Guerra Civil en Timor Oriental

The Early Universe

We have been exploring the period in the history of the universe when the radiation temperature dropped from a little over 10⁴K down to its present value of 2.725 K. We now want to look back to the era when the temperature was greater than about 10⁴K, well before the energy density in radiation fell below that of baryons and cold dark matter. We will carry the story back to when the temperature was above 10¹⁰ K, when electron–positron pairs were abundant and neutrinos were in equilibrium with these pairs, and even farther back, as far as our current knowledge of the laws of physics will take us.

3.1 Thermal history

We first want to work out the history of the falling temperature of the early universe. In this section we will look back only to a time when the temperature was between 10⁴ K and 10¹¹ K, which is low enough so that muon–antimuon and hadron–antihadron pairs were no longer being pro-duced in appreciable numbers.

There are two circumstances that greatly simplify this task. The first is that the collision rate of photons with electrons and other charged particles during this era was so much greater than the expansion rate of the universe that the photons and charged particles can be assumed to have been in ther-mal equilibrium, with a common falling temperature. At sufficiently early times even the neutrinos and perhaps the cold dark matter particles were also in thermal equilibrium with the photons and charged particles; later, when no longer colliding rapidly with other particles, they can be treated separately as free particles. The other circumstance is that the number den-sity of baryons (or more strictly, the number denden-sity of baryons minus the number density of antibaryons) is so much less than the number density of photons that we can ignore the chemical potential associated with baryon number. Baryons will be put back into the picture in the following section.

Also, because the electron/photon number ratio is so small now, it is rea-sonable to assume that the universe has always had a very small net lepton number density (the number density of leptons of all sorts minus that of antileptons) per photon. This means that even at temperatures of order 10¹⁰K, when electron–positron pairs were abundant and the energy density and pressure were not simply proportional to T⁴ and the entropy density was not simply proportional to T³, the entropy density, energy density, and pressure were functions s(T ), ρ(T ), and p(T ) of the temperature alone.

(The possibility of a non-negligible net lepton number is discussed at the end of this section.)

Before studying the history of the universe during this era we will have to take a brief look at the thermodynamics and statistical mechanics of this sort of matter, in thermal equilibrium with negligible chemical potentials. The condition of thermal equilibrium tells us that the entropy in a co-moving volume is fixed

s(T )a³ = constant . (3.1.1)

The second law of thermodynamics says that any adiabatic change in a system of volume V produces a change in the entropy given by

d (s(T )V ) = d (ρ(T ) V ) + p(T ) dV

T . (3.1.2)

Equating the coefficients of dV gives our formula for the entropy density

s(T ) = ρ(T ) + p(T )

T . (3.1.3)

Also, equating the coefficients of V dT and using Eq. (3.1.3) give the law of conservation of energy:

Tdp(T )

dT = ρ(T ) + p(T ) . (3.1.4)

(For instance, for radiation we have p(T ) = ρ(T )/3, so Eq. (3.1.4) gives the Stefan–Boltzmann law ρ = aBT⁴ with a_B a constant that cannot be determined from thermodynamics alone; Eq. (3.1.3) then gives the entropy density for radiation as s(T ) = 4 aBT³/3. This is why in we said in Section 2.2 that the constant σ kB ≡ 4aBT³/3nBmay be interpreted as the radiation entropy per baryon.)

With equal numbers of particles and antiparticles, the number den-sity n(p)dp of a species of fermions (such as electrons) or bosons (like photons) of mass m and momentum between p and p + dp is given by the Fermi–Dirac or Bose–Einstein distributions (with zero chemical potential)

n(p, T ) = 4π gp² (2π ¯h)³

 1

exp(

p²+ m²/kBT ) ± 1



, (3.1.5) where g is the number of spin states of the particle and antiparticle, and the sign is + for fermions and − for bosons. For instance, for photons g = 2 (and of course m = 0), because photons have two polarization states and they are their own antiparticles, while for electrons and positrons g = 4, because they have two spin states and electrons and positrons are distinct

3.1 Thermal history

particle species. The energy density and pressure of a particle of mass m are given by the integrals¹

ρ(T ) =

The entropy density of this particle is then given by Eq. (3.1.3) as s(T ) = 1 In particular, for massless particles Eq. (3.1.6) gives

ρ(T ) = g species and spin of massless fermions makes a contribution to the energy density, pressure, and entropy density that is just the same as for each polar-ization state of photons, except for an additional factor 7/8.

