Download Thermal History of the Universe and the Cosmic Microwave

Thermal History of the Universe and the Cosmic
Microwave Background.
I. Thermodynamics of the Hot Big Bang
Matthias Bartelmann
Max Planck Institut für Astrophysik
IMPRS Lecture, March 2003
◦.
Part 1: Thermodynamics of the Hot Big Bang
1. assumptions
6. photons and baryons
2. properties of ideal quantum gases
7. physics of recombination
3. adiabatic expansion of ideal gases
8. nucleosynthesis
4. particle freeze-out
9. the isotropic microwave background
5. neutrino background
/◦.
1. Assumptions
1. adiabatic expansion
2. thermal equilibrium
3. ideal gases
/◦.
1.1. Adiabatic Expansion
The Universe expands adiabatically.
• isotropy requires it to expand adiathermally: no heat can flow because any
flow would define a preferred direction
• an adiathermal expansion is adiabatic if it is reversible, but irreversible
processes may occur
• the entropy of the Universe is dominated by far by the cosmic microwave
background, so entropy generation by irreversible processes is negligible
/◦.
1.2. Thermal Equilibrium
Thermal equilibrium can be maintained despite the expansion.
• thermal equilibrium can only be maintained if the interaction rate of particles
is higher than the expansion rate of the Universe
• the expansion rate of the Universe is highest at early times, so thermal
equilibrium may be difficult to maintain as t → 0
• nonetheless, for t → 0, particle densities grow so fast that interaction rates are
indeed higher than the expansion rate
• as the Universe expands, particle species drop out of equilibrium
/◦.
1.3. Ideal Gases
Cosmic “fluids” can be treated as ideal gases.
• ideal gas: no long-range interactions between particles, interact only by direct
collisions
• obviously good approximation for weakly interacting particles like neutrinos
• even valid for charged particles because oppositely charged particles shield
each other
• consequence: internal energy of ideal gas does not depend on volume
occupied
• cosmic “fluids” can be treated as possibly relativistic quantum gases
/◦.
2. Properties of Ideal Quantum Gases
1. occupation number
6. entropy density
2. number density
7. results for bosons and fermions
3. energy density
8. numbers
4. grand canonical potential
5. pressure
/◦.
2.1. Occupation Number
• in thermal equilibrium with a heat bath of temperature T , the phase-space
density for a quantum state of energy ε is
−1
ε−µ
h f i = exp −
±1
kT
(“+” for fermions, “−” for bosons)
• in thermal equilibrium with a radiation background, the chemical potential µ =
0: Helmholtz free energy F(T,V, N) = E − T S is minimised in equilibrium for a
system at constant T and V , so from dF = −SdT − PdV + µdN:
∂F
=0=µ
∂N
/◦.
2.2. Number Density
• particles in volume V , number of states in k-space element:
V 3
dk
dN = g
(2π)3
• g is statistical weight, e.g. spin degeneracy factor
• momentum ~p = ~~k, related to energy by
p
ε(p) = c2 p2 + m2c4
• spatial number density in thermal equilibrium:
g
n=
(2π~)3
Z
0
∞
4πp2dp
exp[ε(p)/kT ] ± 1
/◦.
2.3. Energy Density
• mean energy density: number of states per phase-space cell, times
occupation number, times energy per state, integrated over momentum
space:
Z ∞
g
4πp2 ε(p) dp
u=
(2π~)3 0 exp[ε(p)/kT ] ± 1
• integrals can most easily be carried out by substituting a geometrical series:
Z
0
∞
xmdx
=
x
e −1
Z
0
∞ xm e−x dx
1 − e−x
Z
=
∞
m −x
dxx e
0
for fermions, use
∞
∑e
n=0
−nx
∞
=
∑
Z
∞
dxxme−nx = m!ζ(m + 1) ;
n=1 0
1
1
2
=
−
ex + 1 ex − 1 e2x − 1
/◦.
