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Introduction to Cosmology
PROCEEDINGS
Introduction to Cosmology
Abstract: An introductory note of the talk on cosmology. Some of the text are copied from the
talk by G. Lazarides, http://arxiv.org/pdf/hep-ph/9904502v2.pdf. As of the date: May 29,2012.
Keywords: cosmology.
1. Redshift and Hubble Constant
1.1 Redshift due to the recessional veloc-
ity
A B
P
r r ’
r ’’
From homogeneity and isotropy of any expanding
universe, one can show that the recessional veloc-
ity vB of the (galaxies) photon at P observed by
observer B is the same as the recessional velocity
vA of the photon at the same direction observed
by observer B, namely:
vB(r′, t) = vA(r′, t). (1.1)
Assume there is a small relative velocity of B
frame, vA(r′′, t) < c, as seen from the A frame
along the radius vector r′′ = r − r′. Therefore
the velocity vA(r′, t) can be expressed as:
vB(r′, t) = vA(r, t)− vA(r′′, t) = vA(r′, t) .
Therefore,
vA(r′, t) = vA(r, t)− vA(r′′, t) , (1.2)
r′ = r− r′′ (1.3)
is true for every point P in any isotropic and
homogeneous space. This hence implies that:
vA(r, t) = f(t)r . (1.4)
Namely, the velocity-distance ratio is a constant
in r.
Redshift z of the photon spectrum from dis-
tant universe at distance r due to the recessional
velocity v of the photon field is related by:
1 + z =
√1 + v/c
1− v/c. (1.5)
Hence we can show that
v/c ∼ z (1.6)
for v c. For any expanding space with the
relation
v = H0r, (1.7)
the redshift can be shown to follow the relation:
z ∼ H0r
c(1.8)
1.2 Lorentz Transformation
The Lorentz transformation is given by
x′a = Λabxb , (1.9)
with
( Λ )ab =
γ −γβ 0 0
−γβ γ 0 0
0 0 1 0
0 0 0 1
(1.10)
for two reference frames moving apart with ve-
locity v and β = v/c.
Introduction to Cosmology W.F. Kao
In fact all 4-vectors Aa transform as xa =
(ct, x), namely,
A′a = ΛabAb . (1.11)
For example, the 4-momentum vector ka = (ω/c, k)
is also a 4-vector derived from the identity ka =
i∂a = i(∂t/c,∇i) = (ω/c, −k). This also follows
from the fact that the plane wave phase factor
takes the form exp[−ikaxa]. Therefore
k′a = Λabkb . (1.12)
Hence k′ = γ(−βω/c+ k) with k the momentum
3-vector in x-direction. This leads to the result
k′ = kγ(1− β) = k
√c− vc+ v
, (1.13)
following from the fact that c = ω/k. Note that λ
is the wavelength of the photon in the source rest
frame. λ is the photon wavelength in observer’s
rest frame. Note that
1 + z ≡ λobsλsource
=λ′
λ=
k
k′. (1.14)
Hence the identity of red-shift formula follows:
1 + z =
√1 + v/c
1− v/c. (1.5)
1.3 Redshift due to the Recessional Veloc-
ity relative to R(t)
For an expanding universe with a scale factor
R(t), we can define a dimensionless parameter
a(t):
a(t) =R(t)
R0. (1.15)
We will show in a moment that the peculiar ve-
locity and its associated momentum of any mat-
ter field in the FRW space is proportional to
1/R(t). p ∼ 1/R is also true for photon field.
Hence the wavelength of the photon source λ =
h/p ∼ R obeys the relation:
λ0R0
=λ
R(t). (1.16)
In addition, a can be expanded as:
a(t) = 1− a0(t0 − t) (1.17)
by assuming the expansion rate is small. Ex-
act relation will be shown later when we start
to solve the gravitational field equations in the
FRW space with various matter fields incorpo-
rated in a consistent manner.
From the definition of z
z =λ0 − λλ
=1
a− 1 ,
one can show that
z ∼ a0r
c, (1.18)
with the expansion coefficient a0 related to the
Hubble parameter H0:
a0 =R0
R0= H0 . (1.19)
This shows that the definition of redshift (1.5)
agrees with the definition 1+z = 1/a(t) = R0/R(t).
2. Cross section
Cross section is the area of the effective col-
lisional impact. For a classical atom with Bohr
radius a0, the cross section
σ = π(2a0)2 , (2.1)
a0 =h
α mec,
α =e2
(4πε0)hc∼ 1
137.
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Introduction to Cosmology W.F. Kao
The mean free path l of a interacting particle
is the distance between two effective collisions.
Assuming a particle moving with velocity v can
travel a distance L in time interval t such that
L = vt, σl = Vone−collision. lN = L follows if
there are N particles in the volume V = σL. For
a particle with number density n = N/V , one
can thus show that
l =1
nσ. (2.2)
This follows from the fact that:
n =N
V=L/l
σL=
1
σl.
Note that [nσ] = [1/L3][L2] = [1/L] gives the
correct dimension for 1/l.
2.1 Interaction Rate
The interaction rate Γ is defined as the number
of interactions in unit time:
Γ = nσv (2.3)
with v the speed of the particle. Another way to
look at this definition is:
Γl = v . (2.4)
Mean free path times the interaction per unit
time is the velocity of the particle. In addition,
the dimension of the interaction rate is:
[Γ] = [1
L3][L2][
L
t] = [
1
t] (2.5)
which is consistent with its definition.
For example, the photon interaction rate
Γγ = neσT c (2.6)
with σT = 6.65× 10−25cm2 the Thompson cross
section. The cross section of different interac-
tions have to rely on the fundamental quantum
field theories known to most particle physicists.
We will not go into the details of the detailed
derivations. The results will be given. These re-
sults can also be checked out from the well known
data book or any particle physics textbook.
In particular, when Γ > H, particle interac-
tion is still vivid as the interaction rate is greater
than the expansion rate. When Γ < H, particle
interaction will gradually decouple from the heat
bath as we will return to this topic shortly.
3. blackbody radiation
Blackbody radiation assures that all photons are
in thermal equilibrium via collisions with charged
medium and obeys the Planck distribution func-
tion of the form:
Bλ(T ) =2hc2/λ5
exp[hc/λkT ]− 1(3.1)
Express Bλdλ = −Bνdν as a distribution func-
tion of the frequency ν, it is apparent that:
Bν(T ) =2hν3/c2
exp[hν/kT ]− 1. (3.2)
Counting the dimension:
[Bλ] = [hc2/λ5] =
[E
tL3
](3.3)
is the dimension of the power density. Note that
the energy density is uλdλ = (4π/c)Bλdλ, and
[uλdλ] = [hc/λ4] =
[E
L3
]. (3.4)
4. Friedmann-Robertson-Walker met-
ric space
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Introduction to Cosmology W.F. Kao
Homogeneous and isotropic 3-spaces can be
classified into three different classes that can be
parametrized as:
X2 + Y 2 + Z2 +W 2 = A2 (4.1)
X2 + Y 2 + Z2 = B2 (4.2)
X2 + Y 2 + Z2 −W 2 = A2 (4.3)
for a constant scale factor A parameterizing the
radius of the 3-space. B can be arbitrary radius
for the space R3. The first class is a closed space,
the second class is a flat space and the last one
is the open space. It can be parametrized alter-
natively as:
ds23 = gijdxidxj
= a2(t)
[dr2
1− kr2+ r2(dθ2 + sin2 θ dϕ2)
],
(4.4)
with r, ϕ and θ ‘comoving’ polar coordinates.
