学位論文
Decaying dark matter cosmology with arbitrary decay
mass products
(有限な質量の粒子を伴って崩壊する暗黒物質の宇
宙論)
青山尚平
Contents
1 Introduction
1.1 Dark matter . . . . . . . . . . . . . . . . . . . . . .
1.2 Property of the dark matter . . . . . . . . . . . . .
1.3 Cold dark matter (CDM) . . . . . . . . . . . . . . .
1.4 Present temperature of CDM particle . . . . . . . .
1.5 CDM crisis . . . . . . . . . . . . . . . . . . . . . .
1.6 Dark matter models—Candidates for the solutions
problems . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Self-annihilating dark matter (SADM) . . .
1.6.2 Warm dark matter (WDM) . . . . . . . . .
1.6.3 Decaying dark matter (DDM) . . . . . . . .
1.7 Our DDM model . . . . . . . . . . . . . . . . . . .
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2 The effect of DDM on the background geometry
2.1 Formulation . . . . . . . . . . . . . . . . . . . . . .
2.2 Calculation of background distribution functions . .
2.2.1 Mother particle . . . . . . . . . . . . . . . .
2.3 Daughter particle . . . . . . . . . . . . . . . . . . .
2.4 Result and Discussion . . . . . . . . . . . . . . . .
2.4.1 Background energy densities . . . . . . . . .
2.4.2 Constraints from Hubble constant, BAO and
2.4.3 Massless limit of daughter particle . . . . . .
2.4.4 Free-streaming scale . . . . . . . . . . . . .
2.4.5 Comparison with Peter et al. . . . . . . . .
2.5 Summary in this section . . . . . . . . . . . . . . .
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CMB
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3 The effect of DDM on observables of cosmological fluctuations
3.1 Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . .
3.2 Initial condition for the perturbed distribution functions of the
daughter particles . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Radiation-dominated era . . . . . . . . . . . . . . . .
3.2.2 Matter-dominated era . . . . . . . . . . . . . . . . . .
3.3 Time evolutions of perturbations . . . . . . . . . . . . . . . .
3.3.1 A case of non-relativistic decay . . . . . . . . . . . . .
3.3.2 Perturbations crossing the horizon after the decay time
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3.4
3.5
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3.3.3 Perturbations crossing the horizon before the decay time
3.3.4 A case of relativistic decay . . . . . . . . . . . . . . . .
Signatures in cosmological observables . . . . . . . . . . . . .
3.4.1 Effects on Cl . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Effects on P (k) . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Implication on the anomaly in estimated σ8 from Planck
Summary in this section . . . . . . . . . . . . . . . . . . . . .
4 Cosmological parameter estimation including
4.1 Interpolation formula . . . . . . . . . . . . . .
4.2 Result . . . . . . . . . . . . . . . . . . . . . .
4.3 Discussion . . . . . . . . . . . . . . . . . . . .
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DDM
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5 Conclusion
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6 Acknowledgment
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A General Relativistic Cosmology
A.1 General relativity . . . . . . . . . . . . . . . . . . . . . . .
A.2 Introduction to metric . . . . . . . . . . . . . . . . . . . .
A.2.1 The cosmological principle and FLRW metric . . .
A.3 The evolution of space-time and metric . . . . . . . . . . .
A.3.1 Gauge and metric perturbation . . . . . . . . . . .
A.3.2 Scalar, vector, tensor decomposition . . . . . . . . .
A.3.3 Synchronous gauge and conformal Newtonian gauge
A.4 Comic microwave background (CMB) . . . . . . . . . . . .
A.4.1 The physics of CMB temperature anisotropy . . . .
A.4.2 Boltzmann equation . . . . . . . . . . . . . . . . .
A.4.3 CMB photons . . . . . . . . . . . . . . . . . . . . .
A.4.4 Tight coupling approximation . . . . . . . . . . . .
A.4.5 Acoustic oscillation of baryon-photon plasma . . . .
A.4.6 Silk damping . . . . . . . . . . . . . . . . . . . . .
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B Sunyaev-Zel’dovich effect and its number counts
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100
Abstract
The Universe is consisted of a lot of stars and galaxies. These objects
are originated from tiny density fluctuations whose amplitudes in the early
Universe are O(10−5 ). In order to understand the structure formation from
such a tiny density fluctuation to wide variety of structures, the existence of
dark matter is indispensable. The resolution of the property of dark matter
is important for cosmology and elementary particle physics. Dark matter is
the matter which does not have the electro-magnetic and the strong interaction. Although it has passed more than seventy years since dark matter
was discovered, we have little knowledge on the nature of dark matter. It is
widely believed that dark matter consists of particles beyond the Standard
Model. In the standard model of particle physics, many particles decay into
other particles such as electrons and photons. These decay processes are inseparably connected to the interaction which the progenitor particles have.
These decaying process can be characterized by the lifetime. Such a dark
matter is called decaying dark matter (DDM).
We focus on the dark matter which decays into two particles. In this case,
the decaying process can be characterized three parameters regardless of the
Lagrangian. They are the lifetime Γ−1 and the mass ratio of two product
particles to progenitor particle r ≡ mD1 /mM , mD2 /mM . Ichiki et al. (2004)
considered the case that both two product particles are massless particles.
They calculated the effect of dark matter decaying on the power spectrum
of the temperature fluctuation of the cosmic microwave background (CMB)
and constraint on the lifetime of the progenitor particle. On the other hand,
Wang et al. considered the case that one of product particles is created
non-relativistic and the mass of the other is massless and the effect on the
observed Lyman-α forest and set a constraint on the relation between the
life time and the mass ratio by comparing the observation. However previous
studies could not deal with DDM models where mother particles decay into
particles with arbitrary mass at late times. This is what we will explore in
this thesis.
In this thesis, we investigate models of DDM which decays into two daughter particles. We study the time evolution of the background energy density
linear cosmological perturbations in the DDM model without approximations
such that the decay products are massless or highly non-relativistic. Instead,
we directly solve the Boltzmann equations for the mother and daughter particles and follow the time evolutions of their distribution functions in the
4
discretized phase space. In this formulation, one can set masses of both
daughter particles to arbitrary values. While our formulation and numerical
implementation can deal with arbitrary lifetime and/or masses of daughter
particles, to minimize complication, we assume that DDM decays after cosmological recombination into two particles one of which has a finite mass and
the other is massless, i.e. mD2 = 0.
We firstly focus on that the dark matter decaying changes the angular
diameter distance to the last scattering surface of CMB, that to the large scale
structure and the Hubble constant by calculating the background density
evolution of product particles. From the numerical calculation, we find that
the dark matter decaying make the cosmic expansion slower than that of
ΛCDM model. By comparing the calculation with the observational data
obtained by WMAP, we set a constraint on the relation between the lifetime
and the mass ratio.
Next, we calculate the time evolutions of the perturbed quantities of
progenitor and product particles. These calculation enables one to calculate
the power spectrum of temperature fluctuation of CMB and matter power
spectrum. Our calculation reveals that dark matter decaying attenuates the
matter spectrum on sub-horizon scale at present. Due to this attenuation, the
auto-correlation of the temperature fluctuation on large scales are amplified
and these effect is observed as an additional integrated Sachs-Walfe effect on
CMB. In addition, the position of the acoustic peak scale of CMB is shifted
to higher multipole than that of ΛCDM model. These effects are enhanced
as the lifetime becomes shorter or the mass ratio becomes smaller. Because
the standard deviation of the matter density fluctuation, which is called σ8
is sensitive to the free streaming of the product particles, we find that the
lifetime is larger than 200 Gyr in the case that the daughter particles are
generated relativistically.
In the study mentioned above, we fixed the six cosmological parameters
and it may cause the systematic error of the constraints. Thus we estimate
the cosmological parameters including DDM parameter Γ and r by observational data obtained by the Planck satellite. We find that the log likelihood
of the fitting is improved by more than the number of additional degree of
freedom. We reveal that the lifetime is restricted stronger as the mass ratio
becomes smaller. Because the determination of the acoustic peak position
with the Planck satellite is very accurate, the lifetime should be larger than
250 Gyr in the case that both product particles are massless. This constraint
is the strongest constraint so far.
5
1
Introduction
1.1
Dark matter
The scientific investigations of dark matter of the Universe have started from
twentieth century. In 1937, Fritz Zwicky estimated the mass of coma cluster
by the velocity dispersion of the galaxies of this cluster and claimed that the
mass is much larger than the estimated mass of baryons [1]. Three decades
later, in 1970’s, the flat rotation of the Milky Way is discovered by Vera C.
Rubin [2]. These observational results required that the un-luminous matter
and its abundance of the cluster and Milky Way is much larger than that of
baryons. These un-luminous and non-baryonic matter is called dark matter.
1.2
Property of the dark matter
In this section, we explain the properties which the dark mater should have.
The primary three conditions are the followings:
• Dark matter does not have electro-magnetic and strong interactions.
• The abundance (ΩDM ) is enough large (∼ O(1))).
• It is capable to be the source of the current structure in the Universe.
At first, we discuss whether the dark matter is composed of somewhat
compact objects such as brown dwarfs, free flowing planets and black holes
or not. These objects are called as massive compact halo objects (MACHOs)
and the number density of them can be constrained by the gravitational
lensing observations, Big-Bang Nucleosynthesis (BBN) and so on1 . When
one observes a faraway object such as a star or a galaxy, the movement of a
massive object crossing the line of sight produces the distortion of the image
of the object. Especially, this effect is called gravitational micro-lensing.
Observations of the gravitational lensing effect have provided us constraints2
on the abundance of MACHOs (see. [6]). By combining other studies, the
1
Many previous studies assumed that the mass function is proportional to delta function
of some fixed masses.
2
The observation of micro-lensing effect can be used not only setting on constraints
on the abundance of such a massive objects but also discovery of extra-solar planets (e.g.
[3, 4]) and free floating plates [5].
6
weak lensing observation revealed that MACHO which is composed by a
single mass objects cannot explain the abundance of dark matter [7, 8].
In addition, it is difficult to explain that the abundance of dark matter
by the abundance of baryonic matter. The light elements were synthesized in
the BBN. The abundance of light elements such as deuterium, 3 He, 4 He and
7
Li in the current Universe increases as the abundance of baryon increases.
By comparing the theoretical calculations of the BBN with the observation
of the abundance of the light element in the Universe, the abundance of the
baryon is given by [9]
0.021 ≤ Ωb h2 ≤ 0.025 (95%CL) .
(1.1)
It is much smaller than unity and it suggests that the most part of the energy
density of the Universe consists of non-baryonic matter.
Secondary we pick up neutrinos as a candidate. The neutrinos are a
standard model particle and does not have these interactions. In addition,
it is confirmed that the neutrino has a mass by a large underground water
Cherenkov detector [10]. When one considers the thermal production of
neutrino in the early Universe, the abundance in the current Universe (Ων h2 )
depends on the total mass of them as
∑

mν
 .
Ω ν h2 = 
(1.2)
91.4 eV
Thus, in the case that neutrinos have the mass about O(10) eV, neutrinos
become a candidate of the dark matter.
However, this possibility has been ruled out by not only the particle
physics experiment [11] but also the cosmological observations such as large
scale structures [12] and CMB (e.g. [13]). These cosmological studies are
based on the constraints on the free streaming of neutrinos in the Universe.
The free streaming of neutrinos causes the elimination of small scale structures and it should be detected on the precise measurement of large scale
structure. The latter observations suggest that the mass are much smaller
than 1 eV. Thus the current abundance of neutrinos cannot explain the that
of dark matter.
It is also confirmed that such a light particle including neutrinos cannot
play a trailblazing role in the structure formation in the Universe in the
aspect of the growth of the matter density perturbations. The light particle
7
is called hot dark matter (HDM). Because the speed of their motion is close
to the speed of light, the small scale structure compared with the horizon
scales is smoothed and should not be observed in the current Universe. This
prediction does not agree the current observations.
Here we check the explanation above in the aspect of the gravitational
instability. The time evolution of non-relativistic matter density and pressure
is obtained by calculating the conservation of mean energy and momentum
as follows:
∇ν Tµ ν = 0 .
(1.3)
At first we substitute ν = 0 to eq. (1.3). In the static universe, this equation
gives
∂ρ
=0.
(1.4)
∂t
Because the real universe is expanding, the effect of the expansion should be
considered. In order to treat the expansion, we introduce the scale factor
a(t), which describes the time evolution of the distance between two distant
objects (see. §A.2.1). In addition, the conformal time τ and the conformal
Hubble parameter H are defined as functions of the scale factor as
∫ t
1
τ =
dt′ ,
(1.5)
′)
a(t
0
1 da
ȧ
H =
= ,
(1.6)
a dτ
a
where overdot denotes the conformal time derivative of the function.
In order to derive the time evolution equation of the density fluctuation
of matter, we divide the energy-momentum tensor Tµ ν into homogeneous and
ν
isotropic component T µ (τ ), and the other δTµ ν .
Firstly, we consider the zero-th order equation, which describes the conservation law of the homogeneous and isotropic part of energy density ρ as
∂ρ
+ 3Hρ = 0 .
∂τ
(1.7)
Secondly, we derive the first order perturbation equation. The density and
pressure fluctuations is defined as follows:
δ(x, t) ≡
ρ(x, t) − ρ(t)
.
ρ(t)
8
(1.8)
From two equations above, we obtain the equation of motion of the density
fluctuation. By using eqs. (1.3), (1.7) and (1.8),
(
)
δ̈ + 2Hδ̇ − 4πGa2 ρ − c2s k 2 δ = 0 ,
(1.9)
where cs is the sound speed. Here we neglect the fluctuation of entropy per
unit mass. This equation shows that the density fluctuation on the small
scale oscillates in the well of the gravitational potential and cannot grow.
The threshold wave number k is called Jeans scale kJ as
√
a 4πGρ
kJ ≡
.
(1.10)
cs
Only the density fluctuation whose wave number is smaller than the Jeans
scale i.e. (k < kJ ) can grow and the lower limit of mass of astronomical
objects is determined and it is called Jeans mass MJ .
( )−3
kJ
c3
4
1
MJ ≡ πρ
= π 5/2 √ s .
(1.11)
3
πa
6
G3 ρ
When the velocity of dark matter particles are close to unity, it is called hot
dark matter (HDM). In this case, the Jeans mass is approximately
(
)−2
MJ ∼ 1016 Ωm h2
M⊙ .
(1.12)
This mass is much more than that of Milkey Way. Because the Jeans mass
gives the least mass of the structure, there are few objects whose mass is
around or less Milkey way in the Universe and it is disfavored by a number
of observations. When one considers the non-linear evolution of the density
fluctuation, the small scale structure can be created from the large scale ones.
It is top-down scenario of structure formation. However recent observations
strongly suggest that the galaxies were formed earlier than that of clusters.
Thus it is not favored that dark matter is composed of HDM.
Therefore dark matter should be consisted of some particles whose velocity is much smaller than that of light and does not have electro-magnetic and
strong interaction. Such a dark matter is called cold dark matter (CDM).
The existence of CDM is also required in the aspect of the stability of the
observed structure of the Milky Way [14].
9
1.3
Cold dark matter (CDM)
In the previous section, we mention that the most part of dark matter should
be composed of CDM. However the standard model of particle physics does
not include such a particle. Thus CDM is more likely to be composed of some
unknown elementary particles. In the candidates of CDM, weakly interacting
massive particles (WIMPs) is a strong candidate. Due to the enough mass,
the speed of motion of it is suppressed much smaller than speed of light. In
the mass, the speed of motion of it is suppressed much smaller than speed of
light.
The interaction of WIMPs is only weak interaction and the gravitational
interaction. Here we consider the thermal productions of WIMPs in the early
universe. In such a epoch, both pair creation and annihilation occurred in
the universe. However, due to the cosmic expansion, the number density
of the WIMPs decreases and the reaction is frozen out. WIMPs are nonrelativistic before the reaction is frozen. The cross section of this reaction
⟨σv⟩ is sensitive to the current abundance of WIMPs as follows:
(
)
3 × 10−26 cm3 · sec−1
2
Ωχ h =
.
(1.13)
⟨σv⟩
The relation between the mass of a WIMP and cross section ⟨σv⟩ have been
constrained by recent observations of γ ray [15, 16] and CMB [17, 18]. In
the case that the mass of the CDM particle is around 100 GeV, candidates
of WIMPs can be found in the supersymmetry theory. Candidates of CDM
particles are not just limited to WIMPs. Axion is another strong candidate
(see. [19, 20]).
In 1980’s, computer simulations can reproduce the observational results.
It is required that dark matter should be CDM is pointed out by Blumenthal
et al. [21] and Davis et al. [22]. Recently, N -body simulations with CDM
excellently has succeeded in explaining the observational results on large
scales (e.g. [23]).
10
Figure 1: The comparison between N -body simulations of the large scale
structure with corresponding observations [23].
1.4
Present temperature of CDM particle
The temperature of CDM particles at present time T0 can be estimated by
taking into account of the decoupling temperature Td of mother particles
[19]. The temperature which corresponds to the mass of the mother particle
mM = 1 TeV is so high that all species of standard particles such as eight
gluons, W ± , Z 0 , three generations of quarks and leptons, and one complex
Higgs doublet are relativistic at the decoupling of mother particles. We
define g∗ as the total number of effectively massless degrees of freedom. In
the epoch that the mother particles are in thermal equilibrium, we expect
g∗(early) = 106.75, while g∗(now) = 3.36 at present.
The comoving number density of photons when CDM particles are in
11
thermal equilibrium is
nγ =
g∗(now)
nγ0 .
g∗(early)
(1.14)
Here the number density of photons at present nγ0 is 410 cm−3 [24, 19] and
therefore
nγ = 12.9 cm−3 .
On the other hand, the baryon number density can be derived from the
baryon-photon ratio today η = nb /nγ , which is estimated as η = (6.19 ±
0.15)×10−10 (1σ C.L.) [25]. The ratio of the number densities between CDM
mother particles and photons today, nM /nγ0 , can be written as
nM
nM nb
ΩDM mp
=
=η×
,
nγ0
nb nγ0
Ω b mM
(1.15)
where Ωb is the cosmological density parameter of baryon. Here we have
neglected a contribution from helium. By substituting mM = 1.0 TeV,
ΩCDM = 0.222, and Ωb = 0.0446 into eq.(1.15), we obtain
nM
= 2.89 × 10−12 .
nγ0
The time of the thermal decoupling of mother particles can be estimated
through the relation of the Boltzmann factor as,
(
)
nM
mM
= exp −
.
nγ
Td
Thus we obtain the temperature of mother particles as
TM0 = a2d Td ≃ 1.7 × 10−14 K = 1.4 × 10−18 eV ,
(1.16)
where ad is the scale factor when the mother particles decouple from the
thermal bath. We can write ad as
ad ∼
Tγ0
,
Td
where Tγ0 = 2.725 K is the temperature of CMB [24].
12
1.5
CDM crisis
In the previous section, although the nature of dark matter has remained
mystery, we have seen that ΛCDM cosmology has been succeeded in describing large scale structures and temperature fluctuations of CMB. However,
it has been reported that observations on small scales is different from the
theoretical predictions.
The inflation cosmology predicts that the initial power spectrum of matter
distribution P(k) is almost scale invariant and the gravitational interaction
does not have any specific scale. Thus, the number of halos is naively expected to inversely proportional to the mass and many dwarf galaxies should
be observed around the Milky Way [26]. This expectation is discussed quantitatively by Moore et al. and Klypin et al. [27, 28]. They performed
N -body simulation and the cumulative number of the optically observed
satellite galaxies of the Milky Way is ten times smaller than the theoretical
prediction based on ΛCDM model. Springel et al. performed large N -body
simulation and also found many subhaloes of galactic haloes [29]. This problem is called “missing satellite problem”. However, in the case that these low
mass galaxies loose the baryons by galaxy winds and it becomes too faint to
observe, one may underestimate the number of it.
In last several years, due to improvements of observations, structures of
massive galaxy clusters have been gradually revealed. In the ΛCDM cosmology, these clusters are expected to have some subhalos whose masses
are around Milky Way (∼ 1012 M⊙ ) and these are luminous enough to be
detected. In fact, the number of observed halos is much smaller than the
expected value of ΛCDM model. This problem is called “too big to fail problem”. Besides, the N -body simulation suggests that the cuspy core in the
galactic center. The density of the matter has been estimated by observing
the rotation velocity measurement of the HI gas with radio telescopes [30].
However effects of baryon on the structure formation such as AGN feedback
are also important on small scales. Because baryon has electro-magnetic interaction, there are a lot of difficulties in accurate theoretical predictions of
the structure formations on such a scale. Thus the question whether these
discrepancies are caused by some unknown properties of dark matter or not
is not conclusive.
Recently Planck collaborations suggested the expected current standard
deviation of density fluctuation of total matter3 estimated from the CMB
3
This standard deviation is defined as σ8 , which is the amplitude of the density fluc-
13
temperature fluctuation is smaller than that estimated from number counts
of cluster using Sunyaev-Zel’dovich (SZ) effect by more than 2σ confidence
level. If the assumptions which are applied in [31] are valid, they have lost the
two-thirds of existent clusters in this observations or their mass estimates are
wrong by factor 2. In addition, because the typical distance between clusters
is O(1) Mpc or more, this discrepancy is not likely to be caused not by
some feedback effects concerning baryon physics 4 . Ostriker and Steinhardt
suggested that some mechanisms which suppress the structure formation may
solve these problems [26].