During a period of thermal equilibrium, the variation with time of the temperature is governed by the equation (3.1.1) of entropy conservation and the Einstein field equation, with curvature neglected,

˙ (The minus sign is inserted in taking the square root of ˙T², to take account of the fact that the temperature decreases as time passes.) In particu-lar, during any epoch in which the dominant constituent of the universe is a highly relativistic ideal gas, the entropy and energy densities are

1Eq. (3.1.6) follows directly from the definition of n(p, T ), and Eq. (3.1.7) can then be derived from Eq. (3.1.4). Or both Eqs. (3.1.6) and (3.1.7) can be obtained from Eqs. (B.41) and (B.43).

given by

s(T ) = 2N a_BT³

3 (3.1.12)

ρ(T ) = N a_BT⁴

2 , (3.1.13)

where N is the number of particle types, counting particles and antiparticles and each spin state separately, and with an extra factor of 7/8 for fermions.

Then Eq. (3.1.11) becomes t =

3 16π GN aB

1

T² + constant . (3.1.14) With this background, let us now start our history at a time when the temperature was around 10¹¹ K, which is in the range m_µ  kBT  me. Even though it was too cold at this time for reactions like ν_µ+ e → µ + νe

or ν_τ + e → τ + νe, the µ and τ neutrinos and antineutrinos were kept in thermal equilibrium by neutral current reactions, like neutrino-electron scattering or e⁺+ e⁻ ν + ¯ν. Hence the constituents of the universe at this time were photons with two spin states, plus three species of neutrinos and three of antineutrinos, each with one spin state, plus electrons and positrons, each with two spin states, all in equilibrium and all highly relativistic, giving

N = 2 +7

8(6 + 4) = 43

4 , (3.1.15)

so that Eq. (3.1.14) gives, in cgs units:

t =

3c² 172π Ga_B

1

T² + constant = 0.994 sec

 T

10¹⁰K

−2

+ constant . (3.1.16) For instance, with muons ignored and the mass of the electron neglected, it took 0.0098 sec for the temperature to drop from a value 10¹²K to 10¹¹K, and another 0.98 sec for the temperature to drop to 10¹⁰K.

At a temperature of about 10¹⁰ K neutrinos were just going out of equilibrium and beginning a free expansion. The weak interaction cross section for neutrino-electron scattering is roughly σwk ≈ ( ¯hG^wkk_BT )², where G_wk
1.16 × 10⁻⁵GeV⁻²is the weak interaction coupling constant, and the factor ¯h²is included to convert a quantity with the units (energy)⁻² to a quantity with the units (length)²of a cross section. (Recall that we are using units with c = 1.) The number density of electrons at temperatures above 10¹⁰K is roughly given by n_e ≈ (kBT / ¯h)³, so the collision rate of a neutrino with electrons or positrons at such temperatures is

_ν = neσwk ≈ G_wk² (kBT )⁵/ ¯h .

3.1 Thermal history

This may be compared with the expansion rate, which during the radiation-dominated era is of the order

H ≈



G(kBT )⁴/ ¯h³,

with the factor ¯h⁻³included to convert a quantity with the units (energy)⁴ to a quantity with the units mass/length³ of a mass density. The ratio of the collision rate to the expansion rate is thus

_ν

H ≈ G_wk² ( ¯h/G)^1/2(kBT )³

 T

10¹⁰ K

³

Hence neutrinos were scattered rapidly enough to remain in thermal equilibrium at temperatures above 10¹⁰K. This is just a little greater than m_e/kB, so for lower temperatures electrons and positrons rapidly disapp-eared from equilibrium, the collision rate dropped rapidly below G²

wk

(kBT )⁵/ ¯h, and hence the ratio ν/H dropped rapidly below unity. The neutrinos then began a free expansion, in which (as we saw in Section 2.1) the number density distribution n_νcontinued to keep the form (3.1.5), with a temperature T_ν ∝ 1/a.

At lower temperatures we must take into consideration the finite mass of the electron, so the temperature T of the electrons, positrons, and photons (which were still in equilibrium with each other) no longer fell as 1/a. On the other hand, the freely expanding massless neutrinos preserved a Fermi–

Dirac momentum distribution,²with a temperature that continued to drop as 1/a. We must therefore now distinguish between the photon temperature T , and the neutrino temperature T_ν.

