Lecture-03 The Thermal History of the universe Ping He
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Transcript Lecture-03 The Thermal History of the universe Ping He
Lecture-03
The Thermal History of the
universe
Ping He
ITP.CAS.CN
2006.1.13
http://power.itp.ac.cn/~hep/cosmology.htm
1
3.1 Thermal history to study:
(1) T ~ t [temperature ~ time]
(2)
, n, p T (t )
(3) Degree of freedom ~ T, t
(4) Decouple & relic background
(5) Nucleosynthesis
(6) Baryogenesis
They are typical events in the early Universe
2
3.2. Equilibrium Thermodynamics
a(t)r
Piston : L=a(t)r
vp
v p L a(t )r
va
Quasi – static :
vp
va
a (t )r
va
a(t )r (t ) ,
t
vp
(t )
→Thermal equilibrium
(t ) n v
1
t
a(t )
H (t )
a(t )
(t )
va
H (t )
(Eq-3.1)
Reaction rate
Expansion rate
3
This analysis can be applied to cosmology
Eq-2.1 is the kernel of this lecture
Thermal equilibrium :
(t )
(1)
A
H (t )
AC
Coupling : (1) (2)
H (t )
~ A, C equilibrium
TA TC
C
B
(2) AB
H (t )
~ A, B equilibrium
TA TB
(1) AC
Coupling mode :
(2) AB
TA TB TC
4
3.3 Distribution Function in Thermal Equilibrium
xp h 2 2 ,
p
dN grid
let
=c k B 1
d 3 xd 3 p d 3 xd 3 p
3
(2 )
(2 )3
(Eq-3.2)
x
dN particle g f ( p )dN grid
(Eq-3.3)
cosmological principle: f ( x , p ) f ( p )
g: spin-degeneracy factor (inner degree of freedom)
f ( p) : phase-space distribution function (occupancy function )
5
If equilibrium :
1
f ( p)
e
( E )
T
E 2 p 2 m02
1
: Fermi-Dirac
: Bose-Einstein
(relativistic)
If chemical equilibrium :
i j k l
i j k l
(Eq-3.4)
: Chemical potential
kp
E 2 k 2 m02
(Eq-3.5)
From Eq-2.2 & Eq-2.3
dn
dN particle
3
d x
g
3
f
(
k
)
d
k
3
(2 )
Here dn dn(k )
(Eq-3.6)
6
g
3
d E E (k )dn(k )
E
(
k
)
f
(
k
)
d
k
3
(2 )
(Eq-3.7)
y
x
p1
p2
m
l
S
1
dn(k ) S l
6
V
p p f pi 2 pi 2kdN x (k )
kdn(k ) (1)
3
l F S l
V
p F t F
P
(2)
v
S v
v
dn x (k )
1
dn(k ),
6
dN x (k )
m0
v
1
c
E mc 2
k 2 c 2 m02 c 4
k mv
k
v , c 1
E
2
k
kv
g
3
dP(k ) dn(k )
f
(
k
)
d
k
3
3
(2 ) 3E (k )
2
(Eq-3.8)
7
From Eq-2.6 , Eq-2.7 , Eq-2.8 we have
n
g
c kB 1
f (k )d 3k
(2 )
E mc
g
3
E
(
k
)
f
(
k
)
d
k
3
(2 )
3
(Eq-3.9)
2
P
k
g
(2 )
3
3E ( k )
(Eq-3.10)
2
f (k )d 3k
(Eq-3.11)
The above is the general form for relativistic & quantum
cases. In kinetic equilibrium:
kk
f (k ) f (k )
d 3k 4 k 2 dk , k 0 ,
8
3.4 Distributions as a function of E
n
P
g
2
2
g
2
2
g
6
We used
2
m0
m0
m0
1
( E m0 ) 2 E
dE
exp ( E ) 1
T
2
2
1
( E m0 ) 2 E 2
dE
exp ( E ) 1
T
2
2
2
(Eq-3.13)
3
( E m0 ) 2 E
dE
exp ( E ) 1
T
2
(Eq-3.12)
(Eq-3.14)
E 2 k 2 m0 2 EdE kdk
k E 2 m0 2
:
Chemical potential
9
Specifically
(1) relativistic limit, and non-degeneracy
T
m0 , (m0 0)
T
, ( 0)
2
(3)
4
3
gT
(Bose)
gT
(Bose)
2
1
30
; n
; P , (Eq-3.15)
2
3
7 gT 4 (Fermi)
3 (3) gT 3 (Fermi)
4 2
8 30
T
(2) non-relativistic m0
(3) 1.20206...
