Chapter 21 Extending classical big bang theory -...

Chapter 21

Extending classical big bang theory

The big bang is our standard model for the origin of the Universe and

has been for almost half a century. This place in well earned.

• At a broader conceptual level, all of modern cosmology rests

on observations (for example, the very existence of the cosmic

microwave background radiation) that make sense only if there

was a big bang—or something very much like it—in our past.

• At a more nuts and bolts level, the standard big bang model ac-

counts quantitatively for a variety of relatively precise observa-

tional data (for example, those associated with specific proper-

ties of the cosmic microwave background and with the observed

abundances of light elements) that would be difficult to explain

with any competing theory.

However, there are some observations in modern cosmology suggest-

ing that the classical big bang is an essentially correct but perhaps

incomplete picture of the origin and evolution of our Universe. In

this chapter we address some of these issues.

841

842 CHAPTER 21. EXTENDING CLASSICAL BIG BANG THEORY

21.1 Successes of the Big Bang

Let us begin by being more explicit about the successes of the stan-

dard big bang and its place in the modern picture of cosmology. As

we have already discussed to some degree in Chapter 16, The stan-

dard cosmology rests on a relatively few observations and concepts:

1. The redshifts of distant galaxies imply that we live in an expand-

ing Universe described at the simplest level by the Hubble law.

2. On large enough scales (beyond that of superclusters of galax-

ies) the Universe appears to be both homogeneous and isotropic

(cosmological principle).

3. The properties of distant quasars suggest that these powerful en-

ergy sources were once more energetic and more closely spaced

than they are today, implying that the properties of the Universe

on large scales has evolved with time.

4. The cosmic microwave background (CMB) is

• all pervading,

• highly isotropic but with measureable fluctuations at the one

part in 105 level, and

• possesses an almost perfect blackbody spectrum with a tem-

perature of 2.725±0.001 K.

5. The elemental composition of the Universe is (by mass) three

parts H to one part He, with but a trace of heavier elements.

21.1. SUCCESSES OF THE BIG BANG 843

6. Because gravitation and not electromagnetism governs the large-

scale structure of the Universe, it must be charge neutral on large

scales. Conversely, observations indicate that the Universe is

highly asymmetric with respect to matter and antimatter, with

no evidence for significant equilibrium concentrations of anti-

matter.

7. There are many fewer baryons in the Universe than there are

microwave photons.

8. In contrast to the homogeneity of the Universe on very large

scales (cosmological principle), matter on scales comparable to

superclusters of galaxies and smaller exhibits complex and highly

evolved structure,

• This contrasts sharply with the smoothness of the CMB.

• Furthermore, observations indicate that this structure began

to develop very early in the history of the Universe.

9. Detailed analysis of fluctuations in the CMB and of the bright-

ness of distant Type Ia supernovae indicate that

(a) The expansion of the Universe is presently accelerating.

(b) The geometry of the Universe is remarkably flat (euclidean).

10. The bulk of the matter in the Universe is not luminous (dark mat-

ter), being observable only through its gravitational influence.

11. Various observational constraints imply that most of this dark

matter is not baryonic.


The first five observations fit well within classical big bang cos-

mologies:

• Observations (1)–(2) indicate that we live in an expanding,

isotropic universe,

• Observations (3)–(5) indicate that this Universe has

evolved over time and had a beginning in a very hot, very

dense initial state (the big bang).

The remaining observations need not be inconsistent with the

classical big bang, but they require either ad hoc imposition of

particular initial conditions on the Universe, or assumption of

specific properties for the microscopic properties of the mat-

ter and energy fields that the Universe contains that we shall

address in the next section.

21.2. PROBLEMS WITH THE BIG BANG 845

21.2 Problems with the Big Bang

As we have indicated in the previous section, observational

properties (6)–(11) constitute problems for the classical big

bang model.

• They do not necessarily invalidate the big bang, but they

indicate that the big bang in its minimal form may be in-

complete.

• In most cases we shall find that this incompleteness is

likely to originate in an inadequate understanding of how

the particle and field content of the Universe couples to its

evolution.

Let us now discuss these problems.


21.2.1 The Horizon Problem

Observational property (4) [isotopy of the CMB] presents a po-

tential conflict with causality because

• The nearly constant temperature of the CMB in widely

separated parts of the sky is understandable only if those

regions were in causal contact in the past.