In order to solve them, we focus on the properties of dark matter. Because the gravitational interaction is prominently weak compared with other
three known interactions, it is difficult to discover some unknown interactions
whose strength is comparable or weaker than gravitational force in current
accelerator experiments. In the case that the dark matter particles have such
a interaction, the structure formation in the Universe may significantly differ
from that predicted from ΛCDM model.
1.6
Dark matter models—Candidates for the solutions
of these problems
Several dark matter models have been introduced in [26]. In this section,
we present some of them and explain the effect of these properties of dark
matter on cosmological variables.
1.6.1
Self-annihilating dark matter (SADM)
SIDM particles have some unknown interactions with themselves. When
SIDM particle collides another SIDM one, they annihilate to a standard
model particle and its anti-particle such as γ + γ, b + b. In the dense region
such as galactic center or center of the clusters dark matter particles annihilate and, as the result, the structure formation on the scale is suppressed
and the density of the matter is reduced.
tuation of the matter at the scale 8h−1 Mpc.
4
In the analysis of Planck collaboration, the shape of all clusters are assumed to be
sphere. However, studies of density distribution of galaxy clusters by using weak gravitational lensing effect have suggested that the density structures of clusters have strong
individuality and each of them is far from spherically symmetric (see.[32]).
14
The mass of dark matter particle is expected to be around O(102 ) ∼
O(103 ) GeV in several models in theoretical elementary particles. Fortunately, γ-ray telescopes or detectors in space for example Fermi satellite and
Pamela have a detectability of these energy range of γ-ray. In addition, the
dark matter search project with high sensitive photomultipliers is ongoing.
In this project, they prepare a ton of the highly-pure liquefied xenon and are
trying to detect the light from recoiled atomic nuclei by the interactions of
dark matter. They have already set a constraint on the cross section of the
interactions of dark matter.
1.6.2
Warm dark matter (WDM)
In the case that the mass of dark matter particle is O(1) keV, their motion
was ultra-relativistic even in the kinetic5 decoupling. This thermal motion
prevents the structure formation on small scales. Cosmological N -body simulations with WDM have been done by some authors.
1.6.3
Decaying dark matter (DDM)
In the standard model of particle physics, most elementary particles decay
into other stable particles such as photons or electrons. Hence it is possible
that dark matter particles have the property of decaying into other particles.
This kind of dark matter is called decaying dark matter (DDM). Because
particle decays are caused by the interaction which they have, we can obtain
the nature of dark matter and information about a particle theory lying
behind it by constraining the lifetime of DDM. DDM is often discussed as
a solution to such small-scale CDM problems, for example, see Cen [33].
The DDM model, however, should pass the tests from precise cosmological
observations, such as measurements of the CMB and the large scale structure
of the Universe. One of the aims of this thesis is therefore to give a complete
set of perturbation equations of the DDM model with two decay products
having arbitrary masses, and show some evolutions of density perturbations
in the DDM model to compare them with recent cosmological observations.
5
For the matter, there are two decouplings, chemical decoupling and kinetic decoupling.
The former is that the pair creation-annihilation reaction, χ + χ ↔ 2γ, is frozen out due
to the cosmic expansion. After this, the particle χ and χ are no longer created in the
thermal bath. On the other hand, the latter is that the scattering reaction is frozen out.
15
1.7
Our DDM model
In this thesis, we consider a model of DDM (M) which decays into two particles (D1, D2) well after cosmological recombination,
M → D1 + D2 (with Γ−1 ≫ t∗ ) ,
(1.17)
where Γ is the decay rate of DDM. Throughout this paper, we assume a variant of the concordance flat power-law ΛCDM model where CDM is replaced
with DDM, and consider cosmological perturbations.
Cosmological implications of this DDM models have been studied by several authors [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48].
Among them, Kaplinghat [37] and Huo [46] have considered models in which
dark matter decays into two ”daughter” particles6 well before the matterradiation equality epoch. They calculated time evolution of perturbations
and set constraints on the mass ratio and the life time of DDM from the
free streaming scale of daughter particles. The former considered the DDM
which decays into a massive particle and massless one in the early Universe.
They computed the matter power spectrum and discussed the effects of freestreaming of the decay products by considering the phase-space density of the
massive daughter particle. The latter computed the CMB power spectrum
in addition to the matter power spectrum. Moreover, DDM models in which
the mass difference between the mother and the massive daughter particles
∆m is much smaller than the mass of the mother particles are studied in
refs. [44, 45, 48, 49]. In particular, Peter [44] has considered effects of the
recoil energy of the daughter particles on the halo mass-concentration and
galaxy-cluster mass function using N -body simulations and set constraints
on the lifetime of DDM and ∆m in terms of the kick velocity vkick by demanding that the daughter particles in halos with mass M = 1012 M⊙ do
not escape from the halos and destroy the gravitational potential. She also
set constraints by using weak lensing measurements and X-ray observations.
Wang et al. have analyzed N -body simulations with DDM and focused on
the statistical properties of the transmitted Lyman-α (Ly-α) forest flux. By
comparing the data of the Sloan Digital Sky Survey (SDSS) and WMAP 7
with their simulation data, they set a constraint on the life time of DDM as
Γ−1 > 40 Gyr where vkick ≳ 50 km/s [49]. However, previous studies so far
could not deal with DDM models where mother particles decay into particles
6
We call progenitor particle ”mother” particle.
16
with arbitrary mass at late times (i.e. after the matter-radiation equality).
This is what we will explore in this work.
In this thesis, we calculate the time evolutions of mean density and perturbed quantities of mother and daughter particles by calculating the distribution function of the daughter particle. By performing these calculations,
the effects of dark matter dacaying on the geometric observables and growth
history of the density fluctuations of total matter can be estimated. In addition, the calculation of the perturbed quantities provides the estimated power
spectrum of the matter density fluctuation P (k) and the standard deviation
of the fluctuation σ8 . In addition, these calculation enable us to estimate
temperature fluctuations of CMB ClTT and the correlation of temperature
and E-mode polarization ClTE and the auto-correlation of the latter ClEE .
The layout of this thesis is as follows. In §2, we calculate the expansion
history and investigate the effects on the angular diameter distance to the
last scattering surface of CMB, the distance to the large scale structure. The
statements of this section is based on [50]. In §3, we study the effect on
the total matter density fluctuation such as P (k) and show the estimated
ClTT , ClTE , ClEE . This section is based on [51]. We specify the cosmological
parameter in both section. In §4, we estimate the cosmological parameter
including DDM parameter such as Γ−1 , r by an interpolation. §5 concludes
this thesis.
2
2.1
The effect of DDM on the background geometry
Formulation
In this section, we define the decay rate Γ(qD , qM ), which is the function
describing how many daughter particles with comoving momentum qD are
created for a unit time interval from the mother particles with momentum
qD in order to describe the decay process. First, let us write the Boltzmann
equations for the distribution functions of mother and daughter particles.
The Boltzmann equation for the distribution function of the mother particles
fM (qM ) is
(
)
dfM
∂fM dxi ∂fM dqM ∂fM dni ∂fM
∂fM
=
+
+
+
=
,
(2.1)
dt
∂t
dt ∂xi
dt ∂qM
dt ∂ni
∂t C
17
where ni is the unit vector in the direction of the momentum.
The collision term in eq.(2.1) can be expressed as an integration of Γ(qD , qM )fM
with qD as
(
)
∫
∂fM
= − Γ(qD , qM )fM d3 qD .
(2.2)
∂t C
Similarly, the Boltzmann equations of daughter particles are
)
(
dfDj
∂fDj dxi ∂fDj dqD ∂fDj dni ∂fDj
∂fDj
,
=
+
+
+
=
dt
∂t
dt ∂xi
dt ∂qD
dt ∂ni
∂t C
(2.3)
where j is the particle index j = 1, 2.
The daughter particles in this model are created only through the decay
of mother particles. Thus the collision term can be written as
(
)
∫
∂fDj
= + Γ(qD , qM )fM d3 qM .
(2.4)
∂t C
Secondly, we derive Γ(qD , qM ) explicitly. From the definition of CDM, the
thermal motion of mother particles is so slow that we can regard them as
objects at rest. To be specific, the kinetic energy of mother particles is much
smaller than the mass deficit of the decay. Due to the conservation of the
momentum, two daughter particles are emitted in the opposite directions and
the amplitudes of these momenta are the same. So we can write the decay
rate Γ(qD , qM ) as a function proportional to the delta function of qD , which is
determined by the energy momentum conservation. From the conservation,
qM and qD should satisfy the condition as
√
√
√
2 2
2
2
2
2
2
mM a + qM = mD1 a + qD + m2D2 a2 + qD
.
(2.5)
Because qM ≪ amM , the solution of eq.(2.5) leads to
√
1
2
qD ≃
AM a2 + BM qM
.
2
(2.6)
In addition, we assume that these daughter particles are isotropically emitted.
By taking into account the spherical symmetry of the decay, the decay rate
should be written as,
(
)
√
1
Γ
2
2
δ qD −
AM a + BM qM ,
(2.7)
Γ(qD , qM ) =
2
4πqD
2
18
where Γ−1 is the lifetime of the mother particle and a is the scale factor.
Here AM and BM are constants defined as
AM ≡ m2M − 2(m2D1 + m2D2 ) +
BM
(m2D1 − m2D2 )2
,
m2M
(m2D1 − m2D2 )2
≡ 1−
.
m4M
(2.8)
(2.9)
Thirdly, we consider the first order perturbations of the distribution functions fM , fDj (j = 1, 2) as follows. It is convenient to write a distribution
function as a zero-th order distribution, which is the background distribution,
plus a perturbed function Ψ,
(0)
fM ≡ fM (qM , t)(1 + ΨM (xi , qM , ni , t)),
fDj ≡
(0)
fDj (qD , t)(1
+ ΨDj (xi , qD , ni , t)).
(2.10)
(2.11)
By substituting eq.(2.10) and (2.11) into eq.(2.1) and (2.3), respectively, and
comparing the equations order by order, we obtain the following equations
at zero-th order for the mother particles as
∫
(0)
(0)
(0)
unperturbed : f˙M = − Γ(qD , qM )fM d3 qD = −ΓfM .
(2.12)
Here overdot denotes the derivative with respect to the cosmic time t.
For daughter particles, we obtain the following equation,
∫
′
4Γ qD
(0)
(0)
(0) ′
˙
unperturbed : fDj = Γ(qD , qM )fM (qM )d3 qM =
fM (qD
)(2.13)
,
B M qD
√
where
′
qD
≡
2
4qD
− AM a2
.
BM
(2.14)
The two unperturbed equations above can be interpreted as follows. The
equation (2.12) states that all the mother particles should decay with the
decay rate Γ. For daughter particles, on the other hand, the equation (2.13)
means that the daughter particles with momentum qD should be created at a
(0) ′
′
rate proportional to ΓfM (qD
), where qD
is the momentum which the mother
particles should have for the created daughter particle to have the momentum
qD after the decay. Note that because from eq. (2.13) the unperturbed
(0)
(0)
distribution functions of two daughter particles coincide, fD1 = fD2 , and
(0)
hereafter we simply denote them as fD .
19
2.2
2.2.1
Calculation of background distribution functions
Mother particle
The temperature of CDM is low compared with the rest energy of the particle.
If the mother particles do not decay, i.e. Γ = 0, the background distribution
function of the particles is given by the Maxwell-Boltzmann function. In
this case, or in the very early Universe where the decay is negligible, the
(0)
background distribution f˜M is
(
)
2
qM
1
(0)
˜
fM (qM , t) =
exp −
,
(2.15)
(2πmM TM0 )3/2
2mM TM0
where TM0 is the present temperature of mother particles. We consider mM =
1.0 TeV as a working example. Because of the calculation performed in
Appendix 1.4, we obtain the temperature of mother particles as
TM0 ≃ 1.7 × 10−14 K = 1.4 × 10−18 eV ,
(2.16)
And we set the Boltzmann constant kB = 1. In fact, the result depends
only on the mass ratio mD1 /mM , but not on the absolute value of mM . In(0)
cluding the decay, fM is given by the solution of eq.(2.12) as
(
)
2
1
qM
(0)
fM (qM , t) =
exp −
− Γt .
(2.17)
(2πmM TM0 )3/2
2mM TM0
Here we have normalized the distribution function at t = 0 as
∫ +∞
2 (0)
4πqM
fM (qM , t = 0)dqM = 1 .
(2.18)
0
The number density of the mother particles, nM , is determined by
nM = ρc ΩDM /mM ,
(2.19)
where ρc is the critical density of the Universe, ΩDM is the density parameter
of dark matter normalized by the critical density at present. Note that the
parameter ΩDM in the above equation is an extrapolated value of the density
parameter of mother particles without decay. Then the energy density of the
mother particles ρM is given by
∫
√
ρc ΩDM +∞
2
2 (0)
4πqM m2M a2 + qM
fM (qM , t)dqM .
(2.20)
ρM =
4
mM a 0
20
2.3
Daughter particle
In this section we derive the background distribution function of daughter
(0)
particles. By substituting fM in eq.(2.17) into eq.(2.13), we obtain a partial
(0)
differential equation for the unperturbed distribution function fD as,
(
)
√
2
2
−
a
A
4q
M
2
4qD
− a2 AM exp − D
=
exp (−Γt) .
3/2
3/2
2BM mM TM0
π qD B M
(2.21)
Since the daughter particles did not exist in the early stage of the Universe,
(0)
f˙D
√
2Γ
(
1
mM TM0
)3/2
(0)
fD (qD , t = 0) = 0 .
When a mother particle decays, the amplitudes of physical momenta of two
daughter particles are the same, and we denote it as pth . Because the physical
momentum of daughter particles decays as ∝ a−1 as the Universe expands,
the daughter particles which are created in the past should have the comoving momentum smaller than pth . In addition, since the thermal motions of
mother particles are very slow compared with their mass, the time can be
decided uniquely when the daughter particles with the momentum qD were
created. To put it concretely, the redshift zD which corresponds the redshift
when the daughter particles with the present momentum qD were created
should satisfy
1
1
qD =
pth ∼
∆m .
1 + zD
1 + zD
where ∆m is the mass difference between the mother and daughter particles.
In the limit mD1 + mD2 → 0, pth is equal to ∆m.
(0)
Now let us consider the time evolution of fD at a fixed comoving momentum qD . The source term of eq.(2.21) is exponentially suppressed in the
2
very early Universe when a2 ≪ 4qD
/AM , because the typical momentum of
the daughter particles qD is much larger than the temperature of the mother
particles TM0 . The source term, on the other hand, should be zero when
2
a2 ≥ 4qD
/AM , which comes from the energy-momentum conservation law.
Therefore, the source is important only around t ≲ t∗qD where t∗qD is defined
by
2
− a2 (t∗qD )AM = 0 .
4qD
(0)
To take advantage of the rapid convergence of the source term of fD (qD , t)
21
we expand a(t) around t∗qD as
a(t) ≃ a(t∗qD ) + ȧ(t∗qD )(t − t∗qD ) ≡ a(t∗qD ) + ȧ(t∗qD )ε (ε ≤ 0) ,
(2.22)
where ε = t − t∗qD . Here we have omitted the higher order terms in the
(0)
expansion because the source of fD (qD , t) decays exponentially backward in
time for t < t∗qD . Then the evolution equation (eq.(2.21)) can be expanded
as
(
)3/2
√
√
1
2Γ
(0)
(0)
f˙D (qD , t) = f˙D (qD , t∗qD + ε) ≃
−a(t∗qD )ȧ(t∗qD )εAM ,
3/2
3/2
m
T
M M0
π qD BM
((
)
)
AM a(t∗qD )ȧ(t∗qD )
× exp
− Γ ε exp(−Γt∗qD ) ,
(2.23)
BM mM TM0
and the integration of time t can be replaced with that of ε. Furthermore,
we can extend the range of integration as
∫ 0
∫ 0
(0)
(0)
(0)
∗
∗
˙
fD (qD , tqD + ε)dε ≃
f˙D (qD , t∗qD + ε)dε . (2.24)
fD (qD , tqD ) =
−t∗q
−∞
D
This is because the term in the exponential in eq.(2.24) is very large in
AM a(t∗qD )ȧ(t∗qD )
negative value;
(−t∗qD ) ∼ −O(1023 ) ≪ −1 for t∗qD around
BM mM TM0
recombination, and hence the integration of eq.(2.23) for t = [−∞, 0], i.e.,
ε = [−∞, −t∗qD ] is negligible. We are then able to perform this integration of
ε analytically to obtain
√
)3/2
)−3/2 (
πAM a(t∗qD )ȧ(t∗qD ) ( AM a(t∗ )ȧ(t∗ )
1
qD
qD
(0)
fD (qD , t) = Γ
−Γ
3/2
B
m
T
mM TM0
M M M0
BM qD
× exp(−Γt∗qD )θ(t − t∗qD ) .
(2.25)
(0)
We show the shape of the distribution function of daughter particles fD (qD , t)
at present time in figure 2.
We depict the distribution function of daughter particles at two different epochs, at matter-radiation and matter-Λ equalities, in figure 2. We
−3/2
−1
find that the distribution function, fD , is proportional to qD
and qD
for
qD < pth /(1 + zeq ) and pth /(1 + zeq ) < qD < pth /(1 + zΛ ), respectively, where
22
10-20
mM =1.0TeV
mD1=0.98TeV
mD2=0.00TeV
Γ-1 =0.10Gyr
fD(0)(qD,t0)[eV-3]
10-22
10-24
10-26
10-28
10-30 2
10
104
106
qD[eV]
108
1010
Figure 2: Distribution function of daughter particles at present as a function
of comoving momentum of daughter particles qD , with parameters mM = 1.0
TeV, mD1 = 0.98 TeV, and Γ−1 = 0.1 Gyr [50].
23
fD(0)(qD,teq)[eV-3]
10-20
10-22
10-24
10-26
10-28
10-30 2
10
104 106 108
qD[eV]
1010
104 106 108
qD[eV]
1010
fD(0)(qD,tΛ)[eV-3]
10-20
10-22
10-24
10-26
10-28
10-30 2
10
Figure 3: Snap shots of the distribution function of daughter particles (red
lines) at matter-radiation equality teq (left) and matter-lambda equality tΛ
(right). The present distribution function is also given (gray dotted). The
−3/2
−1
green and blue dotted lines are the fitting lines with ∝ qD
and qD , which
represent the particles created in the radiation and matter dominated epochs,
respectively. We find that these fitting lines are in good agreement with the
redlines [50].
24
zeq and zΛ are the redshifts of matter-radiation and matter-Λ equalities, respectively. These dependencies can be understood as follows. First let us
estimate the number density of daughter particles n. When t ≪ Γ−1 , the
number density of daughter particles is given as
Γ
nM .
H
nD ≃ nM (1 − exp(−Γt)) ≃
(2.26)
(0)
The number density nD is also expressed through an integration of fD as,
∫ qD
(0) ′
3 (0)
′
fD (qD , t) .
∼ qD
, t)d3 qD
fD (qD
n=
0
(0)
Thus we can express fD with H and qD as,
(0)
fD
∼
nD
Γ
∼
n .
3
3 M
qD
HqD
(2.27)
In the radiation-dominated epoch, the scale factor a grows proportional to
t1/2 and the Hubble parameter H is proportional to a−2 . In addition, the
masses of the mother and the daughter particles determine the momentum
of daughter particles at their creation to a fixed (constant) value pth . Thus
qD can be written as
qD ≡ apth ∝ a .
(2.28)
Combining the above dependencies altogether, we can derive the qD -dependence
in figure 3 in the radiation-dominated epoch as
−1
fD (qD , teq ) ∝ qD
.
(0)
(2.29)
On the other hand, in the matter-dominated epoch where a ∝ t2/3 and
H ∝ a−3/2 , we obtain the distribution of daughter particles created in the
matter dominated epoch in the same way as
−3/2
(0)
fD (qD , tΛ ) ∝ qD
.
(2.30)
We confirm that these dependencies are indeed found in figure 3.
The energy density of daughter particles can be calculated in the same
way as mother’s. It is given by
∫
√
ρc ΩDM +∞
2
2 (0)
ρD =
4πqD m2D1 a2 + qD
fD (qD , t)dqD
(2.31)
4
mM a 0
∫
√
ρc ΩDM +∞
2
2 (0)
+
4πq
m2D2 a2 + qD
fD (qD , t)dqD .
D
mM a 4 0
25
Finally the time evolution of an homogeneous and isotropic expanding
Universe follows the Friedmann equation,
√
ȧ
8πG
=
(ρM + ρD + ρB + ργ + ρν + ρΛ ) ,
(2.32)
a
3
where ρB , ργ , ρν and ρΛ are the densities of baryon, photon, neutrinos, and
dark energy, respectively.
2.4
2.4.1
Result and Discussion
Background energy densities
(0)
(0)
By integrating the distribution functions fM and fD , we obtain the time
evolution of energy densities of mother and daughter particles, which is shown
in figure 4. As is shown in the figure, if the dark matter particles decay, the
total energy density in the Universe becomes small compared with the standard Λ-CDM model [52, 53]. Thus the time evolution of the scale factor
a differs from that of the Λ-CDM model. This leads to the different angular diameter distances to the last scattering surface of CMB dA (z∗ ) and the
position of BAO dz . The distances to CMB and the position of BAO are
measured precisely by WMAP [54] and SDSS [55], respectively. The uncertainties of these measurements are also available in those papers which can
be used to constrain the decay rate Γ and the mass ratio mD1 /mM , as we
will discuss below.
