The early and late-time accelerated expansion of the universe

Abstract

For decades, since its discovery in the early 20th century, the idea of an expanding universe has been an incredibly active area of research among astronomers and cosmologists. This discovery means that our fundamental idea of how gravity acts fails over cosmological distances. It was previously thought that the universe was static, and so Einstein introduced the cosmological constant term (usually denoted by Λ) into his field equations for general relativity in order to counteract the effect of gravity. Einstein later abandoned this idea after Edwin Hubble's discovery of cosmic expansion in 1929 and so the value of Λ was considered to be zero by the majority of cosmologists until the 1990s. Through observations of distant type Ia supernovae, the accelerated expansion of the universe was discovered in 1998 and has led theorists to search for concepts that explain this phenomenon. In this paper I briefly review the accelerating cosmic expansion and the standard cosmological model, putting particular emphasis on the Friedmann equations and how they can be used to study the evolution of the universe. The observational constraints on dark energy are summarised and I also look at alternative models for dark energy like scalar field models (such as quintessence) and modified gravity, discussing how they work and their possible roles in our universe.

Full text

The early and late-time accelerated
expansion of the universe
Bradley Aldous∗(140172636)
School of Physics and Astronomy,
Queen Mary University of London,
Mile End Road, London, E1 4NS, United Kingdom
(Dated: March 29, 2018)
For decades, since its discovery in the early 20th century, the idea of an expanding
universe has been an incredibly active area of research among astronomers and
cosmologists. This discovery means that our fundamental idea of how gravity acts
fails over cosmological distances. It was previously thought that the universe was
static, and so Einstein introduced the cosmological constant term (usually denoted
by Λ) into his field equations for general relativity in order to counteract the effect
of gravity. Einstein later abandoned this idea after Edwin Hubble’s discovery of
cosmic expansion in 1929 and so the value of Λ was considered to be zero by the
majority of cosmologists until the 1990s. Through observations of distant type Ia
supernovae, the accelerated expansion of the universe was discovered in 1998 and
has led theorists to search for concepts that explain this phenomenon. In this paper
I briefly review the accelerating cosmic expansion and the standard cosmological
model, putting particular emphasis on the Friedmann equations and how they can
be used to study the evolution of the universe. The observational constraints on
dark energy are summarised and I also look at alternative models for dark energy
like scalar field models (such as quintessence) and modified gravity, discussing how
they work and their possible roles in our universe.
Supervisor: Dr Karim Malik
SPA7015U Physics Investigative Project
30 Credit Units
Submitted in partial fulfilment of the requirements for the degree of
MSci Astrophysics from Queen Mary University of London
∗Electronic address: b.aldous@se14.qmul.ac.uk

The early and late-time accelerated expansion of the universe
Contents
List of Figures ii
List of Tables iii
1 Introduction 1
2 The standard model of modern cosmology 2
2.1 TheFriedmannequations .................................... 2
2.1.1 Derivation of the Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Derivation of energy conservation in an expanding universe . . . . . . . . . . . . . . . . . 4
2.3 Solving the governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Time evolution of the scale factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 The critical density and density parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 The evolution of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6.1 Flatuniverse ....................................... 10
2.6.2 Openuniverse ...................................... 10
2.6.3 Closeduniverse...................................... 11
3 Dark energy: evidence and constraints 12
3.1 Evidence.............................................. 13
3.2 Constraints ............................................ 14
4 Alternative models for dark energy 15
4.1 Scalarfields............................................ 15
4.1.1 Quintessence ....................................... 15
4.1.2 K-essence ......................................... 16
4.1.3 Spintessence........................................ 17
4.2 Modifiedgravitymodels ..................................... 17
4.2.1 f(R)theories....................................... 17
4.2.2 Scalar-tensortheories .................................. 18
4.2.3 DGPgravity ....................................... 18
5 Scalar field models 18
5.1 Evolution of the scalar field density parameter . . . . . . . . . . . . . . . . . . . . . . . . 18
5.1.1 Basicequations...................................... 18
5.1.2 Derivations of evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1.3 Evolution of the scalar field density parameter . . . . . . . . . . . . . . . . . . . . 21
5.2 Evolution of the scalar field equation of state parameter . . . . . . . . . . . . . . . . . . . 22
6 Conclusion 24
Acknowledgements 24
References 24
Page i

Bradley Aldous
List of Figures
1 Evolution of the density parameters for matter, radiation and the cosmological constant. . 6
2 Evolution of the densities of matter, radiation and the cosmological constant on logarith-
micscales.............................................. 6
3 Evolution of the scale factor: variations of a flat universe. . . . . . . . . . . . . . . . . . . 8
4 Evolution of the scale factor: the Einstein-de Sitter universe. . . . . . . . . . . . . . . . . 10
5 Evolution of the scale factor: models of open universes containing matter only, and matter
anddarkenergy. ......................................... 12
6 Evolution of the scale factor: models of a closed universe containing matter only. . . . . . 13
7 Evolution of the scale factor: comparison of models for open, closed and flat universes
containingonlymatter. ..................................... 14
8 The different constraints placed on the dark energy equation of state parameter. . . . . . 15
9 Evolution of density parameters: universe containing matter, radiation and a scalar field
with varying initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10 Evolution of density parameters: universe containing matter, radiation and a scalar field
with varying λ. .......................................... 21
11 Evolution of the equation of state parameter for each scalar field model. . . . . . . . . . . 23
Page ii

The early and late-time accelerated expansion of the universe
List of Tables
1 Ages of the universe for models of open universes containing matter only, and matter and
darkenergy............................................. 11
2 Ages of the universe for models of a closed universe containing only matter. . . . . . . . . 12
Page iii

