A dynamical systems analysis of
interacting dark energy models
V Molosi
orcid.org/0000-0001-7770-3589
Dissertation accepted in partial fulfilment of the requirements for
the degree Master of Science in Astrophysical Sciences at the
North-West University
Supervisor: Prof AA Gidelew
Graduation May 2022
32695306
2
Declaration
I, Vuyani Molosi (32695306), declare that this dissertation titled, "A dy-
namical systems analysis of interacting dark energy models" has not been
submitted to this or any other university for any degree or examination,
that the work presented in it is my own and that all the sources I have used
or quoted have been acknowledged by complete reference.
Chapters 4 and 5 of this dissertation are based on the following publica-
tion:
• V Molosi and A Abebe, "A dynamical systems analysis of interacting
dark energy models", SA Inst. Phys. Proceedings SAIP2021 (Under
review).
Signed:
Date: 07/12/2021
3
Abstract
We investigate using dynamical system analysis; the impacts of
various interaction models whereby dark energy is coupled with dark
matter. Examination on the nature of critical points for each interac-
tion model introduced was conducted in order to obtain the cosmo-
logical consequence of each choice of interaction, with all the compo-
nents of the universe considered, namely, the radiation, matter, and
dark energy dominated universes. The existence of unstable radia-
tion epoch, unstable dark matter epoch, and stable dark energy epoch
will be shown for models displaying cosmologically acceptable re-
sults. Using the value of the scale factor at equality we determine
the time at which the matter-dark energy equality occurred for each
model. Constraints on the coupling constant b of the interacting dark
energy models were placed in order to determine the phantom or
quintessence behaviour of the equation of state for dark energy. We
do model comparison and also compare with the ΛCDM, so as to fil-
ter our models for the best possible form of interaction term between
dark matter and dark energy.
Keywords — cosmology; dark energy; dark matter; phase space; dynamical sys-
tems
CONTENTS 4
Contents
Declaration 2
List of Figures 6
List of Tables 6
1 Introduction 8
1.1 Expanding universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 The standard cosmological model 13
2.1 The first Friedmann equation . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Critical density and the density parameter, Ω . . . . . . . . . . . . 15
2.3 The fluid equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 The acceleration equation . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Closed, open and flat universes . . . . . . . . . . . . . . . . . . . . . 18
2.6 Solutions of the Friedmann equations . . . . . . . . . . . . . . . . . 20
2.6.1 Matter dominated universe (non-relativistic) . . . . . . . . 20
2.6.2 Radiation dominated universe . . . . . . . . . . . . . . . . . 21
2.6.3 Matter dominated universe . . . . . . . . . . . . . . . . . . 21
2.6.4 Dark energy dominated universe . . . . . . . . . . . . . . . 22
2.7 Flat universe (k=0, q = 1/2) . . . . . . . . . . . . . . . . . . . . . . . 22
2.7.1 Matter dominated: P = 0, a3ρ = Const . . . . . . . . . . . . 22
2.7.2 Radiation dominated: P = 13 ρ, a
4ρ = Const. . . . . . . . . . 23
2.7.3 Dark energy dominated: P = −ρ, ρ = Const. . . . . . . . . 23
2.8 Closed universe ( k=1, q > 1/2) . . . . . . . . . . . . . . . . . . . . . 23
2.8.1 Matter dominated: P = 0, a3ρ = Const . . . . . . . . . . . . 24
2.8.2 Radiation dominated: P = 13 ρ, a
4ρ = Const. . . . . . . . . . 25
2.8.3 Dark Energy dominated: P = −ρ, ρΛ = Λ. . . . . . . . . . 25
2.9 Open universe ( k= -1, q < 1/2) . . . . . . . . . . . . . . . . . . . . . 26
2.9.1 Matter dominated: P = 0, a3ρ = Const . . . . . . . . . . . . 26
2.9.2 Radiation dominated: P = 1 43 ρ, a ρ = Const. . . . . . . . . . 27
2.9.3 Dark energy dominated: P = −ρ, ρΛ = Λ. . . . . . . . . . . 27
2.10 Solution of the Friedmann equation with total density . . . . . . . 28
3 Dynamical systems in cosmology 31
3.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Defining parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
CONTENTS 5
3.3 Cosmological dynamical systems set up . . . . . . . . . . . . . . . . 34
4 The evolution of interacting DE models 38
4.1 Possible couplings of DM and DE . . . . . . . . . . . . . . . . . . . 38
4.2 Trends in the studied interaction models . . . . . . . . . . . . . . . 48
5 Interactions in the dark: constraints 51
5.1 Theoretical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.1 End of the radiation dominated epoch . . . . . . . . . . . . 51
5.1.2 End of the matter dominated epoch . . . . . . . . . . . . . . 53
5.1.3 End of the matter dominated epoch for different Qs . . . . 53
5.2 Observational constraints . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Discussions and conclusion 58
Acknowledgements 59
References 60
LIST OF FIGURES 6
List of Figures
1 Plot showing velocity versus distance where the distances of the
galaxy were determined using Cepheid variable stars. Credit: [1]. . 9
2 Chart showing contents of the universe, with the Planck 2018 data. 10
3 Graphic showing the timeline of the evolution of the universe.
Credit: NASA/WMAP. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Possible geometries of the universe represented by the curvature
index k. Credit: [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Evolution of the scale factor as function of time in arbitrary units
for the open; flat and closed Friedmann universes. . . . . . . . . . . 19
6 Curve showing the scale factor as function of time for the Concor-
dance model using the constants 2.98 . . . . . . . . . . . . . . . . . . 30
7 Evolution plots for the first interaction case. . . . . . . . . . . . . . 40
8 Evolution plots for the interaction case B. . . . . . . . . . . . . . . . 42
9 Evolution plots for the interaction case C and D. . . . . . . . . . . . 44
10 Evolution of energy densities as a function of redshift . . . . . . . . 49
11 Evolution of deceleration parameter and EoS are plotted as a func-
tion of redshift, for the different interactions. . . . . . . . . . . . . . 50
12 Plots (a) and (b) showing the equality points for radiation-matter
and matter-dark energy, respectively. . . . . . . . . . . . . . . . . . 52
13 (a) shows the radiation and matter densities as function of red-
shift for various Qs, while (b) shows the matter and dark energy
densities as function of scale factor. . . . . . . . . . . . . . . . . . . 54
List of Tables
1 Combinations of eigenvalues λ1 and λ2 showing the stability or
instability properties of an equilibrium point (x0, y0). . . . . . . . . 33
2 In this table we show the equality scale-factor with its correspond-
ing equality redshift and the time at which the equality occurred,
for the different models studied using initials 2.98. . . . . . . . . . 55
3 In this table we present the different models that were investi-
gated, with constraints on b2 and DE EoS. . . . . . . . . . . . . . . . 56
4 Constraint results on the equation of state for dark energy and
matter density, Ωm = 1 − ΩΛ, from different combinations of
datasets. The 2σ limits are obtained from integrals of 2.5% and
97.5% of the marginalized probability. [3]. . . . . . . . . . . . . . . . 57
LIST OF TABLES 7
5 Constraint results on the equation of state for dark energy and
matter density, Ωm = 1−ΩΛ from individual and a combination
of the datasets, at the 3σ confidence limit. [4]. . . . . . . . . . . . . 57
1 INTRODUCTION 8
1 Introduction
Cosmology may be defined simply as the study of the past, present and future of
the universe, or in general, just the universe as a whole. This is a well-established
research field in physics which attempts to provide answers to questions like:
What is the universe composed of? In spatial extent, is it finite or infinite? Is
it going to come to an end at some point in the future? There are still many
uncertainties that need to be investigated further by cosmologists.
1.1 Expanding universe
Observational data from Supernovae Type Ia (SNIa) [5, 6], power spectrum from
numerous galaxies [7] and various other sources; have proposed that our uni-
verse is currently experiencing a phase of accelerated expansion [8, 9]. This ex-
pansion has caused disparities in the governing Friedmann equations and thus it
is a challenging issue for standard cosmology today. One of the techniques used
to determine the expansion of the universe is by calculating the Doppler effect
of faraway objects. Edwin Hubble discovered through observations that galaxies
move away from the earth and the velocity at which they are receding by v is
proportional to the relative distance of the object d, given by the equation
v = H0d (1.1)
which became known as the Hubble law [10], or Hubble–Lemaître law [11]. The
variable H0 is the Hubble constant where the subscript 0 denotes the value as
measured today. This linear relationship is shown on figure 1 [1]. The wave-
lengths λ of objects that are moving towards the observer would be blue-shifted,
whereas those that are moving away from the observer would be red-shifted in
the spectrum. For the objects moving away from the observer, the red-shift can
be given as
λ
z + 1 0= . (1.2)
λ
Hubble presented the value of H0 as 500 km/(sMpc), however, there have been
different estimates around the value of H0 with the improvement of data: On [5]
this value is estimated to be 74.2± 3.6 km/(sMpc), while on [1] this value is 68±
2 km/(sMpc). However, in the most recent results, this value is measured to be
67.74± 0.46 km/(sMpc), as shown by the Planck 2018 data [12]. The Wilkinson
Microwave Anisotropy Probe (WMAP) data was important in establishing the
modern Standard Cosmological Model, resulting in the most precise estimate of
the age of the universe of 13.75 ± 0.11 billion years [13]. Shown on figure 3 is
the full timeline of the universe according to the standard model.
1 INTRODUCTION 9
Figure 1: Plot showing velocity versus distance where the distances of the
galaxy were determined using Cepheid variable stars. Credit: [1].
1.2 Dark matter
Dark matter (DM) is one component of the universe whose fundamental nature
is yet to be known and as a result, over the years, cosmologists have dedicated
most of their time on finding out the matter density of the universe. The matter
density parameter Ω0,m is crucial in determining the universe’s expansion rate
and spatial curvature. The content of matter of the universe is not trivial today,
even if we suppose a non-zero cosmological constant, and in the recent past it
was the dominant component [1].
DM has so far only been detected gravitationally, of which its candidates in-
clude black holes, massive halo objects, and many other non-baryonic particle
models [14, 15]. In recent years, it has been discussed in [16], that the axion has
emerged as a prominent particle candidate for providing the enigmatic dark mat-
ter in the universe. According to the Planck 2018 data, only about 4.6% of the
universe is ordinary matter, while DM makes up approximately 29.5%. The rest
of the content, which is approximately 65.9% is presumed to be dark energy. We
illustrate on Figure 2 the numerical proportion of this content.
1 INTRODUCTION 10
Figure 2: Chart showing contents of the universe, with the Planck 2018
data.
