Boltzmann equation for non-equilibrium particles and its application to non-thermal dark matter production

0.28 MB·English

by  Koichi Hamaguchi

UT-11-39
IPMU-11-0190
November, 2011
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1
Boltzmann equation for non-equilibrium particles and
0
2
its application to non-thermal dark matter production

n
a
J

9
2

] Koichi Hamaguchi(a,b), Takeo Moroi(a,b) and Kyohei Mukaida(a)
h
p
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p a Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
e
h b Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa
[
277-8568, Japan

2
v
4
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5
4 Abstract
.
1
We consider a scalar field (called φ) which is very weakly coupled to thermal bath,
1
1 and study the evolution of its number density. We use the Boltzmann equation derived
1 from the Kadanoff-Baym equations, assuming that the degrees of freedom in the ther-
:
v mal bath are well described as “quasi-particles.” When the widths of quasi-particles
i
X are negligible, the evolution of the number density of φ is well governed by a simple
r Boltzmann equation, which contains production rates and distribution functions both
a
evaluated with dispersion relations of quasi-particles with thermal masses. We pay par-
ticular attention tothecasethatdarkmatter is non-thermallyproducedbythedecay of
particles in thermal bath, to which the above mentioned formalism is applicable. When
the effects of thermal bath are properly included, the relic abundance of dark matter
may change byO(10 100 %)compared totheresultwithouttaking account of thermal
−
effects.
1 Introduction
In particle cosmology, it is inevitable to consider the behavior of quantum fields (or, in other
words, particles) in thermal bath because the universe was filled with hot plasma in the early
epoch. The detailed thermal effects depend on how the particle of our interest, which we
call φ, interacts with degrees of freedom in thermal bath. Importantly, even if φ is so weakly
interacting that itis not inthermal equilibrium, therecan benon-negligiblethermal effects on
its dynamics. This is because the interaction rate of φ surrounded by thermal bath depends
on the properties (in particular, the dispersion relation) of the degrees of freedom in thermal
bath, which can of course be significantly affected by thermal effects.
One important example of such a very weakly interacting particle is non-thermal dark
matter which is produced by the decay of particles in thermal bath. Although the existence
of dark matter is strongly suggested by various cosmological observations [1], particle-physics
properties of dark matter, as well as its production mechanism in the early universe, have not
been understood yet. Various particle physics models including dark matter candidate have
been proposed so far, like supersymmetric models, universal extra dimension models, and so
on [2]. In the future, it is hoped that those models are tested by high energy experiments as
well as by cosmological observations. In particular, the candidate of the dark matter particle
may be discovered and studied by the LHC experiment as well as future linear colliders,
based on which a large class of dark matter models are discriminated. For this program,
precise theoretical calculation of the relic abundance of the dark matter candidate should
be performed by using information about newly discovered particles [3]. The present dark
matter density is very accurately determined by the WMAP collaboration as [4]:
Ω h2 = 0.1126 0.0036, (1.1)
c
±
with h being the Hubble constant in units of 100 km/sec/Mpc, so the dark matter abun-
dance is now known with O(1 %) accuracy. Thus, it is desirable to establish methods of
calculating the dark matter density at the same level of accuracy. For this purpose, detailed
understanding of thermal effects on the dark matter production process is required.
In the present study, we pay particular attention to the case that dark matter particle is
non-thermally produced by the decay of heavier particles in thermal bath. There are many
examples of such non-thermally produced dark matter, like gravitino [5], axino [6], a singlet
field [7], the right handed sneutrino [8], and more generally, the recently proposed “freeze-in”
particles [9]. In such a scenario, the relic density of dark matter is determined at the cosmic
temperature comparable to the mass of decaying particle and is insensitive to the thermal
history before that. In the previous studies, the production rate of dark matter has been
calculated by using the decay rates of particles estimated in vacuum. However, in the actual
situation, the particles decay in the thermal bath, so the production rate taking account of
the thermal effects should be properly used in the calculation of the dark matter density.
As we will see, the thermal effects may significantly change the resultant abundance of dark
matter.