The early and late-time accelerated expansion of the universe
1 Introduction
Cosmic expansion has been an extremely active area of scientific research for the better part of a century
[1][2][3][4][5][6][7], as well as being the topic of Stephen Hawking’s PhD thesis [8], gaining particular
momentum in light of the discovery two decades ago that the expansion of the universe is accelerating.
In 1998, two separate projects studied distant type Ia supernovae - white dwarfs that are tipped over
their stability limit due to the accretion of matter from a companion star - to chart the expansion of the
universe. They found that the further the supernovae was from us, not only were they receding faster but
they were also accelerating away from us, leading to the idea of cosmic acceleration (see Caldwell’s great
review paper [9]). This discovery won Adam G. Riess and Brian P. Schmidt, the leaders of the High-Z
Supernova Team [10], and Saul Perlmutter, the head of the Supernova Cosmology Project [11], the 2011
Nobel Prize in physics. For years scientists believed that due to the attractive nature of gravity, the
expansion of the universe was decelerating. However, this discovery has shown us that our fundamental
idea of gravity breaks down over cosmological distances, prompting us to explore other ideas to explain
the concept of an accelerating expansion.
In 1917, Albert Einstein introduced the concept of a cosmological constant to help obtain a theory
describing a temporally infinite and spatially finite universe known as a static universe. Note that this
definition slightly differs from the original meaning of the term - a temporally and spatially infinite
universe, first proposed in [12]. The cosmological constant took a positive value to oppose the attractive
nature of gravity and to stop the universe from collapsing or infinitely expanding. This model for the
universe came to be known as Einstein’s static universe. However, Einstein later abandoned the idea of
a cosmological constant due to Edwin Hubble’s discovery of an expanding universe in 1929 [13]. From
then up until the aforementioned discovery of cosmic acceleration in 1998, the value of the cosmological
constant was considered by most cosmologists to be zero. In light of this discovery, the value is now
again widely considered to be positive.
Due to several developments in cosmology since the 1990s, like observations based on the Cosmic Mi-
crowave Background (CMB) [14][15][16][17][18], large galaxy redshift surveys [19][20][21][22], the Lyman-
alpha forest [23][24][19] and Baryon Acoustic Oscillations (BAO) [25][26][27][28][14][15][20], scientists
have been able to measure cosmological parameters to an unprecedented degree of accuracy like in the
Wilkinson Microwave Anisotropy Probe (WMAP) mission [29] and, more recently, the Planck Collabora-
tion [30]. These experiments, among other observations, have managed to measure the energy densities
of the constituents of our universe; finding that only around 4.9% of our universe is made up of bary-
onic matter - all of the matter we can see in galaxies and stars throughout the universe. Also, through
observing the rotational velocities of galaxies and calculating the mass of these galaxies from what is
visible, scientists deduced that there must be more matter present that what we can see due to the fact
that these galaxies should fly apart at these velocities, leading to the discovery of dark matter. The
total energy content of the universe that is made up of dark matter has been found to be roughly 26.8%
(84.5% of all mass in the universe). The remainder of the energy budget of our universe is thought to be
made up of dark energy (or a cosmological constant) - over 68%. The dark sector of the universe, dark
matter together with dark energy, makes up over a staggering 95% of the energy content of our universe,
demonstrating why these are incredibly active areas of research within the scientific community.
The cosmological constant is the simplest form of dark energy, essentially being a constant homoge-
neous energy density present everywhere throughout the universe with a value of Λ = 1.11 ×10−52m−2.
Dark energy is an unknown form of energy - described as an exotic negative-pressure fluid permeating
all space, and is the most widely accepted explanation to the accelerating expansion of the universe. The
other models of dark energy differ from the cosmological constant in that they are dynamic quantities
with time-varying energy densities, and their equation of state varies with time also, in contrast to the
cosmological constant which has a constant equation of state (w= constant). These alternative models
fall into two categories: the first being matter-based and the second being a modification of gravity. The
matter-based models focus on phenomena like scalar fields such as quintessence [31], k-essence [32][33]
or spintessence [34][35]. Whereas the second class of models focusses on modifying Einstein’s equations
of general relativity such that the effect of gravity over large distances changes to act in a manner which
matches observations of dark energy [36][37][38][39].
The plan of this paper is as follows: In Sec. 2 I review the standard cosmological model and derive
the relevant mathematical equations needed to explain and model the evolution of the universe. In Sec.
3 I summarise and discuss the evidence for, and constraints on, dark energy, such as constraints from
observations of the CMB, type Ia supernovae and BAO. In Sec. 4 I review the main alternative models
of dark energy, including scalar field models and theories of modified gravity. In Sec. 5 I discuss and
Page 1

Bradley Aldous
model scalar field theories, numerically solving systems of equations to obtain evolutions of the density
and equation of state parameters. In Sec. 6 I summarise my work and outline their implications as well
as looking at the future of this field of cosmology.
2 The standard model of modern cosmology
In this section I begin by reviewing the fundamentals of the standard cosmological model. This model
describes an evolving universe that is homogeneous and isotropic. The metric that describes this universe
is derived from the geometric properties of homogeneity and isotropy, and is termed the Friedmann-
Lemaˆıtre-Robertson-Walker (FLRW) metric, though this abbreviation is not completely consistent in
the literature, often being shortened to FRW, RW or FL - however I will refer to it as the FLRW metric
in this paper. The FLRW metric is given by
ds2=−dt2+a2(t)1
1−kr2dr2+r2dθ2+r2sin2θdφ2,(1)
where dt2is the temporal component of the metric, kis a constant representing the curvature of space, r
is sometimes called the reduced circumference, dr2,dθ2and dφ2are the spatial components of the metric
and a(t) is the cosmological scale factor. The scale factor is a term which represents the relative size of
the universe and is a function of time. It is convention to take t0as the current age of the universe and
to take a(t0) = a0= 1 as the current size of the universe. Note, in the following and throughout this
paper I have assumed natural units such that ~=c= 1.
2.1 The Friedmann equations
In cosmology, the expansion of the universe is governed by a set of evolution equations called the Fried-
mann equations, derived in 1922 by Alexander Friedmann [40]. They are found by assuming the cosmo-
logical principle, the idea that the universe is spatially homogeneous and isotropic (as mentioned above),
which implies that the universe must have a metric taking the form
ds2=a2(t)ds2
3−dt2,(2)
where ds32is just a three dimensional metric which can be one of three possible cases: a sphere of
constant positive curvature, a hyperbolic space with constant negative curvature or just flat space. By
comparison, it can be seen that Eq. (1) is of this form.
2.1.1 Derivation of the Friedmann equations
The Friedmann equations can be derived from Einstein’s field equations for general relativity, for which
the FLRW metric is an exact solution. Einstein’s field equations are a set of equations given by Einstein
in his theory of general relativity [41] which describe how mass and energy can affect the curvature of
spacetime and, in turn, affect how gravity acts. Einstein’s equations are given by
Gµν =Rµν −1
2gµν R−Λgµν = 8πGTµν ,(3)
where Gµν is the Einstein tensor (also called the trace-reversed Ricci tensor), Rµν is the Ricci curvature
tensor, gµν is the metric tensor, Ris the scalar curvature, Λ is the cosmological constant, Gis Newton’s
gravitational constant (which has a value of 6.67408×10−11m3kg−1s−2), and Tµν is the energy-momentum
tensor, which describes the density and flux of energy and momentum in spacetime, and is given by
Tµν = (ρ+P)uµuν−P gµν ,(4)
where ρis the mass-energy density, Pis the pressure and uµ/uνis the four-velocity.
We first use the FLRW metric to calculate the Christoffel symbols. These are numbers that describe
a metric connection and are used to study the geometry of a metric. It is easily seen that the metric
tensor of the FLRW metric is given by
gµν =



−1 0 0 0
0a2
1−kr20 0
0 0 a2r20
0 0 0 a2r2sin2θ




.(5)
Page 2

The early and late-time accelerated expansion of the universe
We can use the terms of the metric tensor to calculate the Christoffel symbols using the definition
Γλµν =1
2gλσ(∂µgσ ν +∂νgσµ −∂σgµν ),(6)
which gives us the following non-zero Christoffel symbols
Γtrr =a˙a
1−kr2,Γtθθ =r2a˙a, Γtφφ =r2a˙asin2θ,
Γrtr = Γrrt = Γθtθ = Γθθt = Γφ
tφ = Γφ
φt =˙a
a,
Γrrr =kr
1−kr2,Γrθθ =−r(1 −kr2),Γrφφ =−r(1 −kr2) sin2θ,
Γθrθ = Γθθr = Γφ
rφ = Γφ
φr =1
r,Γθφφ =−sin θcos θ,
Γφ
φθ = Γφ
θφ =1
tan θ,
(7)
where a dot represents the derivative with respect to time (and a double dot denotes the second derivative
with respect to time).
Next we can calculate the Ricci curvature tensor. This term allows us to measure the amount by
which a volume of the space in question differs from that of standard Euclidean space. The definition
for the Riemann curvature tensor is
Rd
abc =∂bΓdac −∂cΓdab + ΓmacΓdbm −Γmab Γdcm.(8)
However, we only need the Ricci curvature tensor, so we are only interested in the terms where the top
and bottom middle indices are the same, Rm
µmν =Rµν , thus the only non-zero Ricci tensor terms are
Rtt =−3¨a
a,
Rrr =a¨a
1−kr2+2 ˙a2
1−kr2+2k
1−kr2,
Rθθ =r2a¨a+ 2r2˙a2+ 2kr2,
Rφφ =r2a¨asin2θ+ 2r2˙a2sin2θ+ 2kr2sin2θ.
(9)
From this it is easily seen that Rµν is diagonal as there are no terms with differing indices, allowing us
to generalise the spatial part of the Ricci tensor and write it in the form
Rii =−gii
a2(a¨a+ 2 ˙a2+ 2k).(10)
The final term of the Einstein tensor left to calculate is the Ricci scalar, which again tells us the difference
in the curvature of the space we’re working with compared to that of standard Euclidean space, and is
given by
R=gµν Rµν =−6¨a
a−6˙a
a2
−6k
a2.(11)
Lastly, we can find the energy-momentum tensor for a perfect fluid, Tµν , using Eq. (4). A perfect fluid
is isotropic, thus the macroscopic speed of the fluid has only a temporal part
uµ= (1,0,0,0) (12)
giving us the following non-zero terms of the energy-momentum tensor
Ttt =ρgtt,
Tii =−P gii.(13)
Now we have calculated every term in Einstein’s equations we can begin substituting these in and
simplifying. By substituting in only the temporal components of the terms in Einstein’s equations (3),
Rtt,gtt , and ρutut, we get
Page 3

