Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

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9.1 The Universe today 403

−17

−18

−19

−20

−21

−22

101 100 1000

10 1.0 0.1

Wavelength (cm)

Frequency (GHz)

FIRAS

DMR

UBC

LBL-Italy

Princeton

Cyanogen

COBE satellite

sounding rocket

White Mt. & South Pole

ground & balloon

optical

2.73 K blackbody

(W m

−2

−1

)

Fig. 9.1. The observed spectrum of the cosmic (microwave) background radiation

(CBR) [1]. The points at wavelengths < 1 cm come from ground-based experiments.

At shorter wavelengths the Earth’s atmosphere is opaque and measurements must

be made from balloons, rockets or satellites. The high precision points around the

peak of the spectrum were made by the FIRAS instrument of the COBE satellite

which observed from 1989 to 1995 [92]. Compilation courtesy of the Particle Data

Group.

information about the “initial conditions” for structure formation. The spec-

trum of anisotropies is the primary source on the values of the cosmological

densities in Table 9.1. In particular, it constrains Ω

to be very near unity.

9.1.5 Neutrinos

In addition to thermal photons, it is believed that the Universe is ﬁlled with

neutrinos, ν

, ν

and ν

and the corresponding antineutrinos.

Neutrinos interact even less than the CBR photons but they had a suﬃ-

ciently high interaction rate at T>1 MeV to have been thermalized. In Sect.

9.6wewillseethatbythetimethetemperaturedroppedbelowkT ∼ m

relativistic neutrinos had a temperature slightly less than the photon tem-

perature:

=(4/11)

1/3

. (9.17)

This corresponds to a neutrino (+ antineutrino) number density of

404 9. Nuclear Cosmology

=(3/11)n

per species. (9.18)

corresponding to a present day value of 112 cm

−3

The present-day energy density of a neutrino species depends on its mass.

For a massless species, the temperature relation (9.17) would continue to

hold today and consequently the energy density would be a bit less than

that for photons (9.16). This would be approximately true for any neutrino

species with mass much less than the photon temperature, m

 kT

∼

2 × 10

−4

eV.

For a species of mass greater than the temperature, the neutrinos are

currently non-relativistic and the summed neutrino and antineutrino mass

density is

Ω

=0.2 h

−2

10 eV

if m

 10

−4

eV . (9.19)

If one of the neutrinos had a mass > 0.7 eV, it would signiﬁcantly distort

the observed spectrum of CBR anisotropies. Requiring m

< 0.7 eV gives the

upper limit in Table 9.1, Ω

< 0.015 [94].

Evidence for non-zero neutrino masses comes from searches for neutrino

oscillations discussed in Sect. 4.4. In particular, oscillations of neutrinos pro-

duced in the atmospheric interactions of cosmic rays [49] have given results

that are most easily interpreted as

− m

=(3± 1) × 10

−3

⇒ m

> 0.04 eV , (9.20)

where m

is the heaviest of the three neutrinos. This implies:

Ω

∼

> 0.0006 . (9.21)

It is often supposed that neutrinos have a “hierarchical” mass pattern like

that of the charged leptons, i.e. m

 m

. If this is the case, the above

inequality becomes an approximate equality, m

∼ 0.04 eV. It is not possible,

for the moment, to directly verify this hypothesis.

Finally, we note that, because of their extremely weak interactions, there

is little hope of directly detecting the cosmic neutrino background [101].

9.1.6 The vacuum

Perhaps the most surprising recent discovery is that the Universe appears to

be dominated by an apparent “vacuum energy” or “cosmological constant”

Λ:

Ω

∼

∼ 0.7 . (9.22)

The observational evidence for the existence of vacuum energy involves the

apparent luminosity of high redshift objects which can provide information

on whether the universal expansion is accelerating or decelerating (as would

9.2 The expansion of the Universe 405

be expected from normal gravitation). The observations indicate that the

expansion is accelerating and, as we will see in Sec. 9.3, this can be explained

by a positive vacuum energy density.

Vacuum energy is, by deﬁnition, energy that is not associated with par-

ticles and is therefore not diluted by the expansion of the Universe. Unless

the present vacuum is only metastable, this implies the the vacuum energy

density is independent of time. The value implied by Ω

=0.7is

(t) ∼ 3 h

× 10

eV m

−3

. (9.23)

Fundamental physics cannot currently be used to calculate the value of the

vacuum energy even though it is a concept used throughout modern gauge

theories of particle physics. It is expected to be a temperature-dependent

quantity which changes in a calculable manner during phase transitions, e.g.

the electroweak transition at T ∼ 300 GeV when the intermediate vector

bosons, W

and Z

, became massive. While the vacuum energy does not

change in particle collisions, so its existence can usually be ignored, it does

lead to certain observable eﬀects like the Casimir force between uncharged

conductors. Unfortunately, all calculable quantities involving vacuum energy

concern diﬀerences in energy densities and there are no good ideas on how

to calculate the absolute value.

