George M. FullerDepartment of Physics

University of California, San Diego

also known asNUCLEOSYNTHESISNUCLEOSYNTHESIS

from the Big Bang to Todayfrom the Big Bang to Today

Summer School on Nuclear and Particle AstrophysicsConnecting Quarks with the Cosmos

I

Hans BetheHans Bethe

The man who discovered how starsshine made many other fundamental contributions in particle, nuclear, and condensedmatter physics, as well as astrophysics.

In particular, Hans Bethe completelychanged the way astrophysiciststhink about equation of state and nucleosynthesis issues with his 1979insight on the role of entropy.

Bethe, Brown, Applegate, & Lattimer (1979)

There is a deep connection between spacetime curvature and entropy (and neutrinos)

Curvature(gravitational potential well)

Entropy(disorder)

Entropy content/transportby neutrinos

fundamentalphysics of the weak interaction

Entropyentropy per baryon (in units of Boltzmann's constant k)of the air in this room s /k ~ 10entropy per baryon (in units of Boltzmann's constant k)characteristic of the sun s /k ~ 10entropy per baryon (in units of Boltzmann's constant k)for a 106 solar mass star s /k ~ 1000entropy per baryon (in units of Boltzmann's constant k)of the universe s /k ~ 1010

total entropy of a black hole of mass M

S /k = 4π Mmpl

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

≈1077 MMsun

⎛

⎝ ⎜

⎞

⎠ ⎟

2

where the gravitational constant is G =1

mpl2

and the Planck mass is mpl ≈1.221×1022 MeV

EntropyEntropy

a measure of a system’s disorder/order

S = k logΓ

Low EntropyLow Entropy

12 free nucleons 12C nucleus

NucleosynthesisNucleosynthesis

The Big PictureThe Big Picture

Drive toward Nuclear Statistical Equilibrium (NSE)

Freeze-Out from Nuclear Statistical Equilibrium

n/p<1n/p>1

TimeTime

TemperatureTemperature

Weak FreezeWeak Freeze--OutOut Weak FreezeWeak Freeze--OutOut

Alpha Particle FormationAlpha Particle Formation Alpha Particle FormationAlpha Particle Formation

FLRW UniverseFLRW Universe (S/k~1010) NeutrinoNeutrino--Driven WindDriven Wind (S/k~102)

NEUTRONPROTON

T= 0.7 MeV T~ 0.9 MeV

T~ 0.1 MeV T~ 0.75 MeV

Outflow from Neutron StarThe Bang

R. Wagoner, W. A. Fowler, & F. Hoyle

The nuclear and weak interactionphysics of primordial nucleosynthesis(or Big Bang Nucleosynthesis, BBN)was first worked out self consistentlyin 1967 by Wagoner, Fowler, & Hoyle.

This has become a standard toolof cosmologists. Coupled with thedeuterium abundance it gave us the firstdetermination of the baryon contentof the universe. BBN gives us constraintson lepton numbers and new neutrino and particle physics.

BBN is the paradigm for all nucleosynthesis processes which involvea freeze-out from nuclear statisticalequilibrium (NSE).

(from D. Clayton’s nuclear astrophysics photo archiveat Clemson University)

Suzuki (Tytler group) 2006

So where are the nuclei heavierthan deuterium, helium, and lithium made ???

G. Burbidge M. Burbidge

W. A. Fowler

F. Hoyle

B2FH (1957) outlined thebasic processes in which theintermediate and heavy elementsare cooked in stars.

cse.ssl.berkeley.edu/

Photon luminosity of a supernova is huge: L ~ 1010 Lsun(this one is a Type Ia)

Type Ia – C/O WD incineration to NSE

Fe-peak elements, complicated interplay ofnuclear burning, neutrino cooling, and flame frontpropagation

Weaver & Woosley, Sci Am, 1987

Nuclear Burning Stages of a 25 MNuclear Burning Stages of a 25 Msunsun StarStarBurning Stage

Temperature Density Time Scale

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 1 year

Oxygen 200 keV 107 g cm-3 6 monthsSilicon 350 keV 3 X 107 g cm-3 1 day

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

“Bounce” ~ 2 MeV ~1015 g cm-3 ~milli-seconds

Neutron StarNeutron Star < 70 MeV initial~ keV “cold”

~1015 g cm-3 initial cooling ~ 15-20 seconds~ thousands of years

Massive Stars areMassive Stars are

From core carbon/oxygen burning onwardthe neutrino luminosity exceeds the photon luminosity.

Neutrinos carry energy/entropy away from the core!Neutrinos carry energy/entropy away from the core!

