chapter 20 the big bang - university of...

Chapter 20

The Big Bang

The Universe began life in a very hot, very dense statethat we call thebig bang. In this chapter we apply theFriedmann equations to the early Universe in an attemptto understand the most important features of thebig bangmodel,which is the cosmologist’s “standard model” forthe origin of the present Universe.

595

596 CHAPTER 20. THE BIG BANG

20.1 Radiation and Matter Dominated Universes

Because

• the influence of vacuum energy grows with expan-sion of the Universe, and

• vacuum energy is only today beginning to dominate,

we may safely assume that it was negligible in the earlyUniverse (once the inflationary epoch was over). In thatcase, two extremes for the equation of state give us con-siderable insight into the early history of the Universe:

1. If the energy density resides primarily in light parti-cles having relativistic velocities, we say the the Uni-verse isradiation dominated;in that case the equa-tion of state is

P = 13ε (radiation dominated).

2. If on the other hand the energy density is domi-nated by massive, slow-moving particles, we say thethe Universe ismatter dominated;the correspondingequation of state is

P≃ 0 (matter dominated).

In either extreme, the evolution of the Universe is theneasily calculated using the Friedmann equations.

20.1. RADIATION AND MATTER DOMINATED UNIVERSES 597

20.1.1 Evolution of the Scale Factor

The density of radiation and the density of matter scale differently inan expanding universe. If the Universe is radiation dominated,P= 1

3εand

ε +3(ε +P)aa = 0 → ε

ε+4

aa

= 0,

which has a solution

ε(t) ≃ 1a4(t)

(radiation dominated).

If on the other hand the Universe is matter dominated, we haveP≃ 0and

ε +3(ε +P)aa = 0 → ε

ε+3

aa

= 0,

which has a solution

ε(t) ≃ 1a3(t)

(matter dominated).

As we showed in the previous chapter, the corresponding behaviorsof the scale factor are

a(t) ≃

t1/2 (radiation dominated)

t2/3 (matter dominated)


20.1.2 Matter and Radiation Density

In the present Universe the ratio of the number density of baryons tophotons is

nb

nγ≃ 10−9.

However, the rest mass of a typical baryon is approximately109 eV(recall that the rest mass of a proton is 931 MeV), while most photonsare in the∼ 2.7 K cosmic microwave background, with an averageenergy

Eγ ≃ (2.7 K)

(

1 GeV1.2×1013 K

)

≃ 2.3×10−4 eV.

Thus the ratio of the energy density of baryons to energy density ofphotons in the present Universe is

εb

εγ≃ 103−104

and the present Universe is dominated by matter (and by vacuum en-ergy), with only a small contribution from radiation. But,

ε(t) ≃ 1a4(t)

(radiation dominated).

ε(t) ≃ 1a3(t)

(matter dominated).

Thus, as time is extrapolated backwards relativistic matterbecomes increasingly more important, until at some ear-lier time the influence of matter and vacuum energy maybe neglected compared with that of relativistic particles.


0-2-4-6-8-10

-30

-40

-20

-10

0

10

Radiation

Dominated

Matter

Dominated

t ~ 1010 s

T ~ 10 eV

ρmatter

ρ radiation

log ρ

(g c

m-3

)

2

log a

Figure 20.1:Dependence of the energy density of matter and radiation on thescale factor. At this early epoch the influence of vacuum energy and curvatureare negligible and the evolution of the Universe is governedby the competitionbetween radiation and matter.

Fig. 20.1 illustrates. Thus, the early Universe should havebeenradiation dominated.


0-2-4-6-8-10

-30

-40

-20

-10

0

10

Radiation

Dominated

Matter

Dominated

t ~ 1010 s

T ~ 10 eV

ρmatter

ρ radiation

log ρ

(g c

m-3

)

2

log a

Since in this early, radiation-dominated Universe,

ε ≃ a−4 a≃ t1/2 P = 13ε,

the behavior of the density and pressure as time is extrap-olated backwards is

Limt→0

ε(t) = Limt→0

(t1/2)−4 = Limt→0

t−2 = ∞

Limt→0

P(t) = Limt→0

(t1/2)−4 = Limt→0

t−2 = ∞

Furthermore, for a radiation dominated Universe,T ≃a−1 ≃ t−1/2; thus, as time is extrapolated backwards thetemperature scales as

Limt→0

T(t) = Limt→0

t−1/2 = ∞.


These considerations suggest that the Universe startedfrom a very hot, very dense initial state witha(t → 0)→ 0.

• The commencement from this initial state is calledthebig bang.

• If we take t = 0 when a = 0, the transition be-tween the earlier radiation dominated universe andone dominated by matter took place around 50,000years after the big bang (redshift of∼ 3300) whenthe temperature was about 9000 K.

• This matter dominance then continued until about 4-5 billion years ago, when the vacuum energy densitybegan to overtake the matter density.