2.4. Grand Canonical Potential
• extremal principles yield: (thermodynamical potential) = −kT ln(partition sum)
grand canonical potential Φ(T,V, µ) (particle
• adequate potential here:
numbers can change)
• grand-canonical partition sum:
∞
Z=
∑ eµN/kT ∑
N=0
e− ∑ εαNα/kT
{Nα |N}
• grand-canonical potential:
Φ(T,V, µ) = ∓kT
gV
(2π~)3
Z
0
∞
h
i
2
µ/kT −ε(p)/kT
dp4πp ln 1 ± e
e
Fermions
Bosons
/◦.
2.5. Pressure
• Helmholtz free energy:
F(T,V, N) = U − T S
• grand-canonical potential:
Φ(T,V, µ) = F − µN = U − T S − µN
• thermodynamical Euler relation:
• it follows:
Φ
Φ = −PV ⇒ P = −
V
• example: relativistic boson gas in
thermal equilibrium; ε = cp, µ = 0,
π2 (kT )4
PB = g
90 (~c)3
U = T S − PV + µN
/◦.
2.6. Entropy Density
• Helmholtz free energy:
dF(T,V, N) = −SdT − PdV + µdN
• example:
entropy density for
relativistic boson gas in thermal
equilibrium
2
• grand-canonical potential:
S
2π
s = = gk
V
45
kT
~c
3
dΦ(T,V, µ) = −SdT − PdV − Ndµ
• thus, the entropy is:
∂Φ
S=−
∂T
/◦.
2.7. Results for Bosons and Fermions
n
u
P
s
relativistic
Bosons
Fermions
3
3 gF
ζ(3) kT
nB
gB 2
4 gB
π
~c
π2 (kT )4
7 gF
gB
uB
3
30 (~c)
8 gB
π2 (kT )4 uB
7 gF
gB
=
PB
90 (~c)3
3
8 gB
3
2
2π kT
7 gF
gBk
sB
45 ~c
8 gB
nonrelativistic
3/2
kT
2
e−kT /mc
g
2π~
3
n kT
2
n kT
/◦.
2.8. Numbers
note: 1 eV = 1.6 × 10−12 erg correspond to kT = 1.16 × 104 K
•
•
•
3
3
T
kT
nB = 10gB
cm−3 = 1.6 × 1013gB
cm−3
K
eV
4
4
T
erg
kT
erg
−3
uB = 3.8 × 10−15gB
=
2.35
×
10
g
B
K cm3
eV cm3
3
3
sB
T
kT
−3
13
= 36gB
cm = 5.7 × 10 gB
cm−3
k
K
eV
/◦.
3. Adiabatic Expansion of Ideal Gases
1. temperature
2. density
3. matter-radiation equality
/◦.
3.1. Temperature
• for relativistic boson or fermion
gases:
E
u
P= =
3 3V
• first law of thermodynamics,
dE + PdV = 0,
then implies:
dE = −PdV = 3d(PV ) ⇒ P ∝ V −4/3
• since P ∝ T 4 for relativistic gases:
T ∝ V −1/3 ∝ a−1
(a is cosmological scale factor)
• for non-relativistic ideal gas:
T ∝ PV ∝ V −5/3+1 ∝ a−2
(adiabatic index is γ = 4/3)
• for non-relativistic ideal gas: γ = 5/3
/◦.
3.2. Density
• non-relativistic gases:
rest mass energy kinetic energy
mass density:
⇒ ρ ∝ V −1 ∝ a−3
• density of relativistic particles drops
faster than that of non-relativistic
particles:
• dilution due to volume expansion,
plus energy loss due to redshift
• relativistic gases: mass density
u
ρ = 2 ∝ T 4 ∝ a−4
c
/◦.
3.3. Matter-Radiation Equality
• today’s matter density in the Universe:
3H02
ρ0 = Ω0
= 1.9 × 10−29 Ω0h2 g cm−3
8πG
• today’s radiation density in the Universe:
T = 2.73 K ⇒ ρR,0 =
u
−34
−3
=
4.7
×
10
g
cm
c2
• radiation domination:
−3
ρR(aeq) = ρR,0a−4
=
ρ
a
0
eq
eq = ρ(aeq ) ⇒ aeq =
ρR,0
= 2.5 × 10−5 (Ω0h2)−1
ρ0
/◦.