The parameter k is the ‘scalar curvature’ of the
3-space. k = 0, k > 0 or k < 0 correspond to
flat, closed or open universe. The dimensionless
parameter a(t) is the ‘scale factor’ of the universe
normalized by taking a(t) in unit of R0. Equiva-
lently, we are taking R(t) = a(t)R0 ≡ a(t)R(t0).
Here R(t) is the scale factor of the universe and
t0 is the present cosmic time. Hence a0 = 1, i.e.
one unit of R0.
Eq. (4.4) can be proved by parameterizing,
for example, the closed 3-space as;
X = A sinχ sin θ sinϕ, (4.5)
Y = A sinχ sin θ cosϕ, (4.6)
Z = A sinχ cos θ, (4.7)
W = A cosχ. (4.8)
It then follows that
dX2 + dY 2 + dZ2 + dW 2
= A2[dχ2 + sin2 χdΩ
]. (4.9)
This gives the case k = 1 with the parameteriza-
tion r = sinχ. Indeed, dr = cosχdχ, and hence
dr2/ cos2 χ = dχ2 = dr2/(1 − r2). Similarly for
the open space with k = −1 by replacing sinχ
(cosχ) with sinhχ (coshχ).
The four dimensional spacetime in the uni-
verse is then described by the Friedmann-Robertson-
Walker metric
ds2 = dt2 − ds23. (4.10)
5. geodesic equation of a test parti-
cle
For a particle traveling on a metric space given
by:
ds2 = gabdxadxb , (5.1)
the trajectory will make the length of the trajec-
tory as short as possible. Therefore, the trajec-
tory can be derived from the least action prin-
ciple with action given by the length∫ds. It is
also equivalent to varying the Lagrangian given
by
L =
√gab
dxa
dt
dxb
dt. (5.2)
The derivation is straightforward:
δL =1
2L
(∂cgabδx
cvavb + 2gacva d
dtδxc)
=1
2
[∂cgabu
avb − 2d
dt(gacu
a)
]δxc (5.3)
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Introduction to Cosmology W.F. Kao
after an integration-by-part. Note that ua =
(u0, ui)(= γ(1, vi) for the Minkowski space with
metric gab = ηab) with ua = dxa/ds and va =
dxa/dt. Therefore the field equation leads to:
ua +1
2gad(∂bgcd + ∂cgbd − ∂dgbc)ubvc
≡ ua + Γabcubvc = 0 . (5.4)
Note that gab is the inverse of the metric field
satisfying gabgbc = δac. It is also true if we take
another affine parameter λ to replace t:
dua
dλ+ Γabcu
b dxc
dλ= 0. (5.5)
Taking λ as s:
dua
ds+ Γabcu
buc = 0 (5.6)
This equation is referred to as the geodesic equa-
tion.
5.1 covariant derivative
Note that the geodesic equation can also be writ-
ten as
dua + Γabcubdxc
= (∂cua + Γabcu
b)dxc = 0 . (5.7)
Hence the geodesic equation becomes
Dcua = ∂cu
a + Γabcub = 0 (5.8)
with the operator Dcua known as the covariant
derivative of any contra-variant vector ua.
Note that the covariant derivative of any co-
variant vector va is
Dcva = ∂cva − Γbcavb . (5.9)
This can also be derived by requiring that
Dc(uava) = ∂c(u
ava) , (5.10)
Dc(uava) = (Dcu
a)va + ua(Dcva) .(5.11)
Note that the first requirement is demanding that
uava is a 4-scalar. The second requirement is the
Leibniz rule.
Derivative operators ∂c is known as the trans-
lational operators. Indeed, we can write
exp
[iP
h(x+ a)
]= [1 + a∂x] exp
[iP
hx
].
(5.12)
It is equivalent to parallel transport a phys-
ical observable like a vector vb form one point
to another nearby point. If the space is curved,
parallel transport is defined in conjunction to the
curved geometry. Normal derivative will carry
the vector off its original space and no longer re-
mains a well-defined vector in its original space.
Covariant derivative Dc is therefore designed to
remove off-space component of the transported
vector, namely, Γbcavb from the normal deriva-
tive. After removing unphysical component of
the transported vector, we will be able to de-
fined a new transported vector defined on our
own space time.
By requiring
Dcgab = ∂cgab − Γdcagdb − Γdcbgad (5.13)
as if it is similar to the covariant derivative of
a product of two covariant vectors AaBb. Note
that
Dc(AaBb) = (DcAa)Bb +Aa(DcBb) (5.14)
obeying the Leibniz rule. Then it is straightfor-
ward to show that
Dcgab = 0 (5.15)
which is known as the compatibility condition in
Riemannian geometry. We will come back to the
details of the tensor calculus later in this text.
6. particle horizon and velocity fields
ds2 = 0 ( dt2 = a2dr2/(1 − kr2) ) along the
radial path) for a photon field on the FRW space.
Hence ∫ t
0
dt′
a(t′)=
∫ rH
0
dr√1− kr2
. (6.1)
Hence the particle horizon
dH =
∫ rH
0
dr√grr (6.2)
can be shown to be:
dH = a(t)
∫ rH
0
dr√1− kr2
= a(t)
∫ t
0
dt′
a(t′). (6.3)
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Introduction to Cosmology W.F. Kao
Writing in conformal coordinate:
ds2 = gabdxadxb
= a2(η)
[dη2 − dr2
1− kr2− r2(dθ2 + sin2 θ dϕ2)
],
(6.4)
we can write the particle horizon as
dH = a(t)(η(t)− η0). (6.5)
This follows from the fact that a(η)dη = dt.
6.1 Peculiar Velocity
Peculiar velocity is defined as the velocity with
respect to the co-moving frame. Therefore the
peculiar 4-velocity ua = dxa/ds obeys the geodesic
equation:
dua
dλ+ Γabcu
b dxc
dλ= 0 (6.6)
with λ some affine parameter. Note that ua =
(u0, ui) with vi = dxi/dt the 3-velocity.
The 0-component geodesic equation reads:
du0
ds+ Γ0
bcubuc = 0, (6.7)
du0
ds+Hu2 = 0 (6.8)
with u2 = uiujgij . Eq. (6.8) can also be derived
directly from the geodesic equation (5.3)
∂tgabuavb − 2
d
dt(gatu
a) = 0 . (6.9)
Indeed, it reproduces: 2(Hu2 + du0/ds) = 0.
From uaubgab = u0 2 − u2 = 1, we can show
that u0du0 = udu for any matter field. Hence
u
(du
u0ds+Hu
)= 0 . (6.10)
Note that u0ds = dt following the definition of
u0. Therefore, the geodesics equation gives
u
u= −H = − a
a. (6.11)
This implies immediately that
u =p
m∝ 1
a→ 0 (6.12)
at time infinitive for any expanding universe. Note
that p ∝ 1/R is also true for photon field. Hence
we reach the conclusion that the photon wave
length is proportional to the scale factor R(t) or
a(t).
λ ∼ h
mu∼ a(t). (6.13)
Hence the redshift can be defined also as 1 + z =
a0/a = λ0/λ as mentioned earlier.
6.2 luminosity distance
The flux distribution Fλ has a dimension given
by [F ] = [L/A] = [E/tL2]. It is a power per unit
area. Equivalently, the flux follows
Fλdλ =Lλdλ
4πr2= Bλdλ
R2
r2(6.14)
for a radiation source with Bλ radiating from the
source sphere with radius R. r given above is the
distance from the radiation source to the detec-
tor. For an expanding universe, the radiation
source far away is red-shifted by a factor 1 + z.
There is another redshift factor 1 + z represent-
ing the power received by the detector due to the
time-lag effect derived from [F ] = [dE]/[A][dt].