Bradley Aldous
−3¨a
a+3¨a
a+ 3 ˙a
a2
+3k
a2−Λ=8πGρ, (14)
and by simplifying and cancelling terms we obtain the first of the Friedmann equations
˙a
a2
=8πGρ
3+Λ
3−k
a2,(15)
where ρrepresents the sum of the energy densities of the constituents of our universe (matter and
radiation).
To obtain the second of the Friedmann equations we simply need to consider the spatial parts of the
above terms. The same equation is found using any of the three spatial components (r,θor φ), so we
can write it in the general form (10) found above
−gii
a2(a¨a+ 2 ˙a2+ 2k)−1
2Rgii −Λgii = 8πG(−P)gii .(16)
Dividing through by the metric, gii, and simplifying, gives us
¨a
a+1
2˙a
a2
=−4πGP +Λ
2−k
2a2.(17)
We can combine this with the first Friedmann equation (15), to cancel out the common ˙a
a2term, giving
us
¨a
a=−4πG
3(ρ+ 3P) + Λ
3,(18)
which is the second Friedmann equation. Note that there is a non-general relativistic derivation of
the Friedmann equations from Newtonian physics, however these equations were founded from general
relativity by Friedmann and so I have only included this derivation here.
2.2 Derivation of energy conservation in an expanding universe
The final governing equation needed to study the evolution of the universe is the energy conservation
equation, which relates terms of density, pressure and scale factor and can be derived from the first law
of thermodynamics, given in it’s most basic form by
∆E=Q−W, (19)
where Eis the internal energy of a system, Qis the heat and Wis the work done. This equation can be
rewritten in infinitesimal form by taking W=P dV and Q=T dS, giving us
dE +P dV =T dS, (20)
where Eis the energy, Pis the pressure, Vis the volume, Tis the temperature and Sis the entropy.
This is a thermodynamic version of energy conservation and it states that energy in an isolated system
can be converted into other forms but cannot be created or destroyed.
We can consider a sphere in our universe with a comoving radius r= 1. The comoving radius is
defined as r=ax where xis the proper distance, such that at the present time (a(t0) = 1) the ratio
between comoving and proper distances is 1, meaning the comoving distance allows us to factor out the
expansion of space. The volume of this sphere is given by
V=4π
3r3=4π
3a3x3,(21)
and the energy, E=m, where m=ρV , is
E=4π
3a3x3ρ. (22)
We can now differentiate these two equations to get expressions for the terms present in Eq. (20) as
functions of the density and scale factor
Page 4

The early and late-time accelerated expansion of the universe
dE = 4πx3a2ρ˙adt +4πx3
3a3˙ρdt, (23)
dV = 4πx3a2˙adt. (24)
All that is left to do now is to substitute these terms into Eq. (20) and cancel off all common factors,
so that finally, by assuming that the expansion is adiabatic (dS = 0), we obtain the energy conservation
equation
˙ρ+ 3 ˙a
a(ρ+P)=0.(25)
which we can use to derive the scale factor dependence of the density for the different forms of matter
in our universe.
2.3 Solving the governing equations
We know from the Friedmann equations that each form of matter has a different dependency on the
scale factor, and so they each have a dependency on time. It can be seen, from solving the Friedmann
equations, that the expansion of the universe hasn’t been uniform in time and has evolved through
different eras.
The universe began in an inflationary epoch in which the universe went through a rapid exponential
expansion that lasted until 10−32s after the Big Bang. The next stage of the evolution of the universe is
known as the radiation-dominated era, which lasted until roughly 50000 years after the Big Bang. In this
period the universe mainly consisted of relativistic particles such as photons and neutrinos, referred to
more generally as radiation. This period was followed by an era of matter domination, that lasted until
around 9.8 billion years after the Big Bang, which signified the point where the energy density of matter
exceeded that of radiation and the cosmological constant. The last of these phases is known as dark
energy domination and is the current state of our universe. In this phase, the energy densities of matter
and radiation are exceeded by an unknown negative pressure fluid called dark energy. In Figure 1 I have
shown how the density parameters for each of the constituents of our universe evolves with time, which
in turn demonstrates when the points of equality occurred as well as showing the three distinct eras
of the history of the universe (where 1a and 1b are showing two vastly different timescales). In Figure
1a, the era of radiation domination is the period to the left of the dashed line and matter domination
is shown by the period to the right of the dashed line. The dashed line corresponds to around 50000
years (the point of matter-radiation equality), which represents a redshift of roughly 3500, a time where
the universe was a fraction of its current size. The point at which the energy density of dark energy
began to dominate the dynamics of the universe is shown by the dashed line in Figure 1b, where t0is
the current age of the universe. This dashed line corresponds to roughly 9.7 billion years (4 billion years
ago) - a redshift of approximately 0.4. Figure 2 demonstrates the same idea, however it has been plotted
on logarithmic scales of the scale factor and the energy density (not the density parameter). Here the
relevant eras of domination by each species has been labelled.
We now have a complete set of governing equations, given by (15)(18)(25). In order to solve the
governing equations we first need an equation of state - a relation between the pressure, P, and the
energy density, ρ, given by
P=wρ, (26)
where wis the dimensionless equation of state parameter taking different values for different quantities.
For non-relativistic matter w= 0, and so the equation of state for matter is
P= 0.(27)
For ultra-relativistic matter (radiation), w=1
3, and so the equation of state for radiation is
P=1
3ρ. (28)
Finally, for the cosmological constant, the equation of state parameter w=−1, giving the equation of
state for the cosmological constant as
Page 5

Bradley Aldous
0.0000 0.0005 0.0010 0.0015 0.0020
t/Gyr
0.0
0.2
0.4
0.6
0.8
1.0
Ωi
Ωr
Ωm
Ωk
ΩΛ
(a)
0 5 10 15 20
t/Gyr
0.0
0.2
0.4
0.6
0.8
1.0
Ωi
t0
(b)
Figure 1: A plot showing the evolution of the density parameters for radiation (blue line), matter (red line), curvature
(green line) and the cosmological constant (orange line). Plot (a) shows this evolution in the relatively early universe - the
first 2 millions years after the Big Bang. The black dashed line signifies the point of matter-radiation equality corresponding
to roughly 50000 years (z≈3500) after the Big Bang. Plot (b) shows this evolution from the Big Bang up until 20 billion
years after. The black dashed line here represents the point at which the energy density of matter and the vacuum energy
(cosmological constant) were equal - roughly 4 billion years ago (z≈0.4). The black dotted line here represents the current
age of the universe.
10−610−510−410−310−210−1100101102
a
10−11
10−7
10−3
101
105
109
1013
1017
1021
ρi
Radiation
Matter
Dark energy
ρr
ρm
ρΛ
Figure 2: A plot showing the evolution of the densities for each species i (given by radiation: blue, matter: red, and the
cosmological constant: orange) with the scale factor. The left dashed line denotes matter-radiation equality and the right
dashed line denotes matter-cosmological constant equality. Both the density and the scale factor have been plotted on
logarithmic scales.
Page 6

