NUCLEOSYNTHESIS also known as from the Big Bang to Today Summer School on Nuclear and Particle Astrophysics Connecting Quarks with the Cosmos I George M. Fuller Department of Physics University of California, San Diego
The man who discovered how stars shine made many other fundamental contributions in particle, nuclear, and condensed matter physics, as well as astrophysics. In particular, Hans Bethe completely changed the way astrophysicists think about equation of state and nucleosynthesis issues with his 1979 insight on the role of entropy. Bethe, Brown, Applegate, & Lattimer (1979) Hans Bethe
There is a deep connection between spacetime curvature and entropy (and neutrinos) Curvature (gravitational potential well) Entropy content/transport by neutrinos Entropy (disorder) fundamental physics of the weak interaction
Entropy entropy per baryon (in units of Boltzmann's constant k) of the air in this room s/k ~ 10 entropy per baryon (in units of Boltzmann's constant k) characteristic of the sun s/k ~ 10 entropy per baryon (in units of Boltzmann's constant k) for a 10 6 solar mass star s/k ~ 1000 entropy per baryon (in units of Boltzmann's constant k) of the universe s/k ~ 10 10 total entropy of a black hole of mass M S /k = 4π M m pl 2 10 77 M M sun where the gravitational constant is G = 1 2 m pl and the Planck mass is m pl 1.221 10 22 MeV 2
Entropy S = k logγ a measure of a system s disorder/order Low Entropy 12 free nucleons 12 C nucleus
Nucleosynthesis The Big Picture
Drive toward Nuclear Statistical Equilibrium (NSE) Freeze-Out from Nuclear Statistical Equilibrium
FLRW Universe (S/k~10 10 ) The Bang Temperature Neutrino-Driven Wind (S/k~10 2 ) Outflow from Neutron Star Weak Freeze-Out T= 0.7 MeV T~ 0.9 MeV Weak Freeze-Out n/p<1 n/p>1 Alpha Particle Formation T~ 0.1 MeV T~ 0.75 MeV Alpha Particle Formation PROTON Time NEUTRON
The nuclear and weak interaction physics of primordial nucleosynthesis (or Big Bang Nucleosynthesis, BBN) was first worked out self consistently in 1967 by Wagoner, Fowler, & Hoyle. This has become a standard tool of cosmologists. Coupled with the deuterium abundance it gave us the first determination of the baryon content of the universe. BBN gives us constraints on lepton numbers and new neutrino and particle physics. BBN is the paradigm for all nucleosynthesis processes which involve a freeze-out from nuclear statistical equilibrium (NSE). R. Wagoner, W. A. Fowler, & F. Hoyle (from D. Clayton s nuclear astrophysics photo archive at Clemson University)
Suzuki (Tytler group) 2006
So where are the nuclei heavier than deuterium, helium, and lithium made???
W. A. Fowler G. Burbidge M. Burbidge B 2 FH (1957) outlined the basic processes in which the intermediate and heavy elements are cooked in stars. F. Hoyle
Photon luminosity of a supernova is huge: L ~ 10 10 L sun (this one is a Type Ia) Type Ia C/O WD incineration to NSE Fe-peak elements, complicated interplay of nuclear burning, neutrino cooling, and flame front propagation cse.ssl.berkeley.edu/
Weaver & Woosley, Sci Am, 1987
Nuclear Burning Stages of a 25 M sun Star Burning Stage Temperature Density Time Scale Hydrogen 5 kev 5 g cm -3 7 X 10 6 years Helium 20 kev 700 g cm -3 5 X 10 5 years Carbon 80 kev 2 X 10 5 g cm -3 600 years Neon 150 kev 4 X 10 6 g cm -3 1year Oxygen 200 kev 10 7 g cm -3 6 months Silicon 350 kev 3 X 10 7 g cm -3 1 day Core Collapse 700 kev 4 X 10 9 g cm -3 ~ seconds of order the free fall time Bounce ~ 2 MeV ~10 15 g cm -3 ~milli-seconds Neutron Star < 70 MeV initial ~ kev cold ~10 15 g cm -3 initial cooling ~ 15-20 seconds ~ thousands of years
Massive Stars are From core carbon/oxygen burning onward the neutrino luminosity exceeds the photon luminosity. Neutrinos carry energy/entropy away from the core! Core goes from S/k~10 on the Main Sequence (hydrogen burning) to a thermodynamically cold S/k ~1 at the onset of collapse! e.g., the collapsing core of a supernova can be a frozen (Coulomb) crystalline solid with a temperature ~1 MeV!