The entropy density of the photons, electrons, and positrons is s(T ) = 4a_BT³

2This is not exact; even at temperatures under 10¹⁰K, the neutrino distribution was slightly affected by weak interaction processes, such as e⁻+ e⁺ → ν + ¯ν. See A. D. Dolgov, S. H. Hansen, and D. V. Semikoz, Nucl. Phys. B 503, 426 (1997); 543, 269 (1999); G. Mangano, G. Miele, S. Pastor, and M. Peioso, Phys. Lett. B 534, 8 (2002). For a review, see A. D. Dolgov, Phys. Rep. 370, 333 (2002). The weak interactions provide some thermal contact between the neutrinos and the plasma, which is being heated by electron–positron annihilation, so the effect is to slightly increase the neutrino energy density, by an amount usually represented as an increase in the effective number of neutrino species, from 3 to 3.04. This effect is neglected in what follows.

where, recalling that a_B = π²k⁴ The entropy conservation law (3.1.1) gives a³T³S(me/kBT ) constant, and since T_ν ∝ 1/a, this means that T_νis proportional to T S^1/3(me/kBT ). The of T /T_ν without a computer calculation, we note that S(∞) = 1, so for k_BT  me, Eq. (3.1.20) gives

T /T_ν → 

11/41/3

= 1.401 . (3.1.21)

In particular, at the present time, when T = 2.725 K, the neutrino temperature is 1.945 K. Unfortunately there does not seem to be any way of detecting such a neutrino background.

With three flavors of neutrinos and antineutrinos, the total energy density during this period is

ρ(T ) = 6 · We insert Eqs. (3.1.17) and (3.1.22) in Eq. (3.1.11), and find

t =

3.1 Thermal history

where x ≡ me/kBT , and in cgs units t_e ≡

24 π G c⁶a_Bm⁴_e/k⁴_B−1/2

= 4.3694 sec . (3.1.25) The values of T /T_ν and of the time required for the temperature to fall to T (calculated from Eq. (3.1.11)) are given for various values of T in Table 3.1.

After the era of electron–positron annihilation, the energy density of the universe was dominated for a long while by photons, neutrinos, and antineutrinos, all of them highly relativistic, so during this period we have s(T ) ∝ T³, and

ρ(T ) = aBT⁴+7

8·3·aBT⁴

ν = aBT⁴

 1 +7

8 · 3 · (4/11)^4/3



= 3.363 aBT⁴/2 . (3.1.26)

Table 3.1: Ratio of electron-photon temperature T to neutrino temperature T_νand the time t required for the temperature to drop from 10¹¹K to T , for various values of T .

T (K) T /T_ν t(sec)

10¹¹ 1.000 0

6 × 10¹⁰ 1.000 0.0177

3 × 10¹⁰ 1.001 0.101

2 × 10¹⁰ 1.002 0.239

10¹⁰ 1.008 0.998

6 × 10⁹ 1.022 2.86

3 × 10⁹ 1.080 12.66

2 × 10⁹ 1.159 33.1

10⁹ 1.345 168

3 × 10⁸ 1.401 1980

10⁸ 1.401 1.78 × 10⁴

10⁷ 1.401 1.78 × 10⁶

10⁶ 1.401 1.78 × 10⁸

That is, during this period the effective number of particle species is N = 3.363. Using Eq. (3.1.14) then gives, in cgs units:

t =

3c² 3.363 · 16 · π aBG

1

T²+ constant = 1.78 sec

 T

10¹⁰K

−2

+ constant . (3.1.27) For instance, the time required for the universe to cool from a temperature of 10⁹K (where electrons and positrons have mostly annihilated) to a tem-perature of 10⁸ K is 1.76 × 10⁴sec, or 4.9 hours.

According to Eq. (3.1.27), the time required for the temperature to drop to 10⁶ K from much higher values is 1.78 × 10⁸ sec, or 5.64 years. At lower temperatures we must take into account the energy density of non-relativistic matter, and Eq. (3.1.27) no longer applies. We saw in Section 2.3 that for
Mh² = 0.15, it took an additional 360,000 years for the universe to cool to the temperature 3, 000 K of last scattering.

* * *

So far in this section we have been assuming that neutrinos are massless, and that the net neutrino number of each of the three types (that is, the number of neutrinos minus the number of antineutrinos) is much less than the number of photons. In the general case of an ideal gas of particles of mass m, the number n(p) dp of particles of momentum between p and p + dp is given by the Fermi–Dirac and Bose–Einstein distributions

n(p, T , µ) dp = 4π gp²dp (2π ¯h)³

 1

exp[(

p²+ m²− µ)/kBT ] ± 1



, (3.1.28) where µ is the chemical potential for the particle in question, a quantity that is conserved in any reaction occurring rapidly in thermal equilibrium, and again g is the number of spin states of the particle and antiparticle, and the sign is + for fermions and − for bosons. This reduces to Eq. (3.1.5) in the case of zero chemical potential, and it yields the number density (2.3.1) for non-relativistic particles with p  m and kBT  m. During the whole of the era of interest here, electrons and positrons rapidly annihilated into photons, so their chemical potentials were equal and opposite, and since we are assuming charge neutrality and neglecting the tiny number of baryons per photon, we can conclude that the chemical potentials of the electrons and photons were much less than k_BT . At temperatures at which neutrinos and antineutrinos were in thermal equilibrium with electrons, positrons, and photons, reactions like e⁺+ e⁻  νⁱ + ¯νi were occurring rapidly (where i = e, µ, τ label the three types of neutrino), so the chemical