The early and late-time accelerated expansion of the universe
P=−ρ. (29)
If we substitute this into the energy conservation equation, (25), we find that
˙ρ= 0,(30)
from which it can be shown, by simply integrating with respect to time, that
ρ=constant (31)
hence the name cosmological constant.
We can also solve for radiation and matter domination by simply substituting in the relevant equations
of state into (25), and so for matter domination we get
˙ρ+ 3 ˙a
aρ= 0,(32)
which can be rearranged to give
˙ρ
ρ=−3˙a
a.(33)
We can see that both quantities ρand aare in terms of the form f0
f, and from this we can simply
integrate to obtain
lnρ=−3lna+C, (34)
from which we get the following relation for matter domination
ρm=ρm,0a0
a3,(35)
and if we follow the same steps as above but for radiation domination (28), we get
ρr=ρr,0a0
a4.(36)
We can obain a general relation for the energy density and the scale factor from the first law of thermo-
dynamics, from which we find that the energy density scales as
ρ∝a−3(1+w),(37)
which we can substitute the equation of state parameters for the different species to obtain the same
above relations for the energy densities, given by Eqs. (31)(35)(36).
2.4 Time evolution of the scale factor
Now we have the dependence of the scale factor on the energy densities of matter and radiation we can
use the Friedmann equation (15) to get the time evolution of the scale factor for each of these cases,
taking k= 0 for simplicity, such that
˙a
a2
=8πG
3ρ. (38)
We can simply substitute in the relevant equations for the energy densities (35)(36) to obtain a(t) for
matter and radiation domination. So, for matter domination
˙a
a2
=8πG
3ρ0a0
a3,(39)
and it can be easily seen, by simply rearranging, that
√ada =Cdt, (40)
Page 7

Bradley Aldous
−15 −10 −5 0 5 10
t/Gyr
0.0
0.5
1.0
1.5
2.0
a
(a)
−20 −15 −10 −5 0 5 10
t/Gyr
0.0
0.5
1.0
1.5
2.0
a
Ωm,0= 0.1, ΩΛ,0= 0.9
Ωm,0= 0.3, ΩΛ,0= 0.7
Ωm,0= 0.5, ΩΛ,0= 0.5
(b)
Figure 3: A plot showing the evolution of the scale factor for flat universe models. Plot (a) shows the evolution of the
scale factor with time, based on the most up-to-date values of the density parameters taken from data obtained by the
Planck Collaboration [30] (Ωm,0= 0.3089,ΩΛ,0= 0.6911,Ωr,0= 9.24 ×10−5). The current age and size of the universe
have been denoted by the labelled dotted lines. Plot (b) shows this evolution for three different versions of a flat universe:
Ωm,0= 0.1,ΩΛ,0= 0.9 (blue line), Ωm,0= 0.3,ΩΛ,0= 0.7 (orange line) and Ωm,0= 0.5,ΩΛ,0= 0.5 (green line). The
horizontal and vertical black dotted lines correspond to the current size and age of the universe respectively.
where Cis a constant of proportionality. If we then integrate this relation and use the initial conditions
t=t0,a(t0) = a0, we can obtain the following equations for the time dependence of the scale factor and
the energy density
a=a0t
t0
2
3
,
ρm=ρm,0t
t02
.
(41)
We can also run through the same steps again for radiation domination and get
a=a0t
t0
1
2
,
ρr=ρr,0t
t02
.
(42)
From which it is seen that the energy densities of both matter and radiation have the same dependency
on time.
The scale factor and time dependences of the mass-energy densities for matter and radiation are very
useful quantities when trying to solve the Friedmann equations. Figure 3a shows the evolution of the
scale factor for the most current observational data, the Planck Collaboration [30], for a flat universe
containing matter, radiation and a cosmological constant Λ. Notice that the plot starts off in a state
of acceleration, signifying the period of inflation in the very early universe. The expansion begins to
decelerate up until roughly 9 billion years after the Big Bang (where a= 0.75), at which point the
expansion begins to accelerate again due to dark energy now having become dominant over matter.
Figure 3b shows the evolution of the scale factor for different version of a flat universe. Notice that for
greater densities of dark energy (ΩΛ,0), the late-time acceleration is faster.
2.5 The critical density and density parameter
An important quantity to consider when looking at the evolution of the universe is the critical density.
This is the density at which the universe will not collapse in on itself or continue to expand (the density
of a universe with k= 0).
We can find an equation for the critical density, ρc, by taking the Friedmann equation (15) and letting
the normalised spatial curvature, k, equal zero. By also assuming the cosmological constant to be zero
we arrive at the following definition for the critical density
Page 8

The early and late-time accelerated expansion of the universe
ρc=3H2
8πG .(43)
Today, we have ρc(t0)=1.8784h2×10−26 kgm−3by substituting in the values for Newton’s gravitational
constant and the Hubble parameter, where his a constant with a value 0.67729 (calculated using Ωmand
Ωmh2from Table 4 in [30]. The actual density of the universe is found to be incredibly close to this value
and has caused much debate among cosmologists about the so-called flatness problem - a fine-tuning
problem whereby the initial conditions of the universe appear to take very precise values such that any
alterations to these values would result in a very different universe today from the one we know.
There have been a few proposed solutions to this flatness problem. One such solution is to just take
into account the anthropic principle which states that for humans to find ourselves in such a universe
is not a surprise, as human existence is merely a product of the fact that the density of the universe is
so close to the critical value. However, this approach to the problem has been criticised by numerous
scientists [42][43], claiming it to not make any predictions about the universe and to be flawed due to its
lack of scientific form.
Another solution to the flatness problem, and by far the most widely accepted, is inflation. Inflation
is the idea, proposed by Guth in 1981 [44], that the universe expands at an exponential rate for an
incredibly short period of time in the very early universe - from 10−36s to roughly 10−33 s after the Big
Bang. The theory attempts to explain why we observe the flatness problem as well as explaining the
horizon problem - the observation that two widely separated regions of space appear to have the same
temperature despite seemingly having not come into causal contact, which is resolved when applying
inflationary theory. A third issue that inflation attempts to explain is the magnetic-monopole problem,
in which it is predicted that a huge number of heavy, stable magnetic monopoles would have formed in the
very early universe and survived until the present. However, searches for such monopoles have yet to yield
any results and a period inflation is thought to be able to resolve this problem as long as the temperature
doesn’t exceed the minimum needed for the production of these magnetic monopoles. Inflationary theory
takes away the dependence on the initial conditions mentioned above and thus succeeds in explaining
the flatness problem.
The critical density is a useful reference density but a more useful parameter to use when looking at
the evolution of the universe is the density parameter, denoted by Ω, which is simply the ratio of the
actual density to the critical density, given by
Ω(t) = ρ
ρc
=actual density
critical density,(44)
and is a good indicator of how far from spatially flat the universe is.
We can apply this relation to the individual components of the content of the universe to obtain
density parameters for matter and radiation
Ωm=ρm
ρc
,Ωr=ρr
ρc
,(45)
Ωt= Ωr+ Ωm,(46)
as well as for the cosmological constant
ΩΛ=ρΛ
ρc
=Λ
3H2.(47)
The Planck Collaboration [30] has constrained the value of the cosmological constant, in the form of the
dark energy density parameter, to ΩΛ,0= 0.6911 ±0.0062. Other useful cosmological parameters are the
matter density parameter, Ωm,0= 0.3089 ±0.0062, which can be broken down into the baryon density
parameter, Ωb,0= 0.0486 ±0.0010, and the cold dark matter density parameter, Ωc,0= 0.2589 ±0.0057.
The radiation density parameter has a current value of roughly Ωr,0= 9.24 ×10−5, however this value is
negligible. The values from the Planck Collaboration are the most up-to-date constraints on cosmological
parameters and so are the values that I will use throughout this paper (the subscript 0 denotes the present
day value).
Page 9