1.3 Dark energy
One of the most challenging unanswered topics in physics and cosmology is the
existence of dark energy (DE) and the genesis of the universe’s accelerating ex-
pansion [17]. The relatively recent occurrence of near-equality in the past few
billion years between the densities of DE and DM, despite the fact that they must
have formed differently, is a key difficulty for comprehending the rapid expan-
sion of the universe with or without dark energy [18]. The shift from matter
dominance to dark energy dominance is fast, and the fact that observational data
appears to be coherent with the relatively substantial amounts of both DM and
1 INTRODUCTION 11
DE suggests that we have lived through this transition phase. The cosmic coinci-
dence problem refers to the assertion that the current era is unlikely to coincide
with this quick transition phase [19].
There have been attempts to counteract the imbalances in the Friedmann equa-
tions, by either changing the governing equations or by introducing new source
terms. In the framework of standard cosmology, this source is called dark en-
ergy [20]. The cosmological constant Λ, whose equation of state can be given by
p
w = ΛΛ = −1, (1.3)
ρΛ
is the simplest candidate for DE, which gives a vacuum energy background re-
sponsible for the current acceleration of the universe. This cosmological constant
was initially introduced by Einstein and was incorporated in his general relativity
field equations to keep a static universe, it was however later discovered that it
itself can be considered as a component of DE that is also causing the universe’s
late-time acceleration.
As can be shown by 1.3, DE has a negative pressure which distinguishes it
from other types of matter like radiation and baryons, which are also components
of the universe. Though we know about its negative pressure, its fundamental
nature is still hypothetical in modern cosmology. The value of the energy density
for DE is of a very tiny magnitude, ρΛ = Λ/8πG ≈ 10−47 GeV4. If ρΛ were even
a fraction larger, the repulsive force would have forced the universe to expand so
quickly that galaxies would not have had the time to form [21].
The cosmological constant paired with cold1 dark2 matter (CDM) gives rise
to the simplest model in cosmology, known as the Λ cold dark matter, or in short,
ΛCDM. According to many observations, this is the most accepted cosmological
model we have today. However, the ΛCDM model does suffer from a variety
of problems, such as the cosmological constant described in [22]. Understand-
ing what dark energy is and figuring out solutions to its many related problems
remains one of the most challenging issues in modern cosmology.
In the next section we will introduce and derive some of the Friedman equa-
tions and its solutions in the framework of standard cosmology.
1Cold refers to the fact that DM moves slower when compared to the speed of light.
2Dark implies, very weak interactions with ordinary matter and electromagnetic radi-
ation
1 INTRODUCTION 12
Figure 3: Graphic showing the timeline of the evolution of the universe.
Credit: NASA/WMAP.
2 THE STANDARD COSMOLOGICAL MODEL 13
2 The standard cosmological model
The spatially flat standard cosmological model is the most widely used model
in cosmology. Given the enormous success, however, several issues were dis-
covered. Most lately, it was shown that there is a discrepancy (greater than 3σ)
between the cosmological and local Hubble constant measurements [17,23]. Con-
ventional physics, including the theory of general relativity (GR), is assumed in
standard cosmology. At the expansion factor z ∼ 1010, this results in a success-
ful account of the origin of the light elements. The production of light elements
examines the relativistic relationship between expansion rate and mass density,
although it is not a thorough investigation [24].
The modern standard cosmological model, sometimes referred to as the con-
cordance model, presumes that the universe was created in the "Big Bang" from
pure energy; and it is now made up of approximately 68% dark energy, 27% dark
matter, and 5% of ordinary matter. While the ΛCDM model is primarily based on
two theoretical models; GR and the standard model of particle physics, it also de-
pends on other additional assumptions: (1) the universe started from a Big Bang;
(2) mass energy content of the universe is made up of DE, DM, and ordinary mat-
ter; (3) GR describes the gravitational interactions between the aforementioned
components; and that (4) on sufficiently large cosmic scales, the universe is ho-
mogeneous and isotropic. Regrettably, GR is considered incomplete in the sense
that it does not explain the Big Bang cosmology, inflation, the universe’s matter-
antimatter imbalance, or the existence of dark energy [25]. Throughout this text,
the ΛCDM model will be used as the main reference to our results, including the
new information on cosmological parameters such as the one from the Planck
2018 results.
2.1 The first Friedmann equation
The equation that connects the scale factor a(t), curvature constant k, curvature
radius R0 is known as the Friedman equation, named after Alexander Friedmann
[1]. We begin first by deriving a non-relativistic equivalent of the Friedmann
equation from Newton’s law of gravity and the second law of motion. In doing so,
we look at an isolated sphere with radius Rs and mass Ms in a uniform, isotropic3
expansion. From Newton’s law of gravity, the gravitational force experienced by
the test mass is
−GMmF = 2 , (2.1)Rs(t)
3has the same quantity when measured in different directions
2 THE STANDARD COSMOLOGICAL MODEL 14
From the gravitational acceleration at the exterior of the sphere, we can work out
the equation of motion for Rs(t),
d2Rs GM
dt2
= − 2 , (2.2)Rs(t)
where G is the gravitational constant, and M is the total mass. Multiplying both
sides of (2.2) with dRs/dt and(integr)ating, we obtain the energy equation
1 dR 2s GM
= + U, (2.3)
2 dt Rs(t)
where U is just an integration constant, which we physically define as the total
energy per unit mass at the surface of the expanding sphere, that is, the sum
of gravitational potential energy and kinetic energy per unit mass. There are
conditions that occur when U is greater than zero and when U is less than zero,
when;
• U > 0 : the right hand side of eq. (2.3) remains positive, which implies that
the expanding sphere has positive total energy and will expand continu-
ously.
• U < 0 : the right hand side of eq. (2.3) will in the end become zero, and the
sphere will have a negative total energy and will eventually collapse again.
Let us now define the radius of the sphere as follows:
Rs(t) = a(t)rs, (2.4)
where rs is the "comoving4" radius of the sphere which is equal to the physical
radius at the epoch when a(t) = 1. Defining mass of the sphere as
4π
Ms = ρ(t)R3s(t), (2.5)3
the energy equation then becomes
1
r2 ȧ2
4π 2
s = Grs ρ(t)a
2(t) + U. (2.6)
2 3
Thereafter we divide each side of eq. (5) by r2a2s /2 which gives the Newtonian
form of the Friedmann equ(ati)on;
ȧ 2 8πG 2U
= ρ(t) +
a 3 r2a2
. (2.7)
s (t)
4Comoving distance ignores expansion of the universe, giving distance that does not
vary in time.
2 THE STANDARD COSMOLOGICAL MODEL 15
Since ρ(t) is proportional to 1/a3(t), we can tell that if U < 0, the right hand side
of eq. (2.7) will ultimately reach zero, after the expansion reverses. In General
Relativity, the Friedmann(equ)ation is given as
ȧ 2 8πG kc2 1
= ρ(t)−
a 3 R2
, (2.8)
0 a
2(t)
where R0 is the constant of the radius at present and k is the curvature index in
the Friedmann-Robertson-Walker metric [1]. In this text and throughout, we have
used ρ(t) as the total energy density.
2.2 Critical density and the density parameter, Ω
Substituting H(t) = ȧ/a permits us to write the Friedmann equation in terms of
the Hubble parameter,
8πG kc2 1
H(t)2 = ρc(t)−3 R2 a2
. (2.9)
0 (t)
From the above equation we can see that space is flat (k = 0) when the critical
density is;
3H2(t)
ρc(t) = . (2.10)8πG
Most often, we describe the energy density of the universe in Cosmology in terms
of the density parameter Ω, which is the ratio of the total density (ρ) to the critical
density (ρc);
≡ ρ ρ 8πGΩ = 2 2 . (2.11)ρc c 3H
Substituting Ω into the Friedmann equation gives;
kc2 1
H(t)2 = ΩH2(t)− 2 2 , (2.12)R0 a (t)
kc2 1
=⇒ Ω(t)− 1 = 2 . (2.13)H (t)a2(t) R20
The right hand side of this equation 2.13 always vanishes (for the flat universe),
as a result, if
• Ω = 1, then it will be 1 at all times.
In other cases, the value of Ω varies with time, but if
2 THE STANDARD COSMOLOGICAL MODEL 16
• Ω > 1, then it will always be greater than unity;
• Ω < 1, then it will always be less than unity because the sign on the right
hand side cannot change [1].
2.3 The fluid equation
Consider the first law of thermodynamics also known as the law of conservation
of energy,
dE = −PdV + dQ, (2.14)
where dE is the internal energy of a volume of a fluid, PdV is work and dQ is the
heat. As the Universe is assumed to be homogeneous, its expansion is adiabatic5,
therefore:
dE + PdV = 0 =⇒ Ė + PV̇ = 0. (2.15)
For a sphere of comoving radius rs we have
4π
V = r3s a
3(t), (2.16)
3
and its derivative is ( )
4π
V̇ = r3(3a2
ȧ
s (t)) = V 3 . (2.17)3 a
Introducing the total energy density ρ, we have the internal energy of the sphere
as
E = Vρc2,
while the rate of change of the intern(al energy)being
Ė = Vρ̇ + V̇ρ c2. (2.18)
Using equations 2.15, 2.17 and(2.18 we end up w)ith
ȧ ȧ P
V ρ̇ + 3 ρ + 3 2 = 0, (2.19)a a c
hence, ( )
ȧ P
ρ̇ + 3 ρ + 2 = 0. (2.20)a c
Equation (2.20) is the fluid equation, sometimes referred to as the "continuity
equation", which describes the evolution of energy density in an expanding uni-
verse [1, 26].
5a process that occurs without the transfer of heat.
2 THE STANDARD COSMOLOGICAL MODEL 17
In solving the fluid equation, we require an additional equation of state that
relates ρ and P. Assume we write this equation as P = wρ, where w could change
with time, but we will suppose its time derivatives are trivial compared to time
derivatives of the density ρ. The fluid equation then implies,
ρ̇ ȧ
= −3(1 + w) , (2.21)
(ρ ) a ( )
⇒ ρ= ln = − a3(1 + w) ln , (2.22)
ρ0 a0
hence, ( )
a −3(1+w)ρ
= , (2.23)
ρ0 a0
is the solution of the fluid equation.
2.4 The acceleration equation
To derive the acceleration equation, we multiply eq. (2.9) by a2;
2
ȧ2
8πG 2 − kc= ρ(t)a
3 R2
. (2.24)
0
The time derivative yields;
8πG
2ȧä = (ρ̇a2 + 2ρaȧ). (2.25)
3
Dividing both sides of eq. (17) by 2ȧa we get
ä 4πG ( a )
= ρ̇ + 2ρ . (2.26)
a 3 ȧ
From the fluid equation 2.20, substitutin(g )
a − Pρ̇ = 3 ρ +
ȧ c2
, (2.27)
we obtain ( )
ä
= −4πG Pρ + 3 2 . (2.28)a 3 c
From 2.28 we see that if ρ, P > 0, the expansion of the universe decelerates. A
higher P causes stronger deceleration for a given ρ, for instance, a radiation dom-
inated universe decelerates faster than a matter dominated universe.