In the following sections we shall discuss in more detailthe big bang and the early radiation dominated era of theUniverse.

The name “big bang” was actually a termcoined by opponents of this cosmology whofavored the now discredited steady state the-ory. The name stuck, as did the theory.


20.2 Evolution of the Early Universe

Considerations of the preceding section suggest that thebig bang starts off with a state of extremely high densityand pressure for the Universe, and that under those condi-tions the Universe is dominated by radiation.

• This means that the major portion of the energy den-sity is in the form of photons and other massless ornearly massless particles like neutrinos that move atnear the speed of light.

• As the big bang evolves in time, the temperaturedrops rapidly with the expansion and the average ve-locity of particles decreases.

• Finally, about 1000 years after the big bang onereaches a state where the primary energy density ofthe Universe is in non-relativistic matter.

Let us now give a brief description of the most importantevents in the big bang and the evolution from a radiationdominated to matter dominated universe.

20.2. EVOLUTION OF THE EARLY UNIVERSE 603

20.2.1 Thermodynamics of the Big Bang

We have already established that in the initial radiation dominated eraof the big bang,

H2 ≡(

aa

)2

≃ 8πG3

εr = a−4 a≃ t1/2 H =aa

=12t

.

We may assume that the average evolution corresponds approximatelyto that of an ideal gas in thermal equilibrium, for which the numberdensity of a particular species is

dn=g

2π2h3

p2dp

eE/kT +Θ,

wherep is the 3-momentum,g is the number of degrees of freedom(helicity states: 2 for each photon, massive quark, and lepton) and

E =√

p2c2+m2c4 Θ =

+1 Fermions

-1 Bosons

0 Maxwell–Boltzmann

where Maxwell–Boltzmann statistics obtain only if we make no dis-tinction between fermions or bosons in the gas.


Because of the high temperature, let us assume that the gas isultra-relativistic (kT >> mc2 for the particles in the plasma). Then theenergy isE = pc, and the number density is obtained by integratingthe previous expression fordn.

n =∫ ∞

0

dndp

dp=g

2π2h3

∫ ∞

0

p2dp

eE/kT +Θ

=g

2π2h3

∫ ∞

0

p2dp

epc/kT +Θ(20.1)

Integrals of this form may be evaluated using

∫ ∞

0

tz−1

et −1dt = (z−1)!ζ (z)

∫ ∞

0

tz−1

et +1dt = (1−21−z)(z−1)!ζ (z),

whereζ (z) is the Riemann zeta function, with tabulated values

ζ (2) =π2

6= 1.645 ζ (3) = 1.202 ζ (4) =

π4

90= 1.082.

The results for the number density of speciesi are

ni = giζ (3)

π2 T3×

1 Bose–Einstein

3/4 Fermi–Dirac

ζ (3)−1 Maxwell–Boltzmann

where now we introduceh = c = k = 1 units.


Likewise, the energy density is given by

ε = ρc2 =∫ ∞

0E

dndp

dp=g

2π2h3

∫ ∞

0

E p2dp

eE/kT +Θ

=g

2π2h3

∫ ∞

0

E p2dp

epc/kT +Θ(20.2)

which gives

εi = giπ2

30T4×

1 Bose–Einstein

7/8 Fermi–Dirac

90/π4 Maxwell–Boltzmann

The energy density for all relativistic particles is then given by thesum,

ε = g∗π2

30T4 g∗ ≡ ∑

bosons

gb +78 ∑

fermions

gf.

If all species are in equilibrium, the entropy densitys is

s=ε +P

T=

4ε3T

=2π2

45g∗T3,

where we note from comparing this and

ni = giζ (3)

π2 T3×

1 Bose–Einstein

3/4 Fermi–Dirac

ζ (3)−1 Maxwell–Boltzmann

thats≃∑ni. The entropy per comoving volume is constant (adiabaticexpansion),

S≃ sa3 ≃ constant −→ d(sa3)

dt= 0,

provided thatg∗ does not change.


103 102 101 100 10-1 10-2 10-3 10-4 10-5

1

10

100

T (GeV)

Effective D

egre

es o

f F

reedom

g*

Quark

Confinement

~10-6 s

~1 s

Weak

Freezout

Figure 20.2:Variation of the effective number of degrees of freedom in the earlyUniverse as a function of temperature.

In fact, as illustrated in Fig. 20.2, we expectg∗ to be ap-proximately constant for broad ranges of temperature butto change suddenly at critical temperatures wherekT be-comes comparable to the rest mass for a species.

Even though in local processes the entropytends to increase, globally the evolution isdominated by the enormous entropy residentin the cosmic microwave background radia-tion. Thus, cosmologically the expansion isapproximately reversible and adiabatic.