Therefore, the flux received by the detector will
be:
F =L
4πd2L=
L
4πr2(1 + z)2. (6.15)
dL = r(1+z) is known as the luminosity distance.
For a power Bλdλ emitted with an angle θ to the
detector, the total luminosity Lλdλ will be:
[Lλdλ] = [Bλdλ][dA cos θ][dΩ]. (6.16)
Therefore,
Lλdλ =
∫ 2π
0
∫ π/2
0
∫A
[Bλdλ][dA cos θ][dΩ]
= 4πR2Bλdλ (6.17)
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Introduction to Cosmology W.F. Kao
with R the radius of the radiation source sphere.
In addition, ∫ ∞0
Bλdλ =σ
πT 4 , (6.18)
with
σ =2π5k4
15c3h3. (6.19)
7. Hubble’s law
The particle horizon is given by
dH = a(t)
∫ r
0
dr
(1− kr2)1/2· (7.1)
We will write dH = a(t)r, for a flat universe
(k = 0), with r a ‘comoving’ and dH a physical
vector in 3-space. Hence the velocity of an object
is
V = dH =a
adH + a
dr
dt, (7.2)
with over-dots denoting derivation with respect
to cosmic time. The second term in the right
hand side (RHS) of this equation is the ‘peculiar
velocity’, v = a(t)r, of the object. It is the ve-
locity with respect to the ‘comoving’ coordinate
system. For v = 0, Eq.(7.2) reads
V =a
adH ≡ H(t)dH , (7.3)
with H(t) ≡ a(t)/a(t) the Hubble constant. This
is the well-known Hubble law claiming that ev-
erything runs away from each other with velocity
proportional to their distances.
8. Conservation Law of the perfect
fluid
Homogeneity and isotropy of the universe imply
that the energy momentum tensor takes the di-
agonal form
(T ab) = diag(ρ,−p,−p,−p) , (8.1)
with ρ the energy density of the universe and p
the pressure. Energy momentum conservation
DaTab = 0 (8.2)
can be expressed as the continuity equation
dρ
dt= −3H(t)(ρ+ p) , (8.3)
where the first term in the rhs describes the di-
lution of the energy due to the expansion of the
universe and the second term corresponds to the
work done by pressure. Eq.(8.3) can be given the
following more transparent form
d
(4π
3a3ρ
)= −p 4πa2da , (8.4)
which indicates that the energy loss of a ‘comov-
ing’ sphere of radius ∝ a(t) equals the work done
by pressure on its boundary as it expands.
Note that Eq. (8.4) can be interpreted as a
thermal dynamical equation:
dU = −pdV + TdS , (8.5)
with U = ρV , V = 4πa3/3, S the entropy and
T the temperature of the thermal dynamical sys-
tem. dS = 0 for a closed system without energy
loss to its environment. Note that TdS = d−Q.
Also note that the volume of the 3-sphere S3
is 2π2a3 instead of 4πa3/3. This follows from the
metric element shown in Eq. (4.9):
ds2 = a2[dχ2 + sin2 χdΩ
]. (8.6)
The volume of the 3-sphere is∫ π
0
dχ
∫ π
0
dθ
∫ 2π
0
dϕ√g (8.7)
with√g = a3 sin2 χ sin θ. The integral can be
shown directly to give 2π2a3. Hence the New-
tonian approach can not taken seriously at this
point.
9. Friedmann Equation
For a universe described by the Robertson-Walker
metric in Eq.(4.10), Einstein’s equations
Rab −1
2δabR = 8πG T ab , (9.1)
where Rab and R are the Ricci tensor and scalar
curvature tensor and G ≡ M−2P is the Newton’s
constant, lead to the Friedmann equation
H2 +k
a2=
8πG
3ρ (9.2)
with H ≡ a(t)/a(t) the Hubble parameter.
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Introduction to Cosmology W.F. Kao
Writing the equation in the form:
1
2ma2 −mG
[(4π3 a
3)ρ
a
]= −mk
2, (9.3)
one can read this equation as the Newtonian en-
ergy conservation law for a test particle m mov-
ing under the gravitational attraction due to a
sphere of uniform mass density ρ and radius a:
E = T + V = −mk
2. (9.4)
The velocity a is the escape velocity for the case
k = 0. On the other hand, the system is a bound
state for k > 0. The test particle will escape
when k > 0.
10. Expanding Universe
For the equation of state
p = ωρ , (10.1)
we can write
ρ+ p = γρ ,
with the relation
γ = 1 + ω . (10.2)
Eq. (8.3) then becomes
ρ = −3Hγρ ,
which can be solved to give
dρ
ρ= −3γ
da
a
with the solution:
ρ ∝ a−3γ . (10.3)
For matter dominated (MD) universe with p = 0,
thus γ = 1(ω = 0), the solution is
ρ ∝ a−3 . (10.4)
This can be interpreted as a non-relativistic con-
servation law of a fixed number of particles in a
‘comoving’ volume due to the cosmological ex-
pansion.
For a radiation dominated (RD) universe,
p = ρ/3 and, thus, γ = 4/3(ω = 1/3), which
gives
ρ ∝ a−4 . (10.5)
In this case, we get an extra factor of a(t) due
to the red-shifting of all wave-lengths by the ex-
pansion. This can be interpreted as a relativistic
conservation law of a fixed number of particles
in a ‘comoving’ volume due to the cosmological
expansion.
Finally, for the case of vacuum dominated
(VD) universe with γ = 0, we will have
ρ = constant. (10.6)
11. Age of the Universe on a flat
MD FRW space
For a flat FRW MD universe, H2 = 8πGρ/3 with
ρ = ρ0/a3. Therefore,
a2 = H20
1
a. (11.1)
The age of the universe can be integrated from∫dt =
∫da/a. The result is:
t =2
3H0. (11.2)
In addition, the Friedmann equation can also be
written as:
H2 = H20
1
a3, (11.3)
a2 = H20
1
a. (11.4)
Hence the universe expands faster in its earlier
stage for any expanding universe.
In addition, for the VD universe,
H = H0 =
√8πG
3ρ ≡
√Λ
3, (11.5)
with Λ known as the cosmological constant. Hence
a = exp[H0(t− t0) ] . (11.6)
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Introduction to Cosmology W.F. Kao
12. More on the age of the flat uni-
verse
Substituting
ρ ∝ a−3γ
in Friedmann equation with k = 0, we get
a/a ∝ a−3γ/2
and, thus,
a(t) ∝ t2/3γ .
Taking into account the normalization of a(t)
(a(t0) = 1), this gives
a(t) = (t/t0)2/3γ . (12.1)
Hence
Ht =2
3γ. (12.2)
This asserts that the Hubble parameter is larger
at earlier time. Therefore, the scale factor is
a(t) = (t/t0)2/3 ; Ht = 2/3 (12.3)
a(t) = (t/t0)1/2 ; Ht = 1/2 (12.4)
for MD and RD universes respectively. This agrees
with the results shown in previous section.
In summary, we have the following chart for
the expanding flat universe:
ρ a
RD a−4 (t/t0)1/2
MD a−3 (t/t0)2/3
VD constant exp[H0(t− t0)]
If there is ever a small cosmological constant
present in our univserse, it will eventually dom-
inate the energy density of the universe. When
this happens, the universe will enter an fast ac-
celerating expanding phase as the latest observa-
tional evidences provided todate.
13. Thermal equilibrium
According to the definition by Feynman, Ther-
mal Equilibrium (TE) is defined as:
• A system is very weakly coupled to a heat
bath at a given T ,
• if the coupling is indefinite or not known
precisely,
• if the coupling has been on for a very long
time,
• and if all the fast things have happened,
and all the slow things not,
then the system is said to be in TE.