4. Particle Freeze-Out
1. interaction rates
2. freeze-out conditions
3. particle distributions after freeze-out
/◦.
4.1. Interaction Rates
• expansion timescale:
• thermal equilibrium maintained by
two-body interactions, collision rate
for one particle species is
texp ∼ (Gρ)−1/2
ρ is density of dominant fluid
• in early Universe,
dominates, thus
ρ∝a
−4
⇒ texp ∝ a
radiation
2
expansion timescale is shortest at
early times and increases as a2
Γ ∝ n ∝ T 3 ∝ a−3
• thus, the collision time scale is
tcoll ∝ Γ−1 ∝ a3
• for a → 0, tcoll decreases faster
than texp, so equilibrium can be
maintained
/◦.
4.2. Freeze-Out Conditions
• continuity equation in absence of
collisions:
• continuity equation now:
ṅ + 3Hn = −Γn + S
ṅ + 3Hn = 0
• collision rate:
Γ = hσvi n
• source term from thermal particle
creation:
• introduce comoving number density
N = a3n:
d ln N
Γ
=−
d ln a
H
NT
1−
N
• freeze-out can occur if Γ H:
particle number N cannot adapt
S = hσvi n2T
/◦.
4.3. Particle Distributions After Freeze-Out
• relativistic particles:
n ∝ T 3 ⇒ N = a3n = const.
• from freeze-out equation:
d ln N
= 0 ⇒ N = NT
d ln a
independent of Γ!
• relativistic
particles
maintain
thermal distribution even after
freeze-out
• non-relativistic particles, comoving
number density:
2
mc
NT ∝ T −3/2 exp −
kT
• once T mc2, NT drops rapidly,
hence NT N, and
d ln N
Γ
=− →0
d ln a
H
particle production stops: freezeout
/◦.
5. Neutrino Background
1. neutrino decoupling
2. electron-positron annihilation
3. photon heating
/◦.
5.1. Neutrino Decoupling
• neutrinos are kept in thermal equilibrium by the weak interaction
ν + ν̄ ↔ e+ + e−
• this interaction freezes out when the temperature drops to
Tν ∼ 1010.5 K ∼ 2.7 MeV
• neutrinos are ultrarelativistic at that “time”, i.e. their comoving number density
is that of an ideal, relativistic fermion gas
/◦.
5.2. Electron-Positron Annihilation
• the reaction
e+ + e− ↔ 2γ
is suppressed once temperature
drops below
• the entropy of the annihilating
electron-positron gas heats the
photons, but not the neutrinos
• the
microwave
background
is heated above the neutrino
background by the annihilation
T ∼ 2mec2 ≈ 1 MeV ≈ 1010 K
• thus, electrons and positrons
annihilate shortly after neutrinos
freeze out
/◦.
5.3. Photon Heating
• entropy before annihilation equals
entropy after annihilation:
• since s ∝ T 3,
s0e+ + s0e− + s0γ = sγ
• temperature after annihilation:
• before annihilation,
Te+ = Te− = Tγ ≡ T
thus
s0e+
=
s0e−
7
3
2 · + 1 (T 0) = T 3
8
7 0
= sγ
8
0
T=
11
4
1/3
T 0 ≈ 1.4 T 0
• temperature of ν background today:
(ge+ = ge− = gγ = 2)
Tν,0 = 1.95 K
/◦.
6. Photons and Baryons
1. baryon number
2. photon-baryon ratio
/◦.
6.1. Baryon Number
• number density of baryons today:
ρB ΩB 3H02
=
= 1.1 × 10−5 ΩBh2 cm−3
nB ≈
mp mp 8πG
• measurements: abundance ratios of light elements, microwave background
(see later), gas in galaxy clusters:
ΩBh2 ≈ 0.025
• only ∼ 10% − 20% of the matter in the Universe is contributed by baryons
• baryon number density ∝ a−3 ∝ T 3
/◦.
