birth to no more than trace amounts of A = 7 nuclei [1]. It would thus be expected that most neutrons were bound in 4He once BBN had come to a close [1]. This nuclear freeze-out, occurred when kBT ≈ 0.1 MeV corresponding to t ≈ 30 min, after which time no further nuclear reactions took place [1, 2]. Yet, both 7Be and T as well as the neutron were unstable and continued to decay into 7Li, 3He and protons respectively [3]. Following the primordial nucleosynthesis the Universe ought to have contained, based on the previous discussion, primarily hydrogen-1 and helium-4 in addition to small remnants of unburned d, 3He and 7Li [1]. 28 10 0 10 1 10 2 0 1 2 3 4 5 6 7 8 9 Number of Nucleons B in di ng E ne rg y pe r N uc le on [M eV /A ] Stability of Nuclei 4He 8Be d t Figure 5: Maximum binding energies as a function of number of nucleons, A, [27]. Worth noting is that 8Be has a very short half-life, and α-decays almost instantly into two 4He. 29 7.2 The Impact of η With the framework in place it is convenient to turn to the physical param- eters that have the largest impact on the predicted result of the Big Bang Nucleosynthesis. Principally these are the nuclear cross sections that deter- mine the nuclear reaction rates, the neutron lifetime, the baryon to photon ratio as well as the number of neutrino species [1]. Since the latter three have already been under scrutiny, in sections E.8, 5.1 and 4.2 respectively, while giving a meaningful introduction to cross sections is beyond the scope of this text, there shall be no effort to explain the underlying principles for these parameters. The range of plausible values for η has, until very recently, been rather wide compared to those of the other physical properties, previously men- tioned [3]. Indeed, BBN calculations, with the observed values for the pri- mordial abundances used as input data instead of η, used to give the best, if not the only, estimates for the baryon density [2, 3]. It is therefore prudent, not only for historical reasons, to assess the dependence of the final number fractions of the light elements as well as the progression of the primordial synthesis on the baryon to photon ratio. The photodissociation, responsible for delaying the formation of the deuteron and thus BBN as a whole, should in all likelihood have a rather sharp de- pendence on η [2, 9]. Although bold, this statement shall now be justified by analyzing the Boltzmann equation for the system of particles included in (7.1.2) [18]. As with most expressions involving the the cosmic scale factor R(t), the complexity of the underlying theory regretfully means that it will be stated without proof. Furthermore, the form (7.2.1) used in the following derivation is approximate and only applicable for a system composed of particles 1, 2, 3 and 4 [18]. R−3 d dt (n1R 3) = n (0) 1 n (0) 2 〈σν〉 { n3n4 n (0) 3 n (0) 4 − n1n2 n (0) 1 n (0) 2 } (7.2.1) Here ni is the number density of particle type i, while the superscript (0) indicates that the latter should be evaluated at equilibrium conditions [18]. Also, by stating this formula it has been inferred that the only reaction involving the species of interest, 1, is 1 + 2 � 3 + 4 [18]. In (7.2.1) the nature of the interconversion process is taken into account by inclusion of the thermally averaged cross section 〈σν〉 [18]. Though highly significant for modeling not only the system at hand but the Big Bang Nu- cleosynthesis as a whole, giving a qualitative description of these factors is 30 beyond the scope of this text. However, by regarding the particle collisions in a macroscopic sense, at least a basic degree of understanding for the under- lying physics can be achieved [28]. Classically, both the rates and likelihoods of such impacts are dependent on the sizes of the interacting particles, a property measured by for example their cross sectional areas [28]. Even so, the nuclear cross sections are immensely more intricate and depends on a far greater number of physical properties [18]. The term on the left hand side in (7.2.1) ought to represent the total rate of change in the number density for particle 1 [18]. From the definition of R(t) given in the introductory chapter and the results in section C, this term can be seen to take into account the decrease