Despite the lack of ideas, the existence of a vacuum energy density of the

magnitude given by (9.23) is especially surprising. A vacuum energy density

can be associated with a “mass scale” M by writing

∼

(Mc

)

(¯hc)

. (9.24)

Particle physicists are tempted to choose the Planck mass

=(¯hc

/G)

1/2

∼ 10

GeV (9.25)

as the most fundamental scale since it is the only mass that can be formed

from the fundamental constants. This gives ρ

∼ 3 ×10

132

eV m

−3

, i.e about

123 orders of magnitude too large making it perhaps the worst guess in the

history of physics. In fact, the density (9.23) implies a scale of M ∼ 10

−3

which is not obviously associated with any other fundamental scale in particle

physics, though it is near the estimated masses of the neutrinos.

A second problem with an energy density (9.22) is that it is comparable

to the matter density Ω

∼ 0.3. Since the matter density changes with

theexpansionoftheUniversewhilethevacuumenergydoesnot,itappears

that we live in a special epoch when the two energies are comparable. This

coincidence merits an explanation.

9.2 The expansion of the Universe

Modern cosmology started with Hubble’s discovery that galaxies are receding

from us with a velocity dR/dt proportional to their distance R (Fig. 9.2) :

406 9. Nuclear Cosmology

Fig. 9.2. The “Hubble diagram,” of galactic recession velocities versus distance for

a set of galaxy clusters as determined by the Hubble Key Project [97]. The slope

of the line is the Hubble constant H

. The velocities are determined by the galaxy

redshifts and the distances are determined by a variety of methods. For example,

if a supernova of known luminosity L is observed in a galaxy or galaxy cluster, the

distance can be derived from the observed photon ﬂux φ = L/4πR

= H

R. (9.26)

The factor of proportionality, H

, is called the Hubble constant.

In estab-

lishing Hubble’s law, the velocity dR/dt is derived from the “redshifts” of a

galaxy’s photon spectrum:

1+z ≡

∼ 1+v/c , (9.27)

where λ

is the wavelength of photons emitted by the galaxy and λ

is the

wavelength measured later by us. The equivalence of the redshift z with

v/c =(dR/dt)/c is the non-relativistic Doppler eﬀect. As such, the formula

is valid only for z<<1, which, according to Hubble’s law, corresponds to

R<<c/H

∼ 4300Mpc. A more general interpretation is given below (9.36).

In writing (9.26), we ignore the quasi-random “peculiar” velocities v

that are

typically of order v

∼ 10

−3

c ∼ 300 km s

−1

. These are smaller than the Hubble

velocity H

R for R>300 km s

−1

∼ 5Mpc.

9.2 The expansion of the Universe 407

Because photon wavelengths appear

to increase with time, photon ener-

gies (E

= hc/λ

) decrease with time. The energy loss between emission in a

galaxy of redshift z = H

R/c and reception by us is

− E

= hc(λ

−1

− λ

−1

)=E

[(λ

/λ

) − 1] ∼ E

z. (9.28)

Since the time to travel a distance R is R/c = z/H

,wehave

= −H

. (9.29)

CBR photons are not diﬀerent from photons emitted by galaxies, so these

photons must lose energy at the same rate. This implies that the cosmological

photon temperature decreases with time at a rate that is currently given by

= −H

. (9.30)

9.2.1 The scale factor a(t)

It is useful to parameterize the expansion of the Universe by a time-dependent

function that is proportional to the distances between galaxies. This function

is called the “scale factor,” a(t):

a(t) ∝intergalactic distances . (9.31)

The normalization of a(t) can be taken to be arbitrary but we call its value

today

a(t

) ≡ a

≡ today . (9.32)

Hubble’s law (9.26),

R/R = H

informs us that the logarithmic derivative of

a(t) is currently equal to H



˙a



= H

. (9.33)

Substituting this into (9.29) or (9.30) and generalizing to an arbitrary time

we get

= −

. (9.34)

This then says that the energies of photons or their temperature fall simply

with the scale parameters:

∝ a

−1

∝ a

−1

. (9.35)

The photon temperature can therefore be used to deﬁne the scale parameter.