From

s=ε +P

T=

4ε3T

=2π2

45g∗T3,

S≃ sa3 ≃ constant −→ d(sa3)

dt= 0,

sa3 ≃ T3a3 is constant and

T ≃ 1a≃ t−1/2,

where we have used the result that in a radiation dominated universeof negligible curvature,a≃ t1/2.

To summarize, the evolution of the ultrarelativistic, hotplasma characterizing the early big bang is described bythe equations

aa

= −TT

= αT2 t =1

2αT2 =2.4×10−6

g1/2∗ T2

GeV2 s

α =

(

4π3g∗45M2

P

)1/2

MP ≡(

hcG

)1/2

= 1.2×1019 GeV,

whereMP is the Planck mass. These equations are ex-pected to be valid from the end of the quantum gravitationera atT ≃ 1019 GeV up to the decoupling of matter andradiation atT ≃ 10 eV.


20.2.2 Equilibrium in an Expanding Universe

Strictly, we do not expect equilibrium to hold in an expanding uni-verse.

• However, a practical equilibrium can exist as the Universepassesthrough a series of nearly equilibrated states.

• We may expect both thermal equilibrium and chemical equilib-rium to play a role in the expansion of the Universe.

A system is inthermal equilibriumif its phase space num-ber density is given by

dn=g

2π2h3

p2dp

eE/kT +Θ,

E =√

p2c2+m2c4 Θ =

+1 Fermions

-1 Bosons

0 Maxwell–Boltzmann

A system is inchemical equilibriumif for the reactiona+b↔ c+d the chemical potentials satisfyµa + µb = µc +

µd.

We shall illustrate the discussion by considering thermal equilibrium,and will consider the equilibrium to maintained by two-bodyreac-tions (which is the most common situation).


The reaction rate for a two-body reaction may be expressed as

Γ ≃ 〈nvσ〉,

• wheren is the number density,

• v is the relative speed,

• σ is the reaction cross section, and

• the brackets indicate a thermal average.

We may expect that a species will remain in thermal equilibrium inthe radiation dominated Universe as long as

Γ >>aa≡ H ≃ d(t1/2)/dt

t1/2=

12t

.

• In the earliest stages of the big bang, densities, velocities, andcross sections are large and it is easy to fulfil this for most species.

• However, asT and the density drop the number density andvelocity factors will decrease steadily and at certain reactionthresholds the cross sectionσ will become small for a partic-ular species and it can drop out of thermal equilibrium.

Physical reason:if the reaction rates are slow comparedwith the rate of expansion, it is unlikely that the particlescan find each other to react and maintain equilibrium.


e+

e−

Z0

e+

e−

Z0 Z0µ+

µ−

ν

ν

_

_

L

L_

q

q

Figure 20.3:Some weak interactions important for maintaining equilibrium in theearly Universe. Generic leptons are represented byL and generic quarks byq.

20.2.3 Example: Decoupling of the Weak Interactions

As an example of decoupling from thermal equilibrium, let uscon-sider weak interactions in the early Universe.

• At the energies of primary interest to us the weak interactions goquadratically in the temperature.

• Thus, shortly after the big bang the weak interactions are not par-ticularly weak and particles such as neutrinos are kept in thermalequilibrium by reactions likeνν ↔ e+e−.

• Some typical Feynman diagrams are illustrated in Fig. 20.3. Theweak interaction cross sections depend on the square of the weak(Fermi) coupling constant,σw ∝ G2

F, with

GF ≃ 1.17×10−5 GeV−2.

• This may be used to show (Exercise) that the ratio of the weakreaction rate to the expansion rate is

ΓH

≃ G2FT5

T2/MP

≃(

T1 MeV

)3

.

Therefore, weak interactions should have decoupled from thermalequilibrium at a temperature of approximately 1 MeV, which occurredabout 1 second after the expansion began.


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

10-13

1019

1014

102

100

10-3

10-5

10-9

Tem

pera

ture

(G

eV)

Time Since Big Bang

Qua

ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em

Hadrons Leptons Nuclear

Synthesis Photon Epoch Galaxies

Stars Life

Quark-Lepton Soup

Planck time

Guts symmetry breaking

Electroweak symmetry breaking

Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Figure 20.4:A history of the Universe. The time axis is highly nonlinear and1 GeV≃ 1.2×1013 K (after D. Schramm).

20.2.4 Sequence of Events in the Big Bang

The Friedmann equations and considerations of the fundamental prop-erties of matter allow us to reconstruct the big bang. Let us now fol-low the approximate sequence of events that took place in terms ofthe time since the expansion begins (see Fig. 20.4 for an overview).

The primary cast of characters includes:

1. Photons

2. Protons and neutrons

3. Electrons and positrons

4. Neutrinos and antineutrinos


Because of the equivalence of mass and energy, in a radi-ation dominated era

• the particles and their antiparticles are continuouslyundergoing reactions in which they annihilate eachother, and

• photons can collide and create particle and antiparti-cle pairs.