of n1 through the expansion of the Universe [18]. Moving on to the right hand side, n (0) 2 〈σν〉 should qualify as the rate of reaction [18]. The factor within the curly brackets, in turn, vanishes if the particles are in equilibrium, that is if ni = n (0) i ∀ i, and should hence give an estimate of the departure from equilibrium. Since the Hubble parameter H = Ṙ/R, or rather H−1, is a measure of the cosmological time-scale, it is plausible that the magnitude of the term ∂(n1R3) ∂t is in the order of H1n1 [18]. Thus, if the rate of conversion n (0) 2 〈σν〉 is significantly greater than the expansion, the sum of the terms inside the curly brackets must be very small for the equality (7.2.1) to hold. One would in other words require the condition in (7.2.2), often referred to as the definition for chemical equilibrium, to be approximately true [18]. n1n2 n (0) 1 n (0) 2 = n3n4 n (0) 3 n (0) 4 (7.2.2) Because the photons greatly outnumber the nucleons, η ≈ 1010, in the early Universe, it seems reasonable that the number density of the photons nγ ≈ n (0) γ [18]. Equation (7.2.2) therefore takes on the form (7.2.3) for the system at hand [18]. nDnγ n (0) D n (0) γ = nnnp n (0) n n (0) p ⇒ nD nnnp = n (0) D n (0) n n (0) p (7.2.3) In deriving formula (4.3.8) it was shown that the protons and neutrons could, with good approximation, be described by Boltzmann statistics. More- over, this assumption was partly justified by comparing energies equivalent to the masses of these particles and the temperature respectively. For this reason, the same arguments should be applicable to the even more massive 31 deuteron. Consequently, each of the number densities appearing in (7.2.3) could be replaced by (7.2.4), which is a more general form of formula (4.3.8) [18]. More precisely, the former follows from the latter by replacing a factor of 2, given by the number of spin degrees of freedom for the nucleons, with gj and the nucleon chemical potential µ with µj. nj ≈ 2gj(2πkBT )3/2 h3 exp ( µj kBT ) m 3/2 j exp ( −mjc 2 kBT ) (7.2.4) What is more, at equilibrium conditions the chemical potentials vanish and equation (7.2.4) takes on the form (7.2.5) for all relevant species. n (0) j ≈ 2 (2πkBT )3/2 h3 · gjm 3/2 j exp ( −mjc 2 kBT ) (7.2.5) The approximate form (7.2.6) is obtained by introducing of (7.2.5) into the right hand side of (7.2.3). nD nnnp = n (0) D n (0) n n (0) p = 2(2πkBT )3/2/h3 (2(2πkBT )3/2/h3)2 gDm 3/2 D exp ( −mDc2 kBT ) gnm 3/2 n exp ( −mnc2 kBT ) · gpm 3/2 p exp ( −mpc2 kBT ) = 1 2(2πkBT )3/2/h3 gD gngp ( mD mnmp )3/2 exp ( −c2(mD −mn −mp) kBT ) ⇔ nD nnnp = gD gngp h3 2 ( mD/mnmp 2πkBT )3/2 exp ( c2(−mD + mn + mp) kBT ) (7.2.6) This equation can be further simplified, however. As was mentioned in section 4.3 both neutrons and protons are fermions and hence have spin 1/2 corresponding to gn = gp = 2 [18]. The deuteron, on the other hand, has gD = 3 [18]. With these numbers, the ratio of the species spin degrees of freedom that appears on the right hand side of (7.2.6) is easily evaluated to be gD/(gngp) = 3/4. By inserting this result into the previous equation (7.2.6), one yields the expression (7.2.7) [18]. nD nnnp = 3h3 8 ( mD 2πmnmpkBT )3/2 exp ( c2(−mD + mn + mp) kBT ) (7.2.7) 32 10 9 10 10 10 11 10 12 0 1 2 3 4 5 6 x 10 −43 Temperature [K] n D /( n nn p) Ratio of deuterons per neutrons and protons Figure 6: The graph shows the temperature dependence of the ratio nD/(nn · np). It will also be assumed that the small differences in mass, between on one hand the proton and the neutron and on the other the deuteron and its constituents, are negligible, or equivalently that mD = mn + mp ≈ 2mn ≈ 2mp, in the quotient on the right hand side of (7.2.7). In addition, the nominator in the exponential is substituted for the binding energy of the deuterium nuclei, −Q = −mDc2 +(mnc 2 +mpc 2). As such, (7.2.7) is reduced to the form (7.2.8), which corresponds to the graph in figure 6. nD nnnp = 3h3 8 ( 2mp 2πmpmpkBT )3/2 exp ( −Q kBT ) ⇒ nD nnnp = 3h3 8 ( 1 πmpkBT )3/2 exp ( −Q kBT ) (7.2.8) Furthermore, nucleons are baryons by definition and it therefore seems likely that both the proton and neutron number densities are proportional to the number of baryons per unit volume, nb [18]. Also, in section 5.1 an equation, (5.1.9), was