thermodynamics in cosmology nucleosynthesiscosmology.berkeley.edu/classes/s2012/physics_112/... ·...
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Phys 112 (S2012) 9 Cosmology Sadoulet1
Thermodynamics in CosmologyNucleosynthesis
ThermodynamicsExpansionEvolution of temperatureFreeze out
NucleosynthesisProduction of the light elementsPotential barrierPrimordial synthesis calculationsPrimordial abundance measurements
4He, 2H, 3He, 7LiComparison with theory
Phys 112 (S2012) 9 Cosmology Sadoulet3
Universe expansionThe universe is expanding
• Hubble recession of distant galaxiesTo first approximation
• Cosmic microwave background• Abundance of primordial elements
Homogeneous and isotropic universe:Scale parameterBest way to look at expansion: comoving coordinates
Friedman equationThe sum of kinetic energy and potential energy is constant (Newtonian
mechanics)Consider spherical shell of constant comoving radius r
True in General Relativity also
v = H r
r physicalcoordinates
= a t( ) x comoving coordinates
a t( ) = scale parameter Hubble "constant"= H = da t( ) / dta t( ) =
˙ a t( )a t( ) ⇒
˙ r = H r recessionvelocity
+ a t( ) ˙ x peculiarvelocity
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m ˙ a x( )2 − Gm 4π3
ρ ax( )3
ax( )= constant
˙ a a
⎛ ⎝
⎞ ⎠
2
= 8π3
Gρ− 1a2 R2 + Λ
General relativity
Curvature
Cosmological constant
Phys 112 (S2012) 9 Cosmology Sadoulet
How did it started
For all practical purposes, a phase transition: inflationPhase transition with very large vacuum energy (or universe brought far
out of equilibrium)=> exponential expansion
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T decreasesReheating
Slow roll inflationExponential expansion
ϕ inflaton field= order parameter
Landau Free Energy
Phys 112 (S2012) 9 Cosmology Sadoulet
Inflation: justificationsJustification
Uniformity of the Cosmic Microwave BackgroundUnless there is an exponential expansion phase, points of the last
scattering surface further than ≈2 degrees in the sky would not have had the time to communicate.
By productsIf expansion is large enough, space is flat whatever initial conditionsQuantum fluctuations are frozen in and expand with space -> seed
for the large scale structureDilutions of relics such as monopoles
How?Theoretical possibility, e.g. at Grand Unified Theory Sacle
No real mechanismIn practice, delicate implementation (separate bubbles)
What more can we test?A number of potential experimental tests: Tensor/Scalar ratio in
CMBR(gravitational waves seen as B-mode polarization)
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B.SadouletPhys 112 (S12) 9 Cosmology
In class Expansion of the universe
6
Assume a homogeneous and isotropic universeConsider a comoving volume
What is constant in this comoving during the expansion
A: The energy?B: The entropy?C: The temperature?
B.SadouletPhys 112 (S12) 9 Cosmology
In class Expansion of the universe
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Assume a homogeneous and isotropic universeConsider a comoving volume
What is constant in this comoving during the expansion?
B: The entropy
If uniform and isotropic, no heat transferNo generation of entropy if there are no first order phase transition
(not reversible)
Phys 112 (S2012) 9 Cosmology Sadoulet
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Temperature Evolution outside of phase transitions
Entropy per unit comoving volume has to be constantin a homogeneous/isotropic universe (no exchange of heat!)Initially dominated by relativistic particles
No change of number of degrees of freedom
Simple interpretation: comoving number density is constant, energy is redshifted (in agreement with General Relativity).
True for all relativistic species even if they have dropped out of thermal equilibrium.
Change of number of degrees of freedome.g.,
Energy dumped into the photon gas Temperature does not fall as rapidly
u = g * aB2T 4 ⇐ Stefan - Boltzmann
T ∝ a t( )−1
e+ + e- ↔γ +γ : e± disappear when T < 300MeV
⇒σ
Vcomoving
= scomoving ∝ a t( )3g *T 3 = constant ⇒ T ∝ g *− 1
3 a t( )−1
loga t( )
logT t( )
σV
∝ g*T 3 V = a3 t( )Vcomoving where g* = effective degrees of freedom
B.SadouletPhys 112 (S12) 9 Cosmology
In classFormation/Dissociation of objects
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A+C<->ACBinding energy of AC =B
How does the proportion of AC vary with temperature?A: It dependsB: kB T>> B AC does not existC: kB T<< B AC does not exist
e.g., p + e− ↔ H 0
n + p↔ 2H2H + p↔ 3He
B.SadouletPhys 112 (S12) 9 Cosmology
In classFormation/Dissociation of objects
10
A+C<->ACBinding energy of AC =B
How does the proportion of AC vary with temperature?B: kB T>> B AC does not exist
What relation do we have in such equilibrium
A: nA + nC = nACB: µA + µC = µAC
C: µAµC
µAC
= K τ( )
e.g., p + e− ↔ H 0
n + p↔ 2H2H + p↔ 3He
B.SadouletPhys 112 (S12) 9 Cosmology
In classFormation/Dissociation of objects
11
A+C<->ACBinding energy of AC =B
How does the proportion of AC vary with temperature?B: kB T>> B AC does not exist
What relation do we have in such equilibrium
But we have to be careful to take the same origin for the energy scales
B: µA + µC = µAC
⇒ nAnCnAC
= K τ( )
e.g., p + e− ↔ H 0
n + p↔ 2H2H + p↔ 3He
Phys 112 (S2012) 9 Cosmology Sadoulet12
Equilibrium of non relativistic speciesNon relativistic case (m>>T)
By integrating on state density, we have seen that the number density of non relativistic species i in thermal equilibrium with photons is
Statistical equilibrium Exothermic reaction
e.g.