1
In this paper, we raise the question how important the thermal effects are in the pro-
duction process of non-thermal dark matter. To answer this question, we first study the
properties of Boltzmann equation derived from the Kadanoff-Baym equations [10] under the
assumption that the production of dark matter does not affect the thermal bath. In partic-
ular, unless the dark matter production is almost kinematically blocked by thermal masses
during the time when the production of dark matter is most effective, the full Boltzmann
equation canbereduced to a simplified formwhich hasthe same structure asthe conventional
Boltzmann equation#1 but constructed with “thermal masses” of particles in thermal bath.
(As we will see, such a simplified Boltzmann equation is obtained by taking the “zero-width
approximation” of particles in thermal bath.) We evaluate the relic density of dark matter by
solving (i) the full, (ii) zero-width approximated, and (iii) conventional Boltzmann equations.
Comparing the three results, we discuss how important the thermal effects are and when the
zero-width approximation breaks down. We will see that the dark matter abundance may be
reduced by O(10 100 %) compared with the result of calculation where the thermal effects
−
are neglected.
Theorganizationofthispaperisasfollows. InSection2,therelevantformulaetostudythe
evolution of the number density of non-equilibrium particles are summarized. In particular,
properties of the Boltzmann equation to be solved are discussed. Then, in Section 3, we apply
the formalism to the non-thermal dark matter production process. We numerically solve the
Boltzmann equation and discuss how important the thermal effects are. Section 4 is devoted
to conclusions and discussion.
2 Formalism
First, let us introduce the formulaeand equations relevant for our analysis. Althoughmany of
themcanbefoundinliterature(see, forinstance, [10,12,13,14,15,16,17,18,19,20,21,22]),
we summarize the relevant equations to make this paper self-contained for the sake of readers.
We assume that the thermal bath have a common temperature T, which have a large degrees
of freedom, and the back reaction to the thermal bath from the production of φ is negligible.
In this paper, we study the evolution of the number density of a scalar field φ coupled to
scalar fields in thermal bath, which are denoted as χ . We introduce the interaction of the
i
following form:
n
= gφ χ = gφ [χ ,χ , ], (2.1)
int i 0 1
L O ···
i=1
Y
where g is a coupling constant and, for the convenience of the following discussion, we in-
troduced the operator χ . The interaction of φ is assumed to be extremely small,
O ≡ i i
#1 In this paper, the “conventioQnal” Boltzmann equation refers to the Boltzmann equation evaluated with
zero-temperature dispersion relations [11].
2
i.e., gmn−3 1, where m is the mass of φ.#2 In the following, we study the effects which
φ ≪ φ
are leading order in g. For simplicity, we consider the case that is given by a product of
O
scalar fields. However, the extension of the formalism to the case that includes derivatives
O
of scalar fields is straightforward.
The evolution of the number density of φ in the early universe is governed by two effects:
one is the production of φ due to the decay and scattering processes and the other is the
cosmic expansion. We discuss these effects separately.
Because of the weakness of the interaction, φ can be regarded as (almost) free particle,#3
and the number density operator is given by
1 ˙ ˙
Nˆ (t) = : φˆ(t;k)φˆ(t; k)+ω2φˆ(t;k)φˆ(t; k) :, (2.2)
k 2ω − k −
k
h i
where the “dot” denotes the derivative with respect to time, : : is the normal ordering
···
and
ω k2 +m2. (2.3)
k ≡ φ
q
In addition,
φˆ(t;k) L−3/2 d3xe−ik·xφˆ(t,x), (2.4)
≡
Z
where φˆis the field operator for φ and L3 is the volume of the system (where we have adopted
box normalization). For the later convenience, the number density for each momentum
eigenstate is defined. Then, the expectation value of the number density of φ is obtained by
using the density matrix ρˆ:
N (t) Nˆ (t) , (2.5)
k k
≡ h i
where, for an operator Aˆ,
Aˆ tr[ρˆAˆ]. (2.6)
h i ≡
The χ sector is in the thermal bath while φ is always out of thermal equilibrium. In
particular, we are interested in the case that φ is initially absent in the system. Thus, we
assume that the initial density matrix (at t = t ) is given by the direct product of density
i
matrices of two sectors:
ρˆ= ρˆ ρˆ . (2.7)
φ,i χ
⊗
#2 Inthecaseoffreeze-indarkmatterwithaweakscalemass,typicallyacouplingofO(10−13)isnecessary
to obtain the correct dark matter abundance [8, 9].
