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
xp  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
kp
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:
kk
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 
ng
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 109 sec
T  1GeV ,
t 107 sec
(Eq-S3.1)
T  10 0 MeV , t 104 sec
T  10 MeV , t 102 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 sT 3
3
 Ti  7
 Ti 

g s   gi     gi  
 T  8 iF  T 
iB
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 sn
45   (3)
 (3)
3
g
T
, g  2

2

1.80 g sn  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 iF  T 
iB
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
Nn
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 sT 3a 3  const
More over
So, the temperature of the Universe evolves as:
T  g s1/ 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 sT 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  a3
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 s1/ 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 sT 3a 3 ,
g s   gi 
iB
T
me
T  me
7
7
11
g

2


4

 i
8 iF
8
2
e   e   2
g s   gi  2
iB
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 iF  T 
iB

7 4
T  7
T 
g s today   gi  i    gi  i   2  ( )  6  3.91
8 11
 T  8 iF  T 
iB
And
R 
2
g *T 4  8.09  1034 g cm -3
30
 R h 2  4.31 105
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  1029  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 (20 ) 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 1025 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 108
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 105 ( B h 2 )cm 3
N  n, p
(Eq-3.57)
 B c
nB 

mp
mp
B
nB
B
s
(is conserved)  3.81109 ( 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 108 ( B h2 )
n
(Eq-3.59)
40
The primordial nucleosynthesis constrains 
to the interval (4
7) 1010 , that is
 B h 2  (1.5  2.6) 102 ,
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
102
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