Thus, under these conditions the radiation and the matterare in thermal equilibriumbecause they can freely inter-convert.


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

10-13

1019

1014

102

100

10-3

10-5

10-9

Tem

pera

ture

(G

eV)

Time Since Big Bang

Qua

ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 1/100 Second

• T ≃ 1011 K andρ > 109 g cm−3.

• The Universe is expanding rapidly and consists of a hot undif-ferentiated soup of matter and radiation in thermal equilibriumwith an average particle energy ofkT ≃ 8.6 MeV.

• Equilibria:

e−+e+ ↔ photons ν + ν ↔ photons

ν + p+ → e+ +n ν +n→ e−+ p+.

• The number of protons is about equal to the number of neutrons.


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

10-13

1019

1014

102

100

10-3

10-5

10-9

Tem

pera

ture

(G

eV)

Time Since Big Bang

Qua

ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 1/10 Second

• T ≃ 1010 K andρ ≃ 107 g cm−3.

• Free neutron (mnc2 = 939 MeV) less stable than free proton (mpc2 =938 MeV), so,n→ p+ +e−+ ν, with t1/2 ∼ 17 m.

• Thus, the initial∼ equal balance between neutrons and protonsbegins to be tipped in favor of protons.

• By now 62% of the nucleons are protons and 38% are neutrons.

• The free neutron is unstable, but neutrons in composite nucleican be stable, so the decay of neutrons will continue until thesimplest nucleus (deuterium) can form.

• No composite nuclei can form yet because the temperature im-plies an average energy for particles in the gas of about 2.6 MeV,and deuterium has a binding energy of only 2.2 MeV(deuteriumbottleneck).


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

10-13

1019

1014

102

100

10-3

10-5

10-9

Tem

pera

ture

(G

eV)

Time Since Big Bang

Qua

ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 1 Second

• T ≃ 1010 K andρ ≃ 4×105 g cm−3.

• kT ≃ 0.8 MeV and the neutrinos cease to play a role in the con-tinuing evolution (weak freezeout).

• The deuterium bottleneck still exists, so there are no compositenuclei and the neutrons continue to beta decay to protons.

• At this stage the proton abundance is up to 76% and the neutronabundance has fallen to 24%.


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

10-13

1019

1014

102

100

10-3

10-5

10-9

Tem

pera

ture

(G

eV)

Time Since Big Bang

Qua

ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 14 Seconds

• The temperature has now fallen to about3×109 K, correspond-ing to an average energy for the gas particles of about0.25 MeV.

• This is too low for photons to produce electron–positron pairs, sothey fall out of thermal equilibrium and the free electrons beginto annihilate all the positrons to form photons.

e−+e+ → photons.

• This reheats all particles in thermal equilibrium with thepho-tons, but not the neutrinos which have already dropped out ofthermal equilibrium att ∼ 1 s.

• The deuterium bottleneck still keeps appreciable deuterium fromforming and the neutrons continue to decay to protons.

• At this stage the abundance of neutrons has fallen to about 13%and the abundance of protons has risen to about 87%.


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

10-13

1019

1014

102

100

10-3

10-5

10-9

Tem

pera

ture

(G

eV)

Time Since Big Bang

Qua

ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 3 Min 45 Seconds

• Finally the temperature drops sufficiently low (about109 K) thatdeuterium nuclei can hold together.

• The deuterium bottleneck is thus broken and a rapid sequence ofnuclear reactions ensues

n+ p+ → 21H

21H+ p+ → 3

2He+n→ 42He

21H+n→ 3

1H+ p+ → 42He

• Thus, all remaining free neutrons are rapidly cooked into helium.

• Elements beyond4He cannot be formed in abundance becauseof the peculiarity that there are no stable mass-5 or mass-8 iso-topes, and because the density has dropped too low to permitmore complicated reactions like triple-α to produce carbon.


10-43 s 1010 y105 y3 min1 s10-6 s10-11 s10-35 s

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10-9

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pera

ture

(G

eV)

Time Since Big Bang

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ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 35 Minutes

• The temperature is now about3×108 K.

• the Universe consists primarily of protons, the excess electronsthat did not annihilate with the positrons,4He (26% abundanceby mass), photons, neutrinos, and antineutrinos.

• There are no atoms yet because the temperature is still too highfor the protons and electrons to bind together.


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pera

ture

(G

eV)

Time Since Big Bang

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ntum

Gra

vity

?

?

GUTs

Infla

tion

SU(3)cx

U(1)y

SU(2)wx

SU(3)cx

U(1)em



Stars Life

Quark-Lepton Soup

Planck time



Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time ∼∼∼ 400,000 Years

• The temperature has fallen to several thousand K, which is suffi-ciently low that electrons and protons can hold together to beginforming hydrogen atoms.