#3We assume that the self-interaction of φ is absent or weak enough to be neglected. We also assume that
the thermal effects on the expectation value of φ is negligible, which is the case in the example in Section 3.
3
We set t = 0 without loss of generality. The χ sector is in the thermal bath (with the
i
temperature T), so ρˆ is given by
χ
ρˆ = e−Hˆχ/T, (2.8)
χ
with Hˆ being the Hamiltonian for the χ sector. On the contrary, ρˆ determines the initial
χ φ,i
distribution of φ. We assume that it has translational invariance, i.e., [Pˆ,ρˆ ] = 0, where Pˆ
φ,i
˙
ˆ ˆ
is the momentum operator. Furthermore, we assume φ(t = 0) = φ(t = 0) = 0.
h i h i
In order to calculate the evolution of N (t), we define the Hadamard propagator and the
k
Jordan propagator:
Gφ(t,t′;k) φˆ(t;k)φˆ(t′; k) + φˆ(t′; k)φˆ(t;k) , (2.9)
H ≡ h − i h − i
Gφ(t,t′;k) φˆ(t;k)φˆ(t′; k) φˆ(t′; k)φˆ(t;k) . (2.10)
J ≡ h − i−h − i
As can be seen from Eqs. (2.2) and (2.5), the expectation value of the number density is given
by
1
Nk(t) ≡ 4ωk (∂t∂t′ +ωk2)GφH(t,t′;k) t′→t −Ck, (2.11)
h i
where C is normal-ordering constant. In the weak coupling limit, C = 1. The Hadamard
k k 2
propagator and the Jordan propagator satisfy the following equations, namely the Kadanoff-
Baym equations [10, 12]
t
∂2 +ω2 Gφ(t,t′;k) = dτΠφ (t τ;k)Gφ(τ,t′;k), (2.12)
t k J − ret − J
Zt′
(cid:0) (cid:1) t
∂2 +ω2 Gφ(t,t′;k) = dτΠφ (t τ;k)Gφ(τ,t′;k)
t k H − ret − H
Z0
(cid:0) (cid:1) t′
i dτΠφ (t τ;k)Gφ(τ,t′;k), (2.13)
− H − J
Z0
where
Πφ (t;k) = iθ(t) Πφ(t;k) Πφ(t;k) , (2.14)
ret − > − <
Πφ (t;k) = Πφ(t;k(cid:16))+Πφ(t;k). (cid:17) (2.15)
H > <
Here, at the leading order (i.e, O(g2)),
g2
Πφ(t;k) = tr e−Hˆχ/T ˆ(t;k) ˆ(0; k) , (2.16)
> tr[e−Hˆχ/T] O O −
h i
g2
Πφ(t;k) = tr e−Hˆχ/T ˆ(0; k) ˆ(t;k) , (2.17)
< tr[e−Hˆχ/T] O − O
h i
4
with
ˆ(t;k) = L−3/2 d3xe−ik·x ˆ(t,x). (2.18)
O O
Z
Let us define the Fourier transformations
Πφ (ω,k) = dteiωtΠφ (t;k), (2.19)
X X
Z
with Π = Π , Π , Π , and Π . Then, from Eq. (2.14),
X ret H < >
dω′Πφ(ω′,k) Πφ(ω′,k)
Πφ (ω,k) = > − < . (2.20)
ret 2π ω ω′+i0
Z −
By using the relation (ω+i0)−1 = ω−1 iπδ(ω) (with denoting the principal value), we
P − P
obtain
1
Πφ (ω,k) = Πφ(ω,k) Πφ(ω,k) , (2.21)
ℑ ret −2 > − <
h i
and
ω
Πφ (ω,k) = 2coth Πφ (ω,k), (2.22)
H − 2T ℑ ret
(cid:16) (cid:17)
where we have used the so-called Kubo-Martin-Schwinger (KMS) relation [13] Π (ω,k) =
>
exp(ω/T)Π (ω,k) in deriving Eq. (2.22).