• But in the standard big bang it is easy to show that regions

on the sky separated by more than a degree or two in an-

gle could never have been in causal contact (had sufficient

time to exchange light signals) since the big bang.

• That is, they lie outside each other’s horizons, as illus-

trated in Fig. 21.1 on the following page, and thus cannot

have been in causal contact in the past.


ObservableUniverse

EarthEarth

Earth

Distant

observer

Earth's

horizon

Earth

(a)

(c)

(b)

(d)

Earth

Earthhorizon

Horizonfor A

A

Horizonfor B

B

Horizon: greatest distance from which

light could have reached us since the

big bang

Object presently

inside the horizon. It

can be seen from

Earth

Object outside the

horizon. It cannot be

seen from Earth

because light from it

has not had time to

reach us since the

big bang.

The object that

was outside the

horizon now enters

our horizon and is

visible from Earth.

Part of the Universe

that both the Earth

observer and distant

observer can see

Distant

observer's

horizon

Horizon increases

with time

A and B are now outside of each other's horizons. This means they have always been outside of each other's horizons in the standard big bang model.

Figure 21.1: Horizons in an expanding universe. (a) A (cosmological) horizon is

the greatest distance from which light could have reached us since the beginning

of time. (b) Horizons expand with time, so objects currently outside our horizon

may come within our horizon in the future. (c) Cosmological horizons are defined

relative to each observer, so each has her own horizon. (d) Horizon problem pro-

duced by the CMB having identical temperatures on opposite sides of the sky for

an observer: how do A and B know to have the same temperature if they could

never have exchanged signals in the past?


Time

Sca

le f

acto

rTiny change

in initialconditions

Actual flatUniverse

Different tiny

change in initial

conditions

Figure 21.2: The flatness problem: producing a flat Universe today requires re-

markable fine-tuning of the initial curvature for the Universe.

21.2.2 The Flatness Problem

Observational property 9(b) implies that the Universe is euclidean.

• This indicates that the Universe is very near the closure density.

• Consistent with big bang, but this condition can be realized only

if parameters are very finely tuned in the early universe (Fig. 21.2).

• Example (Exercise): the fractional deviation of the density from

the critical density at any time in the evolution of the Universe is

∆ρ

ρ=

ρ −ρc

ρ=

3kc2

8πGa2ρ,

• From this result, unless the flatness is tuned to one part in 1060

at the Planck time we end up with a Universe that is far from flat

today. While possible, this does not seem to be a very natural

initial condition.


Str

en

gth

of

forc

e

Temperature

Pla

nck s

cale

GU

Ts s

cale

Strong

Weak

Gravity

Magneticmonopolesproduced?

E & M

Figure 21.3: The magnetic monopole problem of the standard big bang: where

are the massive relic particles that would be expected to be produced at phase tran-

sitions like grand unification (labeled GUTs)?

21.2.3 The Magnetic Monopole Problem

There are reasons to believe that massive particles like magnetic monopoles

could be produced copiously at phase transitions in the early universe,

as illustrated in Fig. 21.3. This has two adverse consequences:

• Such particles have never been detected in experiments, and

• If they were produced in large numbers they would have halted

the expansion of the Universe and then caused it to collapse.

These problems are removed if we simply declare that such

particles are not produced in large numbers, but why should

such an initial condition be required?


21.2.4 The Structure and Smoothness Dichotomy

Observational properties (4) and (8) (smoothness of CMB vs. exis-

tence of large-scale structure) present a severe compatibility issue:

• The remarkably high smoothness of the CMB implies that the

early Universe was strikingly devoid of density perturbations.

• Where then did the density perturbations responsible for the growth

of rich structure in the present Universe on the supercluster and

smaller scale originate?

The form of the density perturbations required to give the ob-

served structure is known so they could be postulated as initial

conditions, but again we would like to know why.


21.2.5 The Vacuum Energy Problem

Observation 9(a) (accelerated expansion) has a simple explanation

only if the Universe contains dark energy.

• This would be most naturally explained if dark energy is a con-

sequence of vacuum fluctuations.

• However, estimates of the vacuum energy content of the Uni-

verse are spectacularly wrong in comparison with the correspond-

ing observational constraints.

• The accelerated expansion is consistent with the big bang pic-

ture if we simply postulate the existence of dark energy in the

Universe in the required amount.

But it is highly unsatisfying to have no understanding of where

this fundamental influence on the evolution of the Universe

comes from.


