George M. Fuller Department of Physics University of California, San Diego also known as NUCLEOSYNTHESIS NUCLEOSYNTHESIS from the Big Bang to Today from the Big Bang to Today Summer School on Nuclear and Particle Astrophysics Connecting Quarks with the Cosmos I
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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
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)
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.
---- 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
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).
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
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
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.
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