Type II core collapse supernova BLUE - UV GREEN - B RED -I Caltech Core Collapse Project (CCCP) Type Ib/c core collapse supernova www.cfa.harvard.edu/ ~mmodjaz
Fuller & Meyer 1995 Meyer, McLaughlin & Fuller 1998
Primordial Nucleosynthesis (BBN)
Suzuki (Tytler group) 2006
WMAP cosmic microwave background satellite Fluctuations in CMB temperature give Insight into the composition, size, and age of the universe and the large scale character of spacetime. Age = 13.7 Gyr Spacetime = flat (meaning k=0) Composition = 23% unknown nonrelativistic matter, 73% unknown vacuum energy (dark energy), 4% ordinary baryons.
(1) The advent of ultra-cold neutron experiments has helped pin down the neutron lifetime (strength of the weak interaction) (2) The CMB acoustic peaks have given a precise determination of the baryon to photon ratio This has changed the way we look at BBN - New probes of leptonic sector now possible.
Quantum Numbers
baryon number of universe From CMB acoustic peaks, and/or observationally-inferred primordial D/H: three lepton numbers From observationally-inferred 4 He and large scale structure and using collective (synchronized) active-active neutrino oscillations (Abazajian, Beacom, Bell 03; Dolgov et al. 03):
Leptogenesis Generate net lepton number through CP violation in the neutrino sector. Transfer some of this or a pre-existing net lepton number to a net baryon number.
Baryon Number (from CMB acoustic peak amplitudes)
-- Precision baryon number measurement -- Sets up robust BBN light element abundance predictions which, along with observations and simulations of large scale structure potentially enables probes of QCD epoch entropy fluctuations, black holes Early nuclear evolution, cosmic rays, the first stars Neutrino mass physics (leptogenesis( leptogenesis,, mixing, etc.) Decaying Dark Matter WIMPS
Thermodynamic Preliminaries
Thermonuclear Reaction Rates Rate per reactant is the thermally-averaged product of flux and cross section. a+x Y+b or X(a,b)Y rate per X nucleus is λ = ( 1+ δ ax ) 1 σ v ~ 1 E exp b Z Z a X e2 E Rates can be very temperature sensitive, especially when Coulomb barriers are big.
At high enough temperature the forward and reverse rates for nuclear reactions can be large and equal and these can be larger than the local expansion rate. This is equilibrium. If this equilibrium encompasses all nuclei, we call it Nuclear Statistical Equilibrium (NSE). In most astrophysical environments NSE sets in for T 9 ~ 2. T T 9 10 9 K where Boltzmann's constant is k B 0.08617 MeV per T 9
Electron Fraction In general, abundance relative to baryons for species i mass fraction mass number
Freeze-Out from Nuclear Statistical Equilibrium (NSE) In NSE the reactions which build up and tear down nuclei have equal rates, and these rates are large compared to the local expansion rate. Z p + N n A(Z,N) + γ nuclear mass A is the sum of protons and neutrons A=Z+N Z μ p + N μ n = μ A + Q A Binding Energy of Nucleus A Y A ( Z,N ) S1 A Saha Equation [ ]Gπ 7 2 (A 1) 2 1 2 (A 3) T A 3/2 m b 3 2 (A 1) Y Z p Y N n e Q A /T
Typically, each nucleon is bound in a nucleus by ~ 8 MeV. For alpha particles the binding per nucleon is more like 7 MeV. But alpha particles have mass number A=4, and they have almost the same binding energy per nucleon as heavier nuclei so they are favored whenever there is a competition between binding energy and disorder (high entropy).
FLRW Universe (S/k~10 10 ) The Bang Temperature Neutrino-Driven Wind (S/k~10 2 ) Outflow from Neutron Star Weak Freeze-Out T= 0.7 MeV T~ 0.9 MeV Weak Freeze-Out n/p<1 n/p>1 Alpha Particle Formation T~ 0.1 MeV T~ 0.75 MeV Alpha Particle Formation PROTON Time NEUTRON
number density for fermions (+) and bosons (-) g dω E 2 de 2π 2 4π e E /T η ±1 dn g d 3 p 1 ( 2π) 3 e E /T η ±1 where the pencil of directions is dω=sinθ dθ dφ The energy density is then dε g dω E E 2 de 2π 2 4π e E /T η ±1 now get the total energy density by integrating over all energies and directions (relativistic kinematics limit) ρ T 4 2π 2 0 x 3 dx e x η ±1 degeneracy parameter (chemical potential/temperature) η μ T in extreme relativistic limit η 0 x 3 dx = π 4 e x 1 15 0 and x 3 dx = 7π 4 e x +1 120 0 π 2 bosons ρ g b 30 T 4 and fermions ρ 7 8 g f π 2 30 T 4
Statistical weight in all relativistic particles: T g = g b i eff i T i 3 + 7 8 j g j f T j T 3 e.g., statistical weight in photons, electrons/positrons and six thermal, zero chemical potential (zero lepton number) neutrinos, e.g., BBN: g = 2 + 7 ( eff 8 2 + 2 + 6) )=10.75 ν e ν e ν μ ν μ ν τ ν τ
Spacetime Background
Relic neutrinos from the epoch when the universe was at a temperature T ~ 1 MeV ( ~ 10 10 K) ~ 300 per cubic centimeter neutrino decoupling T~ 1 MeV photon decoupling T~ 0. 2 ev Relic photons. We measure 410 per cubic centimeter vacuum+matter dominated at current epoch
Coupled star formation, cosmic structure evolution Mass assembly history of galaxies, nucleosynthesis, weak lensing/neutrino mass Very Early Universe: baryo/lepto-genesis QCD epoch, BBN Neutrino physics Re-ionization: 1 in 10 3 baryons into stars; Nucleosynthesis? Black Holes?