21.2.6 The Matter–Antimatter and Baryogenesis Problem

Observation (8) indicates that the physical universe contains almost

entirely matter with little corresponding antimatter.

• We can impose this as an initial condition but that is bothersome

given that matter and antimatter enter modern elementary parti-

cle physics theory on an equal footing.

• A closely-related problem (because annihilation of baryons with

antibaryons produces photons) is how to account for the large

excess of photons over baryons in the Universe (Observation 7).


21.2.7 Modifying the Classical Big Bang

We shall now discuss some possible resolutions of these problems.

• In attempting to resolve the problems we have to be careful to

preserve the successes of the big bang model.

• The successful predictions of the big bang model rest primarily

on the evolution of the Universe at times later than about one

second after the initiation of the big bang.

• Therefore, any modification of our cosmological model that in-

fluences the Universe at times less than about one second after

the big bang will leave the successes of the big bang intact if they

leave appropriate initial conditions for the subsequent evolution.

We begin with a proposed modification of the evolution of the

Universe operative only in the first tiny fraction of a second

of the big bang that has the potential to resolve the first four

problems listed above in a single stroke.


21.3 Cosmic Inflation

The theory of cosmic inflation is based on a simple but striking idea

that has been discussed already in conjunction with the de Sitter so-

lution:

• In the presence of an energy density that is constant over all

space, the Einstein equation admits an exponentially growing so-

lution.

• If the Universe underwent a short burst of exponential growth

before settling down into more normal big bang evolution, there

would be potentially large implications for the evolution of the

Universe.

• We shall take the essential point of inflation to be this general

idea of the Universe experiencing a period of exponential growth.

• There are many specific versions of inflationary theory that im-

plement this in different ways. For the most part we shall leave

those specifics for the interested reader to pursue in the specialist

literature.

• Our reason is that there is now compelling evidence that the

basic idea of inflation is necessary to explain the evolution of

the early Universe, but no specific version of inflation currently

available gives a completely satisfactory accounting of the cause

and detailed effects of the inflationary period.

21.3. COSMIC INFLATION 855

21.3.1 Inflationary Theory

From earlier discussion of the de Sitter solution,

• A universe with pure vacuum energy expands exponentially,

a(t) = eHt,

where the Hubble parameter H is constant.

• The basic idea of inflationary theory is that shortly after its birth

the Universe found itself in a situation dominated by a constant

(or nearly constant) energy density.

• This drove an exponential expansion for a very short period of

time that cooled the Universe rapidly because of the expansion.

• Then at the end of this period the Universe exited from the infla-

tionary conditions and reheated.

• The mechanism for reheating depends on the version of infla-

tion assumed, but generally involves the rapid conversion of the

constant energy density driving inflation into the mass–energy

of more normal particles).

This then produced a situation dominated by radiation rather

than vacuum energy, which caused the Universe to evolve ac-

cording to a standard (hot) big bang scenario (power law rather

than exponential dependence of the scale factor on time).


In various versions of inflation different reasons are assumed for the

initial conditions that triggered the exponential expansion.

• The original inflationary idea due to Alan Guth assumed that

inflation was driven by a Lorentz scalar field associated with a

first-order phase transition.

• This is conceptually simple, but proved to be incompatible with

observations (as Guth himself realized).

• It was found that the resulting inflation could not halt in a manner

that would give something that looks like the real Universe.

• In subsequent versions of inflation the inflation was often as-

sumed to be driven by a scalar field having a time dependence of

a particular form called a slow rollover transition.

• Although such theories often give a reasonably good account of

data, they suffer from

1. having little connection to scalar fields known already to

exist in elementary particle physics, and

2. requiring extremely fine empirical tuning of parameters to

account well for data.

• In keeping with the philosophy outlined above, we omit discus-

sion of these different forms of inflation and instead concentrate

on the consequences of inflationary expansion.


a

emiTemiT

T 2

~1050

(a) 10-32 s (b)

?

Inflation

Inflation

Reheat

Hot big bang

Hot big bang

?

No

inflation

Figure 21.4: The inflationary scenario. (a) In the inflationary epoch the Universe

expands exponentially, which can increase the scale factor by factors of 1050 or

larger on a timescale of order 10−32 s. (b) The Universe cools as it expands expo-

nentially. At the end of inflation some mechanism, not yet well understood, must

reheat the Universe, which then continues a standard hot big bang evolution.