derived, which shows that the the number density of the photons is proportional to the temperature cubed, nγ ∝ (kBT )3. By implication, these statements together with the above expression (7.2.8) lead 33 to the proportionality in (7.2.9) [18]. nD nnnp ∝ nD nbnb ⇒ nD nb ∝ nbh 3 (kBT )3 (kBT )3 ( 1 (mpkBT )3/2 exp ( −Q kBT ) ⇒ nD nb ∝ h3 nb nγ ( kBT mp )3/2 exp ( −Q kBT ) ⇒ nD nb ∝ ηh3 ( kBT mp )3/2 exp ( −Q kBT ) (7.2.9) Because of the smallness of the baryon to photon ratio, the prefactor will dominate over the exponential as long as the quotient Q/(kBT ) is not very large [18]. For example, at T = 1010 K the exponent Q/(kBT ) ≈ 106 eV/(10−4 · 1010 eV) = 1 [11]. In agreement with previous assessments, this result suggests that the numerosity of the photons hinders the deuterium nuclei from forming until the temperature, kBT , is a few orders of magnitude lower than the binding energy of the deuteron, Q. As mentioned the deuterium nuclei combined to form heavier nuclei shortly after having been created and it might thus be expected that these species would show a similar dependence on η. This is indeed true, at least to some extent. Still, the intricacy of the network of nuclear reactions governing the interactions results in a much more complex dependence. What shall also be mentioned, is that relation (7.2.8) only gives the equilibrium ratio nD/(nnnp), based soley on the reaction (7.1.2), and not the final abundance of D. Nonetheless, it is the dependence of the amounts of the different species produced on the baryon to photon ratio that shall henceforth be discussed. Also, to simplify the comparison with the results obtained from simulations, most importantly figures 9 and 10, the conclusions presented below are sum- marized in table 4. Turning first to 4He, it can be inferred that this species should be rather insensitive to η [4, 9]. This is due to the fact that the binding energies of the light nuclei were independent of the baryon to photon ratio, wherefore most neutrons would have been fused into 4He, regardless of the precise value on η. The extent of the helium-4 production would accordingly have been de- termined, to a large extent, by the total content of neutrons in the Universe when the BBN process was initialized. In turn, this number depends on the temperature at which the neutron freeze-out occurs and thus on the compe- tition between the rate of the n-to-p conversion and that of the expansion [9]. Still, for a large baryon to photon ratio the D-bottleneck ought to have 34 been breached earlier [9]. This results in an increase in both the tempera- ture as well as the total number neutrons at the onset of BBN, since fewer of them would have had time to decay in this case. One would thus expect that slightly more 4He would have been produced for a greater value on the η[9]. However, the previous discussion suggests that the rate of increase, with respect to η, ought to be low. For similar conditions, this implies that lesser amounts would have remained of the species, specifically D and 3He, that collided to form helium-4 nuclei for a higher baryon to photon ratio [4, 9]. The A = 7 nuclides, lastly, are of particular interest as there existed two dominant paths for 7Li production [3, 9]. On one hand, these nuclei could have been created directly via reaction (7.1.7), a route that is assumed to have been favoured for a low baryon to photon ratio [3, 9]. This follows from the fact that a higher baryon density would have resulted in a greater number of protons able to destroy 7Li nuclei through (7.1.11) [3, 9]. On the other hand, 7Be have a higher binding energy per nucleon and should therefore be more stable with regards to such collisions [3, 9]. Consequently it ought to have been produced to a larger extent if η was high [3, 9]. These nuclei were unstable however and later decayed into 7Li through the absorption of an electron [2, 3, 9]. The analysis given is made more complicated by (7.1.12), which suggests that the indirect route for lithium-7 synthesis ought to have been plausible even at a lower baryon to photon ratio [3]. Particularily, (7.1.12) was limited by the amount of neutrons present during the primordial nucleosynthesis, a number that would have been enhanced for lower values on η [3]. Yet, simulations