• Until this point, matter and radiation have been in thermalequi-librium but now they decouple.

• As the free electrons are bound up in atoms the primary crosssection leading to the scattering of photons (interaction with thefree electrons) is removed.

• The Universe, which has been very opaque until this point, be-comes transparent: light can now travel large distances beforebeing absorbed.


20.3 Element Production and the Early Universe

Deuterium serves as a bottleneck until a critical temperature is reachedand then is quickly converted into helium, which is very stable.

• Therefore, the present abundances of helium and deuterium(andother light elements like lithium that are produced by the bigbang in trace abundances) are a sensitive probe of conditions inthe first few seconds of the Universe.

• The oldest stars contain material that is the least alteredfromthat produced originally in the big bang.

• Analysis of their composition indicates elemental abundancesthat are in very good agreement with the predictions of the hotbig bang.

This is one of the strongest pieces of evidence in support of the bigbang theory.

20.3. ELEMENT PRODUCTION AND THE EARLY UNIVERSE 621

Table 20.1: Neutron to proton ratio in the big bangTime (s) T (K) nn/np np per 1000 nn per 1000

nucleons nucleons

2.3×10−8 1×1014 1.000 500 500

2.3×10−4 1×1012 0.985 504 496

2.3×10−2 1×1011 0.861 537 463

2.3 1×1010 0.223 818 182

6.9 5×109 0.221 819 181

37 2.5×109 0.212 825 175

231 1×109 0.164 859 141

20.3.1 The Neutron to Proton Ratio

Nucleosynthesis in the first few minutes of the big bang depends crit-ically on the ratio of neutrons to protons (Table 20.1).

• The neutron is 0.14% more massive than the proton. This favorsconversion of neutrons to protons by weak interactions.

• At very high temperatures the mass difference doesn’t mattermuch and the ratio of neutrons to protons is about one. However,as the temperature drops neutrons are converted to protons andthe ratio begins to favor protons.

• All neutrons would be converted to protons if the neutrons andprotons remained free long enough (a few hours onceT < 1010 K),but neutrons bound up in a stable nucleus like4He or deuterium,are no longer susceptible to being converted to a proton.

• Therefore, as we have seen the neutron to proton ratio dropsasthe temperature drops until deuterium can hold together andtheneutrons can be bound up in stable nuclei.

• This happens at a temperature of about109 K, by which time(preceding table) the neutron to proton ratio is about 16%.


20.3.2 The Production of 4He

Except for generating very small concentrations of3He,7Li, and deu-terium, the essential result of big bang nucleosynthesis is to convertthe initial neutrons and protons to helium and free hydrogen.

• From the preceding table we may estimate how much of each isproduced.

• For example, if we assume that as soon as the deuterium bottle-neck is broken (at aboutT = 1× 109 K) as many free protonsand neutrons as possible combine to make4He, the table entriesmay be used to deduce that the baryonic matter of the Universeshould be about 28%4He by mass, with most of the rest hydro-gen (Exercise).

• Considering the simplicity of our estimate, that is ratherclose tothe 22–24% measured abundance for4He.

More careful considerations than the ones used here give even betteragreement with the observations.


20.3.3 Constraints on Baryon Density

This agreement between theory and observation for light-element abun-dances also constrains the total amount of mass in the Universe thatcan be in baryons.

• That constraint is the basis for our earlier assertion thatmost ofthe dark matter dominating the mass of the Universe cannot beordinary baryonic matter.

• If enough baryons were present in the Universe to make that true,and our understanding of the big bang is anywhere near correct,the distribution of light element abundances would have to differsubstantially from what is observed.

• The implication is that the matter that we are made of (baryonicmatter) is but a small impurity compared to the dominant matterin the universe (nonbaryonic matter).


Mass A

bundance (

%)

22

23

24

25

η

10-4

10-5

10-9

10-10

10-10 10-9

4He

3He3He + d

d

7Li

Figure 20.5:Mass abundances for some light isotopes relative to normal hydrogenas a function of the baryon to photon ratioη . Shaded regions are excluded byobservations and the curves are predicted primordial abundances.

Figure 20.5 compares calculated with observed abundances for thelight elements produced mostly in the big bang (d is deuterium).

• The shaded regions are excluded by observations.

• Example: observations indicate that the abundance of4He in theUniverse can be no more than 24% and no less than 22%.

• Therefore, only the part of the4He curve lying in the unshadedregion is consistent with the observed amount of4He.

• Such considerations allow us to fix with considerable confidencethe quantity on the horizontal axis, which is the ratio of thenum-ber of baryons to number of photons in the present Universe.


Mass A

bundance (

%)

22

23

24

25

η

10-4

10-5

10-9

10-10

10-10 10-9

4He

3He3He + d

d

7Li

• The total number of each kind of particle is not expected tochange in the absence of interactions, so this ratio is also charac-teristic of that at the time when matter and radiation decoupled.