14. partition funtion
The partition function Q of a system in TE
with the heat bath at temperature T is defined
as
Q =
N∑n=1
exp
[−EnkT
](14.1)
with En the energy levels of the system S and k
the Boltzmann constant. And the probability of
finding a system having energy En is
Pn =1
Qexp
[−EnkT
]. (14.2)
Furthermore, the expected value of A is
< A >=1
Q
∑|i>
< i|A|i > exp
[− EikT
].
(14.3)
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Introduction to Cosmology W.F. Kao
15. Chemical Potential
Consider a system of N identical free particles
without interactions. Any two configurations can
be related by the exchange of two or more identi-
cal particles will be considered as the same con-
figuration. Hence the state of the system can
be identified by the number of particles na with
energy Ea.
Therefore we need to compute the partition
function under the constraint
N =∑a
na = constant . (15.1)
In addition, the numbers allowed by statistics is
• na = 0, 1, 2, · · · for bosons,
• na = 0, 1 for fermions.
The states of a system in TE is described
by a set of numbers na allowed by (i) statistic
(ii) the constraint N = constant. The partition
function should be, with β ≡ 1/(kT ),
Q =∑
n1,n2,···exp
[−β∑a
naEa
]. (15.2)
If there is no particle number constraint, we can
write the partition function as
Q =
[∑n1
exp [−βn1E1]
] [∑n2
exp [−βn2E2]
]· · ·
=∏a
∑na
exp [−βnaEa]
. (15.3)
Therefore, we can show that, 1 + x+ x2 + · · · =1/(1− x) with x = exp[−βEa],
Qb =∏a
1
1− exp[−βEa]
(15.4)
for Bose-Einstein gas. On the other hand, we will
have
Qf =∏a
1 + exp[−βEa] (15.5)
for Fermi-Dirac gas. This result cannot, however,
be true if N = constant. To enforce the con-
straint will make the computation of Q very com-
plicate. There is, however, a way to get around
this difficulty by introducing a chemical potential
in the system.
Note that the chemical potential µ is intro-
duced, Ea → Ea − µ, to get around the number
density conservation in this approach. By doing
so, we are imagining that the system S is in ther-
mal contact with a large reservoir H. The parti-
cles in S can be created by extracting a chemical
function of energy from the reservoir such that
the system in thermal equilibrium with the reser-
voir is still well defined. For example, imagining
the system in enclosed by a metal container in
thermal contact with H. The chemical poten-
tial µ is then the work function of the metal for
creating a particle to S.
On the other hand, it is really no need to
explain why we need to include the possibly in-
finite number of particles in the system. The re-
dundant or extra particles included in the system
will be down-graded by the probability function
of the form:
exp[−nβEa] (15.6)
for a large number of particles, n 1, occu-
pying the energy states Ea. It is obvious that
the contribution to the partition function from
large n is physically negligible. Hence the major
contribution of the partition function Q comes
from the lowest energy configuration with most
of the physical particles occupying the lowest en-
ergy states. Summing up all possible configura-
tion with unphysical number of particles occupy-
ing high energy states will simply give us the true
physical configuration of the thermal dynamical
system.
This is somewhat similar to the path integral
formulation of quantum mechanics. All unphysi-
cal paths are included in the computation of the
action function. It is then weighted by the prob-
ability function
exp[ iS ] (15.7)
that will physically eliminate most of the contri-
bution from those unphysical paths in a correct
and magical manner. The major contribution
again comes from the lowest action path after we
summed over all paths of the physical motion.
10
Introduction to Cosmology W.F. Kao
16. partition function with chemical
potential
Once the chemical potential is introduced, the
results of partition function shown in this section
is correct again by replacing Ea with Ea − µ.
By the key principle of statistical mechanics, the
probability of the gas having energy
E =∑a
na(Ea − µ) (16.1)
is proportional to
exp[−βE] . (16.2)
Hence, the partition function Q of a many
particle system is then given by:
Qµ =∑
n1,n2,···exp
[−β∑a
na(Ea − µ)
].
(16.3)
Hence
Qµ = exp[−βG]
=∑
n1,n2,···
× exp −β[n1(E1 − µ) + n2(E2 − µ) + · · ·](16.4)
for Bose-Einstein gas with ni = 0, 1, 2, ... and for
Fermi-Dirac gas with ni = 0, 1. β = 1/kT . It
reduces to:
Qµ = exp[−βG]
=∑n1
exp[−βn1(E1 − µ)]∑n2
exp[−βn2(E2 − µ)]
×∑· · · (16.5)
In addition,∑ni
exp[−βni(Ei − µ)] =1
1− exp[−β(Ei − µ) ]
(16.6)
for Bose-Einstein gas. Hence we can show that
βG =∑i
ln[ 1− exp[−β(Ei − µ)] ] .
(16.7)
Note that Ei denotes the energy levels of ni. It
is the same energy level for each particles.
Similarly we have
βG =∑i
ln [ 1 + exp[−β(Ei − µ)] ]
(16.8)
for Fermi-Dirac gas.
17. Physical origin of the probabil-
ity function in thermal dynamics
E1
E2
H2
H1
S
H
Let E − i be the energy levels of the system.
ImaginingHi are energy levels of the heat bathH
that is quasi-continuous in energy splitting. As-
sume Hi Ej for all i, j. Also assume that the
heat bath and the system are in thermal contact
with an even larger environment such that the
heat bath can tune its energy level through the
interchange of energy with the exterior environ-
ment. Let the collection of S and H be T . En-
ergy conservation assures that E0 = H1 + E1 =
H2 + E2 when energy are interchanged between
S and H. Due to the minor difference in energy
levels of H, its energy levels will not be definite,
but will be E0 − E2 ±∆ with ∆ E0.
Let η(Hi)∆ be the number of states in H
around energy Hi. Then the probability that S is
in a state with E2, P (E2), is proportional to the
number of ways S can have that energy. There-
fore,
P (E2) ∼ η(E0 − E2)2∆ . (17.1)
11
Introduction to Cosmology W.F. Kao
Note again that η(Hi) is the number of states per
unit energy. Hence we reach the result:
P (E2)
P (E3)=η(E0 − E2)
η(E0 − E3). (17.2)
Note that the energy is defined only up to a scal-
ing constant. Hence we have, with E0 → E0 − ε,
P (E2)
P (E3)=P (E2 + ε)
P (E3 + ε)(17.3)
for small ε. When ε Ei, we can expand P (Ei+
ε) = P (Ei)+P ′(Ei)ε. Therefore, we finally have:
P (E2)
P (E3)=P ′(E2)
P ′(E3). (17.4)
This then leads to the result that the probability
of finding S in energy levels Ei obeys the equa-
tion
P ′(E2)
P (E2)=P ′(E3)
P (E3)= β . (17.5)
with a constant β that eventually define the tem-
perature via the relation β = 1/kT . Hence we
prove that
P (Ei) = exp [−βEn] (17.6)
is a reasonable approach in thermal dynamics.
18. System with continuous energy
levels
The Helmhotz free energy F (V, T ) is related to
the internal energy U(V, S) via the Legendre trans-
formation:
F (V, T ) ≡ U(V, S)− TS . (18.1)
We can then show that dU and dF are
dU = TdS − PdV , (18.2)
dF = −PdV − SdT . (18.3)
It can also be shown that the pressure P is re-
lated to F and U via:
P = −∂F (V, T )
∂V= −∂U(V, S)
∂V(18.4)
Writing the partition function Q as a function of
the generating function G, Q ≡ exp[−βG],
exp[−βG] =∑ni
e−βE (18.5)
with E =∑a naEa and Ea → Ea − µ contains
the chemical function that may not be written
explicitly for simplicity. Here summation over
ni is a short handed notation for n1, n2, · · · not
written explicitly.