6.2. Photon-Baryon Ratio
• number density of photons today:
nγ = 407 cm−3
• scales with temperature like T 3 like
baryon number density, hence their
ratio is independent of time
• there is one baryon for about a
billion photons in the Universe
• photon
entropy
completely
dominates entropy of the Universe
• very important
recombination
for
process
of
• constant baryon-photon-ratio:
nB
η=
= 2.7 × 10−8 ΩBh2
nγ
/◦.
7. Physics of Recombination
1. recombination process
2. Saha equation
3. recombination temperature
4. two-photon recombination
5. thickness of the recombination shell
6. expectation on radiation spectrum
/◦.
7.1. Recombination Process
• as the temperature drops, electrons
and protons can combine to form
atoms:
e− + p+ ↔ H + γ
freezes out
• canonical partition function Z,
F = −kT ln Z
with
N
N
ZeNe Zp p ZHH
Z=
Ne! Np! NH!
• How does recombination proceed,
i.e. how does the electron density
change with T ?
• baryon number NB = Np + NH,
electron number Ne = Np, thus
NH = NB − Ne
• solution: minimisation of Helmholtz
free energy F(T,V, N)
• Stirling: ln N! ≈ N ln N − N
/◦.
7.2. Saha Equation
• from ∂F/∂Ne = 0:
• chemical potential:
Ne2
ZeZp
=
NB − Ne
ZH
µ = 0 ⇒ µe + µp = µH
• ionisation potential:
• partition function for single species:
4πgV
Z=
(2π~)3
Z
∞
(me + mp − mH)c2 = χ = 13.6 eV
dpp2e(µ−ε)/kT
0
• x = Ne/NB,
equation:
• nonrelativistic limit:
ε = mc2 +
2
p
2m
nB = NB/V ;
Saha
x2
(2πmekT )3/2 −χ/kT
=
e
3
1−x
(2π~) nB
/◦.
7.3. Recombination Temperature
• nB is the baryon number density; we
had
• 1/η is huge number, so kT χ
required for x to be small
3
• putting x = 0.1 yields kTrec ≈ 0.3 eV,
or
Trec ≈ 3500 K
• insert this into Saha equation:
• high photon number density delays
recombination considerably
nB = ηnγ = η
2
x
0.26
≈
1−x
η
2ζ(3)
π2
2
mec
kT
kT
~c
3/2
e−χ/kT
• want x 1, hence x2/(1 − x) ≈ x2
/◦.
7.3. Recombination Temperature
/◦.
7.4. Two-Photon Recombination
• direct hydrogen recombination
produces energetic photons; last
step to ground state is Lyman-α
(2P → 1S);
3
hν ≥ ELyα = χ = 10.2 eV
4
• abundant Ly-α photons can reionise
the cosmic gas
• How can recombination proceed at
all?
• production
of
lower-energy
photons:
forbidden transition
2S → 1S, requires emission of two
photons; this process is slow
• recombination proceeds at a
somewhat
slower
rate
than
predicted by the Saha equation
• photons cannot be lost as in
clouds; energy loss due to cosmic
expansion is slow
/◦.
7.5. Thickness of Recombination Shell
• recombination is not instantaneous;
“time” interval between beginning
and end?
• time: measured in terms of scale
factor a, or redshift 1 + z = a−1
• optical
depth
recombination shell:
τ=
through
Z
• scattering probability for photons
when travelling from z to z − dz:
p(z) dz = e−τ
dτ
dz
dz
• probability distribution e−τdτ/dz
is well described by Gaussian
with mean z̄ ∼ 1100 and standard
deviation δz ∼ 80
nexσTdr
(σT: Thomson cross section)
/◦.
7.5. Thickness of Recombination Shell
/◦.