in which the baryon to photon ratio is varied, such as figure 12 in section 8.4, reveals a minimum in the 7Li production for intermediate η values [3]. This implies that there should indeed have existed two separate modes for lithium-7 creation in the early Universe and that the relative dominance of these ought to have dependended on the value of η [3]. Alternatively, one can infer, based on the previous discussion on the synthesis of A = 7 nuclei, that 7Li would have been produced primarily via (7.1.7) for low η instead of (7.1.10) for a given temperature. This conclusion stems from the fact that the latter reaction is a two-body collision between particles with higher positive charge, that is one more particle with q = +2e instead of q = +e. 35 Table 4: Prediction of relative abundances at different values of η. η Low Intermediate High 4He slightly lower increases slightly slightly higher d higher decreases lower 3He higher decreases lower 7Li higher minimum higher 7.3 Calculating the Fraction of High Energy Photons Photons follow a black body spectrum, the number of photons per unit vol- ume with energy between E and E + dE is [25] n(E)dE = 8πE2 (hc)3 · 1 exp(E/kT )− 1 dE. (7.3.1) where n(E) is the fraction of photons of energy E. To get the number of photons with energy greater than E◦, (7.3.1) is integrated from said energy to infinity 8π (hc)3 ∫ ∞ E◦ E2 exp(−E/kT )dE = n(E > E◦), (7.3.2) where the approximation 1 exp(E/kT )− 1 ≈ exp(−E/kT ), (7.3.3) has been introduced, which is good for E � kT . The integral in (7.3.2) is solved through integration by parts (hc)3 8π n(E◦ > E) = ∫ ∞ E◦ E2 exp(−E/kT )dE = −kTE2 exp(−E/kT ) ∣∣∣∣R→∞ E◦ + 2kT ∫ ∞ E◦ E exp(−E/kT )dE = kTE2 ◦ exp(−E◦/kT )− 2(kT )2E exp(−E/kT ) ∣∣∣∣R→∞ E◦ +2(kT )2 ∫ ∞ E◦ exp(−E/kT )dE = exp(−E◦/kT )(kTE2 ◦ + 2(kT )2E◦)− 2(kT )3 exp(−E/kT ) ∣∣∣∣∞ E◦ = (kT )3 exp(−E◦/kT ) [( E◦ kT )2 + 2E◦ kT + 2 ] , 36 it follows that n(E > E◦) = 8π (hc)3 (kT )3 exp(−E◦/kT ) [( E◦ kT )2 + 2E◦ kT + 2 ] . (7.3.4) To calculate the fraction of photons with energy greater than E◦, f(E > E◦), here done by dividing the result in (7.3.4) by the integral from zero to infinity in (7.3.1), which has been evaluated with Matlab 8π (hc)3 ∫ ∞ 0 E2 exp(E/kT )− 1 dE = [ x = E/kT ] = 8π (hc)3 (kT )3 ∫ ∞ 0 x2 exp(x)− 1 dx ≈ 1 0.42 8π (hc)3 (kT )3. (7.3.5) Finally (7.3.4) is divided by the result in (7.3.5) f(E > E◦) = 0.42 exp(−E◦/kT ) [( E◦ kT )2 + 2E◦ kT + 2 ] . (7.3.6) 37 8 Simulations 8.1 Calculation of Tfreeze-out Another estimate of interest is the temperature at the time of the neutron freeze-out. In section 4.3 equation (4.3.14), that is restated below for conve- nience, was derived based on the initial assumption that the chemical equi- librium between neutrons and protons ceases when the expansion rate of the Universe equals the n-to-p conversion rate. H(T ) = λn,p(T )( 43πaG 3c2 )1/2 T 2 = 255 τnx5 ( 12 + 6x + x2 ) , x = Q kBT In order to solve the above algebraic equation the textscMatlab routine fsolve.m was applied to the stated problem. As a starting guess Tguess = 1010K was chosen, in agreement with the most frequently occurring estimates of Tfreeze-out in literature [3, 4, 13]. These mathematical statements, together with numerical values on the relevant physical constants, are included in the textscMatlab function-file, 4.3 in Appendix E. On a final note, the solution Tfreeze-out ≈ 7.8965 · 109 K, presented in section 4.3 was used in the program partphotons.m, described in section 8.2, to calculate the number of neutrons remaining at the time of the n-to-p freeze-out. 8.2 Simple Predictions In the program partphotons.m, to be used with Matlab, the abundance of 4He and 1H are calculated according to a very simple model. In this model it is assumed that at freezeout, when neutrons and protons no longer are in thermal equilibrium, the only reaction mechanism at work is the beta decay of the neutrons, see equation (4.3.3) in section E.8 . The program thus only needs to calculate the elapsed time from freezout to the start of BBN in order to determine the number of neutrons that have had time to decay. For BBN to start deuterium nucleus must not only have begun to form, but accumulate as well. This is only possible when the number of photons with energy equal to, or