One can show that the total number of par-
ticles in volume V is
〈N〉 ≡∑ni
exp[−βE]
Q
(∑a
na
)= −∂G
∂µ.
(18.6)
We will be assuming that the particles in the uni-
verse is a collection of ideal gas with continuous
energy levels. Therefore we can show that
βG =∑ni
ln(1− exp[−βE])
→ g
∫ln(1− exp[−βE])
d3p
(2πh)3V ,(18.7)
by summing over all energy levels in the momen-
tum space. Hence we can show that the number
density n = 〈N〉/V is:
n = g
∫exp[−βE]
1− exp[−βE]
d3p
(2πh)3
= g
∫1
exp[−βE]− 1
d3p
(2πh)3. (18.8)
Note that the energy density ρ = U/V= ∂(βG)/(V ∂β)
can also be shown to be
ρ = g
∫1
exp[−βE]− 1E
d3p
(2πh)3. (18.9)
Finally, we can derive the the pressure P of the
system as
P = g
∫1
exp[−βE]− 1
p2
3E
d3p
(2πh)3. (18.10)
This follows from the fact that P = −∂U/∂V =
−∂ 〈E〉/∂V . Therefore, we need to show that
∂E
∂V= − p2
3EV. (18.11)
Indeed, this follows from the fact that: E2 =
p2 + m2. Hence we can show that EdE = pdp
with p the isotropic momentum. In addition,
V ∝ r3 and p ∝ 1/r in an isotropic space and
its conjugated momentum space. The propor-
tional constants are not written explicitly here
12
Introduction to Cosmology W.F. Kao
for simplicity. It will be automatically cancelled
out when we take an inverse of these actions.
What remains to be shown is that
∂E
∂V=
p∂p
E∂V
∝ − p
3E
1
r31
r→ − p2
3EV. (18.12)
This proves the formula (18.10) for the pressure
P .
19. Distribution function
The Fermi-Dirac (+) and Bose-Einstein (−) dis-
tribution functions are given by the probability
functions:
f(p) =1
exp[E−µT
]± 1
. (19.1)
Here the energy E is given by E2 = p2 + m2
for relativistic particles with mass m. µ is the
chemical potential. Note also that we have set
the unit by taking the Boltzmann constant k = 1
for convenience.
The temperature dependence of the number
density n, energy density ρ and momentum den-
sity p are given respectively by:
n =g
(2π)3
∫f(p)d3p , (19.2)
ρ =g
(2π)3
∫Ef(p)d3p , (19.3)
p =g
(2π)3
∫p2
3Ef(p)d3p . (19.4)
These integrals can be written as integral over E
with the help of the relation EdE = pdp:
n =g
2π2
∫ ∞m
√E2 −m2f(p)EdE , (19.5)
ρ =g
2π2
∫ ∞m
√E2 −m2f(p)E2dE ,(19.6)
p =g
6π2
∫ ∞m
√E2 −m2
3f(p)dE . (19.7)
It is straightforward to perform these integrals
under certain limits. For example, in the rela-
tivistic limit with T m and T µ, the results
are:
n =ζ(3)
π2gT 3 , (19.8)
ρ =π2
30gT 4 , (19.9)
p =ρ
3(19.10)
for bosons with g the number of species or the
degree of freedom associated with the particles.
On the other hands, the results are:
n =3
4
ζ(3)
π2gT 3 , (19.11)
ρ =7
8
π2
30gT 4 , (19.12)
p =ρ
3(19.13)
for fermions. Here the zeta function is defined
as ζ(p) =∑∞n=1 1/np. It is easy to show that
ζ(3) ∼ 1.2 ∼ π4/81, and ζ(4) = π4/90. In fact,
for all integer n, ζ(2n) = kπ2n for some rational
numbers k. But it is not clear whether ζ(n) =
k′πn for some rational number k′.
In addition, one can also show that these in-
tegrals read
n =
[mT
2π
]3/2g exp[−m− µ
T] , (19.14)
ρ = nm , (19.15)
p = nT ρ (19.16)
in the non-relativistic limit T m. Note that
the statistics of bosons and fermions will not be
significant in the non-relativistic limit.
20. Funny Integral
Integrals shown in previous section can be inte-
grated mostly by a useful method developed by
Feynman. For example, the integral
ρ =g
2π2
∫ ∞m
√E2 −m2
exp[E−µT
]± 1
E2dE (20.1)
can be approximated as
ρ =g
2π2
∫ ∞m
E3
exp[ET
]± 1
dE (20.2)
in the limit T m and T µ.
Defining x = E/T , one can show that:
ρ =g
2π2T 4
∫ ∞m/T→0
x3
exp [x]± 1dx (20.3)
=g
2π2T 4
∫ ∞0
x3 exp[−x]
1± exp [x]dx . (20.4)
13
Introduction to Cosmology W.F. Kao
Moreover, 1/(1∓ y) can be expanded as a poly-
nomial of y such that
1
1∓ y= 1± y + y2 ± y3 + y4 ± · · ·
=
∞∑n=1
(±1)n+1yn−1 . (20.5)
Therefore, the integral becomes:
ρ =g
2π2T 4
∞∑n=1
∫ ∞0
(∓1)n+1x3 exp[−nx]dx .
(20.6)
Writing z = nx, the integral becomes:
ρ =g
2π2T 4
[ ∞∑n=1
(∓1)n+1
n4
] ∫ ∞0
z3 exp[−z]dz .
(20.7)
The result then follows from the definition of the
Γ function
Γ(n) =
∫ ∞0
exp[−x]xn−1dx . (20.8)
Hence we have
ρ =π2
30gT 4
for RD bosonic gas.
21. One more funny integral
The integral of number density
n =g
2π2
∫ ∞m
√E2 −m2
exp[E−µT
]± 1
EdE (21.1)
in the limit T m is also a good example for
demonstration. This limit is also known as the
non-relativistic limit since rest mass energy is
greater than the kinetic energy proportional to
T . This integral can be approximated as, follow-
ing the largeness of exp[(E − µ)/T ] 1,
n =g
2π2
∫ ∞m
√E2 −m2 × exp
[−E − µ
T
]EdE .
(21.2)
Writing x = E/T and q = m/T , the number
density becomes:
n = gT 3
2π2exp
[µT
] ∫ ∞q
x√x2 − q2 exp[−x]dx .
(21.3)
An integration-by-part will take the integral to
the following form:
n =g
3
T 3
2π2exp
[µT
] ∫ ∞q
(x2 − q2)3/2 exp[−x]dx .
(21.4)
The major contribution of the integrand to the
integral comes from the small x part. This is
due to the decaying power derived from the fac-
tor exp[−x]. Therefore, we can approximate the
integrand as x = y + q and assume that y is
small and can be ignored when y 1. Equiva-
lently, we can say that the major contribution of
the integral comes from the small y q = m/T
contribtutions. As a result, the integral becomes:
n =g
3
T 3
2π2exp
[µT
] ∫ ∞0
(2qy + y2)3/2 exp[−q − y]dy
=g
3
T 3
2π2exp
[µT
] ∫ ∞0
(2qy)3/2 exp[−q − y]dy
=g
3
T 3
2π2
(2m
T
)3/2
exp
[µ−mT
] ∫ ∞0
y3/2 exp[−y]dy
=g
3
T 3
2π2
(2m
T
)3/2
exp
[µ−mT
]Γ(
5
2)
with Γ(5/2) = 3√π/4.