3
m0T
ng
exp (m0 ) T
2
3
m0 n nT , P nT
2
2
(Eq-3.16)
In above calculation , we used the fact that
1
e
( E ) / T
1
e
( E )
T
,
2
p
and E m02 p 2 m0
2m0
Maxwell-Boltzmann
10
(3) For non-degenerate ,relativistic species
average energy /particle
4
30 (3) T 2.701T (Bose)
E
n 7 4
T 3.151T (Fermi)
180 (3)
For a non-relativistic species
3
E m0 T
n
2
E
x
(Eq-3.17)
1
k BT E 3 E
2
x
3
k BT
2
(Eq-3.18)
11
3.5 The excess of fermions over its antiparticle
Particle Particle
2 0
(Eq-3.19)
From thermodynamics and statistical dynamics
F U TS N
dF PdV SdT dN
F
dF stationary
0 for photon
N
From Eq-2.19, we have
(Eq-3.20)
12
The net (the excess of ) Fermion number density
n n n
1
1
2
2 12
E
(
E
m
)
dE
0
e( E ) T 1 e( E ) T 1
2 2 m0
g
gT 3 2
3
(
)
(
)
(T
6 2
T
T
3/ 2
m0
m
T
2 g 0 sinh( )e T (T
T
2
m0 )
(relativistic)
(Eq-3.21)
m0 ) (non-relativistic)
For proton
np
n
gp 2 1 3
8
(
)
1.33(
)
10
g 6 (3) T 2 T
T
(Eq-3.22)
p 0 , n 0 , e 0 , 0
Most of the particle species have
0
13
3.6 Degrees of freedom
2
4
gT
30
From eq-2.15
2
7
gT 4
8 30
(Bose)
(Fermi)
3
From eq-2.16
m0T 2
(m0 )
m0 g
exp
T
2
At the early epoch of the Universe, T is very high. All are in
Relativistic. Non-relativistic exponentially decrease negligible
4
4
4 * 4
Ti 7
Ti
R T g T gi
gi
30
30
i boson T 8 i fermion T
2
2
(Eq-3.23)
14
Here, the effective degrees of freedom:
4
Ti 7
Ti
g gi
gi
T 8 i fermion T
i boson
4
(Eq-3.24)
2 4
PR R / 3
gT
90
(Eq-3.25)
g ~g (T)
(1) T<1 MeV
g 2
1
3
4
ve , ve , v , v , vc , v , g v 1 Tv T
11
15
From eq-2.24
4
3
7
4
g 2 6 3.36
8
11
(2) 1 MeV<T<100 MeV
me 0.5MeV
, v , e , e ,
Te Tv T
7
7
g 2 6 (2 2) 10.75
8
8 ge ge
(3) T>300 GeV
8 gluons, w
, z0 ,
3 generations quarks & leptons
1 complex Higgs doublet.
