Download 6 Departure from thermal equilibrium

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Departure from thermal equilibrium
Entropy conservation implies that the rate of change of the temperature is given by the expansion rate, Ṫ /T ' −ȧ/a = H, except when the temperature in near the mass of some particle.
Thus thermal equilibrium can be maintained only so long as particle interactions are rapid
compared to the Hubble expansion H. Suppose we have a scattering or annihilation process
a + b −→ c + d, which has a cross section σ (which has dimensions of area, or in units with
~ = c = 1, mass−2 ). Then the rate per particle to experience such a collision is roughly given
by Γ ' nσv, where n is the density of initial particles (either a or b) and v is their relative
velocity. You can check that Γ has dimension of inverse time, or mass. Thus for thermal
equilibrium to be maintained, we must have Γ > H.
Just on dimensional grounds it is easy to see that there will be various sorts of cases thermal
equilibrium will not be maintained:
1. In a radiation dominated universe, H ∝ T 2 /Mp , where
−1/2
Mp = G N
= 1.221 × 1019 GeV
is the Planck mass. If the scattering process only involves massless particles, including
in intermediate states, then by dimensional grounds it must go as Γ ∼ α2 T , where
α = g 2 /4π and g is a typical coupling constant in the process. Thus in the very early
universe, when T /Mp & α2 , this scattering process will be out of thermal equilibrium.
2. It can happen that only interaction light particles can experience involves the mediation of
a heavy virtual state with mass M . Neutrinos serve as a particularly important example,
as they only interact through the weak interactions, involving the exchange of massive
W or Z bosons. At temperatures T M the quantum mechanical amplitude for such a
process is proportional to 1/M 2 , and Γ, which is a probability, must be proportional to
1/M 4 . Since Γ has dimension of mass, it follows on dimensional grounds that Γ ∝ T 5 /M 4 .
Thus at low enough temperature, it will always happen that Γ < H and the processes go
out of thermal equilibrium.
3. There are other ways to depart from thermal equilibrium involving phase transitions. In
quantum field theory, the vacuum we live in as very complex, and like any ground state
in a condensed matter system, it can undergo phase transitions as the temperature is
changed. Examples of phase transitions we believe exist include electroweak symmetry
breaking, associated with a temperature Teq ' 100 GeV, where the Higgs gets its vacuum
expectation value and gives mass to W , Z, and fermions in the standard model. The
chiral symmetry breaking phase transition where quarks and anti-quarks condense is
associated with the strong interactions, at a temperature Tstrong ' 1 GeV. One can
invent many other theories containing phase transitions, such as axion models or Grand
Unified Theories (GUTS). If any of these phase transitions are first order, then the
universe can experience supercooling as the universe expands, followed by an explosive
boiling period when the true ground state supplants the metastable state that the universe
finds itself in. Phase transitions are also associated with the production of topological
defects, which behave like heavy particles or extended objects, and which are often out
of thermal equilibrium.
4. Coherent oscillations of scalar fields are sources of possible departure from thermal equilibrium in the early universe. A scalar field takes on a value not at the minimum of
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its potential, and oscillates about the minimum; provided that the field is weakly coupled, its oscillations will decay slowly as its energy is dissipated in particle production.
Examples of such fields that have been contemplated include the axion (a dark matter
candidate), the inflaton (the cause of an epoch of exponential growth in the scale factor a), quintessence (a theory for the dark energy), as well as various fields invented to
explain baryogenesis.
Let us do a rough calculation for the temperature Tdec where neutrinos decouple. Any low
energy 2 ↔ 2 type of interaction involving neutrinos must have an amplitude proportional to
g 2 /M 2 ' GF where g is the weak boson coupling and M is its mass; GF is the Fermi constant,
given by
GF = 1.1664 × 10−5 GeV−2 .
(129)
Therefore Γ ∝ G2F , and on dimensional grounds, Γ ' G2F T 5 . Equating this to H ' T 2 /MP at
T = Tdec we get
Tdec ' MP G2F
1/3
' 1 MeV .
(130)
This estimate is actually very close to the correct answer. The value of Tdec is quite important
for the theory of nucleosynthesis, which we will discuss shortly.
Another type of non-equilibrium phenomenon of interest is the abundance of relic massive
particles. Suppose that the temperature drops below the mass of a stable particle. The
equilibrium number distribution is then ∝ e−m/T , but in order to reduce their population, the
particles and anti-particles must find each other to annihilate. If the annihilation rate is too
slow, then the particles go out of equilibrium, and there will be a residual population of these
heavy particles around today. This is called relic freeze-out, and could be a problem or a
virtue, depending on whether or not the energy density in these particles is similar to what is
required to explain the dark matter we indirectly observe.
Relic freeze-out can be studied by means of the Boltzmann equation. The non-relativistic
Boltzmann equation is given by L̂f = C[f ], where f (x, v, t) is the particle distribution in
phase space, L̂ is the differential operator
L̂ =
d
∂
F
=
+ v · ∇x +
· ∇v ,
dt
∂t
m
(131)
and C is a collision term that describes how particles are scattered into or out of a particular
cell in phase space. We need to come up with a version of this equation which is locally Lorentz
invariant, and which properly accounts for gravitational forces in the L̂ operator. We take f
to be a function of momentum instead of velocity, f = f (xµ , pν ), and replace the terms in mL̂,
by
∂
∂
∂
m
+ v · ∇x −→ pα α ,
F · ∇v −→ −Γαβγ pβ pγ α .