Therefore, we come to the conclusion that:
n =
[mT
2π
]3/2g exp[−m− µ
T] . (19.14)
22. Hydrogen abundance
It is known that the most abundant molecules in
the universe is H and He. In addition, the mass
ratio of Hydrogen is about 3/4. The prediction
of the the ratio XH = nH/nH+He ∼ 3/4 is one
of the great success of the standard cosmology
model and, of course, the incorporated particle
standard model.
Indeed, we will show in a moment that the
number ratio n/p ∼ 1/6 when the temperature
of the neutrino drops to around 0.714 MeV. The
ratio will further be reduced to 1/7 when the uni-
verse cools down even more. Part of the reason is
that neutrino will still interact weakly until the
universe cools even lower. If this is true, the ratio
3/4 follows directly:
XH =p− np+ n
=7− 1
7 + 1=
3
4. (22.1)
14
Introduction to Cosmology W.F. Kao
Note that p − n denotes the number of left-over
protons that can not joint force to form Helium.
The temperature Tν ∼ 1 MeV is the temper-
ature neutrino will decouple from the photon and
electron interactions. Below that energy limit,
e+ and e− pair production or annihilation will
no longer possible. Hence weak decay involv-
ing neutrino will be effectively off the interaction
campus. In other words, when Tγ drops below 1
MeV, neutrino will decouple from the heat bath
of interactions. Neutron will hence stop decaying
to proton any more. This will keep the ratio XH
constant from then on.
The temperature 0.714 MeV comes from the
fact that Tν = (4/11)3Tγ ∼ 0.714 × 1MeV. The
relation of Tν and Tγ will be shown later in this
note. Taking it for granted, we can assert that
n
p= exp
[mp −mn
Tν
]= exp
[−1.293MeV
0.714MeV
]∼ 1
6
(22.2)
following from the number density equation (19.14).
Here n and p denote respectively the number
density functions of neutron and proton.
23. Thermal dynamics of the expand-
ing universe
The universe in its early stages of evolution is
radiation dominated and its energy density is
ρ =π2
30
(Nb +
7
8Nf
)T 4 ≡ c T 4 , (23.1)
where T is the cosmic temperature and Nb (Nf )
is the number of massless bosonic (fermionic) de-
grees of freedom. The combination g∗ = Nb +
(7/8)Nf is called effective number of massless de-
grees of freedom. The entropy density is
s =2π2
45g∗ T
3 . (23.2)
Assuming adiabatic universe evolution, i.e., con-
stant entropy in a ‘comoving’ volume (sa3 =
constant), we obtain the relation aT = constant.
The temperature-time relation during radiation
dominance is then derived from Friedmann equa-
tion (with k = 0):
1
a2∼ T 2 =
MP
2(8πc/3)1/2t· (23.3)
We see that classically the expansion starts at
t = 0 with T = ∞ and a = 0. This initial
singularity is, however, not physical since gen-
eral relativity fails at cosmic times smaller than
the Planck time tP . The only meaningful state-
ment is that the universe under consideration
emerges from a cosmic time ∼ tP with tempera-
ture T ∼MP ∼ 1019 GeV.
24. Entropy Density s = SV
The entropy S of the universe was shown to be
given by:
TdS = pdV + d(ρV )
= d[(p+ ρ)V ]− V dp . (24.1)
We can therefore show that
dS =1
Td[ (ρ+ p)V ]− [ (ρ+ p)V ]
dT
T 2
= d
[(ρ+ p)V
T
](24.2)
if
Tdp = (ρ+ p)dT . (24.3)
This follows from an observation that
∂2S
∂T∂V=
∂2S
∂V ∂T. (24.4)
Indeed, we can show that
V dρ = V∂ρ
∂TdT (24.5)
for ρ = ρ(T ). Hence for S = S(T, V ), we can
derive:
T∂S
∂T= V
∂ρ
∂T, (24.6)
T∂S
∂V= p+ ρ (24.7)
from the equation TdS = pdV + d(ρV ) = (p +
ρ)dV+V dρ.Differentiating above equations again
to obtain
T∂2S
∂V ∂T=∂ρ
∂T, (24.8)
∂S
∂V+ T
∂2S
∂T∂V=∂(ρ+ p)
∂T. (24.9)
Hence the result is:
T
[∂S
∂V+∂ρ
∂T
]= T
∂(ρ+ p)
∂T. (24.10)
15
Introduction to Cosmology W.F. Kao
Consequently, we can show that
T∂S
∂V= T
∂p
∂T. (24.11)
This leads to the result:
Tdp = T∂p
∂TdT = T
∂S
∂VdT
= (ρ+ p)dT .
The last equation follows from Eq. (24.7). There-
fore, we have
Tdp = (ρ+ p)dT . (24.3)
Hence the result (24.2) follows directly.
Our universe is an isolated system, dQ =
TdS = 0. Therefore,
S =(ρ+ p)V
T= constant. (24.12)
Hence we can define the entropy density s =
S/V ∝ (ρ+ p)/T.
For RD, s = 4ρ/3T , therefore s ∝ T 3 as
shown earlier. Note that the above relation is
only valid when all species under consideration
are all in thermal equilibrium with each other.
25. thermal history
During the RD era, there are more than one sin-
gle massless field in thermal equilibrium. The en-
ergy density and entropy density are more com-
plicated:
ρ =π2
30g∗T
4 , (25.1)
g∗ =∑
i=bosons
gi
(TiT
)4
+7
8
∑i=fermions
gi
(TiT
)4
(25.2)
with gi the number of the degrees of freedom of
each particle species. T is the photon tempera-
ture, Ti is the temperature of the i-th particle.
100Mev
1Mev Tν=(4/11)1/3 Tγ (3.86)
Figure 1: g∗(T ) and g∗s(T ).
Figure 2: T and t. www.simagis.org
On the other hand, the entropy density s =
16
Introduction to Cosmology W.F. Kao
Figure 3: GUT timeline. ircamera.as.arizona.edu
S/V = (ρ+ p)/T can be shown to be:
s =2π2
45g∗sT
3 , (25.3)
g∗s =∑
i=bosons
gi
(TiT
)3
+7
8
∑i=fermions
gi
(TiT
)3
(25.4)
g∗ and g∗s is apparently a function of T and
Ti. The counting of the degree of freedom is re-
lated to the massless particles in the high energy
limit. For T 300 GeV, the number of particle
species is g∗ = 106.75.
The counting of degree of freedom is straight-
forward. For all massless gauge bosons (photon,
weak vector mesons, gluons): g = 2; for all lep-
tons: g = 4 [2 (spin up and down) × 2 (parti-
cle/antiparticle) = 4]; for neutrinos: g = 2; for all
quarks: g = 12 [2 (spin up and down) × 2 (parti-
cle/antiparticle) × 3 (colour) = 12]; for complex
Higgs doublet: g = 4.
During the era that standard model is in
charge, we have gluons, W±, Z0, photon, quarks
and leptons before some of them acquire a mass
from SSB effect. In that energy limit, all species
are in thermal equilibrium with each other shar-
ing a same equilibrium temperature T = Tγ .
Hence there are 8 gluons, 1 photon, 3 weak mesons
contribute totally gb = 12×2 = 24. There is also
a complex Higgs doublet contributes gH = 4.
There are additional 3 family of quarks, hence
gq = 6× 12 = 72 taking into account of all anti-
particles and all spins and colors. The leptons
thus contribute gl = 3 × 4 = 12 and neutrinos
contributes gν = 3 × 2 = 6. Hence we can show
that
g∗ = (72 + 12 + 6)7
8+ 24 + 4 = 106.75
(25.5)
when T 300 GeV.
For T ∈ [1, 100] GeV, we have electron,
positron, 3 neutrinos, photon all in thermal equi-
librium that contribute
g∗ = 10× 7
8+ 2 = 10.75 . (25.6)
In this era, Tν = Tγ .
