NUCLEOSYNTHESISNUCLEOSYNTHESIS also known as fromfrom thethe BigBig BangBang toto TodayToday

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 fundamental (disorder) 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 106 solar mass star s/k ~ 1000 entropy per baryon (in units of Boltzmann's constant k) of the universe s/k ~ 1010 total entropy of a black hole of mass M 2 ⎛ ⎞ ⎛ ⎞ 2 M 77 M S /k = 4π⎜ ⎟ ≈10 ⎜ ⎟ ⎝ mpl ⎠ ⎝ Msun ⎠ 1 where the gravitational constant is G = 2 mpl 22 and the Planck mass is mpl ≈1.221×10 MeV EntropyEntropy S = k logΓ a measure of a system’s disorder/order

LowLow EntropyEntropy

12 12 free nucleons C nucleus NucleosynthesisNucleosynthesis

TheThe BigBig PicturePicture Drive toward Nuclear Statistical Equilibrium (NSE)

Freeze-Out from Nuclear Statistical Equilibrium FLRW Universe (S/k~1010) Neutrino-Driven Wind (S/k~102) The Bang Temperature 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

Time PROTON 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

B2FH (1957) outlined the basic processes in which the intermediate and heavy elements are cooked in stars. F. Hoyle

10 Photon luminosity of a supernova is huge: L ~ 10 Lsun (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 NuclearNuclear BurningBurning StagesStages ofof aa 2525 MMsun StarStar Burning Temperature Density Time Scale Stage Hydrogen 5 keV 5 g cm-3 7 X 106 years Helium 20 keV 700 g cm-3 5 X 105 years Carbon 80 keV 2 X 105 g cm-3 600 years Neon 150 keV 4 X 106 g cm-3 1year Oxygen 200 keV 107 g cm-3 6 months Silicon 350 keV 3 X 107 g cm-3 1 day

Core Collapse 700 keV 4 X 109 g cm-3 ~ seconds of order the free fall time

“Bounce” ~ 2 MeV ~1015 g cm-3 ~milli-seconds Neutron Star < 70 MeV initial ~1015 g cm-3 initial cooling ~ 15-20 seconds ~ keV “cold” ~ thousands of years MassiveMassive StarsStars areare

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 Type Ib/c core collapse supernova BLUE - UV GREEN -B RED -I

Caltech Core Collapse Project (CCCP) www.cfa.harvard.edu/ ~mmodjaz Fuller & Meyer 1995 Meyer, McLaughlin & Fuller 1998 PrimordialPrimordial NucleosynthesisNucleosynthesis

((BBNBBN)) 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. QuantumQuantum NumbersNumbers baryon number of universe

From CMB acoustic peaks, and/or observationally-inferred primordial D/H:

three lepton numbers

From observationally-inferred 4He 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. BaryonBaryon NumberNumber

(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, mixing, etc.)

Decaying Dark Matter WIMPS ThermodynamicThermodynamic PreliminariesPreliminaries 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 −1 rate per X nucleus is λ = ()1+ δaX σ v

1 ⎛ Z Z e2 ⎞ ~ exp⎜ −b a X ⎟ E ⎝ 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 T9 ~ 2.

T T ≡ 9 109 K

where Boltzmann's constant is kB ≈ 0.08617 MeV per T9 ElectronElectron FractionFraction

InIn general,general, abundanceabundance relativerelative toto baryonsbaryons forfor speciesspecies ii

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 μp + N μn = μA + QA

Binding Energy of Nucleus A Saha Equation Saha Equation 3 2(A−1) 7 1 ⎛ ⎞ 1−A 2(A−1) 2(A−3) 3/2 T Z N QA /T YAZ(),N ≈ []S Gπ 2 A ⎜ ⎟ Yp Yn e ⎝ mb ⎠ 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~1010) Neutrino-Driven Wind (S/k~102) The Bang Temperature 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

Time PROTON NEUTRON number density for fermions (+) and bosons (-) degeneracy parameter d 3p 1 g ⎛ dΩ⎞ E 2dE (chemical potential/temperature) dn ≈ g 3 E /T −η ≈ 2 ⎜ ⎟ E /T −η ()2π e ±1 2π ⎝ 4π ⎠ e ±1 μ η ≡ where the pencil of directions is dΩ=sinθ dθ dφ T

The energy density is then in extreme relativistic limit g ⎛ dΩ⎞ E ⋅ E 2dE η → 0 dε ≈ ⎜ ⎟ 2π 2 ⎝ 4π ⎠ eE /T −η ±1 now get the total energy density by integrating over all energies and directions (relativistic kinematics limit) T 4 ∞ x 3 dx ρ ≈ 2 ∫ x−η 2π 0 e ±1

∞ x 3 dx π 4 ∞ x 3 dx 7π 4 ∫ x = and ∫ x = 0 e −1 15 0 e +1 120

π 2 ⎛ 7 ⎞ π 2 bosons ρ ≈ g T 4 and fermions ρ ≈ ⎜ g ⎟ T 4 b 30 ⎝ 8 f ⎠ 30 Statistical weight in all relativistic particles:

3 ⎛ ⎞ 3 b⎛ Ti ⎞ 7 f Tj geff = ∑gi ⎜ ⎟ + ∑g j ⎜ ⎟ i ⎝ T ⎠ 8 j ⎝ T ⎠ e.g., statistical weight in photons, electrons/positrons and six thermal, zero chemical potential (zero lepton number) neutrinos, e.g., BBN:

7 geff = 2 + 8 (2 + 2 + 6))=10.75

ν e ν e ν μ ν μ ντ ν τ SpacetimeSpacetime BackgroundBackground Relic neutrinos from the epoch when the universe was at a temperature T ~ 1 MeV ( ~ 1010 K)