2.4.2
Constraints from Hubble constant, BAO and CMB
In this section, we only consider the dark matter which decays after cosmological recombination. For this reason, we fix standard cosmological parameters to the values which agree with the Λ-CDM model in the early Universe
obtained by WMAP7 [54]. Because the decay of mother particles can be
neglected deep in the radiation dominated era, the initial conditions of the
dark matter energy density ρM and the scale factor a(ti ) can be set as in the
Λ-CDM model without decay. in this work we use the relation which holds
in the radiation dominated era, a(t = 0.02 sec ) = 4.60 × 10−12 as our initial
condition.
The ”comoving” angular diameter distance to CMB, dA (z∗ ), is sensitive
to the deviation of a, where z∗ is the shift of recombination. Here dA (z∗ ) is
26
0
10
-2
10
-4
ρ[eV4]
10
-6
10
-8
10
-10
10
-12
10
-3
10
10
-2
a
-1
10
0
10
Figure 4: Evolution of energy densities of the mother (green) and daughter
(blue) particles as a function of scale factor a with model parameters mM =
1.0 TeV, mD = 0.20 TeV, and Γ−1 = 0.1 Gyr. The total energy density of
mother and daughter particles is shown as a red line. The time evolution for
the standard CDM model is also shown for comparison (black dotted line)
[50].
27
determined precisely by WMAP project as
dA (z∗ ) = 14116+160
−163 Mpc .
In the same sense the distance to BAO deviates from that in the ΛCDM model. According to the observation of BAO by SDSS [55], the BAO
constraint is given through the variable defined as dz ≡ rs (zD )/DV (z), where
rs (zD ) is the comoving sound horizon at the baryon drag epoch and zD is the
redshift when photons decouple baryons. According to the report of WMAP7
[54],
rs (zD ) = 153.2 ± 1.7 Mpc .
(2.33)
The distance DV (z) is a function of redshift z defined as [56, 57],
(
DV (z) ≡
(1 + z)2 zd2A (z)
H(z)
)1/3
,
(2.34)
where dA (z) is the angular diameter distance to the point whose redshift is
z. When one specifies a cosmological model and the evolution of the scale
factor a is determined, we can use dz to constraint the parameters of the
model. The distances d0.2 and d0.35 are given by SDSS DR7 as [55],
We define a vector x as
dobs
0.20 = 0.1905 ± 0.0061 ,
obs
d0.35 = 0.1097 ± 0.0036 .
(2.35)
(2.36)
( th
)
d0.20 − dobs
0.20
x=
,
obs
dth
0.35 − d0.35
(2.37)
th
where dth
0.20 , d0.35 are the distances based on the cosmological model to be
constrained. The matrix C −1 , which is inverse of the covariance matrix
C ≡ ⟨x t x⟩, is given by,
(
)
30124 −17227
−1
C =
,
(2.38)
−17227 86977
where t x is transpose of the vector x. We use the value χ2 ≡ t xC −1 x for
our chi-square test. Since we have two model parameters, the regions of 1σ
and 2σ confidence levels correspond to those which satisfy ∆χ2 < 2.18 and
6.30 from the minimum, respectively.
28
Due to the dark matter decay, the total energy density in our model is
always lower than that in the Λ-CDM model if the energy densities of dark
matter in the early Universe are fixed to the same value between the two
models. Thus the Hubble parameter H of our model is always smaller than
that in the Λ-CDM, which we denote as HΛ−CDM . A simple constraint on
our model is therefore put from the current value of the Hubble parameter.
The latest compilation determines the Hubble constant as H0 = 73.8 ± 2.4
km/s/Mpc [58] including systematics. By comparing the Hubble parameter
with the data, we obtain constraints on the lifetime of the mother particle
Γ−1 and the mass ratio mD1 /mM , which are shown in figure 5.
We consider the ”comoving” angular diameter distance toward the last
scattering surface dA (z∗ ), where z∗ is the redshift of recombination. The
distance dA can be written as
∫ z∗
∫ z∗
dz
dz
dA =
>
.
(2.39)
H(z)
HΛ−CDM (z)
0
0
Therefore
dA (z∗ ) > dA(Λ−CDM) (z∗ ) .
(2.40)
In figure 5 we show the constraint on the lifetime of decaying dark matter from the distance to BAO and the last scattering surface of CMB. The
Λ−CDM model corresponds to the limits Γ → 0 and/or (1 − mD1 /mM ) → 0.
As Γ becomes smaller, or mD1 approaches closer to mM , the distance dA (z∗ )
approaches to the value in the Λ-CDM model. In general, constraints from
CMB are stronger than those from BAO. This is simply because CMB data
are more precise than the current data of BAO. In the massless limit of
daughter particles, we find Γ−1 > 30 Gyr from CMB (at 1σ). Because we
(0)
have derived the constraint from the background fD (qD , t) only, this constraint is weaker than that obtained by [34, 40] in which the perturbations
are included. At the limit Γ−1 → 0 we obtain mD1 > 0.97mM from CMB (at
1σ). These values are consistent with the results obtained by the simplified
estimations without using the distribution functions, which are shown in the
following section.
2.4.3
Massless limit of daughter particle
In the case that a mother particle decays into two massless particles, the
energy densities of mother particle ρM and daughter radiation ρD satisfy the
29
2
-1
10
-2
10
BAO
0
10
allowed
region
Hubble
101
CM
B
Γ-1(Gyr)
10
-3
10 -2
10
-1
10
1 - mD1/mM
0
10
Figure 5: Constraints on the lifetime of decaying dark matter and the mass
ratio between the mother and the daughter particles. Solid lines and dashed
lines correspond to 1 σ C.L. and 2σ C.L. of the constraints, respectively. The
names of observational data are also shown in the figure [50].
30
following equations
ρ̇M = −3HρM − ΓρM ,
and
ρ̇D = −4HρD + ΓρM .
These differential equations can be solved to give
ρM = ρM∅ a−3 exp(−Γt) ,
and
ΓρM∅
ρD =
a4
∫
t
a(t′ ) exp(−Γt′ )dt′ ,
0
where ρM∅ is the expected energy density of dark matter without decay. In
this calculation we obtained the following constraints,
Γ−1 ≥ 30Gyr ,
and
Γ−1 ≥ 18Gyr ,
from the angular diameter distances to CMB and the position of BAO, respectively. These values agree with the results in the massless limit obtained
from calculations in the main text, where the distribution functions of daughter particles are directly integrated instead using the above simple equations.
Next we show how one can obtain a limiting value of the mass ratio
mD1 /mM in the limit Γ−1 → 0. In this limit our model corresponds to the
mD1
Λ-CDM model such that ΩM is replaced with
ΩM∅ where ΩM∅ is the
mM
expected density parameter of dark matter without decay. We can deduce a
constraint on the ratio mD1 /mM from a constraint on ΩM from the angular
diameter distances to CMB and BAO. In this way we obtain the following
constraints,
mD1 /mM ≥ 0.97,
and
mD1 /mM ≥ 0.23,
from the angular diameter distances to CMB and BAO, respectively. These
values are roughly in agreement with the values in figure 5, in which the
distribution functions of daughter particles are directly integrated.
31
2.4.4
Free-streaming scale
The daughter particles created by the decay of mother particles move with
large velocity in the expanding Universe. It leads that the depth of the
gravitational potential of mother particles becoming shallower which have
been created by a group of mother particles becomes shallower in time. In
the linear theory of structure formation, the structures smaller than the freestreaming scale lFS are erased. Since it is difficult to fully calculate the
density perturbations of the mother and the daughter particles, here we use
lFS for constraints on Γ−1 and mD1 /mM . We restrict our attention to the case
where Γ−1 is much smaller than the age of Universe at redshift z, Γ−1 ≪ t ∼
H −1 (z). In this case, because almost all the mother particles have decayed
by that time into daughter particles which are responsible for the depth of
gravitational potential of mother particles becoming very shallow, we can
place constraints from a simple argument that the scale of any observed
structure bounded by dark matter should be larger than the free-streaming
scale of daughter particles.
Free-streaming scale lFS (z) at each redshift z can be estimated using the
averaged velocity of daughter particles v(z) as
lFS (z) ∼ v(z) ×
1
.
H(z)
(2.41)
Here we define v(z) as
∫
v(z) ≡
0
pth
(0)
∫
2
4πqD
v(qD , z)fD dqD
pth
,
2 (0)
4πqD
fD dqD
0
where
qD
v(qD , z) ≡ √ 2
.
qD + m2D1 a2
Here v(qD , z) is the magnitude of physical velocity of the daughter particle
whose comoving momentum is qD at redshift z.
We calculate the free-streaming scale lFS of daughter particles at each
redshift z, which is shown in figure 6. We explain these curves in the matter
dominated epoch (1 ≲ z ≲ 3000) as follows. First, when Γ−1 > H −1 the
daughter particles with a constant physical momentum pth are kept being
32
2
lFS[Mpc](physical)
10
mD1
mM =0.98
1
10
0
10
10
-1
100
101
102
1+z
lFS[Mpc](physical)
3
10
mD1
mM =0.50
2
10
101
0
10 0
10
101
1+z
102
Figure 6: Time evolution of the free-streaming scale of daughter particles
for mD1 /mM = 0.98 (left) and 0.50 (right). The different lines in the panels
correspond to the different lifetime of the mother particle: Γ = 0.01 Gyr
(magenta line), 0.1 Gyr (green line), 1.0 Gyr (red line), and 10.0 Gyr (blue
line), respectively. Black dotted lines with steeper slope are the fitting lines to
the case with particles having a constant momentum, which are proportional
to (1 + z)−3/2 , and the others are to the case with non-relativistic particle,
which is proportional to (1 + z)−1/2 . See main text for details [50].
33
1
Γ-1(Gyr)
10
Γ - 1>H - 1(z=3)
0
Free Streaming
lFS>0.25Mpc
10
10
-1
(at z=3.0)
Excluded
Allowed
-2
10 -4
10
10-3
10-2
10-1
100
1- m
m
D1
M
Figure 7: Constraint on the lifetime of decaying dark matter from the freestreaming scale. The range painted over in gray corresponds to Γ−1 >
H −1 (z = 3), at which the gravitational potential from the mother particles
would be significant at z = 3, and hence it cannot be excluded [50].
34
created and the averaged physical velocity of daughter particles becomes
constant. Thus
Const.
lFS ∼
∝ (1 + z)−3/2 ,
(2.42)
H(z)
where we have used the fact that H(z) is proportional to a−3/2 = (1 + z)3/2
in the matter dominated epoch.
Secondly, when Γ−1 < H −1 , i.e., the decay process has finished, the averaged velocity of daughter particles decays as v ∝ a−1 = (1 + z). Therefore
lFS ∝
(1 + z)
∝ (1 + z)−1/2 .
H(z)
(2.43)
One can see that these dependencies well describe the calculated curves in
the figure.
For the dark matter to form structures such as dark halos, the freestreaming scale lFS should be less than the size of the structures. We find
in figure 6 that the free-streaming scale of daughter particles is sensitive to
the lifetime Γ−1 and mD1 /mM . Contrary to the case with massless daughter
particles, the free-streaming scale of daughter particles becomes smaller if
the lifetime of the mother particle becomes shorter. The reason is that the
velocity of massive daughter particles decays faster in the earlier Universe,
because the expansion of the Universe is faster. Hence the velocity of daughter particles decays in shorter timescale if they decay earlier, which leads
to the smaller free streaming scale. As expected, the free-streaming scale
becomes smaller if the mass ratio mD1 /mM becomes smaller. Therefore any
existence of large scale structure by dark matter can be used to constrain
Γ−1 and mD1 /mM through the free streaming scale.
From the observations of Lyman α cloud at z ≲ 3, the density fluctuations
at about 1 Mpc comoving scale have been found, for example, in SDSS
[24, 59]. Therefore, when Γ−1 ≪ H −1 (z = 3) is satisfied, the range of mass
ratio mD1 /mM is excluded if lFS ≥ 1 Mpc (0.25 Mpc in physical scale) at
z = 3. By taking this into account, we obtain the constraint on the lifetime of
the mother particle as shown in figure 7. We find that the free streaming scale
has a constraining power even for (1 − mD1 /mM ) ≲ 0.01, and the constraint
is complementary to those obtained from the geometric distances to CMB
and BAO. In the case Γ−1 ≳ H −1 (z), on the other hand, we can not use this
method since most of the mother particles, and therefore the gravitational
potentials, still remain at the redshift z. A full treatment of cosmological
35
2
10
1
10
Allowed Region
l.
Γ-1(Gyr)
er et a (1σ)
t
e
P
0
B BAO(1σ)
10
CM
10
-1
Free streaming scale
10
-2
lFS>0.25Mpc
10
-3
(at z=3)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
vk(105km/sec)
Figure 8: Comparison with the results from Peter et al. [44] on the life time
and the kick velocity plane. Red, blue and yellow regions are the parameter
space that we exclude in this work [50].
density perturbations will be necessary in this case. Because our Lyman α
constraint is given at z = 3, the region corresponding to Γ−1 ≳ H −1 (z = 3)
(grey region in figure 8) can not be excluded from the free streaming scale
for now.
2.4.5
Comparison with Peter et al.
Before ending this section we compare our result with the constraints obtained by Peter et al. [36]. They considered a dark matter decay and described it in terms of the velocity of daughter particles vk at their creation
(they call it ”kick velocity”). In a virialized dark halo, matter moves slower
than its associated virial velocity vvir [60]. When vk > vvir , dark-matter halos
will be disrupted by these particle decays. They performed several numerical simulations in order to study the detailed evolution of the total mass and
density profile of galaxies composed of particles that undergo such velocity
36
kicks as a function of the kick speed. As a result, vk is strictly restricted
from the stability of the halos. We find that our constraints are comparable
with their results in high kick velocity region, as shown in figure 8. Note
that our constraints are completely independent from theirs and we believe
that our constraints are less uncertain in that they are free from the variety
of galaxies.
2.5
Summary in this section
In this section, we consider a decaying dark matter model in which the massive mother particle decays into two massive and massless daughter particles
after cosmological recombination. We derive a complete set of Boltzmann
equations to describe the evolution of the particles. We obtain constraints
on the lifetime of the mother particle Γ−1 and the mass ratio mD1 /mM with
mD2 = 0 from the Hubble parameter, CMB and BAO. The allowed Γ−1
decreases monotonically as mD1 /mM increases. For the free streaming constraint, on the other hand, we find the opposite dependence. We find Γ−1 >
30 Gyr at the massless limit of daughter particles and mD1 > 0.97mM at the
limit Γ−1 → 0, from the distance to CMB (1σ). We also obtain constraints
from the free-streaming of daughter particles from observations of Lyman
−3/2
α as (Γ−1 /10−2 Gyr) ≲ ((1 − mD1 /mM )/10−2 )
for Γ−1 < H −1 (z = 3).
However, to extend the free-streaming constraint to the range Γ−1 ≳ H −1
or to include the information from density perturbations such as CMB angular power spectrum, a complicated calculation of density perturbations is
necessary.
3
The effect of DDM on observables of cosmological fluctuations
In this section, we calculate the effect of DDM on the CMB and large scale
structure. Throughout this section, we assume a variant of the concordance
flat power-law ΛCDM model where CDM is replaced with DDM, and consider cosmological perturbations. The model is specified with the following
cosmological parameters:
(Ωb , ΩM∅ , h∅ , τreion , ns , As , Γ, mD1 /mM , mD2 /mM ),
37
(3.1)
where Ωb and ΩM∅ are the density parameters of baryons and mother particles, respectively, h∅ is the reduced Hubble constant estimated assuming
that the DDM does not decay, τreion is the optical depth of reionization, and
ns and As are respectively the spectral index and amplitude of the primordial curvature perturbation at k = 0.002 Mpc−1 . Here and in the following,
the subscript ∅ indicates quantities which are estimated assuming Γ = 0.
Following the WMAP 7-year results [25]7 , we fix (Ωb , ΩM∅ , h∅ , τreion , ns , As )
to (0.0454, 0.226, 0.704, 0.088, 0.967, 2.43 × 10−9 ). With the parameterization eq. (3.1), the genuine Hubble constant h and the density parameter
of mother particles ΩM are derived parameters which can be obtained by
solving the background evolution. It should be noted that these values of
parameters are not necessary to be the best fitting values to WMAP-7 year
data in the DDM model. These values are adopted only for a reference. Regarding mD2 /mM , while the formalism we present in this section is applicable
for arbitrary masses of the daughter particles, as is stated in introduction,
we in the rest of this paper assume that the mass of one of the daughter particles can be nonzero and the other one is massless. Thus, Γ and mD1 /mM
are treated as free parameters with mD2 /mM being fixed to zero. Subscripts
”M”, ”D1” and ”D2” indicate the mother, massive daughter, and massless
daughter particles, respectively.
3.1
Boltzmann equations
in this work, we choose the synchronous gauge of the mother particles to
describe cosmological linear perturbations. The line element is given by
{
}
ds2 = a(τ )2 −dτ 2 + (δij + hij (x, τ )) dxi dxj ,
(3.2)
where a is the scale factor and hij represents the metric perturbations. In
terms of Fourier components, hij can be given as
(
)
∫
1
3
hij (x, τ ) = d k hL (k, τ )k̂i k̂j + 6ηT (k, τ )(k̂i k̂j − δij ) exp(ik · x) ,
3
(3.3)
7
Recently, the cosmological parameters derived by Planck have been reported [61].
Since the estimated values of Ωb h2 and Ωc h2 change from those of the WMAP 7-year
results only by several percent, results and constraints presented in this work may not be
affected significantly when the cosmological parameters from Planck are adopted.
38
where k ≡ k k̂ is a wave number vector and k̂ is the unit vector of k. In the
rest of this section, we focus on a single mode of perturbations with k and
often abbreviate dependencies of perturbed variables on k.
The Boltzmann equation which describes time evolution of a phase-space
distribution function f (x, q, τ ) can be written as
( )
∂f
dxi ∂f
dq ∂f
dni ∂f
∂f
,
(3.4)
+
+
+
=
i
i
∂τ
dτ ∂x
dτ ∂q
dτ ∂n
∂τ C
where q is the comoving momentum, which is related to the physical momentum p by q = ap, and q and ni = q i /q are respectively the norm and
the direction of q. For a perturbation mode with k, the second term of the
left hand side can be rewritten as i(k̂ · n) (q/ε) f , where ε is the comoving
energy. In the left hand side, the third term can be rewritten in terms of the
metric perturbations by adopting the geodesic equation
dpµ
p0
+ Γµ αβ pα pβ = 0 .
(3.5)
dτ
In particular, time-component (µ = 0) of eq. (3.5) gives dq/dτ as
)
dq
1(
= η̇T −
ḣL + 6η̇T (k̂ · n)2 .
(3.6)
dτ
2
The fourth term of the left hand side of eq. (3.4) can be neglected to the first
order, because both (dni /dτ ) and (∂f /∂ni ) are first-order quantities in the
flat Universe.
The right hand side of eq. (3.4) is the collision term, which also describes
effects of decay or creation of particles on the distribution function. Aoyama
et al. [50] provided those for the mother and daughter particles. in this
work, we assume that momenta of the mother particles are negligibly small
compared with the mass. Under this assumption in conjunction with our
choice of gauge, the distribution function of the mother particles should be
proportional to the delta function of q. Then we obtain
fM (q, n, τ ) = NM (τ )δ (3) (q) ,
(3.7)
where NM is the comoving number density of the mother particles. Then the
collision terms can be recast into [50]
(
)
∂fM
M :
= −aΓfM ,
(3.8)
∂τ C
(
)
∂fDm
aΓNM
Dm :
δ (q − apDmax ) ,
(3.9)
=
∂τ
4πq 2
C
39
where pDmax is the initial physical momentum of decay particles in the rest
flame of the mother particles, which is given by
[
]1/2
2
2 2
(
)
1
(m
+
m
)
D1
D2
pDmax =
m2M − 2 m2D1 + m2D2 +
.
(3.10)
2
m2M
Note that collision terms are the same for both the daughter particles. For
later convenience, we introduce τq which is the conformal time when the
daughter particles with a comoving momentum q are produced, i.e.,
q = a(τq )pDmax .
(3.11)
From eqs. (3.6), (3.8) and (3.9), the Boltzmann equations for the mother
and the daughter particles can be written as
]
[
)
∂fM
qk
∂fM
1(
2
ḣL + 6η̇T (k̂ · n) = −aΓf
(3.12)
M :
+ i (k̂ · n)fM + q
η̇T −
M ,
∂τ
εM
∂q
2
[
]
)
∂fDm
qk
∂fDm
1(
2
Dm :
+i
(k̂ · n)fDm + q
η̇T −
(3.13)
ḣL + 6η̇T (k̂ · n)
∂τ
εDm
∂q
2
aΓNM
=
δ (apDmax − q) ,
4πq 2
where εM (εDm ) is the comoving energy of the mother (m-th daughter) particles. We divide a distribution function f into the background f and the
perturbation ∆f as
fM (q, k, n, τ ) = f M (q, τ )δ (3) (k) + ∆fM (q, k, n, τ ) ,
fDm (q, k, n, τ ) = f Dm (q, τ )δ (3) (k) + ∆fDm (q, k, n, τ ) ,
(3.14)
(3.15)
where f depends only on q and τ . Due to the isotropy of the background
geometry, ∆f can be expanded in terms of the Legendre polynomials Pl (k̂·n)
with l ≥ 0. Therefore we can define ∆fM(l) (∆fDm(l) ) as the l-th multipole
moment of ∆fM (∆fDm ), i.e.