greater than, the binding energy of deuterium be- comes less numerous than the number of deuterium nuclei that are being formed. This follows because in the formation of a deuteron a photon is cre- ated according to (8.2.1). Another photon can reverse the reaction, however, 38 10 9 10 10 10 11 10 12 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 Temperature [K] R at io o f h ig h en er gy p ho to ns , E > 2 .2 M eV Ratio of high energy photons as a function of temperature ratio=η WMAP Figure 7: The graph shows the temperature dependence of the fraction of photons with energy greater than the binding energy of the deuteron. The asterisk,∗ , marks the point at which this fraction is equal to η. and the onset for BBN is therefor approximately taken as the point in time when the fraction of photons, with enough energy, is equal to the baryon- to-photon ratio. In the model at hand it is also inferred that all neutrons and an equal number of protons combine to form 4He, while the remaining protons end up as 1H. p + p d + γ (8.2.1) Given a value on the baryon-to-photon ratio, the program also calcu- lates the time and temperature at the onset of BBN. This is done in the following manner. First, the fraction of photons with energy greater than the binding energy of deuterium for a particular temperature, f(E > E◦), is calculated, the time evolution of which is depicted in figure 7. More pre- cisely, the program evaluates (7.3.2), without the approximation in (7.3.3). To find f(E > E◦) for a certain temperature, this result is then divided by the integral of (7.3.1), taken from zero to infinity. Lastly, the textscMatlab routine fzero.m is used to solve for which temperature f(E > E◦) equals the baryon-to-photon ration, corresponding to the point marked with an asterisk, ∗, in figure 7. 39 In order to convert the temperature at the onset of BBN to a correspond- ing time, equation (C.2.14) is implemented. Thereafter, the time span from freeze-out to BBN, tspan, can be used to obtain the number ratio of remaining neutrons to protons, given the initial value, Nn,0, for this quotient and the mean life time of the neutron, τ , as was discussed in section 4.3. Nn = Nn,0 exp(−tspan/τ). (8.2.2) In the program Nn,0 is calculated, beforehand, with help of equation (4.3.4). Since the mass of the neutron and proton are almost equal, the mass percent of 4He is taken as two times Nn, while that of 1H is assumed to be given by the remainder, Nn,0 −Nn. The most probable values for the onset of BBN and the abundances of 4He and H ought to be obtained with the value on the baryon to photon ratio presented in section 5.1, mind the simplicity of this pedagogical model. As was pointed out in section 7, however there is much insight to be gained from analyzing the results of such simulations for a range of different values on η. In this case the code described above can easily be used to perform such calculations, for example by evaluating the function in a loop where the baryon to photon ratio is set to a new value at the start of each new iteration. The result of such a procedure, specifically the time for the onset of BBN as a function of η, is represented by the thick line in figure 14. 8.3 Big Bang Nucleosynthesis Using NUC123 It is difficult to find good experimental data to support the previously pre- sented estimations. Therefore, the results are compared to the output of the state-of-the-art software NUC123, written by Lawrence Kawano[29], al- though the majority of the code stems from the work of R.V. Wagoner. NUC123 solves the coupled ordinary differential equations related to the nucleosynthesis reaction network using the Runge-Kutta approximation, which is the same method as Matlab uses to solve ordinary differential equations. All of the reactions within this network that NUC123 takes into account are included in figure 8[29]. Within the program, specifically with help of the user interface, it is also possible to change the model parameters of the standard model, specifically: • Newton’s gravitational constant • the neutron lifetime • the number of neutrino species 40 p d t n 3He 4He 6Li 7Li 8Li 7Be 8Be 8B 10B 11B 12B 11C 12C 13C 14C 12N 13N 14N 15N 14O 15O 16O p,γ α,n α,γ α,p n,γ β+ n,p n,α p,α p,n β− d,p d,n (3He,2p) (n ,p α ) (d ,p α ) (p,dα ) (d,nα ) (p,nα) (n ,2 α ) (2 α ,γ ) (p, 2α ) (α ,nγ ) Figure 8: All of the reactions in the reaction network that NUC123 simulates, that is 26 nuclides and 88 reactions. 