For T 1 MeV, we have 3 neutrino and
photon that contribute
g∗ = 6×(
4
11
)4/3
× 7
8+ 2 ∼ 3.36 (25.7)
counting the effect of
Tν =
(4
11
)1/3
Tγ (25.8)
that will be shown shortly. In this stage, the
neutrinos decoupled from thermal contact with
the photon. Note that 4/11 ∼ 0.36, (4/11)1/3 ∼0.71 and (4/11)4/3 ∼ 0.26.
In addition, the function
g∗s = 6×(
4
11
)3/3
× 7
8+ 2 ∼ 3.91 (25.9)
can also be derived straightforwardly during the
stage T 1 MeV. In fact, the numerical plot of
the T dependence of g∗ and g∗s is shown in Fig.
(1).
26. TR relation after decoupling
We can show that when a particle is in thermal
equilibrium with all other species, the conserva-
tion of entropy S ensures that
g∗sT3R3 = constant . (26.1)
When a massless specie decoupled from the ther-
mal contact of the system, the following identity
holds instead:
TR = constant . (26.2)
17
Introduction to Cosmology W.F. Kao
Alternatively, the following identity
TR2 = constant (26.3)
holds for massive particles after it is decoupled
from the thermal contact with the system.
The difference of T , R relation for massive
and massless particles derived from the fact that
the probability function depends on
exp
[−ET
]. (26.4)
Therefore, the probability remains unchanged if
E ∼ T . On the other hand, we have shown ear-
lier that the momentum p of a particle scales as
1/R. Hence E = pc also scales as 1/T for mass-
less particles. Therefore, we reach the conclusion
TR = constant for massless particle.
For massive particles, E = p2/2m ∼ 1/R2.
Therefore T ∼ 1/R2 that leads to the conclusion
TR2 = constant.
Since R ∼ 1/T , we also have the relation:
H =R
R= − T
T(26.5)
for RD photons.
27. The temperature dropped by
Tν =(
411
)1/3Tγ
due to neutrino decoupling
27.1 effect on g∗s
The entropy will remain constant throughout the
expanding process. Number of the massless par-
ticle species will however change during the pro-
cess of SSB. Therefore, the entropy density will
be shared by massless or relativistic particles be-
fore they decoupled from the thermal contact
with other species.
S = sV ∝ g∗sT3a3 will remain constant.
When the temperature of the universe drops to
around 1 MeV,
Γint = nσv ∼ G2FT
5 (27.1)
will be compatible to the expansing rate
H ∼ T 2
mpl(27.2)
for RD universe. Note that H ∝ T 2 comes from
the result R2 ∼ t, T ∼ 1/R and H ∼ 1/t for RD
universe.
Indeed, the νe → νe sorts of weak interac-
tions come with cross section
σ ∼ G2FT
2 (27.3)
with GF the Fermi constant for the 4-point ef-
fective coupling derived from weak interaction.
In addition, the result also follows from the fact
that n ∼ T 3.
In summary, we have the result:
ΓintH
= G2FT
3mpl ∼(
T
1 MeV
)3
. (27.4)
Therefore, neutrino interactions with electron will
decouple from the heat bath when T ≤ TD = 1
MeV.
After the neutrinos in the early universe went
out of equilibrium, only electrons, positrons and
photons are in thermal equilibrium. The anni-
hilation of the electrons and positrons occurs at
about 1 Mev (5×109 K). To evaluate the effect on
the temperature, one must count the number of
species which goes into the entropy expression.
The effective number is ge = (2 × 2) × 7/8 +
2 = 11/2 when e± remains massless in thermal
contact with photon. The temperature is also
around TD for annihilation process of electron
and positron pairs. Note that electron mass is
around 0.511 MeV. The effective number is how-
ever gγ = 2 when photon is the only massless
particle remain in thermal contact. The entropy
of electrons will then transferred to the entropy
of photons after TD.
Note that at T = TD, we have T = Tγ =
Tν . After neutrino decouples from the heat bath,
TeR = TDRD with RD the scale factor of the
universe at T = TD. In the meantime, the tem-
perature of Tγ will drop according to
gγ(TγR)3 = ge(TDRD)3 = ge(TνR)3 . (27.5)
Therefore, one has
geT3ν = gγT
3γ . (27.6)
Hence
11
2T 3ν = 2T 3
γ , (27.7)
18
Introduction to Cosmology W.F. Kao
lead to the relation
TνTγ
=
(4
11
)1/3
. (27.8)
Therefore, the temperature of photon drops more
slowly than the neutrino temperature, Tγ > Tν ,
after neutrino decoupling.
28. time and temperature
Our universe start out at Planck time t = tp in a
RD phase with all sorts of massless species. At
around t = tM , our Universe kicks off the MD
phase.
Consider the flat universe as an illustration.
In the RD era, t ∼ R2 andR ∼ 1/T . The relation
turns into t ∼ R3/2 and R ∼ 1/T in MD era.
Hence we can write
t = t0
(T0T
)3/2
, (28.1)
R = R0
(t
t0
)2/3
(28.2)
in MD era. On the other hand, we have
t = tp
(TpT
)2
, (28.3)
R = Rp
(t
tp
)1/2
(28.4)
during RD era. The matching condition at t =
TM gives
tM = t0
(T0TM
)3/2
= tp
(TpTd
)2
, (28.5)
RM = R0
(tMt0
)2/3
= Rp
(tMtp
)1/2
.(28.6)
Hence we have
R0 = Rp
(t2/30
t1/2p t
1/6M
), (28.7)
t0 = tp
(T 2p
T3/20 T
1/2M
). (28.8)
Given the Planck energy, photon-decoupling tem-
perature Td and current temperature T0 as:
• Tp ≡√hc5/8πG ∼ 2.43× 1019 GeV,
• Td ∼ 3020 K ∼ 0.26 eV,
• T0 ∼ 2.732 K,
we can extract the time information:
tdtp
=
(TpTd
)2
∼ 8.74× 1057 , (28.9)
t0td
=
(TdT0
)3/2
∼ (1105)3/2 ∼ 36800 .
(28.10)
Here we have vaguely taken tM = td for simplic-
ity. Note that
• 1 eV ∼ 11600 K,
• 1 year = 3.1536× 107 s,
• 1 ly = 9.46× 1012 Km,
• tPl ≡√hG/c5 ∼ 5.39× 10−44 s.
Note also that the effective Planck time tPl 6= tp.
We may start from
td ∼ 3.73× 105 years,
then t0 and tp will be around
t0 ∼ 1.37× 1010 years , (28.11)
tp ∼ 1.35× 10−45 s . (28.12)
Assuming the scale factor now is around the ob-
servable distance R0 = ct0 ∼ 1.3 × 1023 Km, we
can also infer that the scale factor at tM and tpare:
Rd = R0
(T0Td
)∼ 1.2× 1020 Km ,
Rp = R0
(T0Tp
)∼ 1.2× 10−9 Km .
This result is not quite correct, as
• g∗sR3T 3 remains constant during the RD
era with g∗s changes along with the tem-
perature.
• MD starts earlier than T = 0.26 eV. There
is in fact a transition period for the universe
to go from RD to MD phase.
• As time goes, our universe is currently in
its VD phase that accelerate the expanding
speed of the universe.
19
Introduction to Cosmology W.F. Kao
But above result does provide a rough estimate
of the thermal history of our physical universe.
In textbook, the RD era starts around z =
3400. The Last scattering surface is around z =
1100 or around 3.8 × 105 years. The universe
kicks off the Vaccuum donminated era around
t = 5×1010 years. Sometimes around t = 10−33s
to t = 10−32s for about t = 10−36s, there is a
period of inflation that takes the scale factor to
expand around 1025 times its orginal size.