(132)
∂t
∂x
∂p
The second substitution follows by replacing the acceleration d2 xi /dt2 in the non-relativistic
expression for L̂ by d2 xα /dλ2 in the geodesic equation eq. (23), where the parameter λ is taken
to be the proper time τ along the path of the particle, using mdxα /dτ = pα . Recall that a
tenet of GR is that the geodesic equation accounts for the effect of all gravitational forces on
the particle.
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Specializing to the isotropic and homogeneous FRW metric, f becomes a function of only
t and p0 = E, with
∂
2 ∂
mL̂f = E − H|p |
f .
(133)
∂t
∂E
Here H = ȧ/a arises from the FRW value Γ0µν = −Hgij , where gij is the spatial part of the
FRW metric eq. (52) (which comes with a negative signature). This can be simplified further
by considering an equation for the density na of particles of type a:
Z
Z
d3 p
d3 p
∂
2 ∂
E − H|p |
fa
ga
ma L̂fa = ga
(2π)3
(2π)3
∂t
∂E
dna
=
+ 3Hna ,
(134)
dt
where to get the result I integrated by parts and used the expression for the density
Z
d3 p
na (t) = g
fa .
(2π)3
(135)
For the right-hand side of the Boltzmann equation for the process a + b → c + d we write
3
R d3 p
RQ
d pi
4 4
ma Cnr [fa ] −→ −
ga (2π)
3
i=a,b,c,d gi 2Ei (2π)3 (2π) δ (ptot )
h
i
(136)
2
2
× Mab→cd fa fb (1 ± fc )(1 ± fd ) − Mcd→ab fc fd (1 ± fa )(1 ± fb )
where we have replaced the non-relativistic collision term Cnr by its covariant version. The
scattering amplitude M is to be computed from quantum field theory; we have assumed here
that the mean-free-path of particles is short enough here that the amplitudes in flat space can
be used. The (1 + f ) or (1 − f ) terms are the stimulated emission or Pauli blocking terms
relevant for bosons or fermions respectively.
Now I will restrict the discussion to annihilation processes X X̄ → ψ ψ̄ and consider relic
X particles. Consider the X’s to be massive, weakly interacting particles, for example, which
decay into states that are not weakly interacting I will also make a number of simplifying
assumptions:
• time reversal invariance is assumed (equivalent to CP -invariance, in a relativistic field
2
2 theory) so that Mcd→ab = Mab→cd ;
• no chemical potentials for any of the species, and Maxwell-Boltzmann statistics so that
we can replace (1 ± f ) → 1.
• the ψ, ψ̄ products are in thermal equilibrium.
• the X particles are in kinetic equilibrium, but not necessarily in chemical equilibrium.
This means that their distribution fX is proportional to the equilibrium distribution, but
has the wrong normalization. This can occur if X-number conserving interactions are a
lot faster than annihilations, which will always be the case if X density is small, since the
latter process will go as the density of X particles to the second power, since the scarce
X and X̄ particles have to find each other to annihilate, but not to exchange energy with
other particles in the medium.
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With these simplifications we get
Z
Y
2 d3 p i
dnX
eq eq
4 4
gi
+ 3HnX = −
(2π)
δ
(p
)
|M
−
f
f
f
f
tot
X X̄
X X̄→ψ ψ̄
ψ ψ̄ (137)
dt
2Ei (2π)3
i=X,X̄,ψ,ψ̄
Because of energy conservation enforced by the δ-function,
eq eq
fX̄ ,
fψeq fψ̄eq = e−(Eψ +Eψ̄ )/T = e−(EX +EX̄ )/T = fX
(138)
while by our assumption of kinetic equilibrium for the X particles and anti-particles, we have
fX fX̄ =
nX
neq
X
2
eq eq
fX
fX̄ .
(139)
Thus our equation can be written as
dnX
2
+ 3HnX = −hσa vi (nX )2 − (neq
,
X)
dt
(140)
where the thermally averaged annihilation cross section is defined to be
Z
Y
d3 p i
1
gi
(2π)4 δ 4 (ptot ) |MX X̄→ψψ̄ |2 e−(EX +EX̄ )/T . (141)
hσa vi ≡ eq 2
2Ei (2π)3
(nX )
i=X,X̄,ψ,ψ̄
Eq. 140 can be massaged into a more useful form. First, define
YX = nX /s ,
(142)
where s is the entropy density, and use the conservation of entropy d(a3 s)/dt to show that
dnX
+ 3HnX = sẎX .
dt
(143)
ξ = mX /T ;
(144)
Second, define the variable
assuming radiation domination, we know that H = 1/(2t) ∝ T 2 , so that
d/dt = Hξd/dξ .
(145)
Γa ≡ neq
X hσa vi .
(146)
Finally, define the annihilation rate
Then our Boltzmann equation eq. (140) can be expressed as
#
"
YX 2
ξ dYX
Γa
−1 .
=−
H
YXeq dξ
YXeq
You will analyze this equation in your next problem set.
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(147)