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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