<
Using the fact that the Jordan propagator is time translational invariant within our
setup,#4 Gφ(t,t′;k) = Gφ(t t′,0;k), it can be expressed in terms of the spectral density ρ
J J − φ
as#5
dω
Gφ(t;k) Gφ(t,0;k) e−iωtρ (ω,k), (2.23)
J ≡ J ≡ 2π φ
Z
and the solution to Eq. (2.12) is given by
2 Πφ (ω,k)
ρ (ω,k) = − ℑ ret . (2.24)
φ [ω2 ω2 Πφ (ω,k)]2 +[ Πφ (ω,k)]2
− k −ℜ ret ℑ ret
We note that the initial condition is given by Gφ(0;k) = 0 and ∂ Gφ(0;k) = i because of
J t J −
equal time commutation relations.
#4This is because, under no self interactions of φ and truncating the perturbative expansion at (g2), the
O
spectrum of φ is determined only by the thermal bath regardless of the number density of φ. The time
translational invariance of the Jordan propagator is not a general property. See also [12].
#5The spectral density ρ should not be confused with the density matrix ρˆ .
φ φ,i
5
With the initial condition of the Jordan propagator, the Hadamard propagator satisfying
the Kadanoff-Baym equations (2.12) and (2.13) is obtained as follows [12]:
t t′
Gφ(t,t′;k) = Ghom(t,t′;k)+ dt dt Gφ(t t ;k)Πφ (t t ;k)Gφ(t t′;k), (2.25)
H H 1 2 J − 1 H 1 − 2 J 2 −
Z0 Z0
where the homogeneous solution is given by
GhHom(t,t′;k) = − GφH(s,s′;k) s,s′=0∂t∂t′GφJ(t;k)GφJ(t′;k)
(cid:12)
− ∂sGφH(s,s′;k(cid:12)(cid:12)) s,s′=0(∂t +∂t′)GφJ(t;k)GφJ(t′;k)
(cid:12)
− ∂s∂s′GφH(s,s′;(cid:12)(cid:12)k) s,s′=0GφJ(t;k)GφJ(t′;k). (2.26)
(cid:12)
(cid:12)
In the calculation of the production rate of φ i(cid:12)n thermal bath, the most important effect
is the shift of the pole of the spectral density because its imaginary part gives the production
rate. Here, we are interested in the case that the interaction of φ is so small that ω2 Πφ .
k ≫ | ret|
Then, the spectral density can be well approximated by the Breit-Wigner form:
2ωΓ (k)
ρ(BW)(ω,k) = φ , (2.27)
φ (ω2 ω2)2 +(ωΓ (k))2
− k φ
where
Πφ (ω ,k)
Γ (k) ℑ ret k . (2.28)
φ
≡ − ω
k
Although Γ has the argument k in our expression, it depends only on k because of the
φ
| |
rotational invariance of the thermal bath. Here, we neglected the correction to the real part
of the pole, which is expected to be irrelevant. In addition, notice that Γ (k) is of O(g2), and
φ
is much smaller than ω . Then, from Eq. (2.23), the Jordan propagator (for t 0) is well
k
≥
approximated as
sinω t
iGφ(t;k) = k e−Γφ(k)t/2, (2.29)
J t≥0 ωk
(cid:12)
(cid:12)
resulting in the following expression f(cid:12)or the expectation value of the number density defined
in Eq. (2.5), at leading order in Γ /ω ,
φ k
N(BW)(t) = f (ω ) 1 e−Γφ(k)t
k B k −
1
+4ωk (cid:20)(cid:0)GφH(s,s′;k) (cid:1)s,s′=0ωk2 + ∂s∂s′GφH(s,s′;k) s,s′=0 −2ωk(cid:21)e−Γφ(k)t,
(cid:12) (cid:12)
(cid:12) (cid:12) (2.30)
(cid:12) (cid:12)
6
with
1
f (ω) = . (2.31)
B eω/T 1
−
Equivalently, irrespective of the initial condition, one finds
N˙(Coll) = Γ (k)[f (ω ) N ], (2.32)
k φ B k − k
from which we can see that Γ (k) can be regarded as the production rate of φ due to the
φ
decays and scatterings of particles in thermal bath. (Here, the superscript “(Coll)” implies
that this is the collision term in the Boltzmann equation.)