41 • the final baryon to photon ratio • the cosmological constant • the possibility for neutrino degeneracy Most of these are either accurately known today or very hard to motivate changes to based on the present accepted theories. For this reason the only parameter to be varied is the different baryon to photon ratio. More precisely, the program will be evaluated for a range of different values on the former, so as to determine its impact on the final abundances. These calculations are simplified by the fact that modern personal com- puters are many times more powerful than the computers that NUC123 was originally written for. This enables the use of wrapper software and data analysis using Matlab. Specifically, the wrapper script runs NUC123 like any normal user, that is by giving certain commands in the user interface, and varies the final baryon to photon ratio in controlled steps. All avail- able data is written onto the disc and finally analyzed using matlab. These scripts can be found in appendix E. All of the data concerning the light elements generated by NUC123 are presented in number densities relative to the number density of hydrogen, except for 4He and p which are presented in mass percentage. This means that these abundances have to be converted to mass percentage as well. Even though this conversion cannot be done exactly, a good approximation is that all nuclei, created in the physical process, is included in the output of the NUC123 program. 8.4 Simulation of the Time Evolution of BBN Figure 9 presents the results of a run of BBN123 using η = 6.1 · 10−10[12], while all other parameters were set to their standard values. In figure 11 the final abundances relative to hydrogen can be seen and in figure 10 the change in the abundance of hydrogen is visible. As is indicated by these figures, all simulations stop 28 days after the big bang. Even though there is no reason, with regards to the physical process, for the evaluations to seize when t ≥ 28 days, this choice does make it possible to illustrate the abundance of free neutrons before they finally decayed. Furthermore, the values obtained, except 7Li, agree well with observed values [3]. The data previously shown is only valid just after the Big Bang Nucle- osynthesis has ended, however some elements like tritium or free neutrons are unstable. For comparison, a list with final abundances after a majority of the synthesized neutrons and tritium have decayed are presented in table 5. 42 10 0 10 2 10 4 10 6 10 −25 10 −20 10 −15 10 −10 10 −5 10 0 n d t 3He 4He 6Li 7Li 8Li 7Be Particle density relative to Hydrogen, η=6.1E−10 Time [s] P ar tic le s pe r hy dr og en Figure 9: Abundances relative to hydrogen as a function of time. 10 0 10 2 10 4 10 6 0 10 20 30 40 50 60 70 80 90 Hydrogen Free neutrons Helium−4 Mass percentages for light elements, η=6.1E−10 Time [s] M as s pe rc en ta ge % Figure 10: Abundances in mass percentage as a function of time. 43 Table 5: Table of final abundances after decay, η = 6.1 · 10−10 Element Mass percentage Particles per hydrogen H 75.2 1 n 0 0 d 3.90E-03 2.58E-05 t 0 0 3He 2.40E-03 1.04E-05 4He 24.8 0.0825 6Li 5.06E-12 1.12E-14 7Li 1.50E-08 2.85E-11 8Li 7.40E-13 1.23E-15 7Be 2.21E-07 4.20E-10 These results have been obtained from a single run of the NUC123 software by only taking into account the predicted values on the elemental abundances after the last time step. Also, the final time has been chosen sufficiently large for the above mentioned species to have decayed though less than ∼ 300000 years. This upper limit follows from the fact that the 7Be nuclei becomes unstable once formed into atoms, which in turn is assumed to have occurred around this point in time [9, 14]. Estimates, comparable to those in table 5, can also be obtained based on the calculations presented in 8.2. Specifically, the values in table 6 from part of the output from partphotons.m, described in appendix E.7. Table 6: Simple estimates of light element abundances by mass. 1H 81.80 % 4He 18.20 % 8.5 BBN Calculations for a Range of Value on η Since the baryon to