We say “appear” because the photon changes Galilean reference frames as it

travels from galaxy to galaxy. The observer that measures the original high

energy is in a diﬀerent reference frame from the observer who later measures the

redshifted energy.

408 9. Nuclear Cosmology

Relation (9.35) applied to photons emitted by galaxies implies a more

general interpretation of the galactic redshifts:

1+z ≡

a(t

)

a(t

)

, (9.36)

where t

is the time of emission. The redshift of photons coming from a

galaxy of redshift z then gives the factor of expansion of the Universe between

emission and detection.

The scale parameter also determines the time dependences of number

densities of objects whose numbers are conserved. For example, the number

density of galaxies n

gal

is proportional to distances

−3

so, if the number of

galaxies does not change with time, we have

gal

(t)=n

gal

)



a(t)



. (9.37)

While the number density of galaxies is, in fact, not conserved (there were

none in the early Universe), baryon number is conserved to good approxima-

tion so we have

(a)=n

)



a(t)



. (9.38)

This dependence on a is also respected by any species of particles whose

number does not vary with time. Hence, we expect the number density of

cold dark matter particles and photons of cosmological origin to fall like a

−3

The energy densities in the Universe as a function of time can also be

simply expressed in terms of the scale factor. The energy density of non-

relativistic particles like baryons and CDM particles is proportional to the

number density of particles times their mass. It follows that

CDM

(t)+ρ

(t) ≡ ρ

(t)=ρ

)



a(t)



. (9.39)

On the other hand, the energy per photon falls like a

−1

which implies an

extra factor of a

−1

in addition to the three factors associated with the falling

number density:

(a)=ρ

)



a(t)



. (9.40)

Finally, we need to know how the vacuum energy density, ρ

,evolveswith

the expansion of the Universe. The evolution of the matter density is due to

the changing density of particles as the Universe expands. Since there are,

by deﬁnition, no particles associated with the vacuum energy, it seems plau-

sible that the vacuum energy density will remain unchanged as the Universe

evolves:

(a)=ρ

) . (9.41)

9.2 The expansion of the Universe 409

This behavior is conﬁrmed by an analysis using general relativity.

Since the density of relativistic matter (9.40) is proportional to a

−4

while

that of non-relativistic matter (9.39) is proportional to a

−3

, relativistic mat-

ter must come to dominate for a → 0. Equating the two energy densities we

ﬁnd the moment when the photon and matter densities are equal occurred

when the scale parameter took the value

Ω

∼ 2 × 10

−4

, (9.42)

corresponding to a temperature of

∼ 1eV . (9.43)

For temperatures higher than this, radiation dominates the energy of the

Universe.

Since the matter density falls like a

−3

, the future Universe will be increas-

ingly dominated by the constant vacuum energy (if it is truly constant).

We see that the Universe passes through a succession of epochs when

the energy density is dominated by radiation, by non-relativistic matter, and

then by vacuum energy. The energy densities as a function of temperature

are shown in Fig. 9.3.

In Table 9.2, we list some formative events in the history of the Uni-

verse according to this scenario. Non-controversial physics allows us to follow

with conﬁdence the succession of events starting at, say, T ∼ 1GeV when

the Universe was a nearly homogeneous soup of quarks, gluons, and lep-

tons. With time, the Universe cooled and a succession of bound states were

formed, hadrons, nuclei, atoms, and ﬁnally the gravitationally bound stars

and galaxies. The moments of the formation of bound states are called “re-

combinations.” The recombination that resulted in the formation of atoms

caused the Universe to become eﬀectively transparent to photons.

We should add that the nature of the radiation changed with temperature.

Today, the radiation consists of photons and, perhaps, relativistic neutrinos.

At temperatures T>m

, electron–positron pairs could be produced and we

will show in Sec. 9.6 that these pairs were in thermal equilibrium with the

photons and formed a blackbody spectrum similar to that of the photons

and neutrinos. Going back in time, each time the temperature rose above

a particle-antiparticle threshold, a new blackbody component was created.

During this period, the numbers of particles and antiparticles were nearly

equal. The small number of electrons and baryons present today resulted

from the small excess (∼ 10

−9

) of particles over antiparticles present when

T  m

Finally, we note that the two earliest epochs in Table 9.2, those of baryo-

genesis and inﬂation, are speculative and involve physics that is not well-

understood. The existence of these epochs is postulated to solve certain

mysteries in the standard scenario, e.g. the existence of the small particle-

410 9. Nuclear Cosmology

antiparticle asymmetry and the origin of the density ﬂuctuations leading to

structure formation.