As we have mentioned, there is another effect on the evolution of the φ’s number density,
which is the expansion of the universe. Effect of the cosmic expansion can be easily evaluated
by taking into account the red-shift of the momentum and we obtain
∂N
N˙(Exp) = Hk k, (2.33)
k · ∂k
where the superscript “(Exp)” is for cosmic expansion, and H denotes Hubble parameter of
the expanding universe. We assume that the energy density of the thermal bath is much
larger than that of φ, and that the Hubble parameter depends only on the temperature T.
Combining two effects, the Boltzmann equation to be solved is
∂N
N˙ Hk k = Γ (k;T)[f (ω ;T) N ], (2.34)
k − · ∂k φ B k − k
or, for the total number density
d3k
n N , (2.35)
φ ≡ (2π)3 k
Z
the Boltzmann equation is given by
dn
φ (Coll)
+3Hn = n˙
dt φ φ
d3k
Γ (k;T)[f (ω ;T) N ]. (2.36)
≡ (2π)3 φ B k − k
Z
In the above equations, we explicitly show that Γ andf depend onT, which should be iden-
φ B
tified withthecosmic temperature. Thus, themost important quantity tostudy theevolution
of the abundance of φ is the production rate Γ (k;T) given in Eq. (2.28) or, equivalently,
φ
Πφ given in Eq. (2.21). With the operator given in Eq. (2.1), the leading contribution
ℑ ret O
to Πφ is given by
ℑ ret
g2 d4p
Πφ (ω ,k) = i Gχi(p0,p ) (2π)4δ(ω p0)δ(3)(k p )
ℑ ret k 2 (2π)4 < i i k − i i − i i
" #
Z i
Y P P
g2 d4p
i Gχi(p0,p ) (2π)4δ(ω p0)δ(3)(k p ).
− 2 (2π)4 > i i k − i i − i i
" #
Z i
Y P P
(2.37)
7
Here, all the χ -fields are taken to be independent. If there exist identical fields in χ in
i i
{ }
Eq. (2.1), a symmetry factor is needed. In the above expression, the functions Gχi and Gχi
> <
are defined as
1
Gχi(p0,p) = d4xeip0x0−ip·xtr[e−Hχ/Tχ (x0,x)χ (0,0)], (2.38)
> tr[e−Hχ/T] i i
Z
1
Gχi(p0,k) = d4xeip0x0−ip·xtr[e−Hχ/Tχ (0,0)χ (x0,x)]. (2.39)
< tr[e−Hχ/T] i i
Z
These functions satisfy the KMS relation Gχi(p0,p) = ep0/TGχi(p0,p), whereas their differ-
> <
ence, the Jordan propagator, is expressed in terms of the spectral density:
Gχi(p0,p) = Gχi(p0,p) Gχi(p0,p) = ρ (p0,p). (2.40)
J > − < χi
Thus, they are given by
Gχi(p0,p) = (f (p0)+1)ρ (p0,p), (2.41)
> B χi
Gχi(p0,p) = f (p0)ρ (p0,p). (2.42)
< B χi
If the quasi-particle picture is applicable to χ , which we assume in the following analysis,
i
the spectral density can be approximated by the Breit-Wigner form:
2p0Γ (p;T)
ρ(BW)(p0,p) = χi , (2.43)
χi (p02 Ω2 (p;T))2 +(p0Γ (p;T))2
− χi χi
where Ω and Γ are real and imaginary parts of the pole of the propagator, and are related
χi χi
to the real and imaginary parts of the self energy of χ . Contrary to the case of φ, we need
i
to take account of the shirt of the pole for particles which are thermalized. Thus, Ω may
χi
significantly deviate from the frequency satisfying the on-shell condition in the vacuum. In a
large class of models, including the case discussed in the following section, Ω2 (p;T) can be
χi
well approximated as
Ω2 (p;T) = p2 +m˜2 (T), (2.44)
χi χi
where m˜2 (T) is given by the sum of bare and thermal masses.