George Gamow Albert Einstein George LeMaitre A. Friedmann
Birkhoff s Theorem Invoking this requires symmetry: specifically, a homogeneous and isotropic distribution of mass and energy! What evidence is there that this is true? Look around you. This is manifestly NOT true on small scales. The Cosmic Microwave Background Radiation (CMB) represents our best evidence that matter is smoothly and homogeneously distributed on the largest scales.
Homogeneity and isotropy of the universe: implies that total energy inside a co-moving spherical surface is constant with time. total energy = (kinetic energy of expansion) + (gravitational potential energy) mass-energy density = ρ test mass = m a 1 mý a 2 2 a Ý 2 + k = 8 π Gρ a2 3 G [ 4 3 πa3 ρ]m a total energy > 0 expand forever k = -1 total energy = 0 for ρ = ρ crit k = 0 total energy < 0 re-collapse k = +1 Ω = ρ/ρ crit =Ω γ + Ω ν + Ω baryon + Ω dark matter + Ω vacuum 1 (k=0)
Friedman-LeMaitre-Robertson-Walker (FLRW) coordinates defined through this metric...
How far does a photon travel in the age of the universe? (causal horizon) Consider a radially-directed photon ( ) photons travel on null world lines so ds 2 =0
Causal (Particle) Horizon radiation dominated = matter dominated vacuum energy dominated In every case the physical (proper) distance a light signal travels goes to infinity as the value of the timelike coordinate t does. Note, however, that for the vacuum-dominated case there is a finite limiting value for the FLRW radial coordinate as t goes to infinity...
some significant events/epochs in the early universe Epoch T g eff Horizon Length Mass-Energy (solar masses) Baryon Mass (solar masses) Electroweak phase transition 100 GeV ~100 ~ 1 cm ~ 10-6 (~ earth mass) ~ 10-18 QCD 100 MeV 51-62 20 km ~ 1 ~ 10-9 weak decoupling weak freeze out 2 MeV 10.75 ~ 10 10 cm ~ 10 4 ~ 10-3 0.7 MeV 10.75 ~ 10 11 cm ~ 10 5 ~ 10-2 BBN 100 kev 10.75 ~ 10 13 cm (~ 1 A.U.) ~ 10 6 ~ 1 e - /e + annihilation ~ 20 kev 3.36 ~ 10 14 cm ~ 10 8 ~ 100 photon decoupling 0. 2 ev - ~ 350 kpc ~ 10 18 dark matter ~ 10 17 1 solar mass 2 10 33 g 10 60 MeV
The History of The Early Universe: (shown are a succession of temperature and causal horizon scales) ν e + n p + e ν e + p n + e + The QCD horizon is essentially an ultra-high entropy Neutron Star
Co-Moving Entropy Density is Conserved Assume a perfect fluid* stress-energy tensor Energy/momentum conservation in FLRW coordinates but first law of thermo gives *Not true when mixed relativistic/nonrelativistic system, or decaying particles ----- Bulk Viscosity
Cosmic Bulk Viscosity Cosmic Bulk Viscosity only non-adiabatic, dissipative contribution consistent with homogeneity, isotropy rotational, translational invariance Biggest effect when decaying particles have lifetimes of order the local Hubble time, dominate mass-energy! Weinberg 1971; Quart 1930
The Entropy of the Universe is Huge We know the entropy-per-baryon of the universe because we measure the cosmic microwave background temperature and we measure the baryon density through the deuterium abundance and CMB acoustic peak amplitude ratios. S/k = 2.5 x 10 8 (Ω b h 2 ) -1 ~ 10 10 Deuterium, CMB, and large scale structure measurements imply all Ω b h 2 ~ 0.02 Neglecting relatively small contributions from black holes, SN, shocks, nuclear burning, etc., S/k has been constant throughout the history of the universe. S/k is a (roughly) co-moving invariant.