2 THE STANDARD COSMOLOGICAL MODEL 18
2.5 Closed, open and flat universes
The constant k, which represents the spatial curvature of the universe, actually
holds a significant meaning within general relativity. To best describe the geome-
try of the universe, we rewrite the Friedmann equation as:
1 1
ȧ2 + Veff(a) = − k, (2.29)2 2
where Veff(a) = −((4πG)/3)ρa2 is the effective potential. There are then three
possible geometries of the universe that we can describe using the curvature in-
dex:
• k > 0 (closed universe), indicating a negative energy; and the trajectories
are bounded, meaning that the universe will expand up to a point in which
the expansion will eventually reverse and the universe contracts. This is
an elliptic surface and thus the sum of angles of the triangle on this surface
would be greater than 180◦.
• k < 0 (open universe), implies that the energy is positive and the trajec-
tories are unbounded, meaning that the universe will expand forever. As
figure 4 [2] shows, this is a hyperbolic surface and so the sum of angles of
the triangle would be less than 180◦.
• k = 0, a flat universe. On a plane surface, the sum of the triangle angles
sums up to 180◦.
Figure 4: Possible geometries of the universe represented by the curvature
index k. Credit: [2].
On figure 5 the evolution of the scale factor as function of time for a closed, flat
and open universe is indicated. A closed universe means that expansion will
2 THE STANDARD COSMOLOGICAL MODEL 19
be stopped by gravity, causing it to contract, which would result in a massive
contraction of the universe in what we call a "big crunch", from which it could
begin again. The density of the universe would have to be greater than the critical
density ρ > ρc.
On the other hand, an open universe means that gravity would be too weak to
stop the expansion, meaning it would expand forever. The density of the universe
would be less than the critical density ρ < ρc. For a flat universe, the density is
equal to the critical value; consequently the universe will only start to contract
after an infinite amount of time. [27].
Figure 5: Evolution of the scale factor as function of time in arbitrary units
for the open; flat and closed Friedmann universes.
2 THE STANDARD COSMOLOGICAL MODEL 20
2.6 Solutions of the Friedmann equations
From the definition of the Hubble rat(e, H =
ȧ
a , w)e have that
ä
Ḣ = − 2 äH = −H2 1− 2 ≡ −H
2(1 + q), (2.30)
a H a
where we define the deceleration parameter q as:
q ≡ − ä2 . (2.31)H a
2.6.1 Matter dominated universe (non-relativistic)
We model matter-dominated universe by dust approximation6, P = 0. We then
have, from the acceleration equation,
ä 4πG
+ 2 ρ = 0. (2.32)a 3c
Re-writing in terms of H, we have,
− 2 4πGH q + 2 ρ = 0, (2.33)3c
thus, we get and define the density of the universe ρ
3H2
ρ≡ q. (2.34)4πG
Noting that the value q gives the relationship between the density of the universe
ρ and the critical density ρc as
ρ
q = , (2.35)
2ρc
the first Friedmann equation becomes
H2 − 2H2q = − k2 , (2.36)a
=⇒ −k = a2H2(1− 2q). (2.37)
6Dust approximation for a matter-dominated universe means that the matter is ap-
proximated as stationary dust particles which produce no pressure — P = 0.
2 THE STANDARD COSMOLOGICAL MODEL 21
Since both the scale factor a and the Hubble parameter H cannot be zero, the
spatial curvature constant k takes the values+1 closed universe
k =  0 flat universe (2.38)−1 open universe
and for the deceleration parameter we have q = 1/2 for flat universe, q > 1/2 for
closed universe and lastly q < 1/2 for the open universe.
2.6.2 Radiation dominated universe
Radiation dominated Universe is modelled by perfect fluid approximation where
P = 13 ρ. The second Friedmann equation then yields,
ȧ 1 ȧ
ρ̇r + 3 (ρ + ρ ) = ρ̇ + 4 ρ = 0; (2.39)a r 3 r r a r
and thereafter multiplying by a4 we obtain,
a4
d
ρ̇ + 4ȧa3ρ = 0 =⇒ (a4ρ ) = 0 =⇒ a4ρ = a4r r r r 0ρ0 = Const, (2.40)dt
a4
∴ ρr(t) = ρ 00,r 4 . (2.41)a (t)
2.6.3 Matter dominated universe
The second Friedmann equation becomes;
ȧ
ρ˙m + 3 ρm = 0, (2.42)a
and multiplying both sides by a3 to get,
a3ρ˙m + 3ȧa2ρm = 0, (2.43)
=⇒ d (a3ρm) = 0, (2.44)dt
a3ρm = a30ρ0 = Const, (2.45)
a3
∴ ρ 0m(t) = ρ0,m . (2.46)a3(t)
2 THE STANDARD COSMOLOGICAL MODEL 22
2.6.4 Dark energy dominated universe
For dark energy, the equation of state is P = −ρ, then by the fluid equation we
have
ȧ ȧ
ρ̇ + 3 (ρ + P) = ρ̇ + 3 (ρ− ρ) = 0, (2.47)
a a
ρ̇ = 0,
where we have set c = 1 for the speed of light. Thus for dark energy;
ρΛ = Λ = Const. (2.48)
2.7 Flat universe (k=0, q = 1/2)
We derive the solutions of the Friedmann equation when k = 0.
2.7.1 Matter dominated: P = 0, a3ρ = Co(nst)
a 3
a3 a3 ⇒ 0ρ = 0ρ0 = ρ = ρa 0.
From the first Friedmann equation we have;
ȧ2
( )
8πG a 30
= ρ
a2 3 0
, (2.49)
√ a ( )
⇒ da 8πG
a3
= = ρ 00 . (2.50)dt 3 a
At the time of the Big Bang (t = 0), we get that a(t = 0) = 0. Assuming the
convention that a0 = 1 and applying the assumption that the universe is flat,
ρ0 = ρc, we thus have ( )
3 8 G 1/3
( )1/3
π ρ
a(t 0) = t2/3 = 6πGρc t2/3; (2.51)2 3
and substituting ρc = 3H20 /8πG, we(get )
3 2/3
a(t) = H2 t2/30 . (2.52)2
2 THE STANDARD COSMOLOGICAL MODEL 23
2.7.2 Radiation dominated: P = 13 ρ, a
4ρ = Const.
From the first Friedmann equation we obta(in2 )ȧ 8 G a 4π 0
2 = ρ0 , (2.53)a 3√ a ( )
⇒ da 8πG
a4
= = ρ 00 . (2.54)dt 3 a
Looking at the same initial conditions at Big Bang (t = 0) with a0 = 1, we have
that a(0) = 0, thus ( ) ( )
32 G 1/4 32 G 1/4π ρ π ρ
a 0 c(t) = t1/2 = t1/2, (2.55)
3 3
and substituting ρc, we then end up w(ith )1/2
a(t) = 2H t1/20 . (2.56)
2.7.3 Dark energy dominated: P = −ρ, ρ = Const.
Since k = 0, according to equatio(n 2).8, we have,
ȧ 2 8πG
= Λ, (2.57)
a √3( )
⇒ 1 8πG= da = Λ dt, (2.58)
a 3
hence, ( √ )
8πG
a(t) = a0 exp t Λ . (2.59)3
As the scale factor 2.59 is exponential in this case, it implies that the universe will
expand forever, as it is free from the singularity, or collapsing into a "Big Crunch".
2.8 Closed universe ( k=1, q > 1/2)
We derive the solutions of the Friedmann equation when k = 1.
2 THE STANDARD COSMOLOGICAL MODEL 24
2.8.1 Matter dominated: P = 0, a3ρ = Const
From 2.8 we now obtain, ( )
ȧ2 8πG a 30 1
2 = ρ −a 3 0 a a2 , (2.60)√
da 8πGρ0a3
=⇒ = 0 − 1. (2.61)
dt 3a
We now introduce the "arc-parameter measure of time" also called "conformal
time" given as
dt
dη ≡ ,
a(t)
and defining a new constant A as,
≡ 4πGρ0 2 qA = H0 q
0
0 = ,3 2q0 − 1
where we have used equations 2.37 and the fact that a0 = 1 today. This then
results in; ∫ a ( )
− √ 1 a− A 1η η0 = dã = arcsin + π. (2.62)
0 2Aã− ã2 A 2
We require that η =(0 at a = 0)which means that η0 = 0, hence
a− A 1
= sin η − π = − cos(η) =⇒ a = A(1− cos(η)). (2.63)
A 2
Since dt = adη, we hav∫e ∫
t− t0 = adη = A(1− cos(η)dη = A(η − sin(η). (2.64)
At t = 0 we want η = 0 which implies t0 = 0 and thus finally we can express the
scale factor a in terms of the conformal time η:
q
a 0= (1− cos(η)), (2.65)
2q0 − 1
q
t 0=
2q0 −
(η − sin(η)).
1
2 THE STANDARD COSMOLOGICAL MODEL 25
2.8.2 Radiation dominated: P = 13 ρ, a
4ρ = Const.
The first Friedmann equation yields,
2 ( )ȧ 8 G a 4π 0 − 12 = ρa 3 0√ a a2
, (2.66)
da 8πGρ 4⇒ 0
a
= = 02 − 1. (2.67)dt 3a
Once again defining a new constant A ≡ 8πGρ0∫ 1 3 =
2q0
2q (−1 thus0a √ )
,
1
η − η0 = dã = arcsin √
a
. (2.68)
0 A 21 − ã A1
√
With the condition η = 0 at t = 0 s√ets t0 = A1, and hence
2q
a 0=
√ 2q0 −
sin(η), (2.69)
1
2q
t 0= − (1− cos(η)).2q0 1
2.8.3 Dark Energy dominated: P = −ρ, ρΛ = Λ.
The Friedmann equation 2.8 i(mp)lies,
ȧ 2 8πG 1
= ρΛ − 2 (2.70)a √ 3 a( )
=⇒ 8πGȧ = Λ a2 − 1. (2.71)
3
By letting the constant B be defined as√
B ≡ 8πG Λ,
3
we end up with ( )
B−1/2 sinh−1 B1/2a = t, (2.72)
thus, ( )
a(t) = B−1/2 sinh B1/2t . (2.73)
2 THE STANDARD COSMOLOGICAL MODEL 26
2.9 Open universe ( k= -1, q < 1/2)
2.9.1 Matter dominated: P = 0, a3ρ = Const
From the Friedmann equation 2.8, ( )
ȧ2 8πG a 30 1
2 = ρ0 + , (2.74)a 3 √ a a
2
⇒ da
8πGρ0a3
= = 0 + 1. (2.75)
dt 3a
Allowing that a0 = 0 and defining à as à ≡ 4πGρ0 = q0 3 2√q −1 , it then follows that0∫ a √ 1  a + à + a(2à + A)η − η0 = dã = ln , (2.76)
0 2Ãã + ã2 Ã( )
=⇒ η − aη0 = cosh−1 + 1 . (2.77)Ã
Likewise, we require that η = 0 at a = 0 which means that η0 = 0, hence
a + Ã
= cosh(η) =⇒ a = Ã(cosh(η)− 1). (2.78)
Ã
Since dt = adη, we h∫ave ∫
t− t0 = adη = Ã(cosh(η)− 1)dη = Ã(sinh(η)− η). (2.79)
At t = 0 we want η = 0 which implies t0 = 0 and thus finally we can express the
scale factor a in terms of the conformal time η,
q
a 0= − (cosh(η)− 1), (2.80)2q0 1
q
t 0= − (sinh(η)− η).2q0 1
2 THE STANDARD COSMOLOGICAL MODEL 27
2.9.2 Radiation dominated: P = 13 ρ, a
4ρ = Const.