Bradley Aldous
−10 −5 0 5 10
t/Gyr
0.0
0.5
1.0
1.5
2.0
a
(a)
−15 −10 −5 0 5 10
t/Gyr
0.0
0.5
1.0
1.5
2.0
a
(b)
Figure 4: A plot showing the evolution of the scale factor for the Einstein-de Sitter universe. Plot (a) shows the evolution
of the scale factor in the Einstein-de Sitter universe, described by the following values of density parameters and curvature:
Ωm,0= Ωt,0= 1 and k= 0. Plot (b) shows the evolution of the scale factor for two scenarios: the Einstein-de Sitter
universe described above (blue line), and the best fit to current observational data found in [30](orange line). The horizontal
and vertical black dotted lines correspond to the current size and age of the universe respectively.
2.6 The evolution of the universe
The Friedmann equation (15) can be rewritten in terms of the present day density parameters
H2
H2
0
=Ωr,0
a4+Ωm,0
a3+Ωk,0
a2+ ΩΛ,0,(48)
where H0is the Hubble parameter today and the Ω0terms are the density parameters of each constituent
of the universe today. It is this equation that I solve numerically, using data from [30], to obtain plots
of the evolution of the universe. In the following I summarise the different models for the universe and
show how the scale factor evolves with time in each case. The three cases that I consider are: the flat
universe (touching briefly on the Einstein-de Sitter model), the open universe and the closed universe.
Each case corresponds to a different type of curvature.
2.6.1 Flat universe
The simplest of the three cases is the flat universe. This model is described by
Ωt,0= Ωm,0+ Ωr,0+ ΩΛ,0= 1.(49)
and is the closest to current observational data (Ωt,0≈1) out of the three models.
There is a specific (and simpler) variant of the flat universe in which the universe is still spatially flat
but consists of matter with a vanishing cosmological constant and radiation term ΩΛ,0= Ωr,0= 0). This
model was first put forward in 1932 [45] by Albert Einstein and William de Sitter and is described by
Ωt,0= Ωm,0= 1.(50)
Figure 4a shows the evolution of the scale factor with time for the Einstein-de Sitter universe. The age
of the Einstein-de Sitter universe is t0= 9.187 billion years. The end point of an Einstein-de Sitter
universe is known as the ”Big Freeze” and describes a fate in which the universe expands forever at an
increasingly slower rate, asymptotically approaching zero. In Figure 4a, the universe starts off in a state
of acceleration like Figure 3, however once the universe begins decelerating there’s nothing present in the
universe to act against the attraction of gravity, like dark energy, and so the universe keeps expanding
at a continually slower rate.
2.6.2 Open universe
Another potential trajectory that the universe could follow is the one of an open universe model which
has hyperbolic geometry - like a saddle, resulting in a curvature of space in this model that ”curves away
Page 10

The early and late-time accelerated expansion of the universe
from itself”, such that an observer travelling in a straight line would never reach its starting position
again. An open universe is most simply described by
Ωt,0+ ΩΛ,0<1.(51)
In a negatively curved universe without dark energy, k < 0, space still expands with little effect from the
attractive nature of gravity and so, in this case, the universe will end in a Big Freeze. With the addition
of dark energy this expansion accelerates and here, the universe is still not dense enough in order for
gravity to overcome the repellent effects of dark energy and so the universe will expand forever. Stars
will lose their energy as the universe turns cold and dark, and eventually all bound objects will be torn
apart as dark energy overcomes the fundamental forces of nature in a scenario known as the Big Rip,
discussed by Caldwell et al. [46] and Schewe [47]. Figure 5 shows the evolution of the scale factor for an
open universe containing only matter (5a) and an open universe containing 30% matter and 70% dark
energy (5b) with varying curvature. The main difference between these two plots is the present day age
of these universes (the time after the Big Bang at which a= 1). For Figure 5a these ages run from
roughly 9 billion years (Ω0= 0.99) up to 12.5 billion years (Ω0= 0.1), in contrast to the range found
in Figure 5b, which goes from around 13.5 billion years (Ω0= 0.99) all the way up to 42 billion years
(Ω0= 0.1). The difference in the matter and dark energy case is over 8 times greater than the difference
in the matter only case. All of these values for the density parameters in both cases are plotted in Table
1.
Ω0t0/Gyr
0.1 12.3772972365
0.3 11.1461044903
0.5 10.3847820652
0.7 9.82422671986
0.9 9.37952716217
0.99 9.20589013914
(a)
Ω0t0/Gyr
0.1 41.6877
0.3 24.1484
0.5 18.7201
0.7 15.8308
0.9 13.9922
0.99 13.3355
(b)
Table 1: A table showing the age of the universe for each of the cases of an open universe plotted in Figure 5. Table
1a shows the values found for the universe containing only matter and Table 1b shows the values found for the universe
containing matter and dark energy.
Each line in Figure 5a follows a very similar trajectory to the Einstein-de Sitter case in Figure 4a;
acceleration followed by continual deceleration. Whereas the lines in Figure 5b follow a similar trajectory
to the one found in Figure 3, with the line for Ω0= 0.99 following this almost exactly as these case are
nearly identical.
2.6.3 Closed universe
The third possible case for the geometry of the universe is one in which the curvature is positive (k > 0),
meaning the universe is like the surface of a sphere and has a finite size such that an observer travelling
in a straight line would eventually reach its starting position. A closed universe is most simply described
by
Ωt,0+ ΩΛ,0>1.(52)
In this model, the universe is dense enough to stop expansion and allow gravity to collapse the universe
into what is called a Big Crunch [48]. This collapse would start off at an even rate due to the homogeneity
of the universe on large scales, however as the universe reduces in size this collapse becomes more chaotic
due to the fluctuations in the density being more pronounced on these smaller scales. Figure 6 shows the
evolution of the scale factor in a closed universe with varying values of the matter density parameter.
7a shows this plot up to 1500 billion years after the Big Bang to show the eventual end point of a
closed universe depending on the matter content and 7b shows the same plot but focussed in on up to
12 billion years after the Big Bang. From this it can be seen that closed universes, with Ωt,0>1, have
a much shorter lifetime, as in each of these cases the value for the present day age of the universe, t0,
is less than roughly 9 billion years, which is significantly different to the most current measurements,
t0= 13.799 ±0.021 billion years.
Page 11

Bradley Aldous
−10 0 10 20 30
t/Gyr
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a
Ω0= 0.1
Ω0= 0.3
Ω0= 0.5
Ω0= 0.7
Ω0= 0.9
Ω0= 0.99
(a)
0 10 20 30 40 50
t/Gyr
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a
(b)
Figure 5: A plot showing the evolution of the scale factor for models of an open universe, described by Ω0<1 and k < 0.
Here I have plotted for values of 0.1≤Ω0≤0.99. Plot (a) shows this evolution for models of an open universe containing
matter only. Plot (b) shows this evolution for models of an open universe containing matter and a cosmological constant
(with Ωm,0= 0.3Ω0and ΩΛ,0= 0.7Ω0). Note that in plot (a) I have used a timescale such that the zero point corresponds
to the current age of the universe (a= 1). The horizontal and vertical black dotted lines in plot (a) correspond to the
current size and age of the universe respectively. These models are merely here to show the properties of the geometry of
an open universe model - they are, for the most part, in strong disagreement with current observational data.
Ω0t0/Gyr
1.1 9.01115084854
1.15 8.92818282141
1.2 8.84833006354
1.3 8.69712264328
1.5 8.4238579855
2.0 7.86623214832
Table 2: A table showing the age of the universe for each of the cases of a closed universe containing only matter plotted
in Figure 7.
There is a version of this model where the universe follows a cyclic trajectory; immediately after the
universe collapses another Big Bang creates a new universe in what is called a Big Bounce. This process
then potentially repeats for an infinite number of times.
Figure 7 shows the evolution of the scale factor in all three of these cases, with Figure 7a focussing on
large timescales (up to 500 billion years after the Big Bang) and Figure 7b focussing on small timescales
(up to 27 billion years after the Big Bang). From this you can easily compare these models and see how
they differ in terms of their fate, age and evolution.
3 Dark energy: evidence and constraints
Dark energy is described as an exotic form of matter (fluid) with negative pressure permeating the
universe and is our best theory for explaining the accelerating expansion of the universe. However, we
still don’t know the fundamental form of dark energy and there are a number of different concepts that
have been put forward by scientists over the years as candidates for dark energy. The basic properties
of dark energy are that it acts as a type of anti-gravity, prompting cosmic acceleration.
For cosmic acceleration we require an expansion such that the scale factor a grows faster than the
time t. We can take Eq. (37) and substitute it into the Friedmann equation (15), such that
H∝˙a
a∝ρ∝a−3(1+w),(53)
from which it can be seen that, in order for cosmic acceleration to occur, we need
w < −1
3,(54)
Page 12