29. Inflation
Note that at the Planck time tp, the physical
horizon is roughly
dp = ctp ∼ 4× 10−40Km Rp . (29.1)
Therefore, the observable part of our universe
was too large to get all particles in thermal con-
tact. This is the first difficulty in resolving the
physical origin of the isotropic and homogeneous
CMB radiation that represents a perfect exam-
ple of black body radiation. This indicates that
there must be something new unknown to us.
This problem was later resolved by the in-
troduction of an inflationary mechanism. The
universe is probably supported by an scalar field
φ born with incredible energy to induce a strong
expansion phase. The resolution by Guth is that
the universe starts an inflationary phase around
t = t∗ = 4 × 10−45 s to ti = 10−25 s. The uni-
verse then expands 1025-fold during a short pe-
riod ∆t ∼ 10−25 s.
As a result, the scale factor and temperature
of the universe
R∗ = ct∗ ∼ 1.2× 10−34Km , (29.2)
Ri = R∗ × 1025 ∼ 1.2× 10−9 Km ,(29.3)
Ti = 1 KeV . (29.4)
after the inflation ends. This is too cold for the
universe. There is then another period of reheat-
ing induced by the scalar field braking by the
SSB potential when the inflaton scalar field os-
cillate with damping around the bottom of the
potential well V (φ) ∼ (φ2 − v2)2.
30. Levi-Civita Tensor
Levi-Civita tensor is an almost-constant type T(n,
0) tensor defined on any n-D Riemannian spaces
Mn. It is totally skewsymmetric w.r.t all its in-
dices. In what follows, we will introduce it in
Rn under Cartesian coordinates. Definition on a
general Riemannian space Mn will be introduced
later on.
We will start by defining it on R3 and follow
up with a general definition on n-D.
Definition 30.1 εijk is totally skew-symmetric
with ε123 = 1. Here i, j, k = 1, 2, 3.
Definition 30.2 εa1a2...an is totally skew-symmetric
with ε123...n = 1.
There are a number of problems one can work
out rather straightforwardly. These show the
most intrinsic pictures and applications hidden
under the useful Levi-Civita tensor.
Problem 30.1 εij ≡ [ ε ]ij ⇒ ε =
(0 1
−1 0
).
Problem 30.2 εijkεabc = (ijk)+(jki)+(kij)−(i↔ j). Here (ijk) ≡ δaiδbjδck.
Problem 30.3 εijkεabk = δaiδbj − δbiδaj .
Problem 30.4 εijkεajk = 2δai.
Problem 30.5 εijkεijk = 6.
30.1 Matrix Determinant
There are also other applications of Levi-Civita
tensor that are very useful in computing the de-
terminant of a square matrix and in the deriva-
tion of vector cross product. We will present
their definitions along with some useful formu-
lae.
Definition 30.3
detAn×n ≡ εa1a2...anA1a1A2a2 · · ·Anan .
Definition 30.4
Ai ∈ R3 ⇒ (A×B)i ≡ εijkAjBk.
Problem 30.6
εb1b2...bn detA = εa1...anAb1a1Ab2a2 ...Abnan .
20
Introduction to Cosmology W.F. Kao
Problem 30.7
detA = 1n!ε
a1...anεb1...bnAa1b1Aa2b2 ...Aanbn .
Problem 30.8
detA = 1n!ε
a1...anεb1...bnAa1b1Aa2b2 ...Aanbn .
Problem 30.9
detAt = detA.
Definition 30.5 the cofactor Aab of the matrix
component Aab is defined as:
Aab = 1(n−1)!ε
aa2...anεbb2...bn [ Aab ]Aa2b2 ...Aanbnwith [ Aab ] removed from this equation.
Problem 30.10
Aab = Aba
detA with Aab = (A−1)ab such that
AabAbc = δca.
The proof the the last Problem B.10 is quite
straightforward. We will show it for the case of
the 3-D matrix. The proof for the n×n matrices
are straightforward following the proof for the
3-D case. For a 3× 3 matrix,
AabAbc =
∑bijkl
1
2! detAεcijεbkl[ Aab ]AikAjl .
For a 6= c, a must be equal to i or j. Let
a = j, one then has:
AabAbc =
∑bijkl
1
2! detAεcijεbkl[ Aib ]AikAjl = 0
following from the fact that AibAik is symmetric
in bk. On the other hand, if a = c,
AcbAbc =
∑bijkl
1
2! detAεcijεbkl[ Acb ]AikAjl = 1
following directly from the definition of the co-
factor.
Let g ≡ det gab, one can also show that:
Problem 30.11
g−1δg = gabδgab .
We will also show it in the case of 3-D. Indeed,
δg =∑bijkl
1
2!εcijεbkl[ δgcb ]gikgjl
= gcb δgcb = g gcbδgcb . (30.1)
30.2 Structure Constant
Moreover, εijk turns out to be the structure con-
stant of the SU2 and SO3 groups. To be familiar
with algebraic structure of flat 3-D Levi-Civita
tensor is hence very important in studying quan-
tum spin operator as well as spatial angular mo-
mentum operator one has to encounter in Q.M.
Also, from the definition of Pauli matrices, one
is able to link R3 with the algebra su2. Indeed,
we will present the definition of Pauli matrices
along with various applications as problem sets.
Definition 30.6 Pauli matrices:
σi ≡[ (
0 1
1 0
),
(0 −ii 0
),
(1 0
0 −1
) ]
s.t. xiσi =
(z x−x+ z
)with x± ≡ x± iy.
There are some useful formulae one should
know about, e.g.:
σiσj = δij + iεijkσk, (30.2)
ei naσaθ(x) = cos θ + inaσa sin θ, (30.3)[σi, σj
]= 2iεijkσk, (30.4)[
T i, T j]
= iεijkT k (30.5)
with T i ≡ σi
2 .
30.3 Curved Space
Now we are ready to introduce it in any n-D Rie-
mannian spaces. The flat space constant Levi-
Civita tensor will then denoted as ea1a2...an s.t.
εa1a2...an =1√gea1a2...an . (30.6)
One should remark here that ea1a2...an is a
constant function which shall not transform with
coordinate transformation. It is straightforward
to show that (30.6) guarantee that the Levi-Civita
totally skew-symmetric funtion is indeed a type
T(n, 0) tensor. One can further show that there
exists another type T(0, n) tensor εa1a2...an ob-
tained by lowering all upper indices of εa1a2...an .
One can further show that
εa1a2...an =√gea1a2...an . (30.7)
Note that there is a sign omitted in defintion of
(30.7). In our pseudo-Riemannian spaces, g ≡
21
Introduction to Cosmology W.F. Kao
−det gab, one should put a − sign in front of the
L.H.S. of equation (30.7). Indeed, one can easily
show that the form given in equation (30.7) is
correct as a tensor. Note that the formulae listed
in problem sets (8.2) to (8.5) can be generalized
to n-D with appropriate sign added accordoing
to the relevant signature of the manifold under
discussion. For example, one should have
εa1a2...anεa1a2...an = ±n!. (30.8)
30.4 Jacobi Identity
Jacobi identity is a result of Lie derivative obey-
ing the Leibniz rule. For example, we have:
[A, [B ,C] ] = [ [A, B] , C] + [B, [A ,C] ] ,
[A, B ,C ] = [A, B] , C+ B, [A ,C] ,[A, B ,C ] = [ A, B , C]− [B, A ,C ] .
The rule depends on the spin statistic nature of
the operators A, B, and C. The first identity
applies to the case that there is no or one fermion
operator. The second case is the case for two
fermions. The last case is for three fermions.
22