The first Friedmann equation 2.8 yields
2 ( )ȧ 8 4πG a0 1
2 = ρ0 + 2 , (2.81)a 3 √ a a
da 8πGρ 4⇒ 0
a
= = 02 + 1. (2.82)dt 3a
Once again defining a new constant à ≡ 8πGρ0 = 2q(01 3 2q −1 then∫ 0√ √ )
,
a 1 a
η − η −10 = dã = sinh . (2.83)
0 Ã 21 + ã √ Ã1
Given the condition η = 0 at t = 0√setting t0 = Ã1 we end up with
2q
a 0= − sinh(η), (2.84)2q0 1
and √
2q
t 0= − (cosh(η)− 1). (2.85)2q0 1
2.9.3 Dark energy dominated: P = −ρ, ρΛ = Λ.
The equation 2.8 implies ( )
ȧ 2 8πG 1
= ρ
a 3 Λ
+ 2 , (2.86)√( ) a
⇒ 8πG= ȧ = Λ a2 + 1. (2.87)
3
Therefore the constant B̃ is defined as√
≡ 8πGB̃ Λ,
3
then ( )
B̃−1/2 sinh−1 B̃1/2a = t, (2.88)
thus, ( )
a(t) = B̃−1/2 sinh B̃1/2t . (2.89)
2 THE STANDARD COSMOLOGICAL MODEL 28
2.10 Solution of the Friedmann equation with total density
Considering that matter is now a combination of more than two non-interacting
fluids, then the next equation 2.90 holds(separate)ly for each such fluid f;
P
ρ̇f = −3H ρf + f2 , (2.90)c
where in each case ( )
ρ̇f = −3H ρf + wfρf , (2.91)
from which we obtain
ρf ∝ a−3(1+wf). (2.92)
Forming a linear combination of such terms (for matter wf = 0; radiation wf =
1/3; dark energy wf = −1),
ρ −4 −3tot = ρ0,ra + ρ0,ma + ρ0,Λ.
Substituting ρtot into 2.8 gives( )
2 8πG
2
H = ρ a−4 + ρ a−3
kc
+ ρ − . (2.93)
3 0,r 0,m 0,Λ a2
Multiplying and diving the right hand side by the critical density at present,
ρ 2c,0 = 3H0 /8π(G,)we arrive at ( )
ȧ 2 8πG kc2
= ρc,0 Ω −4 −3a 3 0,r
a + Ω0,ma + Ω0,Λ − 2 , (2.94)a
where Ω0,r, Ω0,m and Ω0,Λ are the energy density of radiation, dark matter and
dark energy, respectiv√ely. ( )
⇒ 8πG= ȧ = ρ
3 c,0
Ω0,ra−2 + Ω0,ma−1 + Ω 20,Λa − kc2. (2.95)
Using H0 = 8πG√ρc,0/3(and equation 2.13 we have; )
ȧ = H20 Ω0,ra−2 + Ω
2
0,ma−1 + Ω0,Λa2 + H0(1−Ω0), (2.96)
where,
Ω0 = Ω0,r + Ω0,m + Ω0,Λ.
2 THE STANDARD COSMOLOGICAL MODEL 29
Takin∫g the integral we have,
a [( ) ]−1/2
Ω a−2 + Ω a−1 + Ω a20,r 0,m 0,Λ + (1−Ω0) da = H0t. (2.97)
0
Assuming the currently accepted and mostly used model, the "Concordance"
model, where the universe is said to be nearly flat, we have the following from
the Planck data [12]:
8πG
Ω0,m = 2 ρ0,m = 0.3089± 0.0062;3H0
8πG
Ω0,Λ = 2 ρ0,Λ = 0.6911± 0.0062; (2.98)3H0
8πG
Ω −50,r = 2 ρ0,r = 9.06× 10 ± 0.1233;3H0
where we then get the constraint
Ω0 = Ω0,r + Ω0,m + Ω0,Λ ≈ 1.
In solving the integral 2.97, we use the Trapezoidal rule where the area under a
curve is evaluated by dividing the total area into little trapezoids:
∫b
≈ ∆xf (x) dx Tn = [ f (x0) + 2 f (x1) + 2 f (x2) + · · · + 2 f (xn˘1) + f (x2 n)].
a
(2.99)
Th∫is rule allows us to estimate the area under the curve f (x) to a good approx-imation, therefore we can also estimate the present age of the Universe in theconcordance model:
1 [ ]−1/2
9.06× 10−5a−2 + 0.31a−1 + 0.69a2 + 9.06× 10−5 da = H0t. (2.100)
0
⇒ 0.954828= 0.954828 = H0t =⇒ t = ,H0
∴ t = 13.799± 0.021 Gyrs. (2.101)
where H0 = 0.06928 Gyr−1 is the present value of the Hubble constant [12].
2 THE STANDARD COSMOLOGICAL MODEL 30
Figure 6: Curve showing the scale factor as function of time for the Con-
cordance model using the constants 2.98 .
In the next chapter we will introduce a useful technique in dynamical sys-
tems, which we will utilize to investigate a generalized system of the Friedmann
equations.
3 DYNAMICAL SYSTEMS IN COSMOLOGY 31
3 Dynamical systems in cosmology
The most commonly researched cosmological models have a system of autonomous
ordinary differential equations (ODEs) as their governing equations [28]. A dy-
namical systems method is used since our major purpose is to provide a qualita-
tive description of these models. A short introduction of what dynamical systems
are will be given in the next sub-section.
3.1 Dynamical systems
A dynamical system, in general, can be conceived of as any abstract system made
up of space (phase space or state space) and a mathematical law that relates the
evolution of any point in that space. The state of the system of interest is repre-
sented by a set of critical system parameters; and the state space is the collection
of all possible values for these parameters. A dynamical system has two main
types: continuous and time-discrete.
Continuous dynamical systems are defined by a set of ODEs while time-
discrete ones are defined by difference equations or a map [29]. In this context, we
are only interested in the type of dynamical systems that are continuous, since we
are investigating Friedmann equations, which for an isotropic and homogeneous
space result in an ODEs system.
We express the standard form of a dynamical system as
~̇x = ~f (~x) (3.1)
where ~x = (x1, x2, ..., xn) ∈ X and X is a subset of Rn in a representation of a vec-
tor of the dynamics of a continuous system, with the dot denoting differentiation
with respect to time. The function ~f (is sufficiently sm) ooth
~f : X → X and a vector
filed on Rn
~f (~x) = f1(~x), ..., fn(~x) . (3.2)
The concept of a fixed point is crucial when studying a system’s local behaviour.
We define a fixed point as an equilibrium or constant of a system. A point pro-
duced by the autonomous system 3.1 is a fixed point if and only if, for a continu-
ous system, ~f (~x0) = 0 at a point ~x = ~x0 [30]. We sometimes refer to a fixed point
as the critical point, or stationary point. It is possible for a system in Rn to have
no critical point, only one critical point, numerous critical points, or an infinite
number of critical points.
One of the methods that can be used to investigate properties of the stability
of equilibrium points, is the linear stability theory. To better explain this theory,
we consider a mechanical system in one dimension F(x) = mẍ where F is a force
3 DYNAMICAL SYSTEMS IN COSMOLOGY 32
and suppose there exist a point x0 where F(x0) = 0. To discover the behaviour of
a particle near this point we let x(t) = x0 + δx(t) and suppose that δx(t) is very
small, then ẍ(t) = δ̈x(t) and F(x) = F(x0 + δx) ≈ F(x0) + F′(x0)δ(x) + · · · =
F′(x0)δ(x) + . . . , so that our mechanical system equation near the equilibrium
point turns out to be F′(x0)δ(x) = mδ̈x with F′(x0) being a constant.
Since in this text we aim for working with a system in two dimensions, we
consider an autonomous system
ẋ = f (x, y),
(3.3)
ẏ = g(x, y),
where f and g are functions of x and y. We suppose the existance of an equilib-
rium point (x0, y0) such that f (x0, y0) = g(x0, y0) = 0. Taking fx, fy and gx, gy as
differentiations of x and y, the Jacobia(n matrix)of 3.3 is
J f= x fygx g
. (3.4)
y
The eigenvalues of this matrix can then b√e obtained as1 1
λ1 = ( fx + gy) +2 2√( fx − gy)2 + 4 fygx, (3.5)1 1
λ2 = ( fx + gy)− ( fx − gy)2 + 4 fygx,2 2
which can be analysed on any equilibrium point (x0, y0). The critical points can
be distinguished using the following classification: If the resulting eigenvalues
have negative real parts, then that point can be regarded as stable. If at least
one of the eigenvalues has a real part that is positive, then the critical point in-
volved would be unstable and thus correlate with a saddle point which attracts
and repels trajectories in some directions. Finally, the eigenvalues could all have
positive real parts, in which the critical point would be regarded as unstable and
the trajectories repel [29].
In table 1 we present all the possible cases for an autonomous system in two
dimensions like equations 3.3. The exact same technique described above can be
used when analysing arbitrary dynamical systems, as a result we will utilize it in
the context of cosmology in order to carry out our analysis.
3 DYNAMICAL SYSTEMS IN COSMOLOGY 33
Table 1: Combinations of eigenvalues λ1 and λ2 showing the stability or
instability properties of an equilibrium point (x0, y0).
Eigenvalues Explanation
λ1 > 0 and λ2 > 0 the critical point is not stable and
the trajectories are repelled from the
point (x0, y0). This point may be re-
ferred to as the past attractor.