Nuclear element A,Z
We have kept explicitly mass energyCommon origin of energy
exp µA /T( ) = nAgA
2π2
mAkBT⎛⎝⎜
⎞⎠⎟
3/2
exp mAc2 /T( )
= exp Zµp + A − Z( )µn( ) / kBT⎡⎣ ⎤⎦ =np2
⎛⎝⎜
⎞⎠⎟
2π2
mpkBT⎛
⎝⎜⎞
⎠⎟
3/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
Z
nn2
⎛⎝⎜
⎞⎠⎟
2π2
mnkBT⎛⎝⎜
⎞⎠⎟
3/2⎡
⎣⎢⎢
⎤
⎦⎥⎥
A−Z
exp Zmp + A − Z( )mn( )c2 / kBT⎡⎣ ⎤⎦
ni =1V
1
exp ε s − µi
τ⎛⎝⎜
⎞⎠⎟ ±1
s∑ ≈ exp −
mic2 + εK − µi
τ⎛⎝⎜
⎞⎠⎟∫ D εk( )dεk
ni = gimikBT2π2
⎛⎝⎜
⎞⎠⎟
32exp −
mic2 − µi
kBT⎛⎝⎜
⎞⎠⎟= ginQi exp −
mic2 − µi
kBT⎛⎝⎜
⎞⎠⎟
A + B↔ C + D ⇒ µA + µB = µC + µD
⇒ Law of Mass Action: nAnB = nCnD ×nQAnQBnQCnQD
× exp −mA + mB − mC − mD( )c2
τ⎛⎝⎜
⎞⎠⎟
non degenerate
Phys 112 (S2012) 9 Cosmology Sadoulet
Equilibrium(2)Formation of a bound object:
Product is more bound e.g.Recombination=> Saha equation
Nucleosynthesis
Decay/annihilation of massive objecte.g. dark matter particle
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e + p↔ H 0
A↔ Z p + A − Z( ) nµA = Zµ p + A − Z( )µ n
BH = me + mc − mH 0( )c2
nH 0 =
gH
gpgenpne
mekBT2π2
⎛⎝⎜
⎞⎠⎟−32exp BH
kBT⎛⎝⎜
⎞⎠⎟
⇒nA
npZnn
A−Z = gA2−A A3/2 2π2
mNkBT⎛⎝⎜
⎞⎠⎟
3 A−1( )/2
exp BA / kBT( ) BA = Zmp + A − Z( )mn( ) − mA⎡⎣ ⎤⎦c2
where we have used in prefactor mA
A≈ mp ≈ mn = mN
χ + χ ↔ q + q
mχ >> mq similar expressions but with B < 0
Phys 112 (S2012) 9 Cosmology Sadoulet14
Freeze OutFreese out
In the universe, we will have a similar equilibrium concentration. But not necessarily reached because the universe is expanding. The dynamic evolution of the density of say A is given by
We need enough time: reaction rate should be much bigger than expansion rate.
Not necessarily true: - density decreases - temperature decreases
More difficulty for going above potential barriers => rates usually decrease => Freeze out
dnAdt
= −ΓA→B +C nA + ΓB+ C→A nBnC
where the Γ' s are the reaction rates
− log T( )
comoving density a t( )3nA
Freeze out
exp BkT
⎛ ⎝
⎞ ⎠
− log T( )
comoving density a t( )3nA
Freeze out
exp BkT
⎛ ⎝
⎞ ⎠
Bound object (nuclei)Annihilation
(e.g. dark matter)
Phys 112 (S2012) 9 Cosmology Sadoulet15
Potential barrierPotential energy of nucleus (e.g. 4He)
Need enough energy to “penetrate” barrier + emission of e.g. photon3 temperature regimes
• At high temperature, nucleus cannot exist <- dissociation
• At low energy: nucleus is stable but not enough energy to be formed
• Intermediate: nucleus can be formed and is stable enough to survive
Large abundance of 4He => universe was hot
potential barrier <- Coulomb repulsion,centrifugal barrier
Pote
ntia
l
radius
Binding energy (28MeV)
2 H +2 H →4 He + γ
Epot + Ekin = constant
Phys 112 (S2012) 9 Cosmology Sadoulet16
Primordial Nucleosynthesis
Dependent on expansion rate and Ωb
2H bottleneck
Freeze out
n depleted by low T
Phys 112 (S2012) 9 Cosmology Sadoulet
ConclusionLink between nuclear physics at small scale and the
universe at large scaleNow attempt to explore the links between particle
physics /quantum gravity and the universeInfinitely small <-> infinitely large (Pascal: XVIIth century)Inflation: quantum origin of large scale structure
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