∆fM (q, n, τ ) =
∆fDm (q, n, τ ) =
+∞
∑
l=0
+∞
∑
(−i)l (2l + 1)∆fM(l) (q, τ )Pl (k̂ · n),
(3.16)
(−i)l (2l + 1)∆fDm(l) (q, τ )Pl (k̂ · n) .
(3.17)
l=0
40
• Mother particles
By substituting eq. (3.14) into eq. (3.4), we obtain the Boltzmann equations for the mother particles as follows:
unperturbed : f˙ M = −aΓf M ,
(3.18)
[
]
(
)
∂∆fM
∂f M
1
qk
2
1st order :
η̇T −
+ i (k̂ · n)∆fM + q
ḣL + 6η̇T (k̂ · n)
∂τ
εM
∂q
2
(3.19)
= −aΓ∆fM .
From eqs. (3.7) and (3.18), we obtain
f M (q, t) =
N M (τ )
δ(q)
4πq 2
(3.20)
where N M (τ ) ∝ exp (−Γt) is the mean comoving number density of the
mother particles. Denoting the mean comoving number density of the mother
particles without decay as N M∅ , N M can be given as
N M (τ ) = N M∅ exp (−Γt) .
(3.21)
Note that N M∅ is constant.
According to our gauge choice, the dipole moment of the mother particles is zero. Since the mother particles do not have nonzero momentum
distribution, higher-order multipole moments should vanish, i.e.,
∆fM(l) = 0 (for l ≥ 1) .
(3.22)
On the other hand, from eqs. (3.16) and (3.19), the monopole moment ∆fM(0)
obeys the following equation,
∂∆fM(0)
1
∂f
= ḣL q M − aΓ∆fM(0) .
∂τ
6
∂q
(3.23)
Equations (3.18) and (3.23) can be recast into evolution equations for the
mean energy density ρ̄M and its perturbation ρ̄M δM , which are defined as
∫
1
ρM ≡ 4 dq 4πq 2 εM f M ,
(3.24)
a
∫
1
ρM δM ≡ 4 dq 4πq 2 εM ∆fM(0) .
(3.25)
a
41
Then by integrating eqs. (3.18) and (3.23) multiplied by 4πq 2 εM /a4 , we obtain
d
ρ̄M + 3Hρ̄M = −aΓρ̄M ,
dτ
d
ḣL
[ρ̄M δM ] + 3Hρ̄M δM = − ρ̄M − aΓρ̄M δM ,
dτ
2
(3.26)
(3.27)
where H = ȧ/a is the conformal Hubble expansion rate. In combination,
these two equations lead to
ḣL
δ̇M = −
.
(3.28)
2
We note that eq. (3.28) is the same as that for CDM without decay [62]. We
also note that ρ̄M as ρ̄M (τ ) = mM N M (τ )/a3 since the mother particles are
non-relativistic.
• Daughter particles
By substituting eq. (3.15) into eq. (3.13), one can find
∂f Dm
aΓN M
=
δ(τ − τq ) ,
(3.29)
∂τ
4πq 3 H
[
]
)
∂∆fDm
qk
∂f Dm
1(
2
1st order :
+i
(k̂ · n)∆fDm + q
η̇T −
ḣL + 6η̇T (k̂ · n)
∂τ
εDm
∂q
2
aΓN M
(3.30)
=
δM δ(τ − τq ) .
4πq 3 H
unperturbed :
In deriving these equations, we used a relation
δ(q − apDmax ) =
1
δ(τ − τq ).
qH
(3.31)
As seen from eq. (3.29) and (3.30), the unperturbed and first order perturbation equations for both the daughter particles are identical [37, 48, 50].
Therefore we can write
f D1 (q, τ ) = f D2 (q, τ ) ≡ f D (q, τ ) .
(3.32)
As in previous works [48, 50], by solving eq. (3.29), we obtain
f D (q, τ ) =
aq ΓN M (τq )
Θ (τ − τq ) ,
4πq 3 Hq
42
(3.33)
where aq = a(τq ), Hq = H(τq ) and Θ(x) is the Heaviside function.
Equation (3.30) can be expanded in terms of multipole moments, which
leads to a Boltzmann hierarchy for the daughter particles:
∂
(∆fDm(0) )
∂τ
∂
(∆fDm(1) )
∂τ
∂
(∆fDm(2) )
∂τ
∂
(∆fDm(l) )
∂τ
qk
1
∂f
aΓN M
δM δ (τ − τq ) , (3.34)
∆fDm(1) + ḣL q D +
εDm
6
∂q
4πq 3 H
)
qk (
∆fDm(0) − 2∆fDm(2) ,
(3.35)
=
3εDm
(
)
)
2
qk (
1
∂f
2∆fDm(1) − 3∆fDm(3) −
ḣL + η̇T q D(3.36)
=
,
5εDm
15
5
∂q
(
)
qk
l∆fDm(l−1) − (l + 1)∆fDm(l+1)
(for l ≥
(3.37)
3) .
=
(2l + 1)εDm
= −
By using ∆fDm(0) , ∆fDm(1) and ∆fDm(2) , the perturbed energy density δρ =
ρδ, perturbed pressure δp = pπL , energy flux θ and shear stress σ can be
written as
∫
1
ρDm δDm ≡ 4 dq 4πq 2 εDm ∆fDm(0) ,
(3.38)
a
∫
2
1
2 q
pDm πLDm =
dq
4πq
∆fDm(0) ,
(3.39)
3a4
εDm
∫
k
(ρDm + pDm ) θDm = 4 dq 4πq 2 q∆fDm(1) ,
(3.40)
a
∫
2
1
2 q
(ρDm + pDm ) σDm = 4 dq 4πq
∆fDm(2) ,
(3.41)
a
εDm
where ρDm and pDm are the mean energy density and pressure of the m-th
daughter particles, respectively. Here, πLDm , θLDm and σDm are higher-order
velocity-weighted quantities since phase space integral in these are weighted
by the velocity q/ε or velocity squired (q/ε)2 compared with the density
perturbation δDm .
Equations (3.34)-(3.37) are the same as those for massive neutrinos except
for the last term in eq. (3.34) (see ref. [62]), which is responsible for the
creation of density perturbation of the daughter particles from that of the
mother ones. While the structure of the perturbation equations may not seem
to differ from that of massive neutrinos significantly, in fact their solution is
far more complicated. The complication arises from the source terms (the
second and third terms in the right hand side of in eq. (3.34) and the last
43
term in eq. (3.36)). These terms contain delta functions on τ , which clearly
require a specialized treatment in numerical computation.
In order to solve the perturbation equations and obtain the CMB angular
power spectrum Cl as well as the matter power spectrum P (k) we modified
the publicly available CAMB code [63]. In particular, we need to calculate the
evolution of the phase space distribution of the daughter particles ∆fDm(l)
at discrete values of q. In the following, we describe how we have chosen the
discrete samples of q.
Roughly speaking, we need to choose the range of q where the mean number density of the daughter particles par q, F(q) ≡ q 2 f¯(q, τ0 ), dominantly
contributes to the integrals in Eqs. (3.38)-(3.41). Let us denote the scale
factor at t = min(t0 , Γ−1 ) as aD . Then the location of the maximum of F(q)
is approximately given by q = aD pDmax . In our analysis, we sample q in a
range 10−4 aD pDmax ≤ q ≤ min(pDmax , 5aD pDmax ), with a linearly homogeneous spacing in q. Outside this range, F(q) ∝ q 1/2 exp[−(q/aD pDmax )3/2 ] is
less than 1/100 of the maximum value, and contributions from such a range
of q would little affect the CMB and matter power spectra.8 For a range
of the DDM parameters as 0.1 ≤ mD1 /mM ≤ 1 and 0.01 Gyr ≤ Γ−1 ≤
1000 Gyr, which is of our primary interest in this work, we have found that
it is sufficient to take the number of sampled q and the maximum multipole
l of ∆fDm(l) to be 1000 and 45. We confirm that if we change the number of
q to be 1500 and 2000 and/or the maximum l to be 60 and 100, the results
we will present in Section 3.4 would differ by no more than 0.8%.
The initial conditions of ∆fDm(l) (q, τ ) are set as follows. When τ < τq ,
both daughter particles which have comoving momentum q have never been
generated. Thus
f D (q, τ ) = ∆fD (q, n, τ ) = 0 (for τ < τq ).
(3.42)
In eqs. (3.34)
terms (1/6)ḣL q(∂f D /∂q), (aΓN M /4πq 3 H)δM δ(τ −
( and (3.36), the source
)
τq ) and (1/15)ḣL + (2/5)η̇T q∂f D /∂q contain a delta function δ(τ − τq ),
which makes ∆fDm(0) and ∆fDm(2) arise like a step function at τ = τq . In order to treat these terms, we obtain the initial values by integrating eqs. (3.34)
and (3.36) with τ in a infinitesimal interval around τ = τq . The initial values
of ∆fDm(0) and ∆fDm(2) are provided in eqs. (3.47) and (3.49) in the following
We note that undersampling of the phase space of the daughter particles at a ≲
10 aD little affects cosmological observables as the daughter particles are energetically
irrelevant and hardly affect the metric perturbations at this epoch.
8
−4
44
section, §3.2. For τ > τq , q∂ f¯D /∂q becomes a smooth function of τ and the
time evolutions of the perturbed distribution functions of the daughter particles can be calculated in the same way as massive neutrinos in the standard
cosmology.
3.2
Initial condition for the perturbed distribution functions of the daughter particles
In this section, we first derive the solution of the perturbed phase space
distribution ∆fDm(l) at τ = τq . This gives the initial condition needed in
solving the perturbation evolution numerically. As a side product, we then
also derive the analytic solution of the density perturbations of the daughter
particles at superhorizon scales well before the decay time.
For later convenience, let us rewrite the solution eq. (3.33) for the background distribution function f D (q, τ ) as
f D (q, τ ) ≡ FD (q)Θ(apDmax − q) ,
(3.43)
where
FD (q) =
aq ΓN M (τq )
,
4πq 3 Hq
(3.44)
which is a smooth function of q and does not depend on τ . When the decay
of the mother particles can be neglected, their comoving number density can
be approximated as N M (τ ) = N M∅ . Thus at t ≪ Γ−1 we can approximate
eq. (3.44) as
aq ΓN M∅
FD (q) ≈
.
(3.45)
4πq 3 Hq
Adopting eq. (3.44), eq. (3.34) can be rewritten as
[
]
∂∆fDm(0)
qk
ḣL dFD
ḣL
=−
∆fDm(1) + q
Θ(τ − τq ) + −
+ δM FD δ(τ − τq )(3.46)
.
∂τ
ϵDm
6 dq
6H
By integrating eq.(3.46) in an infinitesimal interval around τ = τq , we obtain
[
]
ḣL (τq )
∆fDm(0) (q, τq ) = −
+ δM (τq ) FD (q) .
(3.47)
6Hq
45
In the same way, eq. (3.36) can be rewritten as
[
]
∂∆fDm(2)
qk
=
l∆fDm(1) − (l + 1)∆fDm(3)
∂τ
(2l + 1)εDm
[
]
[
]
ḣL 2η̇T dFD
1 ḣL 2η̇T
+
q
Θ(τ − τq ) +
+
FD δ(τ −(3.48)
τq ) ,
−
15
5
dq
H 15
5
which leads to
1
∆fDm(2) (q, τq ) =
Hq
[
]
ḣL (τq ) 2η̇T (τq )
FD (q) .
+
15
5
(3.49)
On the other hand, for l ̸= 0, 2, as the right hand side of Eqs .(3.34) and
(3.37) are smooth around τ = τq , we obtain
∆fDm(l) (q, τq ) = 0 (for l ̸= 0, 2).
3.2.1
(3.50)
Radiation-dominated era
Now let us consider the superhorizon solution well before the decay time. In
radiation-dominated era, δM at superhorizon scales, H and a can be related
to τ as follows (see e.g. ref. [62]).
δM ∝ τ 2 ,
H = τ −1 ,
a ∝ τ .
(3.51)
(3.52)
(3.53)
By substituting eqs. (3.51) and (3.52) into eq. (3.28) one can derive a relation
ḣL = −2δ̇M = −4HδM .
(3.54)
Substituting eqs. (3.54) into eq. (3.47), we obtain
5
∆fDm(0) (q, τq ) = FD (q)δM (τq ).
(3.55)
3
Furthermore, by substituting eq. (3.52) into eq. (3.45) and τq ∝ q, one can
find
FD (q) ∝ q −1 ,
dFD
q
= −FD .
dq
46
(3.56)
(3.57)
Then in the limit of kτ → 0, eq. (3.46) in conjunction with eq. (3.55) gives
([
)
]τ
1
∂FD
5
hL q
∆fDm(0) (q, τ ) =
+ FD (q)δM (τq ) Θ(apDmax − q)
6
∂q τq 3
[
]
1
4
= FD (q) δM (τ ) + δM (τq ) Θ(apDmax − q) .
(3.58)
3
3
Using eqs. (3.51) and (3.53), we then obtain
[
( a )2 ]
1
q
Θ(apDmax − q) .
∆fDm(0) (q, τ ) = FD (q)δM (τ ) 1 + 4
3
a
(3.59)
Note that this result does not depend on the particle index m. By substituting eq. (3.59) into eq. (3.38), we can compute the density perturbation
δDm . When the decay products are relativistic pDmax ≫ mDm , we obtain
δDm (τ ) =
17
δM (τ ) .
15
(3.60)
Finally, when the decay products are non-relativistic pDmax ≪ mDm , we
obtain
δDm (τ ) = δM (τ ) .
(3.61)
3.2.2
Matter-dominated era
In the matter-dominated era well before the decay time, δM at superhorizon
scale, H and a can be related to τ as [62]
δM (τ ) ∝ τ 2 ,
H = 2τ −1 ,
a ∝ τ2 .
(3.62)
(3.63)
(3.64)
Then one can find eq. (3.54) should be replaced with
ḣL = −2HδM ,
(3.65)
with which eq. (3.55) should be replaced with
4
∆fDm(0) (q, τq ) = FD (q)δM (τq ).
3
47
(3.66)
By substituting eqs. (3.63) and (3.64) into eq. (3.45), we obtain
FD ∝ q −3/2 ,
dFD
3
q
= − FD .
dq
2
Then from eqs. (3.46) and (3.66), in the limit kτ → 0 we obtain
[
]
1
5aq
∆fDm(0) (q, τ ) = FD (q)δM (τ ) 1 +
Θ(apDmax − q) .
2
3a
(3.67)
(3.68)
(3.69)
In the same way as in radiation-dominated epoch, by substituting eq. (3.69)
into eq. (3.38), we can compute the density perturbation δDm . When they
are relativistic, we obtain
23
δDm = δM ,
(3.70)
21
while when the decay products are non-relativistic, we obtain
δDm = δM .
3.3
(3.71)
Time evolutions of perturbations
In this section, we discuss time evolutions of perturbed quantities. Hereafter
we continuously assume that mD2 is zero, while the mass of the first daughter
particles mD1 can vary in [0, mM ]. For later convenience, we respectively
denote the scale factor at the decay time and the horizon crossing as ad and
ahc , i.e., t(ad ) ≡ Γ−1 and τ (ahc ) ≡ πk −1 .
3.3.1
A case of non-relativistic decay
In this subsection, we consider a case with mD1 ≃ mM , which we refer to
as non-relativistic decay. As a representative value, we here adopt mD1 =
0.999 mM , with which the massive daughter particles are produced with a
velocity kick vkick ≃ 1 − mD1 /mM = 0.001. We expect time evolutions of perturbations depend on whether the scale of perturbations is inside or outside
the horizon, as well as whether the DDM has decayed or not. Therefore in
what follows we separately investigate cases with ad < ahc and ad > ahc .
48
3.3.2
Perturbations crossing the horizon after the decay time
Let us first consider perturbations which cross the horizon after the decay
of DDM, i.e. ad < ahc . Here we adopt Γ−1 = 6 Myr and consider evolutions of perturbations at k = 8 × 10−4 hMpc−1 . This setup corresponds to
ad ≃ 5 × 10−3 and ahc ≃ 0.1. In figure 9, we plot time evolutions of perturbation quantities of the mother and daughter particles, including density
perturbations δi , pressure ones πLi , energy fluxes θi and shear stresses σi .
Note that regarding perturbed quantities of the mother particles, only the
density perturbation δM is nonzero due to the vanishing momentum distribution of the mother particles and our choice of gauge.
Before the decay of DDM a < ad , the density perturbations of the daughter particles δD1 and δD2 grow in proportion to δM . It is because monopole
moments ∆fDm(0) are sourced by the density perturbation of the mother
particles δM and the metric perturbation hL , which also grow during the
matter-domination epoch. In particular, sufficiently prior to the decay time
in the matter-dominated Universe, δD1 and δD2 are related to δM as
δD1 = δM ,
23
δD2 =
δM ,
21
(3.72)
(3.73)
on superhorizon scales, which can be derived analytically as in §3.2. Our
numerical result is consistent with analytic one as seen in the upper left
panel of figure 9.
Since the massive daughter particles are non-relativistic, their pressure
2
perturbations are suppressed as δpD1 ≃ O(vkick
)δρD1 , while that of the massless daughter one is large as δpD2 = δρD2 /3. As can be seen in eq. (3.35)
dipole moments ∆fDm(1) are sourced only by monopole and quadrupole moments via free-streaming. At superhorizon scales k < πτ −1 , therefore dipole
moments ∆fDm(1) and hence the energy fluxes θDm are little generated. On
the other hand, as quadrupole moments ∆fDm(2) are directly generated by
metric perturbations (see eq. (3.35)), σDm can be large. In addition, for
the same reason as the pressure perturbation δpD1 , velocity-weighted quantities of the massive daughter particles, including θD1 and σD1 , are further
suppressed.
After the decay of DDM but still before the horizon crossing ad < a < ahc ,
daughter particles are no longer sourced by the mother particles and the last
term of eq. (3.34) becomes negligible. In this epoch, the Boltzmann equations
49
t~Γ-1
ad 5×10-3
10-3
10-2
10-1
10-3
10-2
10-1
102
100
10-2
10-4
10-6
10-8
10-10
100
10-2
10-4
10-6
10-8
10-10
(1+wi)σi [h-3Mpc3]
wiπLi [h-3Mpc3]
δi [h-3Mpc3]
horizon crossing
ahc 1×10-1
(1+wi)θi [h-2Mpc2]
102
100
10-2
10-4
10-6
10-8
10-10
100
10-2
10-4
10-6
10-8
10-10
t~Γ-1
ad 5×10-3
horizon crossing
ahc 1×10-1
100
Figure 9: Time evolutions of perturbations in the case of non-relativistic decay. Here we show perturbations whose scale k = 8 × 10−4 hMpc−1 crosses
the horizon after the decay time Γ−1 = 6 Myr. Shown are the density perturbations (top left), energy fluxes (top right), pressure perturbations (bottom
left) and anisotropic stresses (bottom right) of the mother (blue solid line),
massive daughter (red dashed line) and massless daughter (green dot-dashed
line) particles. In the upper left panel, the line of the density perturbation
of the massive daughter overlaps that of the mother particle. Two dotted
vertical lines indicate the horizon crossing and the decay time. wi = pi /ρi is
the equation of state of the i-th component. According to our gauge choice,
the dipole and quadrupole moment of the mother particle are zero. Thus the
energy flux and anisotropic stress of mother particle are zero. In addition,
because of our assumption that the momentum of mother particle is negligible compared with its mass, the pressure perturbation of mother particle
becomes zero. Therefore these quantities of mother particle, such as θM , πM
and σM , are zero and not shown in upper right and bottom panels [51].
50
for the daughter particles become the same as that for collisionless freestreaming particles, e.g. massive neutrinos (see ref. [62]). For non-relativistic
particles, the evolution equation of ∆fD1 should be effectively reduced to that
of CDM. Hence the density fluctuation of the massive daughter particles
evolves in the same way as the mother ones. Due to redshifting of physical
momenta, higher-order velocity-weighted quantities such as πLD1 , θD1 and
σD1 decrease afterward. For the massless daughter particles, the evolution
equation of ∆fD2 becomes the same as that of massless neutrinos.
After the horizon crossing a > ahc , the daughter particles start freestreaming. The free-streaming length of the massless daughter particles
equals to the size of horizon and perturbation quantities decay oscillating.
For massive daughter particles, the density perturbation δD1 continues to
grow inside the horizon in the same way as CDM, while the velocity-weighted
quantities continue to decay due to redshifting of momentum.
3.3.3
Perturbations crossing the horizon before the decay time
Now we move to a case where DDM decays inside the horizon. We set the
decay time to Γ−1 = 0.1 Gyr, which corresponds to ad ≃ 3 × 10−2 , and
investigate perturbations on a scale k = 8 × 10−3 hMpc−1 , which crosses the
horizon at ahc ≃ 2 × 10−3 . Time evolutions of perturbed quantities related
to the mother and daughter particles are plotted in figure 10.
By comparing figure 10 with figure 9, we can say that at superhorizon
scales when a < ahc and in this case inevitably before the decay, behaviors of
perturbed quantities of the mother and daughter particles are qualitatively
the same as in the previous case where the horizon crossing occurs after
the decay. In particular, the analytical solutions of eqs. (3.72) and (3.73)
again hold at superhorizon scales, which can be qualitatively confirmed in
the upper left panel of figure 10.