Total g 106.75
16
17
3.7 Time ~Temperature
when radiation dominated
pR R / 3, a(t )
H (T ) 1.66 g 1/ 2
t 0.301g 1/ 2
T
2
R
t 1/ 2
T2
m pl
m pl
T m pl GeV
2
T
1/ 2
2.42 g
MeV sec
T 10 GeV ,
t 109 sec
T 1GeV ,
t 107 sec
(Eq-S3.1)
T 10 0 MeV , t 104 sec
T 10 MeV , t 102 sec
T 1MeV,
t 1sec
decoupling
18
3.8 Evolution of Entropy
In the expanding Universe, 2nd law of thermodynamics
V a 3 (unit coordinate volume)
TdS d ( V ) PdV d ( P)V VdP
More,
2
1
R
g T , PR R
30
3
dP 4 2 4 4
T
g T R R PR
dT 3 30
3
dP P
P
dP
dT
dT
T
T
(Eq-3.26)
4
(Eq-3.27)
19
From Eq-2.26 with Eq-2.27
1
( P)V
( P)V
dS d ( P)V
dT d
const
2
T
T
T
Up to an additive constant , the entropy per commoving volume
a 3 ( P)
S
T
From
(Eq-3.28)
d ( a 3 ) Pda 3 d ( P )a 3 a 3dP
And with Eq-2.26, we have:
TdS 0
S const
20
The entropy per commoving volume is consented during the expansion
of the Universe.
density of
physical volume
Entropy density s
S P
s
V
T
s
2
45
(Eq-3.29)
dominated by
relativistic particles
g sT 3
3
Ti 7
Ti
g s gi gi
T 8 iF T
iB
3
(Eq-3.30)
For most of the history of the Universe
T1 T2 T3 T
g s g
21
For photon number-density n
4
s
g sn
45 (3)
(3)
3
g
T
, g 2
2
1.80 g sn 7.04n
(Eq-3.31)
kB 1
S sa3 const s a 3
Physical entropy density scales as
(Eq-3.32)
a 3
Commoving entropy density is conserved
22
From
eq-2.32
sa 3 const
3 3
2
g
s T a const
2 3
s
gsT
45
(3) 3
gT
2
3 (3) 3
n
gT
2
4
m T 3 2 ( m0 )
g 0 e T
2
n
(Eq-3.33)
Boson - relativistic
Fermions - relativistic
non - relativistic
(3) 3
g nT
2
(Eq-3.34)
3
Ti 3
Ti
g gi gi
T 4 iF T
iB
3
n
23
Define N n
S
45 (3) g n
,
T
4
gs
2
N
3
(m )
45
g m0 2 0T
, T
4 2 5 g T e
s
m, (relativistic)
(Eq-3.35)
m
(non-relativistic)
If the number of a given species in a commoving volume is not changing,
i·e, particles of that species are not being created or destroyed, then
Nn
s
remains constant
N Commoving number density
If no baryon non-conserving mechanism, then
nB nb nb
baryon conserved
s
s
(Eq-3.36)
So, with eq-2.31, we have:
nB nB s
nB
1.8 g S
n
s n
s
(Eq-3.37)
24
nB
is conserved , after e annihilations at T=0.5MeV (t ~ 4sec)
s
g s is constant, so 7.0
nB
s
S sa 3 conserved g sT 3a 3 const
More over
So, the temperature of the Universe evolves as:
T g s1/ 3a 1
Why T g
1
3
s
(Eq-3.38)
a 1 , not just T a 1
Explanation:
When
g
e annihilation, there is a change in s
S g sT 3a 3 conserved , so g s change T change
25
3.9 Decoupling
S0 S A SB SC const
for massless
When A is decoupled
S S B SC S0 ,
and
TA a 1
(1) Assume a massless particle decouples at time t D , temperature TD ,
when the scale factor was
aD , the phase-space distribution at
decoupling is given by the equilibrium distribution:
f ( p, t D )
1
e
E / TD
1
(Eq-3.39)
After decoupling, the energy of each particle ( m0 0)