~ 300 per cubic centimeter

photon decoupling T~ 0. 2 eV neutrino decoupling T~ 1 MeV 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 103 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 4 3 1 2 G[3 πa ρ]m ≈ maÝ ≈− 2 a a 8 aÝ 2 + k = π Gρ a2 3 total energy > 0 expand forever k = -1

total energy = 0 for ρ = ρcrit k = 0

total energy < 0 re-collapse k = +1

Ω = ρ/ρcrit =Ωγ + Ων + Ωbaryon + Ωdark matter + Ωvacuum ≈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 ds2=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

Horizon Mass-Energy Baryon Mass Epoch T g eff Length (solar masses) (solar masses) Electroweak ~ 10-6 phase 100 GeV ~100 ~ 1 cm ~ 10-18 (~ earth mass) transition QCD 100 MeV 51 - 62 20 km ~ 1 ~ 10-9 weak 2 MeV 10.75 ~ 1010 cm ~ 104 ~ 10-3 decoupling weak 0.7 MeV 10.75 ~ 1011 cm ~ 105 ~ 10-2 freeze out ~ 1013 cm BBN 100 keV 10.75 ~ 106 ~ 1 (~ 1 A.U.) e-/e+ ~ 20 keV 3.36 ~ 1014 cm ~ 108 ~ 100 annihilation photon ~ 1018 0. 2 eV - ~ 350 kpc ~ 1017 decoupling dark matter

1 solar mass ≈ 2 ×1033 g ≈1060 MeV The History of The Early Universe:

(shown are a succession of temperature and causal horizon scales)

− + νe + n ↔ p + e νe + 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 CosmicCosmic BulkBulk ViscosityViscosity

only non-adiabatic, dissipative contribution consistent with homogeneity, isotropy – rotational, translational invariance les tic ime, g par ble t ayin b dec cal Hu hen he lo ct w der t effe f or gest es o gy! Big etim -ener e lif ass hav e m inat dom

Weinberg 1971; Quart 1930 TheThe EntropyEntropy ofof thethe UniverseUniverse isis HugeHuge 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.

8 2 -1 10 S/k = 2.5 x 10 (Ωbh ) ~ 10

Deuterium, CMB, and large scale structure 2 measurements imply all Ωbh ~ 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 2π 2 S ≈ g T 3 45 s

proper number density of baryons nb = η nγ

S entropy per baryon s ≈ nb The “baryon number” is defined to be the ratio of the nb − nb net number of baryons η = to the number of photons: nγ

The “baryon number,” or baryon-to-photon ratio, η is a kind of “inverse entropy per baryon,” but it is not a co-moving invariant.

2π 4 1 g η ≈ total S−1 45 ζ ()3 gγ 8 Friedmann equation is aÝ 2 + k = π Gρ a2 and 3 1 G = where = c =1 and the Planck Mass is m ≈1.22 ×1022 MeV m2 h PL PL π 2 1 radiation dominated ρ ≈ g T 4 ~ 30 eff a4 −1 ⇒ horizon is dH()t ≈ 2t ≈ H where the Hubble parameter, or expansion rate is ⎛ 3 ⎞1/2 2 aÝ 8π 1/2 T H = ≈ ⎜ ⎟ geff a ⎝ 90 ⎠ mPL 1/2 ⎛ 10.75⎞ ⎡ MeV⎤2 t ≈ ()0.74 s ⎜ ⎟ ⎢ ⎥ ⎝ geff ⎠ ⎣ T ⎦ The entropy in a co - moving volume is conserved 1/3 1/3 ⇒ geff aT = g ′eff a ′T ′ so that if the number of relativistic degrees of freedom is constant 1 ⇒ T ~ a WeakWeak InteractionsInteractions Weak Interaction/NSE-Freeze-Out History of the Early Universe 2 5 1/2 2 λνe ~ λνν ~ GF T >> H ~ geff T /mpl forces neutrinos into weak interaction Weak Decoupling (flavor) eigenstates T ~ 3 MeV

λνn ~ λen ~ λνp ~ λep >> H

Weak Freeze-Out T ~ 0.7 MeV

λn(p,γ)d = λd(γ,p)n >> H

Nuclear Statistical Equilibrium (NSE) Freeze-Out Alpha Particle Formation e+/e- annihilation T ~ 0.1 MeV (heating of photons relative to neutrinos) 1/3 Tν = (4/11) Tγ

Temperature/Time WeakWeak DecouplingDecoupling

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.

2 2 3 2 5 neutrino scattering rate λν ~G( F T )()T = GF T

−11 -2 where the Fermi constant is GF ≈1.166 ×10 MeV

⎛ 3 ⎞1/2 2 8π 1/2 T expansion rate H ≈ ⎜ ⎟ geff ⎝ 90 ⎠ mPL

weak decoupling temperature ⎛ 3 ⎞1/6 1/6 1/6 8π geff ⎛ geff ⎞ TWD ≈ ⎜ ⎟ ≈1.5 MeV ⎜ ⎟ ⎝ 90 ⎠ 2 1/3 ⎝ 10.75⎠ ()GF mPL 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 Tν WeakWeak FreezeFreeze OutOut

Even though neutrinos are thermally decoupled, there are still ~1010 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

leptonlepton occupationoccupation probabilitiesprobabilities Coulomb correction – Fermi factor attractive Coulomb interaction increases electron probability at the proton, increasing the above phase space factors in which F appears. Neutrinos – if thermal, Fermi-Dirac energy spectra then StrengthStrength ofof thethe WeakWeak InteractionInteraction 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 actual

equilibrium formation of alphas