χi
The effect of non-vanishing Γ will be numerically studied in the next section. Here,
χi
we comment that, if Γ is small enough, the evolution of the total number density of φ is
χi
governed by a simple differential equation. If the interaction is perturbative, it is usually
the case that Γ (p;T) m˜ (T). In such a case, the (Breit-Wigner) spectral density is
χi ≪ χi
well approximated as ρ (p0,p;T) 2πsign(p0)δ(p02 p2 m˜2 (T)), and hence Gχi(ω,k)
χi ≃ − − χi >
and Gχi(ω,k) are also (approximately) proportional to the δ-function. We call this limit as
<
zero-width limit. Then, the collision term in Eq. (2.36) becomes
n˙(Coll) = g2 dΠ(k0>0)(k) dΠ (p ) (2π)4δ(k0 p0)δ(3)(k p )
h φ iΓχi→0 Z φ hY χi i i −Pi i −Pi i
(1+f (p0))sign(p0) f (p0)sign(p0) f (k0) N , (2.45)
B i i − B i i B − k
" #
i i
Y Y (cid:2) (cid:3)
8
where
d4p
dΠ (p ) = i δ(p2 m˜2 (T)), (2.46)
χi i (2π)3 i − χi
and dΠ (k) is defined in the same way. (Here, we have introduced the four-component vector
φ
as p = (p0,p ).) Notice that, in Eq. (2.45), the p0 integration is performed in the region
i i i i
< p0 < , while k0 integration is for k0 > 0.
−∞ i ∞
It is notable that the collision term given in Eq. (2.45) includes all the relevant scattering
and decay processes (and their inverse processes). Regarding χ as scalar particles with
i
masses m˜ (T), the integrand of the collision term becomes non-vanishing if (and only if)
χi
the momentum configurations are kinematically allowed. In addition, Eq. (2.45) contains
the effect of induced emission. For example, if the scattering process χ (p ) χ (p )
1 1 I I
··· ↔
φ(k)χ′(q ) χ′ (q ) is kinematically allowed, the collision term contains
1 1 ··· F F
n˙(Coll) g2 dΠ(p0>0)(p ) dΠ(q0>0)(q ) dΠ(k0>0)(k)
h φ iΓχi→0 ⊃ Z hY χi i ihY χ′f f i φ
(2π)4δ(k0 + q0 p0)δ(3)(k+ q p )
f f − i i f f − i i
[ f (p0)][ (1+f (q0))](1+N ) [ (1+f (p0))][ f (q0)]N ,
i B i Pf PB f kP− i P B i f B f k
(2.47)
(cid:8) Q Q Q Q (cid:9)
where we have used the relation f ( ω) = (1+f (ω)). The right-hand side of the above
B B
− −
equation has the same structure as the collision term in conventional Boltzmann equation;
however, notice that the thermally corrected dispersion relations should be used in evaluating
both the phase-spaces and distribution functions.
3 Application to non-thermal dark matter production
Now, let us apply the formalism to the non-thermal dark matter production scenario, which
is recently called freeze-in scenario, regarding φ as dark matter. In such a scenario, the
dark matter particle is always out of thermal equilibrium because of the weakness of its
interaction, and is produced by the decay of particles in thermal bath, χ . Thus, this is the
i
situation where we can safely use the formalism discussed in the previous section. We pay
particular attention to the question how large the thermal effect can be. For this purpose, we
numerically calculate the production rate and the relic abundance of φ using the formalism
presented in the previous section.
Here, we consider the simplest form of the interaction term, which is
= gφ [χ ,χ ] = gφχ χ . (3.1)
int 0 1 0 1
L O
For simplicity, we concentrate on the case that φ, χ , and χ are all real scalars, with masses
0 1
satisfying m > m + m . In our numerical calculations, we take m = 100 GeV and
χ0 φ χ1 χ0
9