is red-shifted by the expansion:
a(tD )
E (t ) E (t D )
a(t )
(Eq-3.40)
26
In addition
ncom nphy a3 const
so
(Eq-3.41)
n n phy a3
dntD
dn
f dec ( p, t ) 3 3
f equi ( ptD , t D )
d p d ptD
f equi ( p
1
e
E
T
1
a (t )
, tD )
a (t D )
,
1
e
E a (t )
TD a ( t D )
1
aD
here T
TD
a
(Eq-3.42)
aD
T TD
a 1 , for decoupled massless species, while the others
a
still couple with each other, so the temperature scales as:
T g s1/ 3 a 1
27
(2) Massive particle decoupling ,
m0
TD
dn a 3 , p a 1
t D , TD , aD
Decouples at
dnD
dn
f dec ( p, t ) 3 3
f equi ( pD , t D )
d p d pD
f ( p, t )
1
e
( E )
1
,
E
(Eq-3.43)
p2
m0
2m0
1
m0 (t )
p2
exp(
) exp(
) 1
T (t )
2m0T (t )
aD
aD 2
pD , T (t ) 2 TD
So p
a (t )
a (t )
T (t )
(t ) m0 ( D m0 )
TD
1
m0 D
pD 2
exp(
) exp(
) 1
TD
2m0TD
(Eq-3.44)
28
Summary:
In both cases
a (t )
f dec ( p, t ) f equi ( p
, tD )
aD
log(f)
(Eq-3.45)
tD
t
log(p)
29
(3) general cases
For a species that decouples when it is semi-relativistic TD
m
The phase-space distribution does not maintain an equilibrium
distribution. In the absence of interactions:
f ( p, t )
1
e( E ) / T 1
E p m0
2
2
2
aD
p
pD
a
You cannot find a simple relation , for
T f ( a, TD )
So the equilibrium distribution cannot be maintained
30
3.10 Brief Thermal History of the Universe
*Some famous events*
Key: the interaction rate per particle
n v
The correct way to evolve particlen distributions
is to integrate the Boltzmann equation
3.10.1 Neutrino decoupling
vv e e , ve ve , etc
GF2 T 2
n T3
int / H
GF : Fermi constant
int n v
GF 2T 5
T 2 m pl
: decouple
GF 2T 5
T
MeV
at T
3
(Eq-3.46)
1MeV
t 1sec
31
When T 1MeV,
when
T
1
a
me T 1MeV,
e ,
decoupled ve , ve , v , v , v , v
Stot Sv Sother
Stot is conserved Sother is conserved
S g sT 3a 3 ,
g s gi
iB
T
me
T me
7
7
11
g
2
4
i
8 iF
8
2
e e 2
g s gi 2
iB
32
For
: g T
s
3
befor
s
gT
3
after
1
3
TA g ( B)
11/ 2 13 11
)
(
TB g ( A)
2
4
s
s
1
1
TA ; TB
a
a
1
3
(Eq-3.47)
1/ 3
T TB 4
T TA 11
TB
T
(Eq-3.48)
T
TA
Relic neutrino
background
a
: decoupling
T=1 MeV
a
ae
T
a0
4 13
4 13
T ( ) T ( ) 2.75K 1.96 K
11
11
(Eq-3.49)
e annihilation
33
With this value of T / T , we have:
4
4
3
3
7 4 43
Ti 7
Ti
g today gi gi 2 ( ) 6 3.36
8 11
T 8 iF T
iB
7 4
T 7
T
g s today gi i gi i 2 ( ) 6 3.91
8 11
T 8 iF T
iB
And
R
2
g *T 4 8.09 1034 g cm -3
30
R h 2 4.31 105
2 2 3
s
g s T 2970 cm -3
45
2 (3) 3
-3
n
T
422
cm
2
(Eq-3.50)
T0 2.75K
34
3.10.2 Matter-Radiation Equality
m 0 c 1.88 1029 0 h 2 g cm-3
1 m
3
4
m a R a
1 zeg
a R
1 zeq 2.32 104 0 h 2
Teq T0 (1 zeq ) 5.50 0 h 2 eV
teq 1.08 103 (0 h 2 ) 2 years
(Eq-3.51)
In above calculation, we have used:
1
t
teq
(a / aeq 2)(a / aeq 1) 2 2
2 2
1
H eq H 0 (20 ) 2 (1 zeq )
3
2
4( 2 1) 1 4( 2 1) 1 12
3
teq
H eq
H 0 0 (1 zeq ) 2
3
3 2
35