3.1 Thermal history

potential µiof each type of neutrino was equal and opposite to the chemical potential of the corresponding antineutrino. But if we do not assume zero net neutrino (or lepton) number, then there is no a priori reason why the µi

had to be less than k_BT .

If neutrino masses are less than about 1 eV, then they may be neglected at the temperatures of interest in this section. The observations of oscillations between different flavors of neutrinos from the sun, nuclear reactors, and cosmic rays shows³that the two differences in the squares of the masses of the three types of neutrinos of definite mass (which are mixtures of neutrinos of electron, muon, and tau flavor) are 8.0^+0.4

−0.3× 10⁻⁵eV² and between 1.9 and 3.0 times 10⁻³eV². Thus the neutrino masses are all much less than 1 eV, unless they are highly degenerate, which there is no reason to expect. If degenerate, then from the absence of anomalies in the low-energy beta decay of tritium, their common mass must be less than 2 eV.³Whether degenerate or not, it is clear from this that all three neutrino types (if there are only three) have masses very much less than 1 MeV, and were therefore highly relativistic at the time that they went of thermal equilibrium with electrons and positrons, at a temperature of about 10¹⁰K. Once out of equilibrium, their momentum simply decayed as 1/a (as shown in Section 1.1), so if their chemical potential was negligible their momentum distribution remained the same as that of photons, with a temperature less by a factor (4/11)^1/3. Thus once k_BT dropped below the smallest neutrino mass, their energy den-sity became just n_ν

νm_ν = (3/11)n_γ 

νm_ν. (For k_BT much larger than the mass, the integral for the number density of each spin state of fermions is 3/4 the corresponding integral for bosons, and after neutrinos decouple T³

ν = (4/11)T_γ³.) With a non-zero chemical potential the energy density is larger. The agreement between theory (with massless neutrinos) and observation for the cosmic microwave background anisotropies discussed in Sections 2.6 and 7.2 and for the large scale structure discussed in Chapter 8 shows that the sum of the three neutrino masses is less than 0.68 eV (95%

confidence level),⁴ so if they are degenerate the common mass is less than 0.23 eV. This result has been contradicted by the observation of neutrinoless double beta decay in a Heidelberg–Moscow experiment,⁵which suggests a value greater than 1.2 eV for the sum of neutrino masses.⁶ There has not yet been an opportunity to confirm the double beta decay results, and for the present it seems reasonable to continue to neglect neutrino masses.

3W.-M. Yao et al. (Particle Data Group), J. Phys. G. 33, 1 (2006).

4D. N. Spergel et al., Astrophys. J. Suppl. Ser. 170, 377 (2007) [astro-ph/0603449].

5H. V. Klapdor-Kleingroth, I. V. Krivosheina, A. Dietz and O. Chkvorets, Phys. Lett. B 586, 198 (2004).

6A. De La Macorra, A. Melchiorri, P. Serra, and R. Bean, Astropart. Phys. 27, 406 (2007) [astro-ph/0608351].

With neutrino masses neglected, the energy density, pressure, and net lepton number density of neutrinos and antineutrinos of type i is

ρi = 3pi As we have seen, at temperatures above 10¹⁰ K the energy density of the photons and electron–positron pairs is 11 a_BT⁴/4, and the pressure is one-third as great, so the total energy density and pressure are given by

ρ = 3p = T⁴

The equation (1.5.20) of energy conservation tells us that under these circumstances ρa⁴ is constant, while the conservation of each type of neu-trino number also tells us that n_ia³ is constant. Since ρ and ni depend in different ways on the chemical potentials, this requires that as the universe expands in this era the µi/kBT remain constant, and also T ∝ 1/a, just as in the case of zero chemical potential.

As the temperature dropped below 10¹⁰ K the temperature of the photons and electron–positron pairs no longer varied as 1/a, but as we have seen the neutrinos and antineutrinos entered on a free expansion. With each neutrino’s momentum p varying as 1/a, the form of the Fermi–Dirac distributions for each type of massless neutrino was preserved, with a tem-perature T_ν ∝ 1/a and µi/kBT constant, just as before decoupling. We

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