λ1 < 0, λ2 > 0 the equilibrium point is a saddle
point. While some paths will be re-
pulsed, the others will be attracted.
λ1 < 0, λ2 < 0 the critical point is stable and the
trajectories that start close by this
point will advance toward that point
(X0, y0).
λ1 = 0, λ2 > 0 the equilibrium point is not stable.
The positive eigenvalue guarantees
that there’s at least one direction that
is unstable.
λ1 = 0, λ2 < 0 the stability of this combination can-
not be determined using the linear
stability theory as it breaks down,
thus other methods are needed.
λ1,2 = ±iβ the trajectories are oscillatory and we
refer to the point as the centre.
λ1,2 = α± iβ the critical point is stable and spiral if
α < 0 and β 6= 0. But if α > 0 then the
point is unstable.
3 DYNAMICAL SYSTEMS IN COSMOLOGY 34
3.2 Defining parameters
With a spatially flat universe k = 0 filled with radiation, matter and dark energy
assumed, we start first by giving th(e Friedmann e)quation;
2 8πGH = ρ
3 r
+ ρm + ρDE
8πG 8πG 8πG
= ρr + ρ + ρ3 3 m 3 DE (3.6)
8πG 8πG 8πG
1 = 2 ρr + 2 ρm + ρ3H 3H 3H2 DE
= Ωr + Ωm + ΩDE
where
≡ 8πG 8πG 8πGΩr 2 ρ3H r ; Ωm ≡ ρ ; Ω ≡ ρ (3.7)3H2 m DE 3H2 DE
are the fractional energy densities of radiation, matter and dark energy; respec-
tively. Furthermore, because we expect positive energy densities, we have the
conditions 0 ≤ Ωr ≤ 1 and 0 ≤ Ωm ≤ 1. As a result, the condition ΩDE ≤ 1 is
also required in order to satisfy 3.6.
3.3 Cosmological dynamical systems set up
Taking a dimensionless value of the speed of light c = 1, we may rewrite (2.20)
and give the general equation for perfect fluid as
ρ̇ + 3H(1 + w)ρ = 0 (3.8)
where H = ȧ/a. Knowing that for radiation and matter, wr = 1/3 and wm = 0,
we can write,
ρ̇r + 4Hρr = 0 ; ρ̇m + 3Hρm = Q ; ρ̇DE + 3H(1 + wDE)ρDE = −Q, (3.9)
where we have taken into account the Interaction (denoted by Q) between dark
matter and dark energy. The positive sign of Q in (3.9) indicates transition of con-
tents of energy from dark energy to dark matter (baryonic matter, which makes
15.7% of Ω0,m is excluded in this consideration); and for a negative Q, the pro-
cess is reversed [31]. This is also to ensure that total matter is conserved in the
universe. The sum of equations 3.9 gives the total energy conservation in the uni-
verse as ρ̇ + 3H(ρtot + Peff) = 0, where the total equation of state can be written
as
P
w = eff
2Ḣ
eff = −1− . (3.10)ρtot 3H2
3 DYNAMICAL SYSTEMS IN COSMOLOGY 35
where ρtot and Peff are the total energy density and effective pressure, respectively.
Defining the energy density for dark energy [32, 33] as
ρDE = αH + βH2 (3.11)
leads to
ρD˙ E = αḢ + 2βHḢ, (3.12)
where α and β are constants with dimension (mass)3 and (mass)2 respectively.
The model defined by 3.11 is considered since it alleviates the fine-tuning problem
[31]. Substituting in the equations 3.9 leads to;
αḢ + 2βHḢ + 3H(1 + wDE)ρDE = −Q (3.13)
=⇒ 3H(1 + wDE)ρDE + Q = −Ḣ(α + 2βH) (3.14)
⇒ H(1 + wDE)ρDE + Q − Ḣ(α + 2βH)= =
Hρ H2
(3.15)
DE (α + βH)
where on the last equation we have divided by Hρ = αH2DE + βH3 on both sides.
By equations 3.6 we have that
3ḢH
ρ̇r + ρ˙m + ρD˙ E = , (3.16)4πG
hence 3.9 leads to,
− − − − 3ḢH4Hρr 3Hρm + Q 3H(1 + wDE)ρDE Q = 4πG
−4ρr − 3ρm −
3Ḣ
3(1 + wDE)ρDE = 4πG
− 8πG − 3 8πG2 2 ρr 2 ρm −
3 8πG Ḣ
2 ρDE(1 + wDE) = 2 (3.17)3H 2 3H 2 3H H
−[ − 3 3 Ḣ2Ωr Ωm − Ω2 2 DE(1 + wDE]) = H2
−1 Ḣ4Ωr + 3Ωm + 3ΩDE(1 + wDE) = 2 .2 H
Now, first working out the deceleration parameter, q, we have that since H = ȧ/a;
ä − ȧ
2 ä
Ḣ = 2 = − H
2,
a a a (3.18)
− Ḣ − ä= + 1 = q + 1,
H2 aH2
3 DYNAMICAL SYSTEMS IN COSMOLOGY 36
where q = −ä/aH2, thus(the dece)leration parameter is;
Ḣ
q = − 1 + [ ,H2 ] (3.19)
q = − 11 + 4Ωr + 3Ωm + 3ΩDE(1 + wDE) .2
Solving the equation of state (EoS) parameter for dark energy, wDE, by equation
3.15 we have;
− Ḣ(α + 2βH) QwDE = 2 − − 1. (3.20)H (α + βH) HρDE
After substituting eq. (3.20) into (3.17[) and solving for wd we found that; ]
2(α + βH) (4Ωr + 3Ωm)(α + 2βH) 3(α + 2βH)Ωw DE
Q
DE = + − − 1 .2(α + βH)−( 3ΩDE(α +)2βH) 2(α + βH) 2(α + βH) HρDE
Noting that ρDE = 3H2/8πG ΩDE and defining Ωq ≡ (8πG/3H2)Q, we obtain:
α[2Ωq − HΩDE(3(ΩDE + Ωm − 2) + 4Ωr)] + 2βH[Ωq − HΩDE(3(ΩDE + Ωm − 1) + 4Ωr)]wDE = − − ,3HΩDE[α(ΩDE 2) + 2βH(ΩDE 1)]
We define a new system by first defining new dimensionless variables as fol-
lows:
2 2 8πGα 2 8πGβx = Ω ; y = ; m = ; y2 + m2m = ΩΛ, (3.21)3H 3
with the radiation parameter consequently given as Ω = 1−m2r − x2− y2. Using
(3.21), and doing some algebra, we can finally write wΛ, weff and q as:
2m4 + m2(2x2 + 3y2 − 2) + y2(x2 + y2 + 2) + 2 f (x, y)
wΛ = 3(m2 + y2)(2m2 + y2 − , (3.22)2)
2(m2 + x2 − 1) + 5y2 + 2 f (x, y)
weff = , (3.23)3(2m2 + y2 − 2)
and
2m2 + x2 + 3y2 − 2 + f (x, y)
q = , (3.24)
2m2 + y2 − 2
where f (x, y) = Ωq/H. Letting N = ln(a), then the dynamical equations take
the form
′ 1 dx
2
′ 1 dy
2
x = ; y = , (3.25)
2x dN 2y dN
3 DYNAMICAL SYSTEMS IN COSMOLOGY 37
which implies,
′ x
2(2m2 + 2x2 + 5y2 − 2) + f (x, y)(2m2 + 2x2 + y2 − 2)
x = ,
2x(2m2 + y2 − 2)
2 2 2 (3.26)
′ y[(4m + x + 4y − 4) + f (x, y)]y = .
4m2 + 2y2 − 4
The parameter y satisfies the condition 0 < y < 1 (so that 3.26 are continuous).
The dimensionless variable m2 is significant for the early evolution of the uni-
verse, otherwise it is a negligible value. It has been shown by [32] that m2 could
have a fraction energy density of approximately 10 percent in the early universe
hence in this text we will refer to m2 as the early dark energy (EDE) term. That
is to say, the parameter m always satisfies the constraint 0 < m2 ≤ 0.1 . We also
name the y2 part of the ΩDE as late dark energy (LDE).
In the next chapter we look at different terms of interaction Q in order to
investigate the most likely form of the interaction between dark matter and dark
energy; and present the results.
4 THE EVOLUTION OF INTERACTING DE MODELS 38
4 The evolution of interacting DE models
In cosmology, theories in which DM and DE interact play a crucial role, hav-
ing been inspired to tackle the cosmological constant problem and thereafter the
coincidence problem. Due to their capacity to handle the well-known conflicts
between high and low redshift estimates of the Hubble constant H0 and the value
of the amplitude of the power spectrum σ8, interacting dark energy theories have
lately seen a renewed wave of attention [34, 35]. Because they provide for an en-
ergy exchange mechanism between the dark sector components, such interacting
situations are exceedingly generic. As a consequence of the unknown nature and
dynamics of DE and DM, it makes it strenuous to narrate on these components
from first principles with regard to established theories, thus leaving more room
to construct models.
In this chapter, we therefore take a deeper look at the evolution equations by
investigating some of the possible interaction models [15, 31, 36] between dark
matter and dark energy.
4.1 Possible couplings of DM and DE
(A) No interaction: Q = 0
When there is no interaction between dark matter and dark energy, f (x, y) = 0,
the dynamical equations (3.26) are said to be smooth (or continuous) and in this
case there are three acceptable points:
• P1: (x, y) = (0, 0). This point describes the early stages of the universe,
meaning that matter and the late dark energy did not contribute to the con-
tent of the energy of the universe. Both the eigenvalues of the linearization
matrix λ1 = 1/2 and λ2 = 1 are positive, which shows the instability of
this phase. Using eq. (3.23) and (3.24) we see that weff = 1/3, and q = 1
which represents the deceleration expansion of this epoch.
√
• P2: (x, y) = ( 1−m2, 0). From this point we can deduce that it illustrates
the dark matter epoch of the universe, which is basically the matter dom-
inated phase. From the linearization matrix, we obtain the eigenvalues
λ1 = −1 and λ2 = 3/4, implying an unstable phase. We also find that
the effective EoS and deceleration parameter are, weff = 0 and q = 1/2.
√
• P3: (x, y) = (0, 1−m2). Since we refer to y2 as the late dark energy and
remembering that m2 takes very small values at late times, we can say that
the point P3 illustrates the LDE dominated universe. The eigenvalues found
were both negative, λ1 = −3/2 and λ2 = −4 which suggest that this epoch
4 THE EVOLUTION OF INTERACTING DE MODELS 39
is stable. We also found that weff = −1 and q = −1. The deceleration being
q < 0 implies that this phase is undergoing acceleration.
