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Introduction to Cosmology

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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.

2


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

3


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)

4


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)

5


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)

6


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.

7


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)

8


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)

9


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


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


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


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


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


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


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


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


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


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


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


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


−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

Most of the figures are cited from the astrophysics 2e online resources on the Pearson’s website for the book “an introduction to modern astrophysics by Carroll and Ostlie.

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