Core goes from S/k~10S/k~10 on the Main Sequence (hydrogen burning)to a thermodynamically cold S/k ~1S/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 1995Meyer, McLaughlin & Fuller 1998

PrimordialPrimordialNucleosynthesisNucleosynthesis

((BBNBBN))

Suzuki (Tytler group) 2006

WMAP cosmic microwave background satellite

Fluctuations in CMB temperature giveInsight into the composition, size, and ageof the universe and the large scale characterof spacetime.

Age = 13.7 GyrSpacetime = “flat” (meaning k=0)Composition = 23% unknown nonrelativistic

matter, 73% unknownvacuum energy (dark energy),4% ordinary baryons.

(1) The advent of ultra-cold neutron experimentshas helped pin down the neutron lifetime(strength of the weak interaction)

(2) The CMB acoustic peaks have given a precisedetermination of the baryon to photon ratio

This has changed the way we look at BBN -

New probes of leptonic sector now possible.

QuantumQuantumNumbersNumbers

baryon number of universe

three lepton numbers

From observationally-inferred 4He and large scale structureand using collective (synchronized) active-active neutrino oscillations(Abazajian, Beacom, Bell 03; Dolgov et al. 03):

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

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 NumberBaryon Number(from CMB acoustic peak amplitudes)

---- Precision baryon number measurement Precision baryon number measurement ----

Sets up robust BBN light element abundance predictionsSets up robust BBN light element abundance predictionswhich, along with observations and simulations of large scale stwhich, along with observations and simulations of large scale structure ructure potentially enables probes ofpotentially enables probes of

Early nuclear evolution, cosmic rays, the first starsEarly nuclear evolution, cosmic rays, the first stars

Neutrino mass physics (Neutrino mass physics (leptogenesisleptogenesis, mixing, etc.), mixing, etc.)

Decaying Dark Matter WIMPSDecaying Dark Matter WIMPS

QCD epoch QCD epoch –– entropy fluctuations, black holesentropy fluctuations, black holes

ThermodynamicThermodynamicPreliminariesPreliminaries

Thermonuclear Reaction Rates

Rate per reactant is the thermally-averagedproduct of flux and cross section.

a + X → Y +b or X(a,b)Y

rate per X nucleus is λ = 1+ δaX( )−1 σ v

~ 1E

exp −b ZaZXe2

E

⎛

⎝ ⎜

⎞

⎠ ⎟

Rates can be very temperature sensitive,especially when Coulomb barriers are big.

At high enough temperature the forward and reverserates for nuclear reactions can be large and equaland these can be larger than the local expansion rate.This is equilibrium. If this equilibrium encompassesall nuclei, we call it Nuclear Statistical Equilibrium (NSE).

In most astrophysical environments NSE sets in for T9 ~ 2.

T9 ≡T

109 K

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

Electron FractionElectron Fraction

In general, abundance relativeIn general, abundance relativeto baryons for species to baryons for species ii

mass fraction

mass number

FreezeFreeze--Out from Out from Nuclear Statistical EquilibriumNuclear Statistical Equilibrium ((NSENSE))In In NSENSE the reactions which build up and tear down nucleithe reactions which build up and tear down nucleihave equal rates, and these rates are large compared to have equal rates, and these rates are large compared to the local expansion rate.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

Saha EquationSaha Equation

YA Z ,N( ) ≈ S1−A[ ]Gπ72(A−1)2

12(A−3) A3 / 2 T

mb

⎛

⎝ ⎜

⎞

⎠ ⎟

32(A−1)

YpZYn

NeQA /T

Binding Energyof Nucleus A

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 nucleiso they are favored whenever there is a competitionbetween binding energy and disorder (high entropy).

n/p<1n/p>1

TimeTime

TemperatureTemperature

Weak FreezeWeak Freeze--OutOut Weak FreezeWeak Freeze--OutOut

Alpha Particle FormationAlpha Particle Formation Alpha Particle FormationAlpha Particle Formation

FLRW UniverseFLRW Universe (S/k~1010) NeutrinoNeutrino--Driven WindDriven Wind (S/k~102)

NEUTRONPROTON

T= 0.7 MeV T~ 0.9 MeV

T~ 0.1 MeV T~ 0.75 MeV

Outflow from Neutron StarThe Bang

number density for fermions (+) and bosons (-)

dn ≈ g d3p2π( )3

1eE /T −η ±1

≈g

2π 2dΩ4π

⎛ ⎝ ⎜

⎞ ⎠ ⎟

E 2dEeE /T −η ±1

where the pencil of directions is dΩ = sinθ dθ dφThe energy density is then

dε ≈g

2π 2dΩ4π

⎛ ⎝ ⎜

⎞ ⎠ ⎟

E ⋅ E 2dEeE /T −η ±1

now get the total energy density by integrating over allenergies and directions (relativistic kinematics limit)

ρ ≈T 4

2π 2x 3 dx

ex−η ±10

∞

∫

degeneracy parameter(chemical potential/temperature)