The early and late-time accelerated expansion of the universe
0 250 500 750 1000 1250 1500
t/Gyr
0
2
4
6
8
10
12
14
a
(a)
−10 −5 0 5 10
t/Gyr
0.00
0.25
0.50
0.75
1.00
1.25
1.50
a
Ω0= 1.1
Ω0= 1.15
Ω0= 1.2
Ω0= 1.3
Ω0= 1.5
Ω0= 2.0
(b)
Figure 6: A plot showing the evolution of the scale factor for models of a closed universe containing matter only, described
by Ω0>1 and k > 0. Here I have plotted for values of 1.1≤Ω0≤2. Plot (a) shows this evolution over a huge timescale
(up to 1500 billion years after the Big Bang). Plot (b) shows this evolution over a timescale that is comparable to the age
of our universe (up to roughly 20 billion years after the Big Bang). Note that in plot (b) I have used a timescale such
that the zero point corresponds to the current age of the universe (a= 1). The horizontal and vertical black dotted lines
in plot (b) correspond to the current size and age of the universe respectively. These models are merely here to show the
properties of the geometry of a closed universe model - they are, for the most part, in strong disagreement with current
observational data.
corresponding to a negative pressure of P < −1
3ρ.
In this section I summarise the evidence for the existence of dark energy and I briefly discuss some
of the methods and observables that have been used to constrain models of dark energy.
3.1 Evidence
The first hints at the idea of dark energy were made when Einstein introduced the cosmological constant
term into his field equations (3) in order to obtain equations describing a static universe. An idea he
later abandoned upon the discovery of an expanding universe. A similar concept was assumed in the
first inflationary models; it was thought that a negative pressure field could be the cause for inflation in
the very early universe. There are three main pieces of evidence for dark energy.
The first is from observations of distant type Ia supernovae made by two independent projects in 1998
[10][11]. These pro jects used supernovae as standard candles to measure their distance from the observer
and their redshift. From these two measurements they were able to find how fast each supernova was
receding from us and found that the further away from the observer the supernova was, the faster it was
moving away. This meant that not only was the universe expanding, but the expansion was accelerating,
implying that there must be some form of matter or intrinsic property of space that’s causing it.
The second piece of evidence for dark energy comes from observations of CMB anisotropies which
tell us that the geometry of the universe is extremely close to flat (Ω0'1). Also observed from the
CMB spectrum is the total density of matter which tells us that around 30% of the energy content is
made up of matter (baryonic matter and dark matter), meaning that there is 70% of the energy budget
of the universe that is unaccounted for, implying that there is some other form of energy making the
rest up. The WMAP mission [29] has measured these values, and more recently measured by the Planck
Collaboration [30] to a very high degree of accuracy. This has also been suggested by the theory of
large-scale structure.
The third independent piece of evidence for dark energy was made by the WiggleZ survey and follows
a very similar method to the type Ia supernovae method described above, whereby they measured the
redshift of distant galaxies and used voids left by BAO as standard rulers to measure the distance, and
from these measurements they were able to calculate the recession speed of these galaxies, confirming
cosmic acceleration. This obviously also implies the existence of dark energy.
Page 13

Bradley Aldous
0 100 200 300 400 500
t/Gyr
0
5
10
15
20
25
30
a
Ω0= 0.1
Ω0= 0.5
Ω0= 1.0
Ω0= 1.2
Ω0= 1.3
Ω0= 1.5
Ω0= 2.0
(a)
−15 −10 −5 0 5 10 15
t/Gyr
0.0
0.5
1.0
1.5
2.0
a
Ω0= 0.1
Ω0= 0.5
Ω0= 1.0
Ω0= 1.5
Ω0= 2.0
(b)
Figure 7: A plot showing the evolution of the scale factor for models of open, closed and flat universes containing only
matter for 0.1≤Ω0≤2. Plot (a) shows this evolution over a huge timescale (up to 500 billion years after the Big Bang).
Plot (b) shows this evolution over a timescale that is comparable to the age of our universe (up to roughly 30 billion years
after the Big Bang). The horizontal and vertical black dotted lines in plot (b) correspond to the current size and age of
the universe respectively.
3.2 Constraints
There is a number of different cosmological observables that we can study in order to constrain cosmo-
logical parameters, and thus constrain parameters relating to dark energy. Here I briefly touch on some
of the properties of our universe that we can observe and measure.
The most important property of the universe that we can observe is the CMB, this can be seen with
radio telescopes as a faint glow. The CMB formed during the epoch of recombination (around 380000
years after the Big Bang), whereby the universe had cooled down to a temperature of roughly 2700◦C
which is sufficiently cool enough for electrons and protons to recombine and release photons - this is
the CMB that we can observe today. The glow of the CMB is brightest in the microwave region of
the electromagnetic spectrum and there have been numerous missions to probe the microwave region in
order to obtain constraints on cosmological parameters. The first of these experiments was the NASA
Cosmic Background Explorer (COBE) satellite [49], which was put into sun-synchronous orbit in 1989
and ran until 1993. The measurements made by the satellite provided us with two important pieces of
evidence: the first being that the CMB has a near-perfect black body spectrum (and remains the most
perfect black-body spectrum was have observed to date), and the second being that the CMB has faint
anisotropies - both of which support the Big Bang theory. In 2001, the WMAP mission [29] was launched
by NASA to measure these anisotropies to a much greater precision than was achieved by the COBE
mission. The data obtained by the WMAP was important in the formation of the Λ-CDM model of the
universe (the standard cosmological model). This mission was immediately followed by the ESA Planck
Surveyor [30], which ran from 2009 to 2013. This project also measured the temperature anisotropies
of the CMB, the same as the WMAP, and has given us the most precise measurements of cosmological
parameters to date.
Another of the most important cosmological observables that we can use to study and constrain dark
energy are BAO [25]. These are periodic fluctuations in the baryonic matter in the universe and we can
use their clustering as a standard ruler for a cosmological scale of measuring distance. From using BAO
we can obtain measurements of the Hubble parameter and the angular diameter distance from their
radial and tangential directions - these measurements can provide insights into the nature of dark energy
and allow us to constrain certain related cosmological parameters. BAO are only very subtle though,
making the overdensities very difficult to see in a single galaxy separation and so in-depth methods have
been adopted by numerous projects mapping large 3D distributions of galaxies including: the WiggleZ
dark energy survey [51][52], the Sloan Digital Sky Survey [27][53][54][55][28] and the 2 degree field galaxy
redshift survey (2dFGRS) [56][53].
Page 14