• The only values permitted for the baryon to photon ratio by theobserved abundances of the light nuclei included in the plotliein a band that brackets the four vertical dotted lines.

• There are∼ 3-4 billion photons for every baryon in the presentUniverse (but their equivalent mass is∼ 10,000 times less thanthe total mass in visible and dark-matter massive particles).

• There are∼ 4×108 photons in each cubic meter of the Universe,but only about one baryon for every five cubic meters of space.

1. Most of these baryons are neutrons and protons.

2. Most photons are in the cosmic microwave background.


20.3.4 Constraints on Number of Neutrino Families

One of the successes of the hot big bang theory is that the observedabundance of light elements, coupled with the theoretical understand-ing of big bang nucleosynthesis, tells us something about neutrinos.

• The known neutrinos come in three families.

• This number of families is favored in the simplest elementaryparticle theories, but in principle there could be additional fami-lies that are not yet discovered.

• However, the successful predictions of big bang nucleosynthesisrequire that there be no more than four such families total.

• High-energy particle physics experiments have now found moredirectly that (with certain technical theoretical assumptions) thenumber of neutrino families with standard electroweak couplingsis three, confirming the limit placed by big bang nucleosynthe-sis.

20.4. THE COSMIC MICROWAVE BACKGROUND 627

20.4 The Cosmic Microwave Background

There are two important observables in the present Universethat arepresumably remnants of the big bang:

• The cosmic microwave background radiation

• Dark matter

The cosmic microwave background (CMB) is the faint glow leftoverfrom the big bang itself. It was discovered accidentally by Penziasand Wilson in 1964 while testing a new microwave antenna.

• They initially believed the signal that they detected coming fromall directions to be electronic noise.

• Once careful experiments had ruled that possibility out, theywere initially unaware of the significance of their discovery.

• Then it was pointed out that the big bang theory actually pre-dicted that the Universe should be permeated by radiation leftover from the big bang itself, but now redshifted by the expan-sion over some 14 billion years to the microwave spectrum.

Dark matter appears to represent the major part of the mass intheUniverse, but we don’t yet know what it is.

Both the microwave background and the nature of darkmatter provide crucial diagnostics for a fundamental issuein cosmology, theformation of structurein the Universe


2.726 K microwave spectrum (theory and

COBE data agree)

cm-1

Inte

nsity

(10

-4 e

rgs/

cm2

sr s

ec c

m-1

)

0 5 10 15 20 25

0.2

0.0

0.4

0.6

0.8

1.0

1.2

Figure 20.6:The 2.726 K microwave background spectrum recorded by COBE.

20.5 The Microwave Background Spectrum

Measurements by Penzias and Wilson that are relatively crude bymodern standards established that

• The radiation was coming from all directions in the sky, with ablackbody spectrum corresponding toT = 2.7 K.

• More modern measurements using the Cosmic Background Ex-plorer (COBE) satellite confirm an almost perfect blackbodyspectrum, with a temperature of2.726 K, as illustrated in Fig. 20.6.

• The data points and the theoretical curve for a2.726 Kspectrumare indistinguishable.

• This is, by far, the best blackbody spectrum that has ever beenmeasured.

20.5. THE MICROWAVE BACKGROUND SPECTRUM 629

2.726 K microwave spectrum (theory and

COBE data agree)

cm-1

Inte

nsity

(10

-4 e

rgs/

cm2

sr s

ec c

m-1

)

0 5 10 15 20 25

0.2

0.0

0.4

0.6

0.8

1.0

1.2

• By applying basic statistical mechanics to the observed spec-trum, we may deduce a photon density of

Nγ ≃ 410 photons cm−3

in the cosmic microwave background.

• Theory predicts that there is also a cosmic neutrino backgroundleft over from the big bang, but these low-energy neutrinos arenot detectable with current technology.


Frequency

Expansion decreases

number density of

photons

Expansion redshifts

the photons

Near Decoupling

Today

Inte

nsity

Figure 20.7:Schematic evolution of the cosmic microwave background. AstheUniverse expands, the spectrum remains blackbody but the photon frequencies areredshifted and the number density of photons is lowered. The2.7 K cosmic back-ground radiation is the faint, redshifted remnant of the cosmic fireball in whichthe Universe was created. Decoupling occurred at a redshiftaround 1000 (seeFig. 20.8). The photon temperature then of about 3000 K is lowered by the redshiftfactor of 1000 to the presently observed value of a little less than 3 K.

The CMB is the remnants of the big bang itself, redshiftedinto the microwave spectrum by the expansion of the Uni-verse, as illustrated in Fig. 20.7.