η ≡μT

in extreme relativistic limitη → 0

x 3 dxex −10

∞

∫ =π 4

15 and x 3 dx

ex +10

∞

∫ =7π 4

120

bosons ρ ≈ gbπ 2

30T 4 and fermions ρ ≈

78

gf

⎛ ⎝ ⎜

⎞ ⎠ ⎟

π 2

30T 4

geff = 2 + 78 2 + 2 + 6)( )=10.75

Statistical weight in all relativistic particles:

e.g., statistical weight in photons, electrons/positrons and six thermal,zero chemical potential (zero lepton number) neutrinos, e.g., BBN:

geff = gib Ti

T⎛ ⎝ ⎜

⎞ ⎠ ⎟

i∑

3

+78

g jf Tj

T⎛

⎝ ⎜

⎞

⎠ ⎟

j∑

3

ν e ν e ν μ ν μ ντ ν τ

SpacetimeSpacetimeBackgroundBackground

photon decoupling T~ 0. 2 eV

vacuum+matter dominatedat current epoch

neutrino decoupling T~ 1 MeV

Relic neutrinos from the epoch when the universewas at a temperature T ~ 1 MeV ( ~ 1010 K)

~ 300 per cubic centimeter

Relic photons. We measure 410 per cubic centimeter

Re-ionization:1 in 103 baryons into stars;Nucleosynthesis? Black Holes?

Re-ionization:1 in 103 baryons into stars;Nucleosynthesis? Black Holes?

Coupled star formation, cosmic structure evolution –Mass assembly history of galaxies, nucleosynthesis, weak lensing/neutrino massCoupled star formation, cosmic structure evolution –Mass assembly history of galaxies, nucleosynthesis, weak lensing/neutrino mass

Very Early Universe:baryo/lepto-genesisQCD epoch, BBNNeutrino physics

Very Early Universe:baryo/lepto-genesisQCD epoch, BBNNeutrino physics

George Gamow

George LeMaitre

A. Friedmann

Albert Einstein

Invoking this requires symmetry:specifically, a homogeneous and isotropic distributionof mass and energy!

What evidence is there that this is true?

Look around you. This is manifestly NOT true onsmall scales. The Cosmic Microwave BackgroundRadiation (CMB) represents our best evidence thatmatter is smoothly and homogeneously distributedon the largest scales.

Birkhoff’s Theorem

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 = ρtest mass = m

≈ −G 4

3 πa3ρ[ ]ma

≈ 12 mÝ a 2

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

a

(k=0)

Ý a 2 + k =83

π Gρ a2

Friedman-LeMaitre-Robertson-Walker (FLRW) coordinates

defined through this metric . . .

How far does a photon travel in the age of the universe?

Consider a radially-directed photon ( )

photons travel on null world lines so ds2=0

(causal horizon)

=

Causal (Particle) Horizonradiation dominated

matter dominated

vacuum energy dominated

In every case the physical (proper) distance a light signal travels goesto infinity as the value of the timelike coordinate t does.

Note, however, that for the vacuum-dominated case there is a finitelimiting value for the FLRW radial coordinate as t goes to infinity . . .

Epoch T geffHorizon Length

Mass-Energy(solar masses)

Baryon Mass(solar masses)

Electroweakphase transition

100 GeV ~100 ~ 1 cm~ 10-6

(~ earth mass)~ 10-18

QCD 100 MeV 51 - 62 20 km ~ 1 ~ 10-9

weak decoupling 2 MeV 10.75 ~ 1010 cm ~ 104 ~ 10-3

weak freeze out

0.7 MeV 10.75 ~ 1011 cm ~ 105 ~ 10-2

BBN 100 keV 10.75~ 1013 cm(~ 1 A.U.)

~ 106 ~ 1

e-/e+

annihilation ~ 20 keV 3.36 ~ 1014 cm ~ 108 ~ 100

photon decoupling 0. 2 eV - ~ 350 kpc ~ 1018

dark matter~ 1017

some significant events/epochs in the early universe

1 solar mass ≈ 2 ×1033 g ≈1060 MeV

The History ofThe Early Universe:

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

The QCD horizonis essentially anultra-high entropy Neutron Star

νe + n ↔ p + e− νe + p ↔ n + e+

Co-Moving Entropy Density is Conserved

Energy/momentum conservation

in FLRW coordinates

Assume a perfect fluid*stress-energy tensor

but first law of thermo gives

*Not true when mixedrelativistic/nonrelativistic system,or decaying particles ----- Bulk Viscosity

Cosmic Bulk ViscosityCosmic Bulk Viscosity

only non-adiabatic, dissipativecontribution consistent with homogeneity, isotropy –rotational, translational invariance

Weinberg 1971; Quart 1930

Biggest effect when decaying particles

have lifetimes of order the local Hubble time,

dominate mass-energy!