On the other hand, evolutions of perturbations inside the horizon drastically differ from those in the previous case. After the horizon crossing but
still before the decay ahc < a < ad , the massless daughter particles start to
free-stream. However, as we can see in figure 10, the perturbation quantities,
such as δD2 , θD2 and σD2 do not start to oscillate nor decay, contrary to those
in figure 9. This is because the monopole moment of the massless daughter particles ∆fD2(0) is continuously sourced by the density perturbation of
the mother particles, and then the dipole and higher multipole moments
are also continuously sourced by ∆fD2(0) . Therefore perturbation quantities
51
δi [h-3Mpc3]
10
102
100
100
10-2
10-2
10-4
10-4
10-6
10-6
100
100
10
-2
10
-2
10-4
10-4
10-6
10-6
10
-3
10
-2
a
10
-1
10
-3
10
-2
a
10
-1
(1+wi)σi [h-3Mpc3]
wiπLi [h-3Mpc3]
horizon crossing t~Γ-1
ahc 2×10-3 ad 3×10-2
(1+wi)θi [h-2Mpc2]
horizon crossing t~Γ-1
ahc 2×10-3 ad 3×10-2
2
0
10
Figure 10: Same as in figure 9 but for Γ−1 = 0.1 Gyr and k = 8 ×
10−3 hMpc−1 . In the upper left panel, the line of the density perturbation of the massive daughter overlaps that of the mother particle as same as
figure 9 [51].
52
of free-streaming relativistic particles keep on growing even inside the horizon. Moreover, as for the massive daughter particles, higher-order velocityweighted quantities including πLD1 , θLD1 and σLD1 also grow. This is because
the massive daughter particles with the nonzero velocity vkick = 0.001 keep
on being produced, and these particles with relatively large momenta freestream to make higher-order velocity-weighted quantities grow continuously
in addition to the density perturbation δρD1 .
Finally at a > ad , when there are few mother particles to decay, the
source terms in the Boltzmann eqs. (3.34)-(3.37) for the daughter particles
become ineffective and these particles behave as collisionless free-streaming
particles. Perturbation quantities of the massless daughter particles start to
oscillate and decay due to free-streaming. Comparing figure 10 with figure 9,
we can see perturbation quantities of the massless daughter particles decay
more quickly and effects of free-streaming are more prominent than the previous case. This is because most of the daughter particles are produced when
a ≃ ad and these particles immediately free-stream over distances larger than
the perturbation scale. On the other hand, since the massive particles are
non-relativistic, effects of free-streaming is not significant and higher-order
velocity-weighted quantities decrease mostly due to redshifting of momentum.
3.3.4
A case of relativistic decay
In this subsection, we consider a case with mD1 ≪ mM , which we refer to
as relativistic decay. Here we adopt mD1 = 0.1mM , which gives a velocity
kick of the massive daughter particles vkick ≃ 1 − 2(mD1 /mM )2 = 0.98. In
the same way as section 3.3.1, we in the following explore time evolutions of
perturbations in cases with ad < ahc and ahc < ad separately.
• Perturbations crossing the horizon after the decay time
Let us see time evolutions of perturbations whose scale crosses the horizon
after the decay time. We adopt a decay time Γ−1 = 6 Myr and the scale
of perturbations is fixed to k = 8 × 10−4 hMpc−1 . These parameters lead
to ad = 5 × 10−3 and ahc = 5 × 10−2 . Figure 11 shows time evolutions of
perturbation quantities of the mother and daughter particles.
First of all, we expect cosmological effects of the two daughter particles
are identical as long as both of them are sufficiently relativistic. Noting
that momenta of particles scale as a−1 , the time when most of the massive
daughter particles become non-relativistic can be estimated as anr = (vkick /
53
10-1
10-1
10-2
10-2
δi [h-3Mpc3]
10-3 -3
10
10
-2
a
10
-1
10
-3
10
-2
a
10
-1
10-3
(1+wi)σi [h-3Mpc3]
wiπLi [h-3Mpc3]
102
101
100
10-1
10-2
10-3
10-4
(1+wi)θi [h-2Mpc2]
102
101
100
10-1
10-2
10-3
10-4
t~Γ-1 horizon crossing
ad 5×10-3 ahc 5×10-2
t~Γ-1 horizon crossing
ad 5×10-3 ahc 5×10-2
0
10
Figure 11: Same as in figure 9 but for mD1 /mM = 0.1 [51].
√
2
1 − vkick
)ad , which is roughly 2.5 × 10−2 with the parameter values we
adopted here. Therefore when a < anr , perturbation quantities of the two
daughter particles are almost the same, which can be confirmed in figure 11.
One big difference from the case of non-relativistic decay is that the density perturbations δi grow more rapidly after the decay at superhorizon scales
as can be seen in the upper left panel of figure 11. The reason can be understood as follows. Since the decay products are relativistic, the Universe
is effectively dominated by radiation after the decay. Therefore, as in the
radiation-dominated epoch, density perturbations grow as a2 at superhorizon scales instead of a in the matter-dominated epoch (see eqs. (3.51) and
(3.62)). As seen in the ΛCDM model with low Ωm (we refer to, e.g., ref. [64]),
this leads to an enhancement in the matter power spectrum at scales which
crosses the horizon after the matter-radiation equality. This issue will be
discussed further in section 3.4.2.
After horizon crossing, even when a > anr , due to the non-vanishing
velocity and free streaming of daughter particles, at first growth of δD1 is
slower compared with the case of non relativistic decay (see also the upper
left panel of figure 9). This leads that the density perturbation of the mother
particles δM also grows less, for the gravitational potential sourced by δD1
decays inside the horizon. After the massive daughter particles become fully
54
horizon crossing t~Γ-1
ahc 2×10-3 ad 3×10-2
103
102
101
100
10-1
10-2
10-3
100
100
10-1
10-1
10-2
10-2
10
-3
10-3
10-2
a
10-1
10-3
10-2
a
10-1
10
100
-3
(1+wi)σi [h-3Mpc3]
wiπLi [h-3Mpc3]
10
102
101
100
10-1
10-2
10-3
horizon crossing t~Γ-1
ahc 2×10-3 ad 3×10-2
(1+wi)θi [h-2Mpc2]
δi [h-3Mpc3]
3
Figure 12: Same as in figure 10, but for mD1 /mM = 0.1. Some jaggy features
on plots are caused by the numerical error on the calculation [51].
non-relativistic, their density perturbation starts to grow as CDM does.
Other perturbation quantities with higher velocity-weights such as πLD1 , θD1
and σD1 start decreasing almost monotonically without violent oscillations,
which is also seen in cases of non-relativistic decay (see the upper right and
bottom panel of figure 9). The massless particles keep on free-streaming and
their perturbations except for velocity divergence continuously decay.
• Perturbations crossing the horizon before the decay time
Figure 12 shows time evolutions of perturbations for a case where the decay
occurs inside the horizon. Here we adopt Γ−1 = 0.1 Gyr and show perturbations at a scale k = 8 × 10−3 hMpc−1 . These parameters correspond to
ad ≃ 3 × 10−2 and ahc = 2 × 10−3 . In this case, most of √
the massive daughter
2
particles become non-relativistic at around anr = (vkick / 1 − vkick
)ad ≃ 0.15.
At a < anr , as expected, the perturbation evolutions of the two daughter
particles are again almost the same, which can be seen in figure 12. As is
discussed in the case of non-relativistic decay in section 3.3.1, in this case
perturbation quantities continuously grow inside the horizon until the mother
particles completely decay, and then effects of free-streaming become significant. This can be confirmed in figure 12.
55
As the massive daughter particles become non-relativistic at a > anr
their density perturbation δD1 starts to grow and other velocity-weighted
perturbations πLD1 etc. start to decrease with less oscillation than before.
However, the decay of δD1 is so significant, the density perturbation of total
matter becomes much smaller than the density perturbations of the mother
particles δM . This decay of the density perturbation leads to a significant
suppression in the matter power at small scales as will be shown in section
3.4.2. On the other hand, the massless daughter particles continuously freestream.
3.4
3.4.1
Signatures in cosmological observables
Effects on Cl
In this subsection, we consider the CMB power spectrum Cl in the DDM
models with Γ−1 > t∗ . In figure 13, we plot the CMB temperature power
spectrum ClTT with various decay times 0.01 < Γ−1 [Gyr] < 100 and mass
ratios 0.3 < mD1 /mM < 0.9. From the figure, we can see effects of DDM on
ClTT are twofold. First one is the shift of the positions of the acoustic peaks
caused by a change in the background expansion. The other is the integrated
Sachs-Wolfe effect induced by the decay of the gravitational potential.
Let us first investigate the effects on positions of the acoustic peaks.
When Γ−1 is shorter than the age of the Universe at present tage ≃ 14 Gyr
[61] and DDM decays into the relativistic daughter particles, energy density
in the Universe and hence the expansion rate stay below those in the ΛCDM
model. This makes angular diameter distances larger and hence the angular
size of the sound horizon smaller than in the case of the ΛCDM model.
Therefore positions of the acoustic peaks shift toward higher l.
Second, let us move to the effects that arise at large angular scales larger
than the sound horizon at the recombination epoch. At these scales, CMB
temperature anisotropy can be given as [62, 65]
∫ τnow (
)
δT
1
(n) = ψ(x∗ , τ∗ ) +
dτ ϕ̇(x, τ ) + ψ̇(x, τ ) ,
(3.74)
T
3
τ∗
where x = (τnow − τ )n, ϕ(x, τ ) and ψ(x, τ ) are the curvature perturbation and the gravitational potential, respectively. They are given in terms
of the metric perturbations in
as ϕ = ηT − (H/
( the synchronous gauge )
2k 2 )(ḣL +6η̇T ) and ψ = 1/2k 2 (ḧL + 6η̈T ) + H(ḣL + 6η̇T ) [62]. The second
56
TT
l(l+1)Cl /2π[µK2]
104
104
103
103
Γ-1=0.01Gyr
Γ-1=0.1Gyr
102
102
103
103
Γ-1=3Gyr
Γ-1=1Gyr
TT
l(l+1)Cl /2π[µK2]
102 0
10
104
1
10
10
2
3 0
10 10
10
1
10
2
10
3
102
104
103
103
Γ-1=10Gyr
Γ-1=20Gyr
102
102
103
103
Γ-1=40Gyr
Γ-1=100Gyr
2
10
10
0
1
10
l
10
2
3 0
10 10
10
1
l
10
2
10
3
10
2
Figure 13: Effects of DDM on CMB power spectrum of temperature fluctuation ClTT . Line color distinguishes the mass ratio mD1 /mM . Red (longdashed), dark-green (short-dashed), blue (dotted), green (dot-dashed) lines
represent mD1 /mM = 0.3, 0.5, 0.7, and 0.9, respectively. Purple (solid) lines
correspond to ClTT in the ΛCDM model. Each panel shows ClTT with a distinct decay time Γ−1 , which varies from 0.01 (top left) to 100 Gyr (bottom
right) as indicated in each panel [51].
57
term in eq. (3.74) shows that a time derivative of the curvature perturbation
ϕ̇ and that of gravitational potential ψ̇ generate additional CMB temperature
fluctuations on large scales, which is called the integrated Sachs-Wolfe (ISW)
effect. As we have shown in section 3.3, in the DDM models density perturbations of daughter particles and hence the gravitational potentials can
decay at t > Γ−1 , due to free-streaming of the daughter particles. This leads
to a large ISW effect. From figure 13, one can find that the effect is enhanced
as mD1 /mM decreases, which can be easily understood as we have seen that
the suppression of density perturbation δD1 is more significant in case of the
relativistic decay than in the non-relativistic decay in section 3.3. On the
other hand, one can recognize that the largest l where ClTT is enhanced decreases as Γ−1 increases. This is because the ISW effect is only effective at
scales larger than the free-streaming scale of the daughter particles within
which the gravitational potential decays. At smaller scales, photons travel
through a number of peaks and troughs of the gravitational potential and
the net effect of the potential decay becomes negligible.
Finally, let us describe signatures in the CMB polarization spectrum ClEE
and its cross-correlation with the temperature anisotropy ClTE . In figures 14
and 15, we plot ClEE and ClTE , respectively, with the same parameter sets as
in figure 13. First, we can see that the DDM models affect ClEE and ClTE
through the change in the background expansion. Therefore in the same way
as ClTT , acoustic peaks and troughs in ClEE and ClTE shift toward higher l in
the DDM models. Second, in figures 14 and 15 we can also see that there
arises an additional E-mode polarization for Γ−1 < 3 Gyr at l ∼ 10. Because
the decay of dark matter causes an additional ISW effect, temperature fluctuations are created at a superhorizon scale. Just after horizon crossing, the
quadrupole moment of temperature fluctuations is created due to the freestreaming of CMB photons. In addition, free-streaming motion of daughter
particles create a nonzero shear stress and hence induce the anisotropic part
of the metric perturbations, which also generates the quadrupole moment
of the temperature fluctuations. After the cosmological reionization, the
E-mode polarization is created from the quadrupole moment of the temperature anisotropy by the Thomson scattering. When one considers the epoch
well after cosmological recombination, the number density of free electrons
reaches a maximum in the epoch of the cosmological reionization. Thus, the
decay of DDM generates the additional E-mode polarization significantly if
the decay occurs around the reionization epoch at l ∼ 10, which corresponds
to the view angle of the horizon size at the cosmological reionization.
58
102
101
102
-1
-1
Γ =0.01Gyr
Γ =0.1Gyr
101
EE
l(l+1)Cl /2π[µK2]
100
10
100
-1
10
10-2
10-3
101
10-2
Γ =1Gyr
-1
Γ =3Gyr
101
100
100
10-1
10-1
10-2
10-2
101
EE
10-3
-1
10-3 0
10
102
l(l+1)Cl /2π[µK2]
-1
1
10
10
2
3 0
10 10
-1
10
1
10
2
10
3
-1
Γ =10Gyr
Γ =20Gyr
10-3
102
101
100
100
10-1
10-1
10-2
10-2
10-3
101
10-3
-1
Γ =40Gyr
-1
Γ =100Gyr
101
100
100
10-1
10-1
10-2
10-2
10-3
100
101
102
103100
l
101
102
103
l
Figure 14: Same as figure 13 , but for ClEE [51].
59
10-3
103
Γ-1=0.01Gyr
l(l+1) Cl
TE
/2π[µK2]
102
/2π[µK2]
102
101
100
100
10-1
10-1
10-2
10-2
-1
Γ =3Gyr
-1
Γ =1Gyr
102
102
101
101
100
100
10-1
10-1
102
TE
Γ-1=0.1Gyr
101
10-2 0
10
103
l(l+1) Cl
103
1
10
10
2
3 0
10 10
Γ-1=10Gyr
10
1
10
2
10
3
Γ-1=20Gyr
10-2
103
102
101
101
0
0
10
10
10-1
10-1
10-2
102
-1
10-2
-1
Γ =40Gyr
Γ =100Gyr
102
101
101
100
100
10-1
10-1
10-2 0
10
1
10
10
2
3 0
10 10
l
10
1
10
2
10
l
Figure 15: Same as figure 13 , but for ClTE [51].
60
3
10-2
When DDM decays much earlier or much later than the cosmological
reionization epoch, the additional quadrupole moment of the CMB temperature anisotropy at the reionization epoch is small and the signatures of DDM
in ClEE or ClTE become insignificant.
3.4.2
Effects on P (k)
Let us consider effects of the decay of DDM on the matter power spectrum
P (k). As we will see, effects from the decay are more prominent in P (k)
than in Cl , so that we may obtain stronger constraints from observations of
P (k) than from those of Cl .
In figure 16, we plot the matter power spectra P (k) in the DDM models with various lifetimes Γ−1 and mass ratios mD1 /mM . In the same figure
we also plot P (k) for the ΛCDM model as a reference. From the figure,
one can find that there arise two large differences from the ΛCDM model in
P (k) . The first one is the suppression at smaller scales. This suppression is
caused by the free-streaming of the daughter particles, as we have discussed
in section 3.3. The other one is the enhancement at large scales. This enhancement is caused by the growth of density perturbations at superhorizon
scales after the decay of DDM, which we have also discussed in section 3.3.4.
These changes are significant if the decay time is smaller than the age of the
Universe and the mass difference between the mother and daughter particles
are large. On the other hand, if the decay time is larger than the age of the
Universe, a significant fraction of the mother particles, whose density perturbation grows as that of CDM, still survive until today and the deviations
in P (k) from the ΛCDM model become less prominent.
To clarify the cause of the changes in P (k), in figure 17 we plot P (k) for
various decay time Γ−1 = 0.01, 0.1 0.3 and 1 Gyr ≪ tage with a fixed mass
ratio mD1 /mM = 0.7. As a reference, we also plot P (k) of the flat ΛCDM
model in which the CDM density parameter is changed from the fiducial value
Ωc to mD1 /mM × Ωc so that the energy density of dark matter in the present
Universe should be the same as in the DDM models. As we attribute the
suppression at small scales to the free-streaming of the daughter particles, we
in addition plot the free-streaming scales of the massive daughter particles
61
λFSS , which can be approximately given as
∫ τ0
∫ 1
vkick
1
3vkick Γ−1
√
√
dτ v(τ ) ∼
da
∼
λFSS =
,
ad
H0 ΩM ad a−1/2 q 2 + m2 a2
τd
(3.75)
where we assumed that the background expansion does not deviate significantly from that in the reference ΛCDM model, which is a good approximation in the case of mD1 /mM = 0.7. As is expected, we can see that the
scales of the suppressions roughly agree with the free-streaming scales of the
massive daughter particles. We can also see that the matter power spectra
in the DDM models asymptotically become the same as that in the reference
ΛCDM model. This shows that the enhancement in P (k) at large scales
seen in figure 16 is explained by the reduction in the energy density of dark
matter, which effectively changes the matter-radiation equality and growth
of the density perturbations at superhorizon scales.
3.5
Discussion
In order to illuminate the difference of the power spectrum between DDM
and ΛCDM, we plot P (k) in the DDM models normalized by that of the
ΛCDM model, P (k)/PΛCDM (k) with several parameter sets (Γ−1 , mD1 /mM )
in figure 18.
In the figure we find asymptotic plateaus on small scales, especially in
the plots for small mD1 /mM ratios. In the small-scale limit k → ∞, we find
the ratio r to be
(
(
))2
P (k)
tage
r=
∼ exp − −1
.
(3.76)
PΛCDM (k)
Γ
Thus, r is responsible for the energy density of the surviving mother particles.
Therefore precise measurements of the matter power spectrum enable us
to distinguish DDM from the warm dark matter (WDM) models, because
these models predict that the power spectrum monotonically decreases to
zero as k increases on smaller scales9 , although that of DDM approaches
asymptotically to a constant r on these scales.
In order to set a constraint on the parameters of the DDM models, Γ−1
and mD1 /mM , we quantify the effect of DDM on σ(R) , which is the fluctuation
9
For example, Bode et al. mentions that the matter power spectrum in a WDM model
decreases in proportion to k −10 on smaller scales [66].
62
105
105
-1
Γ =0.1Gyr
-1
Γ =0.01Gyr
P(k)[h-3Mpc3]
104
104
103
-1
103
-1
Γ =1Gyr
Γ =3Gyr
104
104
P(k)[h-3Mpc3]
103 -4
10
5
10
10
-3
-2
10
10
-4
Γ-1=10Gyr
10
-3
-2
10
103
105
Γ-1=20Gyr
104
104
103
103
Γ-1=40Gyr
Γ-1=100Gyr
104
103 -4
10
104
10
-3
k[hMpc-1]
-2
10
10
-4
10
-3
k[hMpc-1]
-2
10
103
Figure 16: P (k) for the DDM model with various parameters (Γ−1 , mD1 /
mM ). Red (long-dashed), dark-green (short-dashed), blue (dotted), green
(dot-dashed) lines represent cases with mD1 /mM = 0.3, 0.5, 0.7, 0.9, respectively. Purple (solid) line corresponds to the ΛCDM model [51].
63
P(k)[h-3Mpc3]
105
104
103
10-4
10-3
10-2
k[hMpc-1]
10-1
Figure 17: Matter power spectrum at present for several parameter sets with
Γ−1 ≪ tage . On this graph we fixed mD1 /mM to 0.7 . Red (long-dashed),
green (short-dashed), blue (dotted), purple (dot-dashed) lines represent P (k)
in the case that the lifetime of DDM Γ−1 = 0.01, 0.1, 0.3, and 1 Gyr, respectively. Dotted lines correspond to the free streaming scales kFSS = π/λFSS
calculated form eq. (3.75) in these parameter sets, which correspond to 2.1,
3.1, 4.7, and 10 [×10−3 hMpc−1 ], respectively. Dark-green line shows P (k)
in a case of the ΛCDM model whose dark matter density parameter Ωc is
replaced with mD1 /mM × Ωc [51]. Note that these parameter sets have been
already excluded in our previous work [50].
64
mD1/mM=0.9
-1
k[hMpc ]
10-2 10-4
10-3
-1
10-2
P(k)/PΛCDM(k)
mD1/mM=0.5
1.2
1
1.0
0.8
0.6
0.4
0.2
0
1
1.0
0.8
0.6
0.4
0.2
0
P(k)/PΛCDM(k)
P(k)/PΛCDM(k)
P(k)/PΛCDM(k)
1.2
1.0
0.8
0.6
0.4
0.2 m /m =0.3
D1
M
0
1.0
0.8
0.6
0.4
0.2 mD1/mM=0.7
0
10-4
10-3
k[hMpc ]
Figure 18: The ratio of P (k) to the matter power spectrum in the ΛCDM
model PΛCDM . Lines from bottom to top represent numerically-obtained
values in the case of Γ−1 =10, 15, 20, 30, 40, 50, 100, 200, 400, and 800 Gyr,
respectively. Different panels show P (k) with different mass ratios mD1 /mM
which are 0.3 (top left), 0.5 (top right), 0.7 (bottom left) and 0.9 (bottom
right) [51].