The early and late-time accelerated expansion of the universe
(a)
(b)
Figure 8: A plot showing the different constraints placed on the dark energy equation of state parameter. Plot (a) shows
constraints made on the dark energy equation of state parameter and the density parameter of matter from BAO (green
region), CMB (orange region) and type Ia supernovae (SNe, blue region) - the dotted lines represent the 68.3%, 95.4% and
99.7% confidence-level contours. Taken from Ref. [50]. Plot (b) shows the constraints made on the dark energy equation of
state parameter using the first order Taylor series expansion with the scale factor a, given by: w=w0+ (1 −a)wa. Taken
from the Planck Collaboration [30], where BSH refers to the combination of BAO, SNe and Hubble constant constraints,
WL refers to the weak lensing constraints and RSD refers to the redshift space distortions constraints, all of which are
described in greater detail in [30].
4 Alternative models for dark energy
The simplest model for dark energy is the cosmological constant, Λ, with a constant equation of state
parameter, w=−1. It is thought that the cosmological constant comes from the vacuum energy in
quantum field theory, which predicts the value of this energy to be very large. However there is a huge
discrepancy, as much as 120 orders of magnitude [57], between this and the observed energy scale of
dark energy (the gravitational effects that the vacuum energy should produce are not observed) - this
disagreement is known as the cosmological constant problem [58].
These issues have led scientists to explore other theories for dark energy in a bid to explain the
cosmological constant problem as well as presenting a more viable candidate for solving the cosmic-
acceleration puzzle. In general, these alternative theories for dark energy fall into two classes: the first
being modifications to the right hand side of Einstein’s equations, and the second being modifications
to the right hand side of these equations. The first case corresponds to modified matter models like
quintessence, in which the right hand side is modified by the inclusion of a scalar field term. The
second case corresponds to modified gravity models like f(R) theories, in which the left hand side of the
equations is modified by introducing higher orders curvature into the action. There is also a class of dark
energy models known as phantoms [59][46] which refer to models for dark energy in which the equation
of state parameter is ’super-negative’, w < −1. In this section I review a number of different alternative
theories for dark energy.
4.1 Scalar fields
Most of the main types of alternative theories for dark energy involve scalar fields, which associate a
scalar value to every point in space. Scalar fields have been used to explain other phenomena in physics
such as inflation, which is proposed to be driven by a hypothetical field called the inflaton [60]. Here, I
discuss some of the ways scalar fields have been utilised in explaining the cosmic acceleration.
4.1.1 Quintessence
The simplest of these models of dark energy, and the most popular alternative for the cosmological
constant, is quintessence. Quintessence is described by a canonical scalar field, φ(t), slowly rolling down
Page 15

Bradley Aldous
its potential, V(φ), leading to an accelerating expansion. The total gravitational action for a quintessence
scalar field is given by
S=Zd4x√−g1
2M2
plR−1
2gµν ∂µφ∂νφ−V(φ)+Sm,(55)
where gis the determinant of the metric gµν,Mpl =√8πG is the reduced Planck mass, Ris the Ricci
scalar, gµν is the inverse metric and Smis the matter action. Quintessence is described as minimally
coupled to gravity as there are no terms present in the action that directly connect the scalar field to
gravity.
The idea of a self-interacting, rolling, homogeneous scalar field having an effect on the evolution
of the universe was first introduced by Ratra and Peebles in 1988 [61]. Significant work was done on
this concept by Caldwell et al. in 1998 [62] who first termed it ”quintessence”. There has also been a
huge amount of work done on this concept since then [63][64][65][66][67][68][69]. This differs from the
cosmological constant in that the value of its density and equation of state parameter change with time.
The equation of state for quintessence is given by
wφ=Pφ
ρφ
=
1
2˙
φ2−V(φ)
1
2˙
φ2+V(φ),(56)
where wφ,Pφand ρφare the equation of state parameter, pressure and energy density of quintessence
respectively. In the above 1
2˙
φ2is the kinetic term and V(φ) is the potential term, and so, dependent on
the ratio of these two terms, quintessence has the ability to be attractive or repulsive.
We can obtain the equation of motion for the quintessence scalar field by substituting the equations
for Pφand ρφ, as they appear in (56), into Eq. (25), giving us
¨
φ+ 3H˙
φ+V,φ= 0 (57)
where the ,φdenotes a derivative with respect to φ. The 3H˙
φterm serves as a Hubble-friction term,
preventing the scalar field from rolling straight down to the minimum. From Sec. 3 we know that in
order for cosmic acceleration to occur we need an equation of state parameter of
wφ<−1
3.(58)
This can be obtained through the approximation
˙
φ2< V (φ),(59)
which we can substitute into the equations for Pφand ρφin Eq. (56), giving us
wφ<
1
2˙
φ2−˙
φ2
1
2˙
φ2+˙
φ2=−1
2˙
φ2
3
2˙
φ2=−1
3,(60)
which is the condition for cosmic acceleration, Eq. (58).
4.1.2 K-essence
A popular class of models for dark energy are specific modified versions of quintessence called k-essence.
K-essence models are some of the more general of the dark energy scalar field theories. In these models the
kinetic energy terms are non-canonical and it is thought that the accelerated expansion of the universe
could arise from modifications to these kinetic terms. The idea of a scalar field with non-canonical
kinetic terms was first brought forward as a model for inflation called k-Inflation by Armendariz-Picon
et al. in 1999 [70], however this concept wasn’t applied to dark energy until 2000 by Chiba et al. [71],
with significant work being done again on this theory by Armendariz-Picon et al. in 2000 [72]. The
Lagrangian, Lk(φ, X), for these models is made up of non-linear functions of the scalar field φand
kinetic terms, denoted by
X≡ −1
2(∇µφ)(∇µφ).(61)
There is also a branch of these models called (purely) kinetic k-essence [73][74][75], in which the La-
grangian has only kinetic terms, L=Lk(X). The general action for k-essence is of the form
Page 16

The early and late-time accelerated expansion of the universe
S=Zd4x√−gR
2+Lk(φ, X)+Sm=Zd4x√−gR
2+F(φ, X)+Sm,(62)
where Fis an arbitrary function of φand X. It follows from this that the action for kinetic k-essence is
of the form
S=Zd4x√−gR
2+Lk(X)+Sm.(63)
For normal k-essence theories, with the action given by Eq. (62), the pressure and density of the scalar
field are given by Pφ=Fand ρφ= 2XF,X −Frespectively. Thus, the equation of state parameter for
k-essence theories is given by
w=Pφ
ρφ
=F
2XF,X −F.(64)
So if |2XF,X ||F|is satisfied, the value of the equation of state parameter can be close to −1, in
agreement with current observations of w.
4.1.3 Spintessence
Spintessence [34][35] refers to another class of alternative theories for dark energy based on scalar fields,
and, like k-essence, is also a variant of quintessence. Here, the scalar field is complex with a U(1)
symmetry and is made up of two real fields. The field is spinning in a U(1)-symmetric potential V=
V(R), and is described by
φ(x, t) = φ1(x, t) + iφ2(x, t)≡R(x, t)exp[iΘ(x, t)],(65)
where φ1and φ2represent the two real scalar fields and Θ is the spin frequency. If the spin frequency is
large enough, such that ˙
ΘH, then its the rotation, and not the expansion rate H, that’s stopping the
field from rolling instantly down to its minimum. However there are issues associated with spintessence
models.
It has been shown that spinning complex scalar fields for spintessence potentials that result in cosmic
acceleration deform almost completely into non-topological soliton states known as Q-balls [76], such
that the equation of state eventually resembles that of matter or radiation. Thus making it difficult to
find workable spintessence models that explain the accelerated expansion.
There has also been work done on the concept of a scalar field oscillating in a potential of the
form V(φ) = a|φ|, termed oscillescence, where ais a constant. This idea was first introduced by M.
Turner in 1983 [77] where it was applied to the inflationary universe scenario - the energy produced
by the oscillations was proposed to contribute significantly to the energy density of the universe. In
these models the scalar field is thought to act like a fluid with equation of state wφ= (n−2)/(n+ 2),
giving wφ<−1/3 for n < 1. However, it has been argued [78] that oscillating fields with w < 0
are dynamically unstable to small-scale perturbations, making these models unsuitable candidates for
explaining the accelerated expansion.
4.2 Modified gravity models
Aside from scalar field models, which correspond to a modification of the energy-momentum tensor in
Einstein’s equations (3), there are models based on the idea of modifying the left-hand side (Gµν ) such
that the effect of gravity changes. These models are referred to as modified gravity [36][37][38][39] and
are discussed in the following.
Note that the Einstein-Hilbert action is given by
S=1
16πG Zd4x√−gR. (66)
4.2.1 f(R)theories
A popular class of alternative-gravity theories are f(R) theories [79][80][81][82], which refer to models that
include fas an arbitrary function of the Ricci scalar R[83] (see Ref. [84] for an extremely detailed review
Page 17