9.3 Gravitation and the Friedmann equation

While the temperature of the Universe T ∝ 1/a is simply determined by the

scale parameter a(t), in order to calculate the end product of cosmological

particle or nuclear reactions, we will need to know how long the Universe

spends near a given temperature, i.e. we need to know the time dependence

of a(t).

In the absence of gravitation, the recession velocities of galaxies would

be constant, implying ¨a = 0. In the presence of the attractive eﬀects of

gravitation, we might expect that the expansion would be decelerated, ¨a<0.

The correct equation for ¨a can only be found within the framework of a

relativistic theory of gravitation, i.e. general relativity. The standard Newto-

nian formalism is inadequate for two reasons. First, the Newtonian integrals

GUT epoch

Log

(eV m

−3

)

−25

100

125

15 10

nucleosynthesis

−5

−10

LogT (eV)

recombination

today

equality

matter−radiation

Fig. 9.3. The energy density of matter, radiation and vacuum (assumed constant)

as a function of temperature. The temperature scale starts at the expected “grand

uniﬁcation” scale of ∼ 10

GeV. We suppose that the CDM particles have masses

greater than ∼ 10 GeV so the line of ρ

starts at 10 GeV.

9.3 Gravitation and the Friedmann equation 411

Table 9.2. Some formative events in the past. The values of t

, t

rec

, and t

depend

on (h

,Ω

) and we have used (1, 0.3, 0.7).

t kT

(eV) event

∼ 1.5 × 10

yr 2.349 × 10

−4

today

∼ 10

yr ∼ 10

−3

formation of the ﬁrst structures,

rec

∼ 5 × 10

yr 0.26 “recombination” (formation of atoms),

Universe becomes transparent

∼ 5 × 10

yr 0.8 matter-radiation equality

3 min 6 × 10

nucleosynthesis (formation of light nuclei,

A =2, 3, 4, 6, 7)

1s 10

−

→ γγ

4 × 10

−6

s ∼ 4 × 10

QCD phase transition (formation of

hadrons from quarks and gluons)

< 4 × 10

−6

s > 10

baryogenesis (?) (generation of baryon–

antibaryon asymmetry)

inﬂation (?)

(generation of density ﬂuctuations)

used to calculate the force on a particle or the gravitational potential at a

particular point are not deﬁned in an inﬁnite medium of uniform density.

Second, the source of Newtonian gravitation is the mass density. We can

expect that a relativistic theory will have a more general source, just as in

electromagnetism both charge densities and charge currents can lead to ﬁelds.

General relativity treats gravitation geometrically. One way of approach-

ing the problem is is illustrated in Fig. 9.4. The ﬁgure shows a small region

of a two-dimensional homogeneous Universe with a coordinate grid attached

to free test particles (galaxies). The test particles are initially equally spaced,

as determined by light propagation times between galaxies. With this choice,

over a small region, the distance dS between test particles separated by co-

ordinate diﬀerences (dx, dy) is just

=dx

+dy

. (9.44)

As the Universe expands, the test particles withdraw from each other.

Compared to the original situation, all intergalactic distances are multiplied

by a scale factor a(t). It is convenient to keep the coordinates of the galax-

ies constant during the expansion, as shown in Fig. 9.4. Such coordinates

412 9. Nuclear Cosmology

are called “comoving” coordinates. Since the coordinate separations do not

change with time, at a later time t the relation between dS and the coordinate

diﬀerences is given by

==a(t)

(dx

+dy

) . (9.45)

For this homogeneous Universe, all the eﬀects of gravity should be stored in

the geometric scale factor a(t). The equations of general relativity will tell us

how a(t) evolves with time.

=dx +dy

=a(t)

(dx

+dy

)

Fig. 9.4. A small region of a homogeneous two-dimensional Universe. A coordinate

grid has lines of constant x and y that intersect at equally spaced test particles

(galaxies). As the Universe expands, the distances dS between galaxies increases [by

afactora(t)], but we choose to keep the coordinates of the galaxies ﬁxed (comoving

coordinates).

We note that the geometrization of gravitation is possible only because

of the Principle of Equivalence which insures that gravity aﬀects all test

particles in the same way, independent of their mass and composition. This

means that if, in Fig 9.4 we place two test particles at the same point with

vanishing relative velocity, then in the future they will stay at the same point.

If this were not the case, then the expansion of the coordinate grid would not

be unique.