65
amplitude at scale R in units of h−1 Mpc. Given P (k), σ(R) can be obtained
as
∫ ∞
2
σ(R) = 4π
dk k 2 W 2 (kR)P (k) ,
(3.77)
0
where W (kR) is defined as
W (kR) ≡
3
(sin(kR) − kR cos(kR)) ,
(kR)3
(3.78)
by employing the top hat window function.
We calculate the fluctuation amplitude at 8 h−1 Mpc, σ8 , in the DDM
model and compare it to observations [67, 68, 69, 70]. Recently, σ8 is reported
as [69]
σ8 = 0.80 ± 0.02 .
(3.79)
(th)
We rule out the parameter region where σ8 deviates from eq. (3.79) by
more than 2σ confidence level as shown in figure 19. In the same figure, we
also depict constraints from Peter [44] and Wang et al.[49], which we referred
to in introduction.
To understand the constraint in figure 19, it is convenient to consider
cases Γ−1 > tage and Γ−1 < tage separately. First, in the case of Γ−1 > tage ,
not all of the mother particles have decayed by now. As Γ−1 increases, more
mother particles survive in the present Universe and the deviation from the
ΛCDM model becomes less. Therefore, depending on the fractional mass
difference mD1 /mM , small Γ−1 is excluded. In particular, when the decay is
to some extent relativistic, that is, mD1 /mM ≲0.9, Γ−1 ≳ 200 Gyr is allowed.
We think the reason why the lower bound on Γ−1 hardly depends on mD1 /mM
is that the daughter particles have a velocity kick which is close to the speed
of light. This result is consistent with previous works [34, 40, 44, 50]. On
the other hand, when the decay is highly non-relativistic, mD1 /mM >0.998,
the lifetime of DDM is not constrained, since the evolution of the decay
products is indistinguishable from that of CDM. Second, in the case of
Γ−1 < tage , all the mother particles decay into the daughter particles. In this
case how P (k) is suppressed can be understood in terms of the free-streaming
length λFSS , as is discussed in the literature, e.g. ref. [50]. Therefore a
parameter region with a large mass difference 1 − mD1 /mM is excluded in
figure 19. As we have mentioned in section 3.4.2, given a fixed mD1 /mM , λFSS
decreases as the mother particles decay earlier and the suppression becomes
66
103
Γ-1[Gyr]
102
tage
101
100
10-1
10-3
10-2
102
101
100
excluded
region
Wang et al.
10-2 -4
10
103
Ichiki et al.
0.1
0.3
1-(mD1/mM)
Γ-1[Gyr]
allowed Peter
region
10-1
1
10-2
Figure 19: Constraint on the lifetime of DDM and the mass ratio of the
massive daughter particle to the mother particle. The blue shaded region
represents the excluded parameter region at 2 σ confidence level from a constraint of σ8 [69]. The green (dashed) line presents the age of the Universe
at present tage = 13.8 Gyr. The purple, green and red shaded regions are the
parameter regions which Peter [44], Wang et al. [49] and Ichiki et al. [34]
have excluded, respectively [51].
less significant. Therefore the constraint on the mass difference becomes
weaker as the lifetime Γ−1 becomes smaller.
Let us remark on constraints from CMB data. As we have shown, DDM
models affect the CMB temperature power spectrum effectively in two ways:
shifting the angular scale of the acoustic oscillation and enhancing the ISW
effect. The constraint from the former effect was derived in Aoyama et al.[50],
and we have found that it is less strong than one we here derived from σ8 .
On the other hand, we also expect that the latter effect would not be so
powerful as σ8 due to the cosmic variance.
We should note that the constraint on Γ−1 and mD1 /mM we derived here
is overestimated, as we fixed other cosmological parameters which may degenerate with these two parameters. We will pursue this issue in future
works.
3.5.1
Implication on the anomaly in estimated σ8 from Planck
Planck collaboration reports that the estimated σ8 from their cluster number count through the SZ effect is σ8 = 0.78 ± 0.01, which is smaller than
67
103
Γ-1[Gyr]
10-1
102
101
100
Wang et al.
10-2 -4
10
10-3
10-2
Γ-1[Gyr]
102
tage
101
100
103
Ichiki et al.
Peter
10-1
0.1
0.3
1-(mD1/mM)
1
10-2
Figure 20: The parameter region which can explain the tension in the estimated σ8 from the SZ effect cluster number count and from ClTT at 1σ
confidence level [51]. The parameter regions excluded in previous studies are
also shown in the figure. The purple, green and red regions are the same as
in figure 19.
that from the anisotropy of the CMB σ8 = 0.834 ± 0.027 by more than 2σ
confidence level [71]. Because the number of clusters reflects the matter perturbation in the late-time Universe, this discrepancy in the estimated σ8 may
indicate some mechanisms which suppress the matter perturbation at small
scales after cosmological recombination. The DDM model may reconcile the
discrepancy, because the decay of DDM with the lifetime slightly larger than
the age of the Universe can suppress the matter power at present keeping the
CMB power spectrum almost unchanged. In figure 20, we plot a parameter
region which can explain the estimated σ8 obtained from the cluster number
count in this DDM model at the 1σ confidence level with the cosmological
parameters obtained by Planck [61], i.e. (Ωb , Ωc , h∅ ,τopt , ns , As )= (0.04900,
0.2671, 0.6711, 0.0925, 0.9675, 2.215 × 10−9 ).
We see in figure 20 that Γ−1 ≃ 200 Gyr is favored if 1 − mD1 /mM ≳ 10−1 ,
in which case DDM decays into two relativistic particles. On the other hand,
Γ−1 < tage is favored if mD1 /mM ∼ 1 − 10−2.5 , in which case the massive
daughter particles are non-relativistic when they are produced. However
this parameter region has been already excluded by Peter [44] and Wang et
al.[49]. Peter is due to observations of the halo mass-concentration and
galaxy-cluster mass function. Wang et al. is due to small scale structures
observed by Lyman-α Forest, which should not be destroyed by decaying of
68
1.5
1
φφ
l2(l+1)2 Cl /2π×107
2
0.5
0
-0.5
101
102
103
l
Figure 21: The effect of DDM on the CMB lensing potential power spectrum
Clϕϕ . The shaded boxes are the data from the Planck experiment [72]. Red
(dashed) and green (dot-dashed) lines represent cases with mD1 /mM = 0.3
and 0.9, respectively. In both cases, the lifetime of DDM is set to be Γ−1 =
200 Gyr. For comparison, the black (solid) line represents the case in the
ΛCDM model.
dark matter.
While the DDM model with Γ ≳ 200 Gyr and 1−(mD1 /mM ) ≳ 0.1 may be
able to solve the discrepancy in σ8 estimated from the CMB power spectrum
and the SZ cluster counts of Planck, one may wonder such models can be
constrained by the CMB lens power spectrum Clϕϕ . In figure 21, we compare
Clϕϕ in the DDM model with the observational result which was reported
by the Planck paper [72]. The figure shows that the dark matter decaying
suppresses Clϕϕ compared with that in the ΛCDM model. However, it seems
that the current data is not constraining enough for such the parameter region
to be excluded. We defer a more quantitative analysis to future work.
69
3.6
Summary in this section
In this section, we considered cosmological consequences of the DDM model
where the cold mother particles decay into massive and massless particles.
In particular, we focused on evolutions of cosmological perturbations in the
model and their signatures in the CMB power spectrum Cl and the matter
power spectrum P (k). While similar kinds of models had been studied by
various authors, we for the first time explored cases with an arbitrary mass
ratio between the daughter and mother particles mD1 /mM . For this purpose,
we solved the phase space distributions of the decay products. To summarize, the main effect of the decay of DDM is that the free-streaming of the
daughter particles suppresses structure formation at scales smaller than the
free-streaming length.
As for CMB, the DDM model mainly affects the temperature anisotropy
at large angular scales through the ISW effect. A constraint on Γ−1 and
mD1 /mM from the peak shift of ClTT has been set on Aoyama et al.[50].
Their constraint from it is Γ−1 > 30 Gyr at mD1 /mM ≪ 1. However, since
measurements of Cl at large angular sales are fundamentally limited by the
cosmic variance, CMB may not be a promising probe of the DDM model.
On the other hand, the matter power spectrum P (k) is affected by dark
matter decaying significantly even when Γ−1 ≫ tage . Indeed, by using the
observational data of σ8 [69], we succeeded in excluding parameter region
in the 2D plane of the fractional mass difference between the daughter and
mother particles and the decay time. If the decay product is relativistic, the
decay time Γ−1 should be longer than 200 Gyr, and if the decay time is shorter
than the age of the Universe, the fractional mass difference 1 − mD1 /mM
should be smaller than 10−2.5 The tension between estimated σ8 from the
SZ effect and the CMB angular power spectrum in the recent Planck data
may be explained by the DDM model if Γ−1 is around 200 Gyr and the
decay products are relativistic.
4
Cosmological parameter estimation including DDM
In §2 and §3, we discuss the effects of dark matter decaying on the cosmological observables and set the constraints on the relation between the life
time and the mass ratio by fixing the cosmological parameters such as (Ωb h2∅ ,
70
Ωc h2∅ , h∅ , τopt , ns , As ). However, dark matter decaying at late time changes
the expansion history of the Universe and the background geometry from
that estimated from ΛCDM cosmology and this fixing may cause a systematic error of setting the constraint. Hence, in this section, we estimate the
cosmological parameters including DDM parameters such as the lifetime Γ−1
and the mass ratio r.
It is important how to estimate them with rapidity and accuracy. Firstly
we consider the strategy in which one calculates all parameter sets by brute
force in the parameter space and calculate χ-square of the result one by one.
When you require the 1% accuracy of the estimation, it is required to perform
(1/(0.01))8 = 1016 calculations. In the case that it takes only three seconds
to calculate the estimated observables such as Cl and P (k) from a parameter
set, it will cost approximate ten billion years, which is approximately equal
to the age of Universe, to obtain one estimation result. It is too long to
estimate the calculation.
In order to solve this problem, it is often used to do it by the Markov
Chain Monte Carlo (MCMC) method. It has been known that one can reduce
the required time for the estimation from ten billion years to a few days by
using this method. The public code CosmoMC [73] has been often used in
several projects and authors.
However it cannot be directly applicable to our calculations. In our code
for DDM cosmology, because we calculate the time evolution of perturbed
distribution function of daughter particles in these phase space, it takes approximate seven hours to obtain the estimated observables for one parameter
set. Thus the computational time is too long to obtain the estimation result
in realistic time.
In this section, we focus on that the uncertainties of several cosmological
parameters are suppressed several percent. Thus we calculate ClTT by interpolation from discrete points in seven cosmological parameter {Xi } and the
decaying rate Γ. Here Γ is a decay rate, not the lifetime. The component of
{Xi } is defined as X ≡ (Ωb h2 , Ωc h2 , h, τopt , ns , As , r). The reason why Γ is
not included in X is that the case such as Γ = 0 corresponds to ΛCDM cosmology regardless of the value of r. We numerically calculate ClTT at these
sampling points. We select them in parameter space as
Xi = Xifid , Xifid ± ∆i ,
Γ = 0, 0.01 [Gyr−1 ]
71
(4.1)
(4.2)
where Xi , σi(Planck) are defined as:
Xifid = (0.02207, 0.1196, 0.674, 0.097, 0.9616, 2.3 × 10−9 , 0.4)
(4.3)
The fiducial values of the cosmological parameters in the ΛCDM model are
the best fit values in the observations by the Planck satellite, which are
appeared in [13]. ∆i are defined as
∆i ≡ 2.5σi(Planck) (i ̸= τopt )
∆τopt ≡ 2στopt (Planck)
∆r ≡ 0.3 ,
(4.4)
(4.5)
(4.6)
(4.7)
where σi(Planck) is set from the uncertainty of the parameter estimation of the
analysis of the Planck collaboration [13]. In concrete terms,
∆i = (8.25 × 10−4 , 0.00775, 0.035, 0.076, 0.0235, 0.4 × 10−9 , 0.3) .
The prior ranges are set to be
Xifid − ∆i ≤ Xi ≤ Xifid + ∆i ,
0 ≤ Γ ≤ Γ1 ,
(4.8)
(4.9)
where Γ1 = 0.01 [Gyr−1 ].
It is an important question whether the curvature of the Universe is finite
or zero. In fact, in the last decade, numerous observations concerning both
CMB and the large scale structure have strongly suggested that the Universe
is globally flat (e.g. [74, 75, 76, 77, 78, 13, 79]). Therefore we also set K = 0
in this section.
4.1
Interpolation formula
In this section, we explain how to obtain the interpolated value of ClTT (X, Γ).
It is obtained from
ClTT (X, Γ) = (1 − Γ/Γ1 )ClTT (X, Γ = 0) + (Γ/Γ1 )ClTT (X, Γ = Γ1 ) (4.10)
The interpolated value ClTT (X, Γ) is calculated from
ClTT (X, Γ) = ClTT (X fid , Γ) + Ai l (Γ)∆qi + B ij l (Γ)∆qi ∆qj ,
72
(4.11)
where ∆qi is the difference between the input cosmological parameter and
the sampled one, i.e. Xi −Xifid . Ai l (Γ) and B ij l (Γ) are obtained by difference
calculations among the sampled points as
ClTT (Xi+ , Γ) − ClTT (Xi− , Γ)
,
(4.12)
2∆i
−2ClTT (Xifid , Xjfid , Γ) + ClTT (Xi+ , Xjfid , Γ) + ClTT (Xi− , Xifid , Γ)
B ii l (Γ) =
(4.13)
∆2i
2ClTT (Xi+ , Xj+ , Γ) − ClTT (Xi+ , Xj− , Γ) − ClTT (Xi− , Xi+ , Γ)
ij
B l (Γ) =
(4.14)
,
∆i ∆j
Ai l (Γ) =
where we define Xi± ≡ Xifid ± ∆i .
We fit our DDM model into observed ClTT data obtained by Planck. In
this analysis, we include the gravitational lensing effect. In order to qualify
the confidence level, we compute the χ2 statistic and obtain the log likelihood
as − ln L.
4.2
Result
We perform the parameter estimation with MCMC by using the interpolation
formula written above. The best fit values and these standard deviations are
shown in table 4.2.
parameter best fit value 68% limits
Ωb h2
0.02193
0.00022
2
ΩM∅ h
0.1191
0.0009
H0
67.4
0.5
ns
0.9606
0.0048
tauopt
0.0902
0.0089
109 As
2.30
0.05
Γ
0.0013
0.0010
r
0.39
0.18
Table 1: Cosmological parameter values for the eight-parameter base DDM
model. They give results for the Planck temperature data with Planck lensing
[13]. The unit of H0 and Γ are km/sec/Mpc and Gyr−1 , respectively.
73
In order to introduce the our DDM model, the log likelihood − ln L at
the best fit sample is slightly improved by 2.983 from that of ΛCDM model:
4950.437.
We plot the constraint on the relation between cosmological parameter
Ωb h2 , ΩM∅ h2 , h, ns , τopt , As and the decaying rate Γ and the relation between
Γ and the mass ratio r as shown in the figure 22. The inner and outer contours
are presented 1σ and 2σ confidence region, respectively.
We found that the estimated values of the cosmological parameters included in the ΛCDM model are nearly-unchanged by including DDM parameters. The observation constraints the physical processes which cause the
additional integrated Sachs Walfe effect such as dark matter decaying. We
obtain Γ ≲ 0.003 Gyr−1 with 2σ confidence level. The allowed decay rate Γ
is almost independent of the mass ratio r.
4.3
Discussion
The reason that estimated values of the cosmological parameters in the
ΛCDM are nearly unchanged is that the decaying rate of the mother particle
is restricted strictly by the observation of temperature fluctuation on large
scales. In addition it suggests that the constraint of the lifetime of DDM is
determined by the gravitational potential originated from survived mother
particle. This constraint corresponds to the lifetime should be larger than
300 Gyr and it is consistent to Ichiki et al.[34]. It is the strongest constraint
of the lifetime of the DDM from CMB observation.
According to the discussions in §2 and §3, it is expected that the Hubble
constant is estimated smaller as the estimated decaying rate becomes larger.
On the other hand, the abundance of mother particle ΩM∅ is expected to be
estimated larger as the estimated decay rate Γ becomes larger.
The reason of the former tendency is the following. The dark matter
decaying causes the decrease of the size of acoustic peak scale due to the
decelerating of cosmic expansion caused by the dark matter decaying. When
one compares the estimated ClTT to the observed one, the estimated value of
Hubble constant becomes larger in order to cancel out the shift on the ClTT .
The reason of the latter is the following. The dark matter decaying causes
the more decrease of the abundance of dark matter at the current Universe
as the decay rate becomes larger. In the CMB measurement, this decrease
is measured as the decrease of the original abundance of mother particles.
However the lifetime is constrained strictly, these tendencies cannot be seen
74
0.008
0.006
0.006
Γ/Gyr−1
Γ/Gyr−1
0.008
0.004
0.002
0.000
0.004
0.002
0.000
0.02160.02200.0224
Ω bh 2
ΩM∅h2
0.008
0.1215
0.006
ΩM∅h2
Γ/Gyr−1
0.1200
0.1185
0.000
0.02160.02200.0224
Ωb
h2
0.008
0.008
0.006
0.006
0.004
0.002
0.004
0.002
2.2
2.3
2.4
0.000
2.5
0.008
0.008
0.006
0.006
0.004
0.952 0.960 0.968
ns
Γ/Gyr−1
Γ/Gyr−1
109 As
0.002
0.000
67 68 69 70 71
H0
Γ/Gyr−1
Γ/Gyr−1
0.004
0.002
0.1170
0.000
0.1175 0.1200 0.1225
0.004
0.002
0.000
0.0600.0750.0900.105
τ
75
0.15 0.30 0.45 0.60
r
Figure 22: The constraint of the decaying rate of the DDM. The inner and
outer contours enclose 1σ and 2σ confidence region.
in the figure 22.
5
Conclusion
In this thesis, we consider a decaying dark matter model in which the massive mother particle decays into two massive and massless daughter particles
after cosmological recombination. We derive a complete set of Boltzmann
equations to describe the evolution of the particles. We obtain constraints
on the lifetime of the mother particle Γ−1 and the mass ratio mD1 /mM with
mD2 = 0 from the Hubble parameter, CMB and BAO. The allowed Γ−1
decreases monotonically as mD1 /mM increases.
Firstly, for the free streaming constraint, we find the opposite dependence. We find Γ−1 > 30 Gyr at the massless limit of daughter particles and
mD1 > 0.97mM at the limit Γ−1 → 0, from the distance to CMB (1σ). We
also obtain constraints from the free-streaming of daughter particles from
−3/2
observations of Lyman α as (Γ−1 /10−2 Gyr) ≲ ((1 − mD1 /mM )/10−2 )
for
−1
−1
Γ < H (z = 3). However, we find that the calculation of density perturbations such as CMB angular power spectrum, a complicated calculation of
density perturbations is necessary.
Secondly, we focused on evolutions of cosmological perturbations in the
model and their signatures in the CMB power spectrum Cl and the matter power spectrum P (k). While similar kinds of models had been studied
by various authors, we for the first time explored cases with an arbitrary
mass ratio between the daughter and mother particles mD1 /mM . For this
purpose, we solved the phase space distributions of the decay products. To
summarize, the main effect of the decay of DDM is that the free-streaming of
the daughter particles suppresses structure formation at scales smaller than
the free-streaming length. As for CMB, the DDM model mainly affects the
temperature anisotropy at large angular scales through the ISW effect. A
constraint on Γ−1 and mD1 /mM from the peak shift of ClTT has been set on
Aoyama et al.[50]. Their constraint from it is Γ−1 > 30 Gyr at mD1 /mM ≪ 1.
However, since measurements of Cl at large angular sales are fundamentally
limited by the cosmic variance, CMB may not be a promising probe of the
DDM model. On the other hand, the matter power spectrum P (k) is affected
by dark matter decaying significantly even when Γ−1 ≫ tage . Indeed, by
using the observational data of σ8 [69], we succeeded in excluding parameter
region in the 2D plane of the fractional mass difference between the daughter
76
and mother particles and the decay time. If the decay product is relativistic, the decay time Γ−1 should be longer than 200 Gyr, and if the decay
time is shorter than the age of the Universe, the fractional mass difference
1 − mD1 /mM should be smaller than 10−2.5 The tension between estimated
σ8 from the SZ effect and the CMB angular power spectrum in the recent
Planck data may be explained by the DDM model if Γ−1 is around 200 Gyr
and the decay products are relativistic.
Thirdly we consider the cosmological parameter estimation including DDM
parameters by interpolation from the observational data obtained by Planck
satellite. It is to avoid the systematic errors which are caused by that the cosmological parameter is specified. We find that the estimated values included
in ΛCDM model do not differ from the fiducial values based on the model.
We also find that additional integrated Sachs-Walfe effect restricts the large
decay rate of dark matter and set a constraint on the relation between the
decaying rate and the mass ratio of the massive daughter particle to mother
particle.
In our future work, the systematic error caused by the interpolating formula should be reduced. In order to do this, we have to find the method
of the computation without calculating the distribution function of daughter
particles directly and perform the parameter estimation by MCMC method
without interpolation. As a prospect of our future work, the dark matter
which decays into more then two particles will be studied. In this case, the
physical momentum of the daughter particles is not determined by the mass
difference and the distribution is expected to have some non-trivial distributions and it is expected to take much more time to calculate the time
evolutions of the perturbed quantities. However, in the many particle decaying processes in the standard model particles, the number of daughter
particles is more than two, we have to find the reduction method to calculate
them.