In figure (7), we show the phase plane for the case where Q = 0, that is, when
there is no interaction. As can easily be seen on this figure, all the field lines end
at the (x, y) = (0, 1) point; which is consistent with the dark energy dominated
universe. We can label the paths of figure (7) as true cosmological paths as they
begin at the unstable radiation phase (x = 0, y = 0), passing through unstable
matter phase (x ≈ 1, y = 0) and ending at the stable dark energy dominated
point (x = 0, y = 1).
Figure 7a shows the phase space evolution for the case Q = 0, where all
three fixed points are well represented. In 7b the cosmic evolution of the energy
density Ωi is presented. The initial conditions are chosen at present as Ω0,r =
9.06× 10−5, Ω0,m = 0.31 and Ω0,Λ = 0.69. Figure 7c shows the evolution of the
equation of state together with the deceleration parameter.
(B) Linear interaction: Q = 3b2Hρtot
Since f (x, y) = Ωq/H and Ωq = (8πG/3H2)Q, we have
8πG 8πG
f (x, y) = 3 Q = 2 (ρr + ρm + ρΛ)3b
2 = 3b2.
3H 3H
We then find that the dynamical equations (3.26) now take the form
x2(2m2′ + 2x
2 + 5y2 − 2) + 3b2(2m2 + 2x2 + y2 − 2)
x =
2x(2m2 + y2 − ,2)
(4.1)
′ y[(4m
2 + x2 + 4y2 − 4) + 3b2]
y = .
4m2 + 2y2 − 4
Investigating (4.1) we obtain the following points:
√
• P1: (x, y) = ( 1−m2, 0).The following eigenvalues correspond to this
point: λ1 = −3b2 − 1 ≈ −1 and λ2 = −3b2/4 + 3/4 ≈ 3/4, from which
we conclude that this point is unstable. The effective EoS and decelera-
tion parameters were calculated as, w 2 2eff = −b /(1 − m ) ∼ 0 and q =
1/2(1− 3b2/(1−m2)) ∼ 1/2 (since both m2 and b are small parameters),
respectively. This point is consistent with the matter dominated epoch of
the universe.
√
• P2: (x, y) = (b, 1− b2 −m2). With the eigenvalues λ1 = −4 and λ2 =
3(b2 − 1)/(b2 − 1) ∼ −3, since at late times the parameter m2 is small, we
4 THE EVOLUTION OF INTERACTING DE MODELS 40
(a) Phase space evolution
(b) Evolution of energy density as a function of redshift
(c) Deceleration and EoS parameter vs. redshift.
Figure 7: Evolution plots for the first interaction case.
4 THE EVOLUTION OF INTERACTING DE MODELS 41
conclude that this equilibrium point is stable. This point describes the late
dark energy scaling solution. We also discover that q = −1 and wDE =
1/(b2 − 1) < −1.
Performing the same analysis as in the first model, on the phase space of figure
8a, we cannot make similar conclusions with the model (B), since we do not have
a unique starting point for the paths shown. Therefore according to this linear
interaction solution, the model suffers from the absence of radiation dominated
era at the early times, which is due to the discontinuity of the system 4.1 near the
point (0, 0).
Figure 8a shows the evolution of phase space for the linear interaction (B)
where we have chosen b = 0.2. Initial conditions are chosen as Ω0,r = 1 ×
10−5, Ω0,m = 0.04 and Ω0,Λ = 0.96. This model breaks down at the radiation
dominated fixed point. Due to lack of a fixed point, 8b only shows the cosmic
evolution for matter and dark energy. The evolution of EoS and the deceleration
parameters are shown in 8c.
(C) Non-linear interaction: Q = 3b2HρDEρm/ρtot
With this non-linear interaction case, we find, from equations 3.7 and 3.21 that;
f (x, y) = 3b2x2(y2 +m2), which 3.26 then leads to a new form of dynamical equa-
tions:
′ x[y
2(b2(9m2 + 6x2 − 6) + 5) + 2(3(bm)2 + 1)(m2 + x2 − 1) + 3(by2)2]
x =
4m2
,
+ 2y2 − 4
2 2
′ y[m (3(bx) + 4) + 3(bxy)
2 + x2 + 4(y2 − 1)]
y = .
4m2 + 2y2 − 4
(4.2)
In solving 4.2, three physically acceptable critical points are obtained:
• P1 : (x, y) = (0, 0). The calculated eigenvalues λ1 = 1 and λ2 = 1/2 indi-
cates the instability of this point. Using equations 3.23 and 3.24, we identify
EoS and deceleration parameter as weff = 1/3 and q = 1, respectively. We
can then say that this is a radiation dominated era.
√
• P2: (x, y) = ( 1−m2, 0). According to the pair of eigenvalues, λ1 = 3/4
and λ2 = −1, this is an unstable equilibrium point. We also find that weff =
−(bm)2 > −1/3 and q = 1/2(1 − 3(bm)2) > 0 implies deceleration of
matter. This point correlates with the dark matter dominated epoch.
√
• P3: (x, y) ≈ (0, 1−m2). A stable fixed point is implied by the corre-
sponding eigenvalues of this point: λ1 = −4 and λ2 ∼ −3/2. Recalling
4 THE EVOLUTION OF INTERACTING DE MODELS 42
(a) Phase space evolution
(b) Evolution of energy density as a function of redshift
(c) Deceleration parameter and EoS versus redshift.
Figure 8: Evolution plots for the interaction case B.
4 THE EVOLUTION OF INTERACTING DE MODELS 43
that the parameter m2 is very small at late times, we have P3 correlating to
the dark energy dominated epoch. The following values are also obtained;
weff = wΛ = −1 and q = −1.
(D) Non-linear interaction: Q = 3b2Hρ2m/ρtot
With this particular interaction, equation 3.26, and noting (3.7) and 3.21, leads us
to a set of new dynamic[al equations, namely ]
′ 1 2(3(bx
2)2 − 4m2 + x2 + 4)
x = x 2 2 − + 3(bx)
2 + 5 ,
2 2m + y 2
(4.3)
y[3(bx2)2 + 4m2 + x2 + 4(y2 − 1)]
y′ = .
2(2m2 + y2 − 2)
We now analyse (4.3) at these three physically acceptable critical points;
• P1 : (x, y) = (0, 0). Corresponding to this critical point are the eigenvalues
λ1 = 1 and λ2 = 1/2, which suggest an unstable point. By using 3.23
and 3.24 we then get weff = 1/3 and q = 1. Radiation scaling phase is
demonstrated by this critical point.
√
• P2: (x, y) = ( 1−m2, 0). With this critical point we find that weff =
−b2(1−m2) and q = 1/2(1− 3b2(1−m2√ )). In determining a matter dom-inated phase, we know that weff > −1/3 and q > 0 which leads to the
constrain b < 1/ 3(1−m2). This is an unstable (due to λ1 = −1 and
λ2 = 3/4) matter dominated epoch.
√
• P3: (x, y) = (0, 1−m2). The eigenvalues λ1 = −3/2 and λ2 = −4 sug-
gest this point is stable. As usual, using equation 3.23 and 3.24, the pa-
rameters weff = −1 and q = −1 are obtained, which is consistent with the
standard ΛCDM solutions. This is a late dark energy phase shown by this
point.
With interactions (C) and (D) having similar sets of critical points, and displaying
exactly the same paths; in figure 9a we represent their phase space evolution,
where b = 0.2 is chosen and initial conditions were chosen as Ω0,r = 0.0, Ω0,m =
0.02 and Ω0,Λ = 0.98. In 9b, the cosmic evolution Ωi is presented. Figure 9c
represents the evolution of the deceleration and EoS parameters.
4 THE EVOLUTION OF INTERACTING DE MODELS 44
(a) Phase space evolution
(b) Evolution of energy density as a function of redshift
(c) Deceleration and EoS parameter vs. redshift.
Figure 9: Evolution plots for the interaction case C and D.
4 THE EVOLUTION OF INTERACTING DE MODELS 45
(E) Non-linear interaction: Q = 3b2Hρ2DE/ρtot
A new form of equation 3.26 is obtained as,
x2′ (2m
2 + 2x2 + 5y2 − 2) + 3b2(x2 + y2 − 1)2(2m2 + 2x2 + y2 − 2)
x = 2 2 − ,2x(2m + y 2)
2 2 (4.4)
′ y[3b (x + y
2 − 1)2 + 4m2 + x2 + 4(y2 − 1)]
y = .
4m2 + 2y2 − 4
Investigating 4.4 at the three different physically acceptable critical points, we
have the following,
√
• P1: (x, y) = ( 1−m2, 0). From λ1 = 3/4 and λ2 = −1, we read that this
equilibrium point is unstable. We also get that the deceleration parameter is
q = 1/2 + 3(bm2)2/(2(m2 − 1)); and effective EoS as weff = −(bm2)2 ∼ 0.
By this point, EDE phase is shown.
√
• P2: (x, y) ≈ (bm2, 1−m2). Because of the pair of eigenvalues λ1 = −3/2
and λ2 = −4, this a stable fixed point. For this point, we also find that
weff = −3/3 = −1 and q = −1. As the constant m2 is small at late times,
this critical point demonstrates a dark energy dominated epoch of the uni-
verse.
• P3: Due to this point being being very messy to include in this text, it is
omitted, but upon checking, it also displayed trends of the dark energy
epoch.
(F) Non-linear interaction: Q = 3b2Hρ3 /ρ2DE tot
For the function f we get f (x, y) = 3b2(m2 + y2)3, thus from 3.26 the following
system of equations is obtained for this interaction:
x2[2(m2 + x2 − 1) + 5y2] + 3b2(m2 + y2)3[2(m2 + x2 − 1) + y2′ ]x = 2 2 − ,2x(2m + y 2)
3 2 2 2 2 2 4 2 6 2 (4.5)
′ y[3(bm ) + (3bm y) + m (9b y + 4) + 3b y + x + 4(y
2 − 1)]
y =
4m2 + 2y2 − .4
Analysing 4.5 at various acceptable points we have the following:
• P1 : (x, y) ≈ (0, 0). The corresponding eigenvalues are λ1 = 0 and λ2 = 1
which implies that this equilibrium point is not stable. A radiation dom-
inated phase by this point is implied, from which we also get weff = 1/3
and q = 1.
4 THE EVOLUTION OF INTERACTING DE MODELS 46
√
• P2: (x, y) = ( 1−m2, 0). Corresponding to this fixed point are the eigen-
values λ1 = 3/4 and λ2 = −1 which confirm the instability of this point.
With this point we notice dark matter phase where weff = −b2m6 ∼ 0 and
q = 1/2 + 3b2m6/(2(m2 − 1)) > 0.