Figure 21.4 illustrates the behavior of the scale factor and the temper-

ature in highly schematic fashion during inflation and the following

big bang evolution.

• During inflation the Universe expanded at a much higher rate

than in normal big bang evolution.

• At the same time, the temperature dropped rapidly in the expo-

nentially expanding Universe.

• Finally, when the period of inflation halted the Universe first

rapidly reheated and then began to decrease in temperature ac-

cording to the standard big bang scenario.

The question marks represent our substantial lack of knowledge

concerning the Universe prior to the inflationary period.


Causally-connected,

homogenous region

1 2

Eventual location

of Earth

1

2

1

2Reenter

horizon

Inflation

Standard big

bang evolution

Horizon

expands

faster

than

big bang

universe

Inflationary

universe

expands

faster

than the

horizon

In inflation an original small, homogeneous

region expands much faster than does the

horizon distance. Then when inflation ends

and normal big bang evolution begins, the

horizon expands faster than the Universe.

Horizon

Horizon

Figure 21.5: Solution of the horizon problem in the inflationary universe.

21.3.2 Taking the inflationary cure

The inflationary scenario provides (in principle) a solution to the four

fundamental problems posed above.

Solution of the Horizon Problem

The solution of the horizon problem is illustrated in Fig. 21.5.

• The tremendous expansion means that regions that we see

widely separated in the sky now at the horizon were much

closer together before inflation.

• Thus, they could have been in contact by light signals.


Solution of the Flatness Problem

The tremendous expansion greatly dilutes any initial curvature.

• Think of standing on a basketball. It would be obvious that

you are standing on a two-dimensional curved surface.

• Now imagine expanding the basketball to the size of the

Earth.

• As you stand on it now, it will appear to be flat, even

though it is actually curved if you could see it from large

enough distance.

• The same idea extended to four-dimensional spacetime

accounts for the present flatness (lack of curvature) in the

space of the Universe.

Out to the greatest distances that we can see the

Universe looks flat on large scales, just as the Earth

looks approximately flat out to our horizon.


Solution of the Monopole Problem

The rapid expansion of the Universe tremendously dilutes the

concentration of any magnetic monopoles that are produced.

• Simple calculations indicate that they become so rare in

any given volume of space that we would be very unlikely

to ever encounter one in an experiment designed to search

for them.

• Nor would they have sufficient density to alter the gravity

and thereby the normal expansion of the Universe follow-

ing inflation.


Tim

e

Vacuum

Vacuum

Subatomic scale

Particle Antiparticle

Figure 21.6: Quantum fluctuations can create a particle-antiparticle pair from the

vacuum for a fleeting instant. In inflation such microscopic fluctuations can be

stretched to macroscopic scales, producing density perturbations that can later seed

the formation of large-scale structure.

Inflation and the Formation of Structure

Perhaps the most important consequence of inflation is that it pro-

vides a possible solution to the origin of large-scale structure. The

inflationary explanation is in fact rather remarkable.

• During inflation quantum fluctuations such as that illustrated in

Fig. 21.6 (which must be present because of the uncertainty prin-

ciple) end up being stretched from microscopic to macroscopic

dimensions by the exponential expansion.

• Because this process occurs during the entire time of inflation,

one ends up with density fluctuations of macroscopic size vary-

ing over many length scales.


• Technically, this produces what is termed a scale-invariant spec-

trum of density fluctuations, and it is known empirically that this

is the spectrum of density perturbations that is most likely to lead

to the observed large-scale structure in the Universe.

• These density perturbations will generally be expanded beyond

the horizon during inflation.

• But after inflation is over and normal big bang evolution sets in,

the horizon grows with time.

• Eventually these density perturbations come back within the hori-

zon where they can serve as nucleation centers for the formation

of structure.


2 10 40 100 200 400 600 800 1000 1200 1400 1600 1800

Multipole order

0

20

40

60

80180

Tem

pera

ture

flu

ctu

ation (

µK

)

20 5 2 1 0.5 0.2

Angular wavelength (degrees)

Inflation

with dark

energy

Open universe,

no inflation, no

dark energy

Inflation with

no dark energy

WMAP

CBI

ACBAR

BOOMERANG

Figure 21.7: Temperature fluctuations in the cosmic microwave background from

satellite and high-altitude balloon observations compared with various theoretical

models.