6
Acknowledgment
I would like to express my gratitude to the supervisor Naoshi Sugiyama for
his sophisticated and suitable ideas and advice. Thanks to his advice, many
difficulties on my investigations have been solved. In addition, he introduced
me not only to the theoretical astrophysicists but also to people who specialize in a lot of different areas such as theoretical elementary particle physics.
77
He also provided a lot of chances to introduce my studies to researcher in
other universities.
My thanks also goes to Kiyotomo Ichiki. He devoted much time to teach
the basics of the cosmology and linear perturbation theory in my master
course. He always accept to discuss my new and halfway ideas and help me
to improve these ideas so much. His efforts and encouragements enable me to
study not only the decaying dark matter but also the gravitational waves and
modified theories. Daisuke Nitta contributed to the knowledge and share a
lot of surprising and excellent ideas and proscriptions to solve the difficulties
on my studies. Tsutomu T. Takeuchi taught the basics of the observations
and statistics. He always accept to answer my questions and discuss the
problem which I could not solve or got ideas. His detailed, accurate and
wide-ranging knowledge helped a lot to my study and investigations. His
comments on the foreground problems of CMB observations such as Planck
and COBE helped to understand the observational data properly.
Toyokazu Sakiguchi and Shohei Saga shared me the computational method
and usage of the CAMB and CosmoMC. Their contribution enabled me to
implement DDM into these public codes.
A
A.1
General Relativistic Cosmology
General relativity
Gravity is the fundamental interaction in the Universe. In 1916, Einstein proposed general relativity, in which the gravitational interaction is due to the
distortion of the space-time [80]. This torsion is originated from the energy
density, pressure and shear of the matter and radiation, which are described
by the energy momentum tensor T µ ν . This theory can explain the gravitational lensing and the perihelion procession of Mercury. In 1919, at the time
of total solar eclipse, the gravitational lensing caused by the gravitational
field of the sun is observed as approximately same as the general relativity
predicted. In order to describe the structure of the space quantitatively, it
is necessary to introduce the metric tensor gµν . It turns coordinate distance
into physical distance.
78
A.2
Introduction to metric
In order to understand the concept of the metric, we firstly consider the
distance dl between two different points whose coordinate distance is represented as (dx1 , dx2 ) in flat space. According to the Pythagorean theorem,
the relation between dl and dx1 , dx2 can be written as
dl2 = (dx1 )2 + (dx2 )2 .
Eq.(A.1) can be rewritten as
( 1) (
)
dx
1 0 ( 1 2)
2
dx dx ≡ gij dxi dxj .
dl =
dx2
0 1
(A.1)
(A.2)
Because the space is flat, gij becomes the unit matrix of a square matrix of
size two, i.e.
(
)
1 0
gij =
.
(A.3)
0 1
Generally speaking, the space is not always flat10 . In order to introduce the
metric for a non-flat space, we change the coordinate to polar coordinate
(x, θ) consider the two different points on the three-dimensional spherical
surface. On the figure 23, the position on the surface is quantified by x, θ
and the radius of the sphere Rc . The difference vector between these points
can be written as (dx, dθ) and the line element on the surface dL2 is
dL2 = dx2 + r2 dθ2 ,
(A.4)
1
=
dr2 + r2 dθ2 ,
(A.5)
1 − Kr2
( )
x
where r = Rc sin
and K ≡ Rc−2 . When one considers a real sphere,
Rc
K is always positive. In this case, the curvature of this two-dimensional
space is positive. However, mathematically, K ≤ 0 is possible. The case
such as K = 0 is realized in the limit Rc → +∞ and the curvature is zero.
In addition, in the case such as K < 0, the curvature is negative and the
morphology of the surface is like a saddle. On the curved space, the sum of
the internal angles of a triangle is not 180◦ . As shown on the figure 24, in
the case such as K > 0, the sum is more than 180◦ . Besides the sum is less
than 180◦ in the case K < 0.
10
Here “flat” means that the geometry is Euclidean.
79
x
Rc
Figure 23: The definition of the variables (x, θ). This figure is based on [19]
(a) flat (K=0)
(b)positive curvature
(K>0)
(c)negetive curvature
(K<0)
Figure 24: The effect of the sign of the curvature on the sum of the internal
angles of a triangle α + β + γ [81].
80
By normalizing the coordinate r, the range of K is limited to the following
three values and names of the structure of these spaces are assigned as


1 (closed) ,
K = 0 (flat) ,
(A.6)


−1 (open) .
Let us extend this discussion above to the four-dimensional space-time.
In the four-dimensional space-time, the squared line-element for photon ds2γ
is set zero. Hence, in the flat space, the line element is can be written as
3
∑
ds = −dt +
(dxi )2 .
2
2
(A.7)
i=1
Here we set the time t the zero-th component of the space, i.e. x0 . In this
case,
ds2 = gµν dxµ dxν ,
(A.8)
where indices µ and ν run from zero to three. The metric gµν becomes


−1 0 0 0
 0 1 0 0

gµν = diag[−1, 1, 1, 1] = 
(A.9)
 0 0 1 0 .
0 0 0 1
In the curved space, as the same as the two-dimensional case, we consider
the three-dimensional polar coordinate with time (t, r, θ, φ), and the metric
becomes
1
, r2 , r2 sin2 θ] .
(A.10)
gµν = diag[−1,
2
1 − Kr
A.2.1
The cosmological principle and FLRW metric
Here we introduce the cosmological principle and the time evolution of the
metric in the Universe. The cosmological principle is a hypothesis that the
Universe is homogeneous and isotropic when it is viewed on a large enough
scale. This principle is strongly supported by several observations on the
large scale structure and the cosmic microwave background (CMB) and large
scale structure. In the completely homogeneous and isotropic Universe, the
81
expansion of the Universe depends only on the time t. Due to quantification
of the expansion, we introduce the scale factor a(t).
This scale factor is defined by the distance between a pair of objects which
do not interact with each other, e.g. two distant clusters. Here we define
d(t) as the distance at the time t. The distance at any arbitrary time t can
be written as
a(t0 )d(t0 ) = a(t)d(t) ,
(A.11)
where t0 is the current cosmic time. Traditionally, we set a(t0 ) as unity.
Let us consider the cosmic expansion in terms of the scale factor. Friedmann, Lemaitre, Robertson and Walker proved that the metric for the expanding Universe is generally written as
(
)
1
2
2
2
2
2 2
2
2
ds = −dt + (a(t))
dr + r θ + r sin θdφ .
(A.12)
1 − Kr2
In the expanding Universe, the conformal time τ is convenient to calculate
the expansion history and perturbation growth.
∫ t
1
dt′ .
(A.13)
τ≡
′
0 a(t )
By taking partial derivatives of the metric, one can obtain geometric
quantities such as the Christoffel symbols Γα µν , Riemann tensor Rµ ναβ , Ricch
tenor Rµν and curvature R as,
1 αβ
g (gµβ,ν + gνβ,µ − gµν,β ) ,
2
Rµ ναβ = Γµ βν,α − Γµ αν,β + Γµ αλ Γλ βν − Γµ βλ Γλ αν ,
Rµν = Rα µαν ,
R = g µν Rµν .
Γα µν =
A.3
(A.14)
The evolution of space-time and metric
The cosmic expansion of the Universe is given by the Einstein equation. By
substituting eq. (A.12) and K = 0 into eq. (A.14), the result is followings.
(
)
2
R00 = 3Ḣ, Rij = Ḣ + 2H δij .
(A.15)
)
(
R = 6a−2 Ḣ + H2 .
(A.16)
82
Thus we obtain the time evolution equation for the background space time.
8
πGa2 ρtotal ,
3
(
)
4
Ḣ = − πGa2 ρtotal + P total ,
3
H2 =
(A.17)
(A.18)
where ρtotal and P total represent the mean total energy density and pressure,
respectively. The former equation is called Friedmann equation.
A.3.1
Gauge and metric perturbation
In this section, we start to consider the perturbation of the metric. The
perturbation of the metric on the flat Universe is generally written as
)
(
−1 − 2A
−Bi
2
gµν = a (τ )
,
(A.19)
−Bi
(1 − 2D)δij + 2Eij
where Eij is a trace-less matrix such as
δ ij Eij = 0 .
(A.20)
When one decomposes the metric into the background part g µν (t) and
the perturbed part δgµν (t, x), they become
g µν = a2 (τ )diag[−1, 1, 1, 1]
(
)
−2A
−Bi
δgµν =
,
−Bi −2Dδij + 2Eij
(A.21)
(A.22)
Here A and Bi are called the lapse function and the shift vector, respectively.
A.3.2
Scalar, vector, tensor decomposition
In order to consider the relation between the density fluctuation and the
perturbation, we decompose the gµν into scalar, vector, and tensor part.
Firstly we consider the scalar perturbation. Scalar perturbations is written by the spatial derivative of the scalar function. In concrete terms, the
scalar part of shift factor Bi and Eij , which are BiS , EijS respectively, can be
written with scalar functions B(x, τ ), E(x, τ ) as
BiS = ∂i B(x, τ ) ,
(
)
1
S
2
Eij =
∂i ∂j − δij ∇ E(x, τ ) .
3
83
(A.23)
(A.24)
Hereafter the argument (x, τ ) are omitted. These time evolution of the scalar
mode such as A, B, D, E is described by the density fluctuation (see.[82,
62]).
The vector part of Bi , Eij , which are BiV , EijV respectively, are defined as
BiV = Bi − BiS = Bi − ∂i B
1
EijV = − (∂j Ei + ∂j Ei ) ,
2
(A.25)
(A.26)
where the divergence of the vector function BiV , Ei are set to zero. Thus
these matrices are divergence-less, thus
∂i BiV ≡ ∇ · B V = 0 ,
∂i EiV ≡ ∇ · E V = 0 .
(A.27)
(A.28)
The tensor part EijT is defined as
(
)
EijT ≡ Eij − EijS + EijV ,
(A.29)
where the matrix EijT is a transverse-trace-less matrix. In concrete terms, the
constraint equations are given as the followings.
δ ki ∂k EijT = 0
δ
ij
EijT
(A.30)
= 0.
(A.31)
The number of degree of freedom of an arbitrary 3-dimensional Hermite
matrix is six and these constraint equations reduces it by four. Thus the
basis of matrix can be generally represented as two matrix as




1 0 0
0 1 0
e1 = 0 −1 0 , e2 = 1 0 0 .
(A.32)
0 0 0
0 0 0
This gauge is called TT gauge and often used in the expression of the gravitational waves.
Then we show that the metric perturbation δgµν can thus be divided into
the scalar, vector and tensor modes as
scalar mode
vector mode
tensor mode
A, B, C, D
Bi , Ei
Eij
84
n.o.f
4
4
2
(A.33)
(A.34)
(A.35)
(A.36)
The sum of the number of degree of freedom is ten and arbitrary perturbation
δgµν can be decomposed by these three modes.
A.3.3
Synchronous gauge and conformal Newtonian gauge
In this section, we discuss the gauge fixing. We consider the spatially flat
background spacetime with isotropic scalar metric perturbations. In the
previous section, there are four functions A, B, D and E, which are represented the scalar perturbations. However the number of physical quantities
which are relevant to scalar perturbations are only two, which correspond to
gravitational potential ψ the curvature perturbation ϕ. In order to reconcile this discrepancy, we have to make a correlation between these functions
A, B, D, E and ψ, ϕ. This procedure is called gauge fixing. In this thesis, we introduce the two gauge, which are the synchronous gauge and the
conformal Newton gauge.
Firstly, we introduce the synchronous gauge, in which the components
g00 and g0i are unperturbed, and the line element is given by
(
)
ds2 = a2 (τ ) −dτ 2 + (δij + hij ) dxi dxj .
(A.37)
As we mentioned in §A.3.2, a 3-dimensional matrix such as hij can be deT
composed to scalar (hSij ), vector (hV
ij ) and tensor part (hij ). In this section,
because we consider the scalar perturbation, we set
T
hV
ij = hij = 0 (∀i, j) .
(A.38)
Thus hij can be represented by two scalar functions hL ≡ hL (k, τ ) and ηT ≡
ηT (k, τ ) as
))
(
∫ (
1
hij ≡
hL k̂i · k̂j + 6ηT k̂i · k̂j − δij
exp (ik · x) d3 k .
(A.39)
3
Secondly we adopt the conformal Newtonian gauge, in which the non-diagonal
elements of the metric are unperturbed, and the line element is represented
by
(
)
ds2 = a2 (τ ) − (1 + 2ψ) dτ 2 + (1 + 2ϕ) δij dxi dxj ,
(A.40)
where ψ, ϕ are the gravitational potential and the curvature perturbation,
respectively.
85
By performing the gauge transformation between two gauges (e.g.[62]),
the gravitational potential ψ and the curvature perturbation ϕ are related to
the synchronous potentials hT and ηL in the momentum space by
{
] ȧ [
]}
1 [
ψ =
ḧ + 6η̈ +
ḣ + 6η̇
,
(A.41)
2k 2
a
]
1 ȧ [
ϕ = η − 2 ḣ + 6η̇ .
(A.42)
2k a
A.4
Comic microwave background (CMB)
In the late 1920’s, Edwin Hubble and Georges Henri Lemaitre independently
found the expansion of the Universe by the measuring the recession velocity
of galaxies and the age of the Universe is around ten-billion years. The
discovery of the expansion of Universe indicates that the existence of the hot
beginning of the Universe, which is called Big-Bang.
When the beginning of the Universe, the temperature is high enough
that protons are no longer able to hold electrons and photons cannot move
freely. As the Universe expands adiabatically and the temperature drops
approximately 3000 Kelvin, electrons are captured by protons and photons
can travel freely for the first time of the Universe. This epoch is called the
cosmological recombination. George Gamow predicted the existence of the
black body background radiation as the remnant of it at present. Because
the temperature of this radiation was expected around five Kelvin, it would
be detected as radio waves. It is called the comic microwave background
(CMB). In 1960’s, Arno Penzias and Robert Wilson discovered this radiation
by accident in the course of the observation of hydrogen atoms in Milky Way
with radio waves.
In 1970, P. J. E. Peebles and J. T. Yu claimed that the density fluctuations
of matter and radiations are ascribed in the temperature fluctuation of CMB
[83]. The temperature fluctuations of CMB is discovered by the Cosmic
Background Explorer (COBE) at first. COBE measured the spectrum and
the anisotropy of the cosmic microwave background (CMB) on the large
scales such as l < 20. COBE satellite revealed that the spectrum of CMB in
excellent agreement with the black body spectrum and the mean temperature
is 2.725 Kelvin11 .
11
Precisely speaking, the spectrum of CMB is not in complete agreement on the black
body spectrum due to Silk dumping
86
It is also reveal that the amplitude of temperature anisotropy is approximately one of a hundred-thousand. Several authors had been revealed that
the amplitude of temperature fluctuations of CMB is three-fourths times
of the amplitude of the density fluctuation of matter at the recombination.
Therefore, at the begging of the Universe, the amplitude of the density fluctuation of the matter is order 10−5 . Therefore it was revealed that the large
scale structure of the Universe at present is originated by such a tiny fluctuations and has formed only for ten-billion years. This tiny fluctuation is
originated by the quantum fluctuation of cosmological inflation field, which
are composed by unknown scalar particle called inflaton. In 1990’s, the
method to extract cosmological parameters from temperature fluctuations of
CMB had been formulated by several authors. Here the cosmological parameters are the quantities which describes the components and the geometrical
structure of the Universe.
A.4.1
The physics of CMB temperature anisotropy
A.4.2
Boltzmann equation
The Boltzmann equation is the differential equation which describes the time
evolution of the distribution function in six-dimensional phase space even for
the non-equilibrium system. The solution of it expresses the statistical behaviors of thermal dynamical characteristics. Several author’s have been
revealed that many kinds of cosmological quantities can be extracted from
the temperature fluctuation of CMB. In addition, they have found that the
time evolution of CMB in expanding and spatially-distorted Universe can be
calculated by solving the Boltzmann equation. Its calculation result excellently agrees with observational results which are observed in COBE, WMAP
and Planck satellite. This equation is stood by the principle of the conservation of particles in the six-dimensional phase space (three dimensional space
and three dimensional comoving momentum) and derived by the Gauss’s
theorem.
We consider a continuous distribution function f˜ in a finite domain in
the phase space Ω. In this space, the equation of continuity is
∫
Ω
∂ f˜
dV6 +
∂t
∫
f˜w · ndσ5 = 0 ,
(A.43)
dΩ
where w is the time derivative of the variables of the phase space, i.e. dx/dt
87
and dq/dt. n, dΩ, dσ5 are the 5-dimensional surface of Ω, a normal vector
to the surface and the surface element of the dΩ, respectively.
By applying the Gauss’s theorem, the second term of eq. (A.43) becomes
∫
∫ {
( )}
f˜w · ndσ5 =
∇6 · f˜w dV6
(A.44)
dΩ
Ω
∫ {
( )
( )}
=
∇x · f˜ẋ + ∇q · f˜q̇ dV6 .
(A.45)
Ω
Here the time derivative of x and p do not depend on x, p, respectively.
Thus
( )
( ) dx ∂ f˜ dq ∂ f˜
i
i
∇x · f˜ẋ + ∇q · f˜q̇ =
·
+
·
.
(A.46)
dt ∂xi
dt ∂qi
Because of the arbitrary of the determination of the domain Ω we obtain the
following equation by eqs. (A.43),(A.45),(A.46).
∂ f˜ dxi ∂ f˜ dqi ∂ f˜
+
+
=0.
∂t
dt ∂xi
dt ∂qi
(A.47)
In fact, the conservation of the number of particles in phase space is violated
by interactions. In order to treat the interactions in this equation, we add
the collision term C[f˜] to the right hand side of eq. (A.50). Thus,
df˜
= C[f˜] ,
dt
(A.48)
where C[f˜] is described not only collisions or repulsion with other particles
but also the decaying and annihilating of the particles. In order to respect the
isotropy of the Universe, we divide the comoving momentum q into the norm
q and its directional vector n. Here length of n is set unity i.e. n · n ≡ 1.
One can write the Boltzmann equation also with xi , q and ni as
∂ f˜ dxi ∂ f˜ dq ∂ f˜ dni ∂ f˜
+
+
+
= C[f˜] .
∂t
dt ∂xi
dt ∂q
dt ∂ni
(A.49)
In order to solve this partial differential equation of f˜, the Fourier transformed function f is useful. Here we try to write the Boltzmann equation
(eq. (A.50)) in terms of f . In the case
)
qk (
dq ∂f
dni ∂f
∂f
+
k̂ · n̂ f +
+
= C[f ] .
(A.50)
∂t
ε
dt ∂q
dt ∂ni
88
Here, we divide it into the two functions such as
f (x, q, τ ) = f (q) (1 + Ψ (k, q, τ )) .
(A.51)
Because the inhomogeneity in the Universe at a large scale is so small that
|Ψ| ≪ 1, we neglect the second order (Ψ2 ) or more higher perturbed quantities (Ψn (n ≥ 3)). In addition, both the acceleration direction vector dni /dt
and the direction dependence of the distribution function ∂f /∂ni are first
order quantity and the scalar product of them is second order quantity and
we neglect it.
Let us consider the the expression of terms on eq. (A.50) in Fourier space.
dq/dt is derived from the geodesic equation as
—Conformal Newtonian gauge
(
)
dq/dτ = q ϕ̇ − εni ∂i ψ = q ϕ̇ − ε k̂ · n̂ ψ
(A.52)
—Synchronous gauge
(
)
)2
1
ḣ + 6η̇ (
dq/dτ = − q ḣij ni nj = η̇ −
k̂ · n̂
2
2
(A.54)
(A.53)
(A.55)
Then the Boltzmann equation for Ψ in phase space becomes
—Conformal Newtonian gauge
(
)
(
(
) ) ∂f
∂Ψ qk
1
k̂ · n̂ Ψ + q ϕ̇ − ε k̂ · n̂ ψ
C[f ]
+
=
∂τ
ε
∂q
f
—Synchronous gauge
[
]
)
)2 ∂f
∂Ψ qk (
ḣ + 6η̇ (
1
+
k̂ · n̂ Ψ + η̇ −
k̂ · n̂
=
C[f ]
∂τ
ε
2
∂q
f
(A.56)
, (A.57)
(A.58)
, (A.59)
where we use that the range of the background distribution function of pho1
tons f is always positive and hence 0 < < ∞ (∀q).
f
89
A.4.3
CMB photons
In order to solve the Boltzmann equation for CMB photons with accuracy
and rapidity, we define new variables as
∫ +∞
f (q)Ψq 3 dq
Fγ (k, n̂, τ ) ≡ ∫0 +∞
(A.60)
3
f (q)q dq
0
Because the isotropy of the Universe, one can expand Fγ by using the Legendre polynomials Pl (k̂ · n̂) as
Fγ (k, n̂, τ ) ≡
+∞
∑
(−i)l (2l + 1)Fγl (k, τ )Pl (k̂ · n̂) .
(A.61)
l=0
The density fluctuation δγ , the divergence of the fluid θγ and the shear stress
σγ can be written as
1
3
δγ = Fγ0 , θγ = kFγ1 , σγ = Fγ2 .