20.5. THE MICROWAVE BACKGROUND SPECTRUM 631

z = infinite

z =1000

Universe Opaque

Earth

Universetransparent

Observable

Universe

Last

scattering

surface

~ 9000 Mpc

Figure 20.8:Last scattering surface for the CMB.

• The photons detected in the CMB by modern measurements cor-respond to photons emitted from thelast scattering surfaceillus-trated in Fig. 20.8.

• The last scattering surface lies at a redshiftz∼ 1000and repre-sents the time when the photons of the present CMB decoupledfrom the matter (roughly 400,000 years after the big bang).

• At earlier redshifts the Universe becomes opaque to photons,because that represents a time early enough in the history oftheUniverse when matter and radiation were strongly coupled.


COBE

WMAP

Figure 20.9:The COBE and WMAP microwave maps of the sky.

20.6 Anisotropies in the Microwave Background

COBE and WMAP measured angular distribution of CMB (Fig. 20.9).

• Isotropic down to a dipole anisotropy at the 10−3 level corre-sponding to a Doppler shift associated with motion of the Earthrelative to the microwave background.

• Once the peculiar motion of the Earth with respect to the CMBis subtracted, the background is isotropic down to the 10−5 level.

• COBE measured an anisotropy that corresponds to

δTT

= 1.1×10−5.

Even more precise measurements of the CMB anisotropies havebeen made by WMAP.

20.7. PRECISION MEASUREMENT OF COSMOLOGY PARAMETERS633

OpenFlat Closed

Figure 20.10:Influence of spacetime curvature on WMAP microwave fluctua-tions.

20.7 Precision Measurement of Cosmology Parameters

The WMAP observations in particular have begun to yield preciseconstraints on the value of important cosmological parameters.

• This is because the detailed pattern of CMB fluctuations is ex-tremely sensitive to many cosmological parameters.

• For example, Fig. 20.10 illustrates schematically that lensing ef-fects on the CMB distort it in a way that depends on the overallcurvature of the Universe.


Multipole Moment (L)

0 10 40 100 200 400 800 1400-1

0

1

2

3

0

1000

2000

3000

4000

5000

600090o 2o 0.5o 0.2o

Angular Scale

L(L

+1

)CL/2π

(µ

K2)

(L+

1)C

L/2π

(µ

K2)

WMAP

ACBAR

CBI

ReionizatonTE Cross Power

Spectrum

TT Cross Power

Spectrum

Figure 20.11:(Top) Angular power spectrum of temperature fluctuations inthecosmic microwave background radiation. (Bottom) Cross-power spectrum of cor-relation between the cosmic microwave background temperature fluctuation andthe polarization.

Fig. 20.11 illustrates the power spectrum of CMB fluctuations.

• Multipole moments onx axis correspond to angular decompo-sition of the CMB pattern in terms of spherical harmonics ofdifferent orders.

• Roughly speaking, a multipole moment is sensitive to an angularregion (in radians) equal to one over the multipole order.

20.7. PRECISION MEASUREMENT OF COSMOLOGY PARAMETERS635

Multipole Moment (L)

0 10 40 100 200 400 800 1400-1

0

1

2

3

0

1000

2000

3000

4000

5000

600090o 2o 0.5o 0.2o

Angular Scale

L(L

+1

)CL/2π

(µ

K2)

(L+

1)C

L/2π

(µ

K2)

WMAP

ACBAR

CBI

ReionizatonTE Cross Power

Spectrum

TT Cross Power

Spectrum

• Thus, the low multiples in the above figure carry informationabout the CMB on large angular scales and the higher multipolecomponents carry information in increasingly smaller angularscales.

• Detailed fits to such power spectra using cosmological theoriesplace strong constraints on those theories, and permit cosmolog-ical parameters to be determined with high precision.


Table 20.2:Cosmological parameters.

Parameter Symbol Value

– Global Parameters (10) –

Hubble parameter† h 0.72±0.07

Deceleration parameter q0 −0.67±0.25

Age of the universe t0 13±1.5 Gyr

CMB temperature T0 2.725±0.001 K

Density parameter Ω 1.03±0.03

Baryon density ΩB 0.039±0.008

Cold dark matter density ΩCDM 0.29±0.04

Massive neutrino density Ων 0.001–0.05

Dark energy density Ωv 0.67±0.06

Dark energy equation of state w −1±0.2

– Fluctuation Parameters (6) –

Density perturbation amplitude√

S 5.6+1.5−1.0×10−6

Gravity wave amplitude√

T <√

S

Mass fluctuations on 8 Mpc σ8 0.9±0.1

Scalar index n 1.05±0.09

Tensor index nT

Running of scalar index dn/d(lnk) −0.02±0.04† H0 = 100h km s−1 Mpc−1

Some values of cosmological parameters extracted fromWMAP data are displayed in Table 20.2. The precisionwith which cosmological parameters are now being de-termined from WMAP and from high-redshift supernovaeis unprecedented and is rapidly turning cosmology into aquantitative science constrained by precise data.