(G) Interaction: Q = 3b2Hρm
With this choice of interaction, since f (x, y) = Ωq/H and Ωq = (8πG/3H2)Q,
we have the following: ( )
8πG 8πG 9 8πG
f (x, y) = 3 Q = 3 (3b
2Hρ ) = b2 ρ = 3b2x2. (4.6)
3H 3H m 8 3H2 m
We then find that the dynamical equations (3.26) now take the form
′ x(2m
2 + 2x2 + 5y2 − 2) + 3b2x(2m2 + 2x2 + y2 − 2)
x = ,
2(2m2 + y2 − 2)
y[(4m2 + x2
(4.7)
′ + 4y
2 − 4) + 3b2x2]
y = .
2(2m2 + y2 − 2)
We now analyse 4.7 at these three physically acceptable critical points;
• P1 : (x, y) = (0, 0). Using equations (3.23) and (3.24) we see that weff =
1/3 and q = 1. This is a radiation dominated epoch. Furthermore, the
eigenvalues λ1 = 1 and λ2 ∼ 1/2 are both positive, thus confirming the
instability of the radiation era.
√
• P2: (x, y) = ( −m2 + 1, 0). From (3.23) and (3.24) we read that weff =
−(3/8)b2 ∼ 0 and q = 1/2− (9/16)b2 ∼ 1/2 since b is very small. Since the
eigenvalues λ1 = −1 and λ2 ∼ 3/4 are positive and negative, the instability
of the matter dominated era is confirmed.
√
• P3: (x, y) = (0, 1−m2). The calculated eigenvalues were λ1 = −3/2
and λ2 = −4, implying that the epoch of DE is stable with the EoS being
weff = −1. The deceleration parameter was also found as q = −1.
(H) Interaction: Q = 3b2HρDE
For this type of interaction Q, we(have the f)unction f (x, y) = Ωq/H as follows:
8πG 8πG
f (x, y) = 23 Q = 3b 2 ρDE = 3b
2Ω = 3b2(y2Λ + m2).H 3H
4 THE EVOLUTION OF INTERACTING DE MODELS 47
The new set of dynamical equations is then;
x2(2m2 + 2x2′ + 5y
2 − 2) + 3b2(y2 + m2)(2m2 + 2x2 + y2 − 2)
x =
2x(2m2 + y2 − ,2)
y[(4m2 + x2
(4.8)
′ + 4y
2 − 4) + 3b2(y2 + m2)]
y = .
4m2 + 2y2 − 4
Analysing this system, the following fixed points are obtained:
√
• P1: (x, y) = ( −m2 + 1, 0). The corresponding eigenvalues of this fixed
point; λ1 = 3/4 and λ2 = −1, implies an unstable matter-dominated uni-
verse. The EoS was approximated as weff ≈ 0 since the m2 is small. The
deceleration for this point was q = 1/2.
√
• P2: (x√, y) = (
1 1 2
b2 1 , b2 1 − m ). Eigenvalues λ1 = −5 + 41/2 and λ+ + 2 =
−5− 41/2. This is a stable late dark energy epoch. Calculation of the EoS
gives weff = −5/3 while deceleration gives q < −17/10.
(I) Interaction: Q = ηρm
We now analyse another type of interaction which is different from the other in-
teraction models that we have already studied in the previous subsections. Here,
we introduce the transfer rate η = γH0 where γ is a dimensionless constant.
Therefore, for the function f , the follo(wing equ)ation is obtained;
8πG η 8πG η η
f (x, y) = 3 Q = 2 ρm = Ω = x
2
m . (4.9)H H 3H H H
To keep the dimensionality the same, we introduce a new dimensionless param-
eter v given as;
H
v 0= (4.10)
H + H0
where 0 ≤ v < 1. We now obtain a new function (f given)as
η
f (x, y) = x2
H0 v
= γ x2 = γ − x
2. (4.11)
H H 1 v
Thus, the set of new dynamical equations is:
(1− v)x2(2m2 + 2x2 + 5y2 − 2) + γvx2(2m2′ + 2x2 + y2 − 2)x = ,
2(1− v)x(2m2 + y2 − 2)
2 2 2 2 (4.12)
y′
y[(1− v)(4m + x + 4y − 4) + γvx ]
=
2(1− v)(2m2 .+ y2 − 2)
From the above system 4.12 we arrive at a set of three acceptable critical points;
4 THE EVOLUTION OF INTERACTING DE MODELS 48
• P1 : (x, y) = (0, 0). The corresponding eigenvalues λ1 = 1 and λ2 =
−(γv− v + 1)/(2(−1 + v)) ∼ 1/2, suggest that we have an unstable fixed
point. The EoS is weff = −(−2 + 2v)/(6(1− v)) = 1/3 while deceleration
is q = 1.
√
• P2: (x, y) = ( 1−m2, 0). The eigenvalues λ1 = (γv− v + 1)/(−1 + v) ∼
−1 and λ2 = γv/(4(−1 + v)) + 3/4 > 0 show an unstable matter domi-
nated epoch . The EoS and deceleration are weff = −γv(3(1− v)) ∼ 0 and
q = −γv + 1/2 ∼ 1/2, respectively.
√
• P3: (x, y) = (0, 1−m2). Eigenvalues were found as, λ1 = −4 and λ2 ∼
−3/2 which show that this point is stable. The equation of state is weff =
−(3 + 2v)/(3(1− v)) ≤ −1, and for the the deceleration, q = −1 was cal-
culated for this fixed point.
(J) Interaction: Q = 3b2H(ρDE + ρm)
For this late interaction term, we obtain the following
f (x, y) = 3b2(y2 + x2 + m2), (4.13)
and thus a set of dynamical equations of the form
′ x
2(2m2 + 2x2 + 5y2 − 2) + 3b2(y2 + x2 + m2)(2m2 + 2x2 + y2 − 2)
x = ,
2x(2m2 + y2 − 2)
y[(4m2 + x2 + 4y2 − 4) + 3b2(y2 + x2 + m2
(4.14)
′ )]y =
4m2 + 2y2 − .4
Calculating and analysing the critical points from the set above, we draw the
following conclusions:
√
• P1: (x, y) = ( 1−m2, 0). The eigenvalues for this epoch λ1 = 3/4 and
λ2 = −1 show that this point is unstable, with the EoS and deceleration
parameter calculated as weff = 0 and q = 1/2, respectively.
√
• P2: (x, y) = (b, 1− b2 −m2). With the corresponding eigenvalues being
λ1 = −4 and λ2 ∼ −3, we confirm the stability of this point. The EoS and
deceleration were weff = −1 and q = −1, respectively.
4.2 Trends in the studied interaction models
In this section we compare and explore the trends in our interaction models. As
shown on the first interaction, model (A), the cosmological paths which we ex-
pected, we were able to observe on the phase space plot. The fractional energy
4 THE EVOLUTION OF INTERACTING DE MODELS 49
density plot showed dominance of the radiation era in the early times of the uni-
verse and then it became less dominant when the next epoch of matter took over.
The last epoch to dominate is the dark energy era, which is currently dominating
our universe. We demonstrate this behaviour for the first three of our models on
figure 10. Similar behaviours with the interaction models C, D, F, G, and I were
also observed. However, we also analysed models, such as models B, E, H and J,
Figure 10: Evolution of energy densities as a function of redshift
which failed to demonstrate the normal behaviour as according to the standard
model. These models started their cosmological evolution from the matter dom-
inated epoch and ended at the dark energy dominated epoch, which are not the
cosmologically acceptable paths. The possible cause and commonality around
these models is their linearity, and the inability to show the radiation epoch in
the early times of the universe. From the first models mentioned before, those
with acceptable cosmological paths (C, D, F, G, and I), we also saw from figure
11 that radiation and matter dominated eras were decelerating (q > 0) while the
DE epoch was accelerating (q < 0).
4 THE EVOLUTION OF INTERACTING DE MODELS 50
Figure 11: Evolution of deceleration parameter and EoS are plotted as a
function of redshift, for the different interactions.
5 INTERACTIONS IN THE DARK: CONSTRAINTS 51
5 Interactions in the dark: constraints
In the last decade, cosmological data from various observations has been quickly
updating, not only in terms of quantity but also in terms of quality. This allows for
more trustworthy scientific discoveries and stricter limits on theoretical models in
cosmology. The most recent Planck results [12] provide significant contributions
to a number of theoretical cosmological investigations and have reduced more
uncertainties than the older results. In this chapter we will use this newer and
more precise cosmological data to aid and analyse the DM and DE interaction
models.
5.1 Theoretical constraints
5.1.1 End of the radiation dominated epoch
When the radiation dominated epoch ended, and its density ρr was dropping as
per 2.41, the matter epoch began to dominate where its density ρm fell off at a
much lesser rate 2.46, which resulted in a point of equality where the two densi-
ties were equal. At the point where the two densities matched, we had;
a4 a3
ρr(t) = ρ 0 00,r 4 = ρa t 0,m( ) a3
= ρm(t), (5.1)
(t)
that is,
a(t) ρ0,r Ω0,r
= = = 2.93× 10−4, (5.2)
a0 ρ0,m Ω0,m
where the corresponding redshift was,
a
z 0= − 1 = 3408.27. (5.3)
a(t)
Using (2.97) we find the time at which the radiation era ended,
t(a = 2.93× 10−4) = 50953 years.
On figure 12a we show the point of intersection, or zeq, where matter then began
to dominate.
5 INTERACTIONS IN THE DARK: CONSTRAINTS 52
(a) Radiation and matter densities versus redshift.
(b) Matter and dark energy densities versus scale factor.
Figure 12: Plots (a) and (b) showing the equality points for radiation-
matter and matter-dark energy, respectively.
5 INTERACTIONS IN THE DARK: CONSTRAINTS 53
5.1.2 End of the matter dominated epoch
At the end of matter dominated era, as the density was falling as per 2.46, the dark
energy epoch started to dominate and the matter-dark energy densities matched
at a point where
a3
ρm(t) = ρ 00,m a3
= ρ , (5.4)
(t) 0,Λ
that is, ( )
a t 1/3
( )1/3
( ) ρ0,m Ω0,m
= = = 0.76458± 0.01681, (5.5)
a0 ρ0,Λ Ω0,Λ
with the corresponding redshift being,
a
z 0= − 1 = 0.31, (5.6)
a(t)
and the time at which this era ended, using (2.97), we found,
t(a = 0.7646) = 10.19 Gyrs.
Due to the matter-dark energy equality redshift being small values and not being
visible enough on a density versus redshift plot, we convert to scale factor where
the corresponding value is represented on the density versus scale factor graph,
as shown by figure 12b. Since the radiation density Ωr is sub-dominant at the
late times for the spatially flat universe (Ωm ≈ 1−ΩΛ), we will only look at the
matter-dark energy equalities in the next sub-section.
