Thermodynamics in Cosmology Nucleosynthesiscosmology.berkeley.edu/Classes/S2012/Physics_112/... ·...

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 Sadoulet

History of the universe

2


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

12

m ˙ a x( )2 − Gm 4π3

ρ ax( )3

ax( )= constant

˙ a a

⎛ ⎝

⎞ ⎠

2

= 8π3

Gρ− 1a2 R2 + Λ

General relativity

Curvature

Cosmological constant


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

4

T decreasesReheating

Slow roll inflationExponential expansion

ϕ inflaton field= order parameter

Landau Free Energy


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)

5

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?


In class Expansion of the universe

7

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)


8

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


In classFormation/Dissociation of objects

9

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



10


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



11


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


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


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

13

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


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)


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


Primordial Nucleosynthesis

Dependent on expansion rate and Ωb

2H bottleneck

Freeze out

n depleted by low T


17

Baryonic Density

Fields &SarkarAstro-ph/0601504


A surprising but consistent picture

Ωmatter

ΩΛ


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

19


ConclusionA taste of how statistical physics can be put at the

service of frontier science

20

Looking beyond: Camille Flammarion

Thermodynamics in Cosmology Nucleosynthesiscosmology.berkeley.edu/Classes/S2012/Physics_112/... ·...

Documents

Transcript of Thermodynamics in Cosmology Nucleosynthesiscosmology.berkeley.edu/Classes/S2012/Physics_112/... ·...