The Entropy of the Universe is HugeThe Entropy of the Universe is HugeWe know the entropy-per-baryon of the universe becausewe measure the cosmic microwave background temperatureand we measure the baryon density through the deuterium abundance and CMB acoustic peak amplitude ratios.

S/k = 2.5 x 108 (Ωbh2)-1 ~ 1010

Deuterium, CMB, and large scale structuremeasurements imply all Ωbh2 ~ 0.02

Neglecting relatively small contributions fromblack holes, SN, shocks, nuclear burning, etc.,S/k has been constant throughout the history of the universe.

S/k is a (roughly) cois a (roughly) co--moving invariant.moving invariant.

entropy per baryon in radiation-dominated conditions

entropy per unit proper volume

S ≈2π 2

45gs T

3

proper number density of baryons nb = η nγ

entropy per baryon s ≈Snb

The The ““baryon number,baryon number,””or baryonor baryon--toto--photon ratio,photon ratio, η is a is a kind of kind of ““inverse entropy per baryon,inverse entropy per baryon,””but it is but it is notnot a coa co--moving invariant.moving invariant.

η ≈2π 4

451

ζ 3( )gtotal

gγ

S−1

The “baryon number”is defined to be the ratio of the net number of baryons to the number of photons:

η =nb − nb

nγ

Friedmann equation is Ý a 2 + k =83

π Gρ a2 and

G =1

mPL2 where h = c =1 and the Planck Mass is mPL ≈1.22 ×1022 MeV

radiation dominated ρ ≈π 2

30geffT

4 ~ 1a4

⇒ horizon is dH t( ) ≈ 2t ≈ H−1

where the Hubble parameter, or expansion rate is

H =Ý a a

≈8π 3

90⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

geff1/ 2 T 2

mPL

t ≈ 0.74 s( ) 10.75geff

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2MeV

T⎡ ⎣ ⎢

⎤ ⎦ ⎥

2

The entropy in a co - moving volume is conserved⇒ geff

1/ 3aT = ′ g eff1/ 3 ′ a ′ T so that if the number of relativistic degrees of freedom is constant

⇒ T ~ 1a

Weak InteractionsWeak InteractionsWeak Interactions

Weak Interaction/NSEWeak Interaction/NSE--FreezeFreeze--Out Out History of the Early UniverseHistory of the Early Universe

Temperature/Time

λνe ~ λνν ~ GF2T5 >> H ~ geff

1/2 T2/mpl

λn(p,γ)d = λd(γ,p)n >> H

e+/e- annihilation(heating of photons relative to neutrinos)

Tν = (4/11)1/3 Tγ

λνn ~ λen ~ λνp ~ λep >> H

Weak DecouplingT ~ 3 MeV

Weak Freeze-OutT ~ 0.7 MeV

Nuclear Statistical Equilibrium (NSE)Freeze-OutAlpha Particle Formation

T ~ 0.1 MeV

forces neutrinos into weak interaction(flavor) eigenstates

Weak DecouplingWeak DecouplingThis 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 exchangeenergy with the photon-electron plasma.

neutrino scattering rate λν ~ GF2 T 2( )T 3( )= GF

2 T 5

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

expansion rate H ≈8π 3

90⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

geff1/ 2 T 2

mPL

weak decoupling temperature

TWD ≈8π 3

90⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 6geff

1/ 6

GF2 mPL( )1/ 3 ≈1.5 MeV geff

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 Tν

Weak Freeze OutWeak Freeze OutEven 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 maintainedbetween leptons and nucleons.

Eventually, weak interaction-driven isospin flip rate fallsbelow expansion rate, neutron/proton ratio “frozen in,”------- this is Weak Freeze Out

Neutron-to-proton ratio is set bythe competition between the rates of these processes:

threshold

threshold

threshold

neutron-proton mass difference

Charged Current Weak Interaction Rates for Neutrons and ProtonsCharged Current Weak Interaction Rates for Neutrons and Protons

Coulomb correction – Fermi factorattractive Coulomb interaction increases electronprobability at the proton, increasing the above phase space factorsin which F appears.

lepton occupation probabilitieslepton occupation probabilities

Neutrinos – if thermal, Fermi-Diracenergy spectra then

Strength of the Weak InteractionStrength of the Weak Interactionradiative corrections

Determine this by using the measured free (vacuum) neutron lifetime

Any effect which increases this phase space factorwill decrease the overall weak interaction strength,leading to earlier (hotter) freeze out, more neutronsand, 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 ratesare large enough, andexpansion slow enough,system can approachSteady State EquilibriumSteady State Equilibrium

valid at high Twhere we can neglectfree neutron decayand the three-body reverse process

Steady State Equilibrium

Chemical Equilibrium --- the Saha equation

equality holds when leptonshave thermal, Fermi-Diracenergy distribution functions

equilibrium

actual

formation of alphas