• Although there are many specific theories of inflation, it is en-

couraging that simulations of large-scale structure give reason-

able results when the effects of inflation are included.

• Furthermore, the fluctuations in the CMB observed by WMAP

are described best by theories that include the effect of inflation.

– For example, Fig. 21.7 compares the angular fluctuations

in temperature for the CMB with various models with and

without inflation and with and without dark energy.

– Models with both dark energy and inflation are clearly fa-

vored.


21.3.3 Inflation Doesn’t Replace the Big Bang

Inflation is not a theory in competition with the big bang:

• The theory of inflation modifies only the first tiny instants

of creation. After the completion of the brief period of

inflation, it is assumed that big bang evolution continues

as described earlier.

• Thus, inflation should be viewed as a modified form of the

big bang that accounts for effects due to the properties of

elementary particles that are not included in the standard

big bang.

21.4. THE ORIGIN OF THE BARYONS 865

21.4 The Origin of the Baryons

In the standard big bang the preponderance of matter over anti-

matter and of photons over baryons in the Universe have to be

introduced through initial conditions without fundamental jus-

tification. Sakharov first enumerated the ingredients required

to generate baryon asymmetries within the standard big bang

model:

1. There must exist elementary particle interactions in the

Universe that do not conserve baryon number NB.

2. There must exist interactions that violate both charge con-

jugation symmetry (C) and the product of charge conju-

gation and parity (P) symmetries (the product is denoted

CP).

3. There must be departures from thermal equilibrium during

the evolution of the Universe.


In these requirements

• Baryon number NB takes the value +1 for a baryon and

−1 for an antibaryon. Baryon number is then the algebraic

sum of these numbers for all the particles in a reaction.

• Conservation of baryon number (observed in every exper-

iment done so far) means that this number does not change

in the interaction.

• Charge conjugation symmetry (C) is symmetry under ex-

change of a particle with its antiparticle.

• Parity symmetry (P) is symmetry under inversion of the

coordinate system through the origin.

• CP symmetry is symmetry under inversion of the coordi-

nate system and exchange of particle with antiparticle.

Most interactions conserve these symmetries to high precision

but the weak interactions are known to violate P, C, and CP.


• Departures from thermal equilibrium are likely to have oc-

curred at various times in the evolution of the Universe.

• At least the weak interactions are known to violate both C

and CP symmetry.

Thus (in principle) all ingredients exist to account for baryon

asymmetry except for baryon non-conserving reactions.

• Experimentally, baryon non-conservation has never been

observed.

• However, there are theoretical reasons to believe that

baryon conservation might not be an exact symmetry but

just one that has not yet been observed to be violated.

• For example baryon non-conservation may occur only on

a energy scale not yet reached in laboratory experiments

but that could have occurred in the early Universe.


One class of elementary particle theories that features baryon-number

violating interactions prominently and thus could have played a role

in producing the baryons is that of Grand Unified Theories (GUTs).

• In the Standard Electroweak Theory of elementary particle physics

the electromagnetic interactions and the weak interactions have

been (partially) unified.

• This means that at high enough energy (in this case a scale of

about 100 GeV, where a GeV is 109 eV) the weak and electro-

magnetic interactions take on the same properties.

• A GUT attempts to extend this idea to unify weak, electromag-

netic, and strong interactions into a single unified theory.

• The characteristic GUT energy scale is very high (1014−15 GeV

is a common estimate), but on that scale GUTs typically violate

baryon number strongly.

At one time GUTs were favored as the likely way to account for

baryogenesis but there have since been shown to be difficulties with

this approach.

• In particular, it seems that a baryon asymmetry generated by a

GUTs phase transition would likely be wiped out by the cosmic

inflation described earlier.

• Thus a viable theory of baryon asymmetry may require a baryon-

violating phase transition at a lower energy scale than the GUTs

scale.


A possible alternative mechanism is leptogenesis.

• Leptogenesis postulates that perhaps the baryon asymme-

try was generated by strongly CP-violating processes in

the electroweak sector.

• But it is not clear that this can account for the observed

baryon asymmetry of the Universe because the known

electroweak interactions do not exhibit interactions with

the required characteristics.

• However, the correct electroweak theory could be more

general than the present one (which is not tested exhaus-

tively at energies of 100 GeV or larger, and is now known

to be contradicted by the small but finite masses observed

for neutrinos).

• Thus, an improved electroweak theory eventually might

be able to account for the baryon asymmetry.

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