3.10.3 Photon Decoupling and Recombination
Thomson cross-station
Radiation-Matter decoupling
ne T
T 6.65 1025 cm2
H
nH :
number density of free hydrogen
np :
number density of free protons
ne :
number density of free electrons
n p ne (charge neutrality)
nB n p nH ( n4 He )
In thermal equilibrium, at T mi
3
i mi
miT
T
ni gi
e
, i e, p, H
2
p e H p e H
2
36
gH
meT
nH
n p ne
g p ge
2
3
2
B: binding energy of hydrogen ,
e
B
T
(Eq-3.52)
B m p me mH
Define : the fractional ionization (ionization degree)
0 : neutral. Ionization=0
Xe
nB 1 : ionization total
np
g e g p 2, g H 4
nB n
(Eq-3.53)
( s s1 s2 0, 1)
: baryon-to-photon ratio
( B h 2 )2.68 108
T T0 (1 z )
T0 2.75K
From eq-2.52 the equilibrium ionization fraction
1 X eeq 4 2 (3) T 3/ 2 B / T
( ) e
eq 2
(Xe )
me
(Eq-3.54)
37
1 z
X e 0.1
1200 1400
Depends on
B h2
1 zrec 1300 Trec T0 (1 zrec ) 3575K 0.308eV
(Eq-3.55)
38
When recombination , matter–dominated
trec
2 1 12
H 0 0 (1 zrec ) 3/ 2 4.39 1012 (0 h 2 ) 1/ 2 sec
3
Decoupling:
H 1 zdec
1100 1200
on 0 , B
1 zdec 1100
Tdec T0 (1 zdec ) 3030 K 0.26eV
tdec
2 1 1/ 2
H 0 0 (1 zdec ) 3/ 2 5.64 1012 (0 h 2 ) 1/ 2 sec
3
Summary :
1 zdec
X e 0.1
1380( B h 2 )0.023 1240 1380
1 zdec 1100 (0 / B ) 0.018 1100 1200
tdec 1.8 105 ( 0 h 2 ) 1/ 2 yr
4.7 105 yr
(for 0 0.3, h 0.7)
(Eq-3.56)
39
3.10.4 The baryon number of the Universe
nB nb nb
nB :
baryon number density
nB nN 1.13 105 ( B h 2 )cm 3
N n, p
(Eq-3.57)
B c
nB
mp
mp
B
nB
B
s
(is conserved) 3.81109 ( B h 2 )
(Eq-3.58)
B is defined to be the baryon number of the Universe
since the epoch of
s
7.04n
7 B,
e
annihilation
n : photon density
nB
2.68 108 ( B h2 )
n
(Eq-3.59)
40
The primordial nucleosynthesis constrains
to the interval (4
7) 1010 , that is
B h 2 (1.5 2.6) 102 ,
B
(6 10) 10 11
1
s
(1 ~ 2) 1010
B nB
(Eq-3.60)
entropy per baryon
To compare with, in a star, the entropy is n / nN
102
So the entropy of the Universe is enormous !!!
41
T 0.1MeV
t 3min
T 10MeV
t 10-2 s
t pl
Primordial
nucleosynthesis
teq 1.08 103 (0 h2 ) 2 years
zeq 2.32 104 0 h2
t0 1.38 1010 yr
10 43 s
T 1MeV
t 1sec
decoupling
T 0.5MeV
t 4sec
e annihilation
tdec 5.64 1012 (0 h2 ) 1/ 2 sec
1.8 105 (0 h2 ) 1/ 2 yr
zrec 1100 ~ 1200
trec 4.39 1012 (0 h2 )1/ 2 sec
zrec 1240 ~ 1380
A human age:
one day
100 years
42
Key points:
(1) f ( p, t ) phase-space distribution function
(2)
S sa 3 const
(3)
n v interaction rate per particle
H
H
Based on
/ H
entropy is concerned
thermal equilibrium
decouple
argument:
Qualitative and semi-quantitative
Full-quantitative treatment: solve collisional Boltzmann Equation
43
References
• E.W. Kolb & M.S. Turner, The Early
Universe, Addison-Wesley Publishing
Company, 1993
• T. Padmanabhan, Theoretical
Astrophysics III: Galaxies and
Cosmology, Cambridge, 2002
44