20.8. SEEDS FOR STRUCTURE FORMATION 637

20.8 Seeds for Structure Formation

The fluctuations in the CMB presumably reflect conditions when mat-ter and radiation decoupled, and presumably reflect the initial densityperturbations that were responsible for the formation of structure inthe Universe.

• If the CMB were perfectly smooth, it would be difficult to un-derstand how structure could have formed.

• Fluctuations at this level at least make it possible to considertheories for structure formation, though such theories have notbeen very successful yet in correlating both the observed visiblematter and the microwave background.

• As we shall see in Ch. 21, a period of exponential growth in thescale factor of the early Universe calledcosmic inflationmayhave been central to producing these density fluctuations.

• Dark matter may have played an important role in the initiationof structure formation.

1. Because dark matter does not couple strongly to photons, itcould begin to clump together earlier than the normal mat-ter.

2. Because there is so much more dark matter than normalmatter, it could clump more effectively.

• Thus, it is likely that dark matter provided the initial regions ofhigher than average density that seeded the early formationofstructure in the Universe.


20.9 Summary: Dark Matter, Dark Energy, and Structure

Let us conclude this chapter by summarizing present understandingof dark matter, dark energy, and the formation of structure.

• If inflation were correct (see Ch. 21) and the cosmological con-stant were zero, the matter density of the Universe would beexactly the closure density, which would lead to flat geometry.

• Current data indicate that the Universe is indeed flat, as predictedby inflation, but that it does not contain a closure density ofmat-ter because there is a non-zero cosmological constant.

1. Instead, about 30% of the closure density is supplied bymatter and about 70% by dark energy (vacuum energy ora cosmological constant).

2. Luminous matter contributes a small fraction of the closuredensity, implying that the vast majority of the mass densityis dark matter.

3. Thus, the present Universe is dominated by dark matter anddark energy.

• The known neutrinos are relativistic (that is, they are hotdarkmatter) and therefore they erase fluctuations on small scales.

1. They could aid the formation of large structures like super-clusters but not smaller structures like galaxies.

2. Thus, they are not likely to account for more than a smallfraction of the dark matter.

3. WMAP indicates that light neutrinos contribute less than2% of the total energy density at decoupling.

20.9. SUMMARY: DARK MATTER, DARK ENERGY, AND STRUCTURE639

• On the scale of galaxies and clusters of galaxies, 90% of thetotalmass is not seen.

1. In this case, a significant fraction of the dark matter couldbe normal (that is, baryonic) and be in the form of small,very low luminosity objects like white dwarfs, neutron stars,black holes, brown dwarfs, or red dwarfs.

2. However, microlensing observations and searches for sub-luminous objects generally have not found enough of these“normal” objects to account for the mass of galaxy halos.

• Data indicate a small mass for neutrinos, but not one large enoughto dominate the mass density of the Universe.

• Further, strong constraints from big bang nucleosynthesis com-pared with the observed abundances of the light elements indi-cate that most of the dark matter is not baryonic.

1. Thus, a significant fraction of the dark matter is likely tobe nonbaryonic and not neutrinos, and to be cold (that is,massive so that it does not normally travel at relativistic ve-locities).

2. Current speculation centers on not yet discovered elemen-tary particles as the candidates for this cold dark matter.


• Large-scale structure and its rapid formation in the earlyUni-verse is hard to understand, given the smallness of the cosmicmicrowave background fluctuations implied by COBE and WMAP,unless cold dark matter plays a central role in seeding initialstructure formation.

• The models of structure formation most consistent with currentdata are probably the class ofΛCDM modelsthat combine a cos-mological constant (denoted byΛ) with cold dark matter(CDM)to give an accelerating but flat universe with cold dark matter toseed structure formation.

• As a bonus, the finite cosmological constant (with associatedacceleration of the cosmic expansion) that is implicit in thesemodels also makes the age of the Universe greater than we wouldestimate otherwise, which may help erase with any remainingdiscrepancies between the age of the Universe and the age of itsoldest stars.

20.9. SUMMARY: DARK MATTER, DARK ENERGY, AND STRUCTURE641

These observations taken together appear to justify several generalstatements.

• First, the Universe is flat and is presently dominated by

1. dark energy (finite cosmological constant)

2. dark matter.

This strongly favors the validity of theinflationary hypothesis.

• Second, cold dark matter probably was central to the formationof structure.

• Third, most of the dark matter is probably not “ordinary matter”(not baryonic).

• Thus, the growing evidence is that we live in a Universe domi-nated by dark energy and (non-baryonic) dark matter.

1. We have as yet no strong clues as to the source and detailednature of either because neither has been captured in a lab-oratory.

2. At present, we know about dark matter and dark energy onlyfrom observations on galactic and larger scales in the cos-mos.

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