5.1.3 End of the matter dominated epoch for different Qs
Our models showed multiple (and mostly different) redshifts of equality, or when
the densities of the matter and dark-energy dominated era matched (mathemat-
ically by equation 5.4, and so on this section we interpret these redshifts for the
studied models, and the times at which the equality occurred. The results are
shown on Table 2. Figures 13a and 13b, shows graphically the end (intersects) of
the radiation and matter epochs, respectively.
The relative expansion of the universe appears to have been delayed for most
of the models, due to their forms of the interaction term as their scale factors at
equality is a bit larger, that is aeq > 0.76, as compared to the result from the Planck
data equation 5.5 [12]. There is however a model (model I), where the scale factor
at equality occurs as early as 9.79 Gyrs when compared to the Planck value of
10.19 Gyrs. This model has been studied further in [36].
In order to ensure stability conditions on the investigated models, we put con-
straints on the coupling parameter b and the results are presented on Table 3. For
5 INTERACTIONS IN THE DARK: CONSTRAINTS 54
(a)
(b)
Figure 13: (a) shows the radiation and matter densities as function of red-
shift for various Qs, while (b) shows the matter and dark energy densities
as function of scale factor.
5 INTERACTIONS IN THE DARK: CONSTRAINTS 55
Table 2: In this table we show the equality scale-factor with its correspond-
ing equality redshift and the time at which the equality occurred, for the
different models studied using initials 2.98.
Model Q aeq zeq Time (Gyr)
B 3b2Hρtot 0.96 0.04 13.20
C 3b2HρΛρm/ρtot 0.79 0.27 10.59
D 3b2Hρ2m/ρtot 0.80 0.25 10.75
E 3b2Hρ2Λ/ρtot 0.83 0.20 11.22
F 3b2Hρ3 /ρ2Λ tot 0.80 0.25 10.75
G 3b2Hρm 0.85 0.18 11.53
H 3b2HρΛ 0.89 0.12 12.15
I ηρm 0.74 0.35 9.79
J 3b2H(ρΛ + ρm) 0.96 0.04 13.20
precision cosmology, the general strategy in investigating dark energy properties
is to presume that there exist a fundamental equation of state, pi = wiρi, such
that the field i is described, which is believed to be responsible for the universe’s
accelerated expansion. [37]. If the EoS falls in the interval −1 < wi < −1/3, then
the field i is referred to as standard quintessence, but if wi < −1, then it is "phan-
tom" since this likelihood is said to be non-canonical in Quantum Field Theory,
that is, it imposes a negative kinetic energy, and also breaks the weak energy con-
dition7. We can see that models H and I mimic the phantom behaviour as they
both have an EoS for DE that is less than negative one. It has been shown that
such models with w < −1, although unstable, may be constructed and might be
phenomenologically realistic if seen as effective field theories that are valid only
up to a specific momentum cut-off [38]. The Dominant Energy Condition (DEC)
implies that w ≥ −1 and since for cosmological reasons we are looking for a
source with ρ > 0, we can see on Table 3 that models B through G correspond
with this condition.
7The weak energy condition stipulates that for all time-like vector fields, the observed
matter density by the corresponding observers is always non-negative.
5 INTERACTIONS IN THE DARK: CONSTRAINTS 56
Table 3: In this table we present the different models that were investi-
gated, with constraints on b2 and DE EoS.
Model Q Constraints EoS for DE
B 3b2Hρtot b2 < 1 −1 < weff < 0
C 3b2HρΛρm/ρtot b < 1 −1 < weff < −1/3
D 3b2Hρ2 /ρ b2 < 1/3(1−m2m tot ) −1 < weff < −1/3
E 3b2Hρ2 /ρ 2 2 2Λ tot b < (1−m )/m −1 < weff < 0
F 3b2Hρ3 /ρ2 b2Λ tot < (1−m2)/m3 −1 < weff < 0
G 3b2Hρm b2 < 8/9 −1 < weff < 0
H 3b2Hρ 2Λ b < 1 weff < −1
I ηρm v < 1 weff < −1
J 3b2H(ρΛ + ρm) b2 < 1 weff < −1
5.2 Observational constraints
In order to constrain the dark energy equation of state, a combination of datasets
from different cosmic microwave background (CMB) experiments have been stud-
ied in [3, 4] to show the constraints on EoS for DE, at certain confidence levels.
Thus in this context, we will use such results and others to analyse and compare
with our theoretical findings, in order to draw some conclusions. The result that
EoS for DE may be less than −1 is not completely ruled out at present by cosmo-
logical data as it seems, to some degree, is preferred by the joint analysis of the
supernovae (SN) and CMB data [37].
In Table 4 we see different combinations of datasets from CMB, Hubble Space
Telescope (HST), Big Bang Nucleosynthesis (BBN), supernovae type Ia (SN-Ia)
and finally also data from the 2dF survey [3]. The SN-Ia data enables cosmo-
logical parameters to be determined by investigating the current values of the
deceleration parameter q0 and the Hubble constant H0 [39]. It is evident on this
table that, the more combinations of datasets we use, the slimmer the constraint
values of the EoS of the DE component and thus the better the result. This trend
is also similar in constraining the matter density Ωm. With more recent data as
in [4], we see an improvement on the constraint of weff, shown on Table 5. It is
apparent from the table results that the individual datasets constrain the DE weff
very poorly and these limits are improved when combining the datasets.
5 INTERACTIONS IN THE DARK: CONSTRAINTS 57
Table 4: Constraint results on the equation of state for dark energy and
matter density, Ωm = 1 − ΩΛ, from different combinations of datasets.
The 2σ limits are obtained from integrals of 2.5% and 97.5% of the
marginalized probability. [3].
Datasets EoS for DE Constraint on Ωm
CMB + HST −1.65 < weff < −0.54 0.19 < Ωm < 0.43
CMB + HST + BBN −1.61 < weff < −0.57 0.20 < Ωm < 0.42
CMB + HST + SN-Ia −1.45 < weff < −0.74 0.21 < Ωm < 0.36
CMB + HST + SN-Ia +2dF−1.38 < weff < −0.82 0.22 < Ωm < 0.35
Table 5: Constraint results on the equation of state for dark energy and
matter density, Ωm = 1−ΩΛ from individual and a combination of the
datasets, at the 3σ confidence limit. [4].
Datasets EoS for DE Constraint on Ωm
SN-Ia −1.57 ≤ weff ≤ −0.66 0.05 ≤ Ωm ≤ 0.43
BAO −2.19 ≤ weff ≤ −0.42 0.19 ≤ Ωm ≤ 0.36
H(z) −1.78 ≤ weff ≤ −0.72 0.20 ≤ Ωm ≤ 0.35
SNIa + BAO + H(z) −1.13 ≤ weff ≤ −0.95 0.25 ≤ Ωm ≤ 0.31
6 DISCUSSIONS AND CONCLUSION 58
6 Discussions and conclusion
An investigation was performed, using dynamical systems, on the impacts of in-
teractions between dark matter and dark energy on the evolution of the universe.
In the first chapter of this work an introduction on the expansion of the universe
was presented. With more and more accurate data generated, we saw that the
Hubble constant, which is the proportionality constant in the Hubble–Lemaître
law 1.1, has changed and improved over time. The current value of this constant,
as presented in [12] is 67.74± 0.46 km/sMpc. We also introduced the dark sector
components, namely, dark matter and dark energy. The cosmological constant
Λ is the simplest candidate for DE. More recently, it has been discussed in [16]
how the axion has emerged as a DM candidate and how it was produced in the
early phases of the universe. It has been said that ever since the discovery of
the universe’s current accelerating expansion in 1998, dark energy has played a
significant role in cosmological studies [40].
Despite the ΛCDM model’s success in explaining the formation and evolu-
tion of large scale structure in the universe; the state of the early universe, the
abundance of various forms of matter and energy; and its predictive power, ac-
cording to the majority of the cosmology community, it presents several chal-
lenges [22, 41]. Among the most popular of these includes the “cosmic coinci-
dence problem” and the “cosmological constant fine tuning problem”. In the
framework of the aforementioned model, we derived Friedmann equations and
solutions, considering three spatial curvatures of the universe: closed, open and
flat universes. Using data from [12] we calculated the age of the universe today
as 13.78 billion years.
Dynamical systems were introduced in the third chapter of this work. Since
our main purpose was to provide a qualitative description of the interaction mod-
els investigated, this method came in handy. Friedmann equations were reduced
to dimensionless variables, from which we then used the aforementioned tech-
nique to generalize the equations into ODEs in two dimensions. The eigenvalue
combinations and explanations were presented in Table 1, which we used to aid
our analyses of the stability and classification of the fixed points for each model
studied.
We introduced and studied different possible interaction terms between the
dark sector components in chapter four. We observed a behaviour similar to the
ΛCDM model when there is no interaction, Q = 0, that is, where the trajectories
on the phase space plot started from the unstable radiation dominated era, passed
through the also unstable matter dominated epoch, and ended at the stable dark
energy dominated epoch. This was a display of the acceptable cosmological paths
we desired. Such behaviour was observed for multiple other models (C, D, F, G,
6 DISCUSSIONS AND CONCLUSION 59
and I). However, there were some interaction terms (B, E, H, and J) between DM
and DE that we noticed a system’s break down in the early times of the uni-
verse (the system 3.26 was divergent), and so they only displayed trajectories
that started from the matter dominated epoch up to the dark energy dominated
epoch.
The equality redshift (and scale factor), for when there was a radiation-matter
and matter-dark energy match in densities using the Planck 2018 data, was pre-
sented; from which we saw that these events occurred at redshifts zeq = 3408.27
and zeq = 0.31, respectively. Since we considered DM-DE interactions, we also
worked out the DM-DE equality redshifts for our models and presented the re-
sults on Table 2. To ensure stability of the models studied, we placed constraints
on the coupling constant b2 which allowed us to constrain the effective EoS for
dark energy. The results were presented on Table 3.
Whilst making sure we preserve the dominant energy condition and that our
results are comparable to the ΛCDM model, we discovered models C through
G to be in good agreement. With these models being possible candidates of the
interaction term between dark sector components, however, model C emerged as
the best possible candidate to helping us answer the natural question: what is the
form of the interaction term between the two dark sector components? Obser-
vational tests on the studied models can make significant improvement in better
constraining and providing tighter bounds on the parameters, and proving better
solutions to the said question.
Acknowledgements
VM acknowledges funding through the National Astrophysics and Space Science
Program (NASSP) and National Research Foundation (NRF) scholarship.
REFERENCES 60
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