7.6. Expectation on Radiation Spectrum
• last-scattering shell has finite width: CMB photons received today were
released at different redshifts
• cosmic plasma cooled during recombination, photons were released at
different mean temperatures: T = T0 (1 + z), thus
δT = T0δz ≈ 200 K
• photons were redshifted after release: those released earlier, i.e. from hotter
plasma, were redshifted by larger amount: E = E0 (1 + z)
• these effects cancel if plasma temperature depends on scale factor like T ∝
a−1; then: Planck spectrum of single temperature expected
/◦.
8. Nucleosynthesis
1. formation of the light elements
2. Gamow criterion
3. prediction of the microwave background temperature
/◦.
8.1. Formation of the Light Elements
• high temperature in the early
Universe allows nuclear reactions
like in stars; density is much
lower, so higher temperatures are
required:
Tnuc ∼ 109 K
• before:
neutrons and protons
formed; equilibrium through
p + e− ↔ n + ν , p + ν̄ ↔ n + e+
• relative abundance freezes out
once weak interaction becomes too
slow;
Tfreeze−out ∼ 1.4 × 1010 K
afterwards, free neutrons decay
• nucleosynthesis proceeds through
strong interactions until these
freeze out, e.g. deuterium formation
n + p ↔ D + γ stops at
TD ∼ 7.9 × 108 K
/◦.
8.2. Gamow Criterion
• deuterium fusion is the most important step towards the fusion of higher
elements; e.g. Helium:
D + p → 3He + γ , 3He + n → 4He
• deuterium needs to be produced in sufficient abundance for higher elements
to form, but if all neutrons are immediately locked up into deuterium, no higher
elements can form either
• George Gamow noticed in 1948 that deuterium formation has to proceed at
“just right” rate:
nB hσvit ∼ 1 ,
i.e. collision rates for baryons should not be too small or too large
/◦.
8.3. Prediction of CMB Temperature
• for deuterium formation, t ∼ 3 min
• baryon density drops like T 3; thus:
• from a theoretical estimate for
hσvi, Gamow estimated the baryon
density at deuterium formation:
nB ∼ 1018 cm−3
nB
=
nB,0
nB,0 ∼ 1.1 × 10−5 ΩBh2 cm−3
TD
T0
3
• using TD = 7.9 × 108 K:
• today’s baryon density is
T0 =
nB,0
nB
1/3
TD ∼ 4 K
CMB temperature is predicted by
the Big-Bang model!
/◦.
9. The Isotropic Microwave Background
1. temperature and spectrum
2. limits on chemical potential and Compton parameter
3. the dipole
/◦.
9.1. Temperature and Spectrum
• microwave
background
was
detected, but not recognised,
by Penzias and Wilson in 1965
• NASA’s
Cosmic
Background
Explorer
(COBE)
precisely
measured its spectrum (and
other things, see below)
• its temperature is
T0 = (2.726 ± 0.002) K
• its spectrum is the best black-body
spectrum ever measured
/◦.
9.2. Limits on µ and y
• finite width of the last-scattering
surface: it is important that the CMB
has a black-body spectrum
• shape of the spectrum allows
constraints on the chemical
potential:
|µ| ≤ 9 × 10−5
• hot gas between the last-scattering
surface and us can distort the
spectrum
through
Compton
scattering
• Compton-y parameter:
kT
y=
mec2
Z
neσTdl
(typical energy change
scattering probability)
times
• constraint
from
COBE-FIRAS
spectrum strictly contrains heat
input in young Universe:
y ≤ 1.5 × 10−5
/◦.
9.3. Dipole
• “freely
falling”
observers
in
the Universe define comoving
coordinates
• the CMB is at rest with respect to
this coordinate frame
• Earth moves around the Sun, Sun
around the centre of the Galaxy,
Galaxy within the Local Group etc.
• COBE has measured the dipole:
v = (371 ± 1) km s−1
towards
l = (264.3 ± 0.2)◦ , b = (48.1 ± 0.1)◦
• motion causes temperature dipole:
v
T (θ) = T0 1 + cos θ + O (v2/c2)
c
/◦.