photon ratio,η, is well determined[12], it might seem unnecessary to regard it as a variable parameter. Even so, the final elemental abundances for a sweep over η is presented in figure 11. While on the subject, the graph in figure 11 corresponding to 7Li is displayed by itself in figure 12 to ease the comparison with the discussion in section 7. Since the abundances 44 10 −11 10 −10 10 −9 10 −8 10 −25 10 −20 10 −15 10 −10 10 −5 10 0 n d t 3He 4He 6Li 7Li 8Li 7Be η WMAP Particle density relative to Hydrogen 28 days after bang. Baryon to photon ratio, η P ar tic le s pe r hy dr og en Figure 11: Abundances relative to Hydrogen as a function of baryon to pho- ton ratio. of 4He and H are significantly greater than those of the other elements, the results for these species are also shown separately in figure 13. The latter diagram also includes graphs, that is the dashed lines, corresponding to the abundances of 4He and H respectively calculated with help of partphotons.m for a similar range of values on η. As became apparent in section 7.3, the nucleosynthesis should start at an earlier time for a greater value on baryon to photon ratio. Remarkably, if the onset of nucleosynthesis is defined as the point in time where there is the highest quantity of deuterium this trend becomes apparent even though there is a large oscillation around the onset time, see figure 14. The thick line in figure 14 represent the corresponding onset times given by the simple estimations done in partphotons.m, see section 8.2. 45 10 −11 10 −10 10 −9 10 −8 10 −10 10 −9 10 −8 7Li η WMAP Particle density relative to Hydrogen 28 days after bang. Baryon to photon ratio, η Li th iu m p ar tic le s pe r hy dr og en Figure 12: Abundance of 7Li relative to Hydrogen as a function of baryon to photon ratio. 46 10 −11 10 −10 10 −9 0 10 20 30 40 50 60 70 80 90 100 Hydrogen Baryon to photon ratio, η M as s P er ce nt ag e % Mass percentages 28 days after bang. Helium−4 η WMAP NUC123 Simple estimate Figure 13: Abundances of 4He and H in mass percentage as a function of baryon to photon ratio, where the dashed and solid lines represent the outputs from partphotons.m and NUC123 respectively. 47 10 −11 10 −10 10 −9 10 −8 250 300 350 400 450 500 Baryon to photon ratio, η S ec on ds to n uc le os yt he si s on se t [ s] Time until start of nucleosynthesis NUC123 onset with error Simple theoretical onset Figure 14: The onset time of BBN as a function of baryon to photon ratio. The thick line is the result of the simple calculation from partphotons.m (see appendix E.7). 48 9 Discussion Generally, any efforts to describe and make predictions on physical phenom- ena, assumptions are necessary both in order to get a qualitative understand- ing of the process and to be able to perform relevant calculations. Obviously, this is also the case for this report. However, to assess the validity for a given simplification one is required to have a deeper understanding of the process or to possess a significant quantity of data, measured with a precision that makes it possible to make valid comparisons with the simulations. For the model at hand, there exists some evidence, namely the 7Li discrepancy dis- cussed below, to suggest that some of the simplifications might be flawed. However, the degree of understanding of the authors of this report is con- fined to the physics that has been presented so far. It is therefore difficult to point towards particular simplifications that would be less likely to hold if put under scrutiny. Still, an effort shall be made to indicate which assumptions was found to be the most astonishing. With regards to the primordial nucleosynthesis, the most profound as- sumption that is probably that all particles, that have not yet decoupled, are in thermal equilibrium. Additionally, no effort has been made to take into account possible concentration gradients and thus the entire Universe has, in fact, been inferred to act as a perfectly stirred reactor. These assump- tions seem plausible however given the high kinetic energies of the particles involved in the process and the smallness of the anisotropy in the cosmic background radiation models as well as the fact that none of the alterna- tive models presented so far significantly improves on the standard