4
2
(A.62)
Let us consider the collision term of the photons C[f ]. In order to solve
the Boltzmann equation of photons in terms of Fγ ,
Thomson scattering takes the central role in the interaction between photons and baryons. Because Thomson scattering depends on the polarization
of the photons, we define the new function Gγ (k, n, τ ), which is the difference
of the two linear polarization components of incoming photons. As same as
Fγ , we define the Gγl as,
Gγ (k, n, τ ) ≡
+∞
∑
(−i)l (2l + 1)Gγl Pl (k̂ · n̂) .
(A.63)
l=0
With Fγ and Gγ , their collision terms become ([84, 85, 86] see also. [62, 65,
87, 88])
(
]
)
[
∂Fγ
1
= ane σT −Fγ + Fγ0 + 4n̂ · v e − Π ,
(A.64)
∂τ
2
]
(
)
[
∂Gγ
1
(A.65)
= ane σT −Gγ + Π(1 − P2 ) ,
∂τ
2
90
where Π ≡ Fγ2 + Gγ0 + Gγ2 and σT is Thomson cross section. In terms of
associated moments Fγl and Gγl , eqs. (A.67) and (A.68) become
[
(
)
∂Fγ
4i
= ane σT − Fγ + Fγ0 + (θγ − θb ) P1
(A.66)
∂τ C
k
]
(
)
+∞
∑
1
+ Fγ2 − Π P2 +
(−i)l (2l + 1) Fγl Pl , (A.67)
10
l=3
[
]
(
)
+∞
∑
∂Gγ
1
l
= ane σT Π(1 − P2 ) −
(−i) (2l + 1) Gγl Pl , (A.68)
∂τ
2
l=3
By performing the integration of both ∫eqs (A.57) and (A.59) which are
multiplied q 3 f over q and dividing them by q 3 f dq, the Boltzmann hierarchy
for photons becomes
—conformal Newtonian gauges
4
δ̇γ = − θγ + 4ϕ̇ ,
(A.69)
3(
)
1
θ̇γ = k 2
δγ − σγ + k 2 ψ + ane σT (θb − θγ ) ,
(A.70)
4
)
(
4
3
1
σ̇γ =
(A.71)
θγ − kFγ3 − ane σT σγ + Π , ,
15
10
20
]
k [
Ḟγl =
lγ(l−1) − (l + 1)Fγ(l+1) − ane σT Fγl (for l ≥ 3),(A.72)
2l + 1
[
Ġγ0 = −kGγ1 − ane σT
]
1
Gγ0 − Π ,
2
1
k (Gγ0 − 2Gγ2 ) − ane σT Gγ1 ,
3
]
[
1
1
=
k (2Gγ0 − 4Gγ2 ) − ane σT Gγ2 − Π ,
5
10
1
k (lGγ0 − (l + 1)Gγ2 ) − ane σT Gγ1 (for l ≥ 3),
=
3
(A.73)
Ġγ1 =
(A.74)
Ġγ2
(A.75)
Ġγl
91
(A.76)
—Synchronous gauges
2
4
δ̇γ = − θγ − ḣ ,
(A.77)
3(
3 )
)
2 (
1
θ̇γ = k 2
δγ − σγ +
ḣ + 6η̇ + ane σT (θb − θγ ) ,
(A.78)
4
15
(
)
4
1
3
σ̇γ =
(A.79)
θγ − kFγ3 − ane σT σγ + Π ,
15
10
20
]
k [
Ḟγl =
lFγ(l−1) − (l + 1)Fγ(l+1) − ane σT Fγl (for l ≥ 3) (A.80)
,
2l + 1
[
Ġγ0 = −kGγ1 − ane σT
]
1
Gγ0 − Π ,
2
(A.81)
1
k (Gγ0 − 2Gγ2 ) − ane σT Gγ1 ,
(A.82)
3
]
[
1
1
=
k (2Gγ0 − 4Gγ2 ) − ane σT Gγ2 − Π ,
(A.83)
5
10
1
=
k (lGγ0 − (l + 1)Gγ2 ) − ane σT Gγ1 (for l ≥ 3) .(A.84)
(2l + 1)
Ġγ1 =
Ġγ2
Ġγl
In order to solve this hierarchy actually, we have to truncate this hierarchy
at l = lmax ≫ O(1). In the many public code for calculating the Boltzmann
hierarchy, forms of truncation are
lmax + 1
Fγ(lmax ) − ane σT Fγ(lmax ) ,
(A.85)
(2lmax )
τ
1
lmax + 1
=
kGγ(lmax −1) −
Gγ(lmax ) − ane σT Gγ(lmax ) (for l(A.86)
≥ 3)
(2lmax )
τ
Ḟγl =
Ġγl
1
kFγ(lmax −1) −
• Fluctuations on large scales
On the large scale such as the super-horizon at the recombination τ = τ∗ ,
the terms which are multiplied by k n (n ≥ 1) are negligibly smaller than
other ones. In this condition, the difference between the divergence of the
velocity of electrons and that of photons θγ − θb is much smaller than other
quantities in a large scales. In addition the temperature anistropy Θ ≡ ∆T /
T is related to the density fluctuation δγ as
δγ = 4Θ .
92
(A.87)
Hence the Boltzmann hierarchy of photons in the conformal Newtonian gauge,
which corresponds to eqs. (A.69) — (A.72), becomes
Θ̇γ = −ϕ̇ ,
F˙γl ≃ 0 (for l ≥ 1),
(A.88)
(A.89)
Θγ (k, η∗ ) = ϕ(k, η∗ ) + ϕ∅ (k) ,
(A.90)
Thus we obtain
where ϕ∅ is the initial condition of the curvature fluctuation. As a consequence of the inflation, one gets ϕ∅ = (3/2)ϕ(k, η∗ ) (e.g.[64]). Hence we
obtain the temperature anistropy at the recombination on a large scales as
1
Θγ (k, η∗ ) = ϕ(k, η∗ ) .
2
(A.91)
• Fluctuations on a small scales and the acoustic oscillation
In this section, we consider the behavior of photons on small scales. When
one considers the interaction of baryons with photons, the relativistic momentum conservation of photon-baryon fluid is given by
(
)
ργ + pγ δVγ = ρb δVb ,
(A.92)
where δVγ and δVb are the amount of change of the velocity of the photon
and that of the baryon. On the other hand, because the velocity of photons
Θ1 are changed by the Thomson scattering, one can obtain
δVγ = ane σT (Θ1 − vb ) ,
(A.93)
where Θ1 is defined the dipole moment of the temperature fluctuation by
using Legendre polynomials as the followings together with other higher moment l ≥ 2.
∫ 1
1
−l
Θl = (−i)
Pl (x) Θ(x)dx .
(A.94)
2
−1
By combining eqs. (A.92) and (A.93), one can obtain
δVb =
(ρb + ρb )
ane σT (Θ1 − vb ) .
ρb
93
(A.95)
On the other hand, by considering the continuity equation and the Euler
equation, we obtain the time evolution equation of the density contrast δB
and the velocity vb . Thus
δ̇b = −kvb + 3ϕ̇ ,
ane σT
v̇b + Hvb = kψ +
(Θ1 − vb ) ,
R
(A.96)
(A.97)
where R is defined as
R≡
ργ + ρb
= 3.04 × 104 Ωb h2 (1 + z)−1 .
ρb
(A.98)
From the BBN prediction, the abundance of matter can be obtained as
Ωb h2 = 0.022 ± 0.001 .
(A.99)
This result is consistent with the CMB observations [74, 13].
When one writes eqs. (A.96) and (A.97) in terms of the velocity divergence of the baryon and photons θb , θγ , they becomes
δ̇b = −θb + 3ϕ̇ ,
ane σT
θ̇b + Hθb = k 2 ψ +
(θγ − θb ) ,
R
A.4.4
(A.100)
(A.101)
Tight coupling approximation
We recall here photons had been tightly coupled with electrons in the early
Universe and the mean collision time is much smaller than the age of the
Universe (see.[62]). Hence
ane σT ≡ τc−1 ≫ H ,
(A.102)
where τc ≡ (ane σT )−1 is the conformal mean free path of photons.
Firstly, we simplify the Boltzmann hierarchy of photons by this approximation. In this approximation, ane σT σγ and ane σT Fγl are much larger than
other terms in the right hand side in eqs. (A.79) and (A.80), respectively. In
concrete terms,
σ̇γ ≃ −ane σT σγ
Ḟγl ≃ −ane σT Fγl (for l ≥ 3) ,
94
(A.103)
(A.104)
Hence σγ and higher multipole of photons Fγl (l ≥ 3) exponentially decrease
and we can neglect them. In terms of the temperature fluctuation Θl , which
is defined in eq. (A.94), it corresponds to
Θl ≃ 0 (l ≥ 2) .
(A.105)
Secondly, we derive the equation of the acoustic oscillation. Although
photons are tightly coupled with electrons, there are slight difference between
the velocity divergence of photons and electrons. Here we write the difference
with τc as
θb = θγ + τc f ,
(A.106)
where f is the coefficient function of the expansion and f˙ is negligible. By
meaning of f , the Euler equation of baryon can be written
(
)
1
˙
˙
θγ + Hθγ + τc f + Hτc f
= k2ψ − f .
(A.107)
R
Because we consider the free streaming scale is much smaller than the horizon
scale, one can obtain Hτc ≪ 1 and thus Hτc f is a second order quantity and
negligible compared with first ones. Then eq. (A.107) becomes
θ̇γ + Hθγ = k 2 ψ −
f
.
R
(A.108)
When one applies the tight coupling approximation to eq.(A.70), this equation becomes
1
θ̇γ = k 2 δγ + k 2 ψ + f .
(A.109)
4
By combing eqs. (A.71), (A.108), we can eliminate f and obtain
θ̇γ +
k2
HR
1
HR
θγ +
θγ = −ϕ̈ −
ϕ̇ − k 2 ψ .
1+R
3(1 + R)
1+R
3
(A.110)
By substituting eq. (A.69) into (A.110),
δ̈γ +
4RH
HR
4
δ̇γ + c2s k 2 δγ = −4ϕ̈ +
ϕ̇ + k 2 ψ ,
1+R
R+1
3
(A.111)
1
is the sound speed of the baryon-photon plasma. In
3(1 + R)
order to solve this equation, it is important that the recombination is occurred
where cs ≡
95
in the matter dominated Universe. In this epoch, the gravitational potential
and the curvature fluctuation had not changed for the matter dominated
Universe i.e. ϕ̇ ≃ 0 and ϕ̈ ≃ 0. In addition, R is much smaller then unity
before the recombination
and Ṙ/R is negligible. Hence the sound speed is
√
approximately 1/ 3. This equation is an forced oscillation equation 12 in
the subhorizon scale at the time and the solution is written as
1 ˙
δγ (τ ) = [δγ (0) + (1 + R)ψ] cos (kds )+
δγ (0) sin (kds )−(1+R)ψ , (A.112)
kcs
∫
where ds (τ ) ≡ cs (τ ′ )dτ ′ gives the sound horizon at the conformal time τ .
When one observes the CMB temperature fluctuation, the fluctuations of
photons at the recombination τ = τ∗ ≃ 2.7 × 109 second are only observed
because the visibility function is sharp. Because the sound of speed is known
precisely, the sound horizon scale can be also predicted with high accuracy
and it can be used as a standard ruler.
A.4.5
Acoustic oscillation of baryon-photon plasma
Before the recombination, as mentioned in eq. (A.105), the distribution of
photons have only monopole and dipole moment. Thus higher multipole
moments are generated from the free streaming of photons after the recombination.
The Boltzmann hierarchy of the CMB photon can be written in terms of
the temperature fluctuation as the following (see [64]).
d
(Θ exp (ik · n̂τ − τopt )) = S̃,
dτ
(A.113)
where κ ≡ ane σT and τopt are the opacity and optical depth, respectively. S̃
is the source function of the CMB photon as
]
[
(
)
1
S̃ ≡ −ϕ̇ − ik · n̂ψ − κ Θ0 − Θ + k̂ · n̂ vb − P2 (k̂ · n̂)Π . (A.114)
2
Θ̇ + (ik · n̂ − κ) Θ = exp (ik · n̂τ − τopt )
By multipling exp (ik · n̂τ − τopt ), one obtains
d
(Θ exp (ik · n̂τ − τopt )) = S̃ exp (ik · n̂τ − τopt ) .
dτ
12
(A.115)
In other word, the solution of this oscillation is dominated by the simple harmonic
motion which is added the external force term originated from the gravitational potential
see. [65]
96
By performing the integration of eq. (A.115) over τ , the temperature fluctuation of CMB at present can be written as
Θ (τ0 ) = Θ (τinit ) exp (ik · n̂ (τ0 − τinit )) exp (−τopt (τinit )) (A.116)
∫ τ0
+
S̃ (τ ′ ) exp (ik · n̂ (τ − τ0 ) − τopt (τ ′ )) dτ ′ ,
τinit
∫τ
where we set τopt (τ0 ) ≡ 0 a(τ ′ )ne (τ ′ )σT dτ ′ = 0. When one consider that
the initial time τinit is well before the recombination, the optical depth τopt
is much more than unity and the first term in eq. (A.117), which is relevant
to the initial condition of temperature fluctuation, becomes negligibly small
compared with the second term. Hence we obtain
(
) ∫ η0
Θ k, k̂ · n̂, τ0 =
S̃ (τ ′ ) exp (ik · n̂ (τ − τ0 ) − τopt (τ ′ )) dτ ′ . (A.117)
0
In this case, the solution of the multipole of the temperature fluctuation Θl
becomes
∫ τ0
l
Θl (k, τ0 ) = (−1)
S̃ (k, τ ′ ) jl [k (τ ′ − τ0 )] exp (−τopt (τ ′ )) dτ ′ , (A.118)
0
where jl (x) is the spherical Bessel functions and we use the mathematical
formula such as
∫
1 1
Pl (x) exp (ikx (τ − τ0 )) dx = (−1)−l jl (k (τ − τ0 )) .
(A.119)
2 −1
When we define S(k, τ ) as
[
(
)]
[
(
)]
1
d
ivb κ
3 d2
S (k, τ ) ≡ exp (−τopt ) −ϕ̇ − κ Θ0 + Π +
exp ψ −
− 2 2 [exp (−τopt ) κΠ] ,
4
dτ
k
4k dτ
(A.120)
the Θl becomes
∫ τ0
Θl =
S(k, τ )jl [k(τ0 − τ ′ )] dτ ′ ,
(A.121)
0
where we use κ ≡ τ̇opt . When one introduces the function such as
g(τ ) = κ exp (−τopt ) ,
(A.122)
one can simplify the transformed source function S(k, τ ) as
(A.123)
97
S(k, τ ) = g (τ ) [Θ0 + ψ]
(
)
d ivb (k, τ )g(τ )
+
dτ
k
[
]
+ exp(−τopt ) ψ̇ − ϕ̇
(A.124)
g (τ ) is named the visibility function. The visibility function has a sharp
peak at the recombination z = 1090.
This peak tells the observed temperature fluctuation of CMB is originated
only from the density fluctuations of photons at around z = 1090. most of
CMB photons can travel from the last scattering surface to the observer
without collisions with electrons or high energy photons for 14 Gpc and they
have information about the expansion history and geometrical structure of
the Universe.
A.4.6
Silk damping
As mentioned in eq. (A.105), the distribution of photons have only monopole
and dipole moment and no anisotropic stresses. It is caused by frequent collisions between electrons and photons. Silk pointed out that the movement of
photons in the tight-coupled plasma can be understood as a random walk and
the fluctuation of this plasma on small scales are dissipated by the random
walk just before the end of the recombination.
The free streaming scale of the photons in the plasma λc is related to the
opacity κ, which are the conformal time derivative of the optical depth τ̇opt ,
as the follows.
λc = κ−1 = (ane σT )−1 .
(A.125)
The diffusion scale λD is determined the number of collisions of photons
N = t∗ /λc , where τ∗ is the conformal time at the recombination, i.e. approximately 280 Mpc. Because this diffusion process is one of random walk
process,
√
λD = N λ c .
(A.126)
In the case that all atoms are ionized just before the end of recombination,
one can set
ρb
,
(A.127)
ne = nb ≃
mH
98
where mH is the mass of a hydrogen atom ≃ 938 MeV. Then the diffusion
scale can be written as
√
1
8
mH (1 + z∗ )−9/4
≃ 1.2 Mpc ≃
λD =
πG
τ∗ ,
(A.128)
1/2
1/4
3/2
3
σT H0 Ω Ω
200
b∅
m∅
Thus small temperature fluctuation of CMB such as l ≳ O(102.5 ) had been
exponentially attenuated. This damping tail has been clearly detected on
the ACT, SPT and Planck satellite [76, 75, 89, 13].
This damping scale corresponds to the mass scale, which is called Silk
mass MD .
(
)3
4
1
MD = πρB
λD ∼ 1013 M⊙ .
(A.129)
3
2
It indicates that the fluctuation whose mass corresponds to that of the Milky
Way had erased.
This fact is a serious problem of the purely baryonic Universe. In the
Einstein-De Sitter Universe, the density fluctuation of the matter δm grows
proportionally to the scale factor a (see [82]). In addition, the density contrast in the cluster is much more than unity. When one considers the scale
factor at the recombination is approximately 1/1090 , the matter density
contrast at the recombination should be more than 10−3 . However the observational data COBE, WMAP and Planck satellite suggest that the amplitude
of the density fluctuation of photons at the recombination is O(10−5 ). In the
adiabatic expansion Universe, the relation between density fluctuation of
baryon and that of photon at the recombination is
1
1
δb = δγ .
3
4
(A.130)
Hence the amplitude of the density fluctuation of baryon is O(10−5 ) and it
is too small to explain the structure formation on the cluster scale at the
present. If the matter which does not have electro-magnetic interactions
exists, because the growth of density fluctuation of the matter is not affected
by the diffusion damping and can grow after the matter-radiation equality, it
may reconcile this problem. This matter is called dark matter. Here “dark”
means that this matter does not emit or absorb the electromagnetic waves.
99
B
Sunyaev-Zel’dovich effect and its number
counts
The CMB photons which are observed around the Earth have passed through
many galaxy clusters. The cluster is a virialized object and the temperature of the gas Te in it is around 108 Kelvin. During the passing over the
cluster, CMB photons interacts these high-temperature electrons via inverseCompton scattering and the CMB spectrum is distorted. This effect is called
Sunyaev-Zel’dovich effect [90, 91].
The time evolution of the distribution function f of photons is described
by the Boltzmann equation. The collision term is originated from the Thomson scattering. Thus,
[
]
∂f
ȧ ∂f ne σT ∂ 4
∂f
= q
{q Te
+ f (1 + f ) }
(B.1)
∂t
a ∂q me q 2 ∂q
∂q
When one considers the distortion caused by the crossing the cluster, the
effect of cosmic expansion during the passing of cluster on the distribution
function is negligible. In order to simplify the eq. (B.1), we define two
variables x, y as
q
,
Te
∫
σT t
y ≡
Te ne dt′ .
me
x ≡
By using these variables, we obtain Kompaneets equation as
[ (
)]
∂f
∂f
1 ∂
4
= 2
x
+ f (1 + f )
.
∂y
x ∂x
∂x
(B.2)
(B.3)
(B.4)
∂f
is originated from the energy-derivative of the distribution function
∂x
i.e. Te ∂f /∂q. Because the temperature of cluster is high, f (1+f ) is negligible
∂f
compared with
and we obtain
∂x
)]
[ (
∂f
1 ∂
∂f
4
= 2
+ f (1 + f )
.
(B.5)
x
∂y
x ∂x
∂x
Here
100
Because this equation is a linear differential equation, the analytical solution
f (x, t) can be obtained as
( 2)
∫ +∞
1
s
f (x, y) = √
f (x exp (2s + 3y) , 0) .
(B.6)
exp −
πy −∞
y
The original spectrum of CMB f0 is almost exactly Plankian i.e. 1/ (exp(x)).
Our interest is the difference between f (x, y) and f0 (x, y) normalized f0 (x, y),
i.e. ∆f /f0 ≡ f (x, y) − f0 (x, y))/f0 (x, y). It becomes
( )
(
)
∆f
xy
1
=
x coth
x −4
(B.7)
f0
1 − exp (−x)
2
Here we estimate the SZ effect on the Rayleigh-Jeans part and Wien part
of the CMB spectrum. On the former part, the energy of photons is much
smaller than the temperature of the electron, i.e. x = E/T ≪ 1. In this
case,
∆f
→ −2y .
(B.8)
f0
On the other hand, on the latter part e.g. x ≫ 1, we obtain
∆f
→ x(x − 4)y .
f0
(B.9)
From eqs. (B.8) and (B.9), the CMB spectrum is distorted.
SPT collaboration firstly discovered unknown clusters by detecting this
distortion of CMB [92]. On Planck mission, 1227 clusters and the candidates
have been found by detecting SZ effect. 13 [93].
According to Press-Schechter theory [94], one can estimate the density
fluctuation of matter from counting of the number of clusters. By assuming
that the shape of cluster is sphere and using the mass function which is
obtained by Tinker et al. [95], Planck collaboration has reported as
σ8 (Ωm /0.27)0.3 = 0.78 ± 0.01 .
(B.10)
This value is lower than the estimated from the temperature fluctuation of
CMB by more then 2 σ confidence level.
13
In this catalog, 683 clusters are already-known clusters.
101
References
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Astrophysical Journal 86 (Oct., 1937) 217–+.
[2] V. C. Rubin, W. K. Ford, Jr., Rotation of the Andromeda Nebula from
a Spectroscopic Survey of Emission Regions, Astrophysical Journal 159
(Feb., 1970) 379–+.
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