entropy per baryon in radiation-dominated conditions entropy per unit proper volume S 2π 2 45 g s T 3 proper number density of baryons n b = η n γ entropy per baryon s S n b
The baryon number is defined to be the ratio of the net number of baryons to the number of photons: η = n b n b n γ The baryon number, or baryon-to to-photon ratio, η is a kind of inverse entropy per baryon, but it is not a co-moving invariant. η 2π 4 45 1 ζ() 3 g total g γ S 1
Friedmann equation is a Ý 2 + k = 8 π Gρ a2 and 3 G = 1 where h = c =1 and the Planck Mass is m 2 PL 1.22 10 22 MeV m PL radiation dominated ρ π 2 30 g T 4 eff ~ 1 a 4 horizon is d H () t 2t H 1 where the Hubble parameter, or expansion rate is H = a Ý a 8π 3 90 t 0.74 s 1/2 ( ) 10.75 g eff 1/2 T 2 g eff m PL 1/2 MeV The entropy in a co - moving volume is conserved g 1/3 eff at = 1/3 g eff T 2 a T so that if the number of relativistic degrees of freedom is constant T ~ 1 a
Weak Interactions
Weak Interaction/NSE-Freeze Freeze-Out History of the Early Universe Weak Decoupling T ~ 3 MeV λ νe ~ λ νν ~ G F2 T 5 >> H ~ g 1/2 eff T 2 /m pl forces neutrinos into weak interaction (flavor) eigenstates λ νn ~ λ en ~ λ νp ~ λ ep >> H Weak Freeze-Out T ~ 0.7 MeV λ n(p,γ)d = λ d(γ,p)n >> H Nuclear Statistical Equilibrium (NSE) Freeze-Out Alpha Particle Formation T ~ 0.1 MeV e + /e - annihilation (heating of photons relative to neutrinos) T ν = (4/11) 1/3 T γ Temperature/Time
Weak Decoupling This occurs when the rates of neutrino scattering reactions on electrons/positrons drop below the expansion rate. After this epoch the neutrino gas ceases to efficiently exchange energy with the photon-electron plasma. neutrino scattering rate λ ν ~ ( G 2 F T 2 )( T 3 )= G 2 F T 5 where the Fermi constant is G F 1.166 10 11 MeV -2 expansion rate H 8π 3 90 weak decoupling temperature T WD 8π 3 90 1/6 1/2 1/2 T 2 g eff m PL 1/6 g eff ( G 2 F m ) 1.5 MeV g eff 1/3 PL 10.75 1/6
As pairs annihilate, their entropy is transferred to the photons and plasma, not to the decoupled neutrinos. Product of scale factor and temperature is increased for photons, constant for decoupled neutrinos: current epoch? scale factor T ν
Weak Freeze Out Even though neutrinos are thermally decoupled, there are still ~10 10 of them per nucleon. Weak charged current lepton-nucleon processes flip nucleon isospins from neutron to proton to neutron to proton... If this isospin flip rate is large compared to the expansion rate, then steady state, chemical equilibrium can be maintained between leptons and nucleons. Eventually, weak interaction-driven isospin flip rate falls below expansion rate, neutron/proton ratio frozen in, ------- this is Weak Freeze Out
Neutron-to-proton ratio is set by the competition between the rates of these processes: threshold threshold threshold neutron-proton mass difference
Charged Current Weak Interaction Rates for Neutrons and Protons lepton occupation probabilities Neutrinos if thermal, Fermi-Dirac energy spectra then Coulomb correction Fermi factor attractive Coulomb interaction increases electron probability at the proton, increasing the above phase space factors in which F appears.
Strength of the Weak Interaction radiative corrections Determine this by using the measured free (vacuum) neutron lifetime Any effect which increases this phase space factor will decrease the overall weak interaction strength, leading to earlier (hotter) freeze out, more neutrons and, hence, more 4He.
Define the total neutron destruction rate Define the total proton destruction rate Then the time rate of change of n/p is If the weak rates are large enough, and expansion slow enough, system can approach Steady State Equilibrium valid at high T where we can neglect free neutron decay and the three-body reverse process
Steady State Equilibrium equality holds when leptons have thermal, Fermi-Dirac energy distribution functions Chemical Equilibrium --- the Saha equation
equilibrium actual
formation of alphas