model predictions [4]. As of yet though, it is not possible to entirely disregard the plausibility that some of these simplifications are inherently flawed. Another interesting simplification is the assumption that the baryonic content of the Universe only consists of protons and neutron. What is more, this is inferred without giving any compelling reason for why it ought to hold. The same is true for leptons, since the only species under consideration in addition to electrons are the neutrinos and anti-neutrinos. Moving on to the presented results, what is the most surprising is the accuracy with which the amounts of most light elements created in the nu- cleosynthesis can be calculated. The degree of correspondence between the very simple calculations, described in sections 8.2, and the more accurate sim- ulations performed by the NUC123 software is also remarkably high, given the roughness of the simplifications in the former. Specifically, a comparison of tables 6 and 5 in section 8.4 reveals that the error in the estimates of 4He and H are in the order of a few mass percent. This is not surprising since the NUC123 simulations, presented in figure 9 and table 5, shows that 49 elements other than H and 4He were only synthesized to a very small extent. In addition, the graphs in figures 13 and 14, in 8.5, that represent calcula- tions with NUC123 and partphotons.m respectively are in good agreement as well, perhaps to a less degree with regards to the latter diagram. It is also important to note that the predicted mass percent of 4He at the end of the BBN process is underestimated in the simplified calculations, even though the opposite might be expected given the assumption that all neutrons are fused into 4He nuclei. One plausible explanation for this result could be that textttfreezeout.m predicts the n-to-p freeze-out to have occurred at a later time compared to NUC123, in which case fewer neutrons survive long enough to take part in the nucleosynthesis process. While on the subject, one shall not fail to mention that the conclusions drawn regarding the dependencies of the elemental abundances in η discussed in section 7, particularly with respect to table 4, also agrees with the behav- iors shown in figure 9 and 12. In table 7 the output from the NUC123 software are presented together with the primordial abundances given by observations and recent calculations with the latest data on nuclear reaction rates, more precisely cross section measurements [30]. Table 7: Observed and calculated abundances. Coc et al. 2010 Calculated Observed Factor 4He 0.2486± 0.0002 .248 0.232-0.258 ×100 D/H 2.49± 0.17 2.58 2.82+0.20 −0.19 ×10−5 3He/H 1.00± 0.07 1.04 0.9-1.3 ×10−5 7Li/H 5.24+0.71 −0.67 0.285 1.1± 0.1 ×10−10 When the columns in the above table are compared, the most noticeable discrepancies are seen to concern 7Li. Most importantly, the abundance of 7Li obtained as an output to NUC123 lies one order of magnitude lower than both the results presented by Coc and Vangioni, as well as the observational averages [30]. Yet, it is also obvious that even these most recent simulations, with the baryon-to-photon ratio taken from the WMAP observations, still fail to comply with the observed 7Li abundances. This oddity is very important as it may point to errors in NUC123 or even the standard model itself. The poor agreement between the two simulations however, could perhaps be explained by the fact that the results given in the second column are based on more recent estimates of the relevant physical parameters, most importantly the nuclear cross sections [30]. Additionally, the system of differential equation 50 was solved using a more sophisticated computer program, specifically with the help of Monte-Carlo calculations [30]. On a final not, it might appear as if the standard model for the Big Bang, as in the case of classical physics in the 19th century, gives an almost complete description of the underlying physics of the processes it describes. It is furthermore often indicated that the few flaws that do exist are only minor in nature. Still, one is compelled to draw a comparison between the 7Li deficiency and the problem with the black body radiation in classical physics. 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