Bradley Aldous
of f(R) theories and Ref. [85] for very recent constraints based on gravitational wave observations). This
idea was first introduced by Buchdahl in 1970 [86]. The action for f(R) theories is given by
S=1
16πG Zd4x√−gf (R) + Sm,(67)
where f(R) is a function of Rwhich defines the specific theory. Note that the value for Newton’s
gravitational constant, G, in these models may differ from our measured value. Here we can see that the
Einstein Hilbert action (66) is just the simplest form of the f(R) action, when f(R) = R.
4.2.2 Scalar-tensor theories
Some of the most popular modified gravity theories are the scalar-tensor theories [87][88][89][90][91],
which have been constrained in light of the recent gravitational wave observations from the neutron star
merger GW170817 [85][92][93]. These theories can be most easily described as models of gravity with a
variable gravitational constant. The action for such theories is given by
S=Zd4x√−gb(λ)R−1
2h(λ)gµν (∂µλ)(∂νλ)−U(λ) + LM(gµν , ψi),(68)
where λ(~x, t) is the scalar field, b(λ), h(λ) and U(λ) are functions of the scalar field that define the
theory, and LM(gµν , ψi) is the matter Lagrangian, where ψirefers to the matter fields. We can see that
in this action that Ris being multiplied by the b(λ) term, which varies in space, implying that Newton’s
gravitational constant is dependent on this term in these theories.
4.2.3 DGP gravity
The alternative-gravity models discussed so far are based on making a modification to GR, introducing
a new scalar degree of freedom. However, there are more exotic models involving higher-dimensional
spacetimes known as braneworld scenarios. The Dvali-Gabadadze-Porrati (DGP) gravity (introduced in
2000 [94]) is one such model, assuming the existence of a 4+1 dimensional Minkowski space embedded
with ordinary 3+1 Minkowski space. The action for DGP gravity is given by
S(5) =Zd5x√−gR
16πG(5) +Zd4xp−g(4) R(4)
16πG +LSM ,(69)
where G(5) is the 5-d gravitational constant, gis the 5-d determinant of the metric gµν and g(4) is the
induced determinant of the metric gµν on the brane. Here, the second term is the standard Einstein-
Hilbert action (66) and the first term is this same action extended to five dimensions.
Braneworld models are very interesting as candidates for explaining cosmic acceleration, but they only
very loosely agree with current observations as well as having numerous other complications [95][96][97].
However they are still important research areas, relating cosmology with areas of other fields of physics.
5 Scalar field models
5.1 Evolution of the scalar field density parameter
5.1.1 Basic equations
In order to show how to derive the necessary equations needed to model scalar fields like quintessence
we take the starting equations adopted by Copeland et al. in [98] for their phase-plane analysis. Here
they define
x≡κ˙
φ
H√6,(70)
y≡κ√V
H√3,(71)
z≡κ√ρr
H√3.(72)
We also take the following evolution equations of the energy density and scalar field
Page 18

The early and late-time accelerated expansion of the universe
˙ρi=−3H(ρi+Pi),(73)
¨
φ=−3H˙
φ−V,φ,(74)
which are just rearranged versions of Eqs. (25) and (57) respectively, where idenotes the species -
either matter or radiation. A third evolution equation we need is that of the Hubble parameter, which
is derived from Eq. (15), given here as
H2=κ2
3(ρφ+ρi),(75)
where the only differences are that we have replaced the ρΛ(cosmological constant energy density) term
with the ρφ(scalar field energy density) term and we have let κ2≡8πG. We can rearrange this and
differentiate with respect to time to give us
d
dt(3H2)=6H˙
H=κ2( ˙ρφ+ ˙ρi),(76)
which can be rearranged further, after substituting in the relevant terms, to give us
˙
H=−κ2
2 3H2
κ2+˙
φ2
2+1
3ρr−V!(77)
In this model, I use a scalar potential of the exponential form, given by
V(φ) = V=V0exp(−λκφ).(78)
5.1.2 Derivations of evolution equations
Now that we have the basic equations, we can begin to derive the necessary equations for modelling a
scalar field. We start off by differentiating Eq. (70) with respect to N≡lna(e-foldings) using the chain
rule, which gives us
x0≡dx
dN =da
dN
dt
da
d
dt κ˙
φ
H√6!
=κ
h√6"1
H
d
dt ˙
φ−˙
φ
H2˙
H#.
(79)
We can then substitute in Eqs. (74) and (77), as well as the derivative of (78) with respect to φ- which
is simply V,φ=−λκV , giving us
x0=−κ
H√6"3˙
φ−λκV
H+˙
φ
H2 −κ2
2 3H2
κ2+˙
φ2
2+1
3ρr−V!!#,(80)
which simplifies down to
x0=−3x+λy2r3
2+3
2x(x2−y2+ 2z2+ 1).(81)
We can approach deriving the same equation for yusing a similar method, the only differences being
that after differentiating ywith respect to N, we get
y0=κ
H√31
H
d
dt(√V) + √Vd
dt 1
H,(82)
meaning now we have to substitute in V1
2,t=−λκ ˙
φ
2V1
2(where ,tdenotes a derivative with respect to
time t), and
d
dt 1
H=−˙
H
H2,(83)
to get
Page 19

Bradley Aldous
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
1.0
Ω
Ωφ
Ωr
Ωm
(a)
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
1.0
Ω
(b)
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
1.0
Ω
(c)
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
1.0
Ω
(d)
Figure 9: A plot showing the evolution of the density parameters for a universe containing matter, radiation and a scalar
field with varying initial conditions and λ= 1. Plot (a) shows this evolution for x0= 0.6, y0= 0.6 and z0= 0.2. Plot
(b) shows this evolution for x0= 0.4, y0= 0.4 and z0= 0.4. Plot (c) shows this evolution for x0= 0.2, y0= 0.2 and
z0= 0.6. Plot (d) shows this evolution for x0= 0.01, y0= 0.01 and z0= 0.8. Note that plot (d) is the only plot which
has a domination sequence that is the same as the one for our universe (radiation-matter-dark energy).
y0=κ
H√3"−1
H
λκ ˙
φ√V
2+κ2√V
2 3H2
κ2+˙
φ2
2+1
3ρr−V!#,(84)
which simplifies down to
y0=−λxyr3
2+r3
2y(x2−y2+1
2z2+ 1).(85)
Lastly, the derivation of z0is the simplest one of the three, following the same method as the x0derivation,
except after differentiating zwith respect to N, giving us
z0=κ
H1
H−4Hρr
2√2ρr+rρr
3−1
H2˙
H,(86)
we don’t need to substitute in some derivative of the potential, only ˙
H, which gives us
z0=−2κ
Hrρr
3+κ
Hrρr
3
κ2
2H2 3H2
κ2+˙
φ2
2−V+1
3ρr!,(87)
simplifying down to
Page 20

The early and late-time accelerated expansion of the universe
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
1.0
Ω
Ωφ
Ωr
Ωm
(a)
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
Ω
(b)
0246810
ln(a)
0.0
0.2
0.4
0.6
Ω
(c)
0246810
ln(a)
0.0
0.2
0.4
0.6
0.8
Ω
(d)
Figure 10: A plot showing the evolution of the density parameters for a universe containing matter (blue line), radiation
(orange line) and a scalar field (red line) with varying λand initial conditions: x0=y0= 0.01 and z0= 0.8. Plot (a)
shows this evolution for λ= 1 and plot (b) shows this evolution for λ= 2. Note that in both of these cases the scalar field
ends up dominating in late time. Plot (c) shows this evolution for λ= 3 and plot (d) shows this evolution for λ= 4. Note
that in both of these cases the scalar field never comes to dominate in late time - instead we have matter domination.
z0=z
2(3x2−3y2+z2−1).(88)
5.1.3 Evolution of the scalar field density parameter
We now have a system of coupled non-linear ordinary differential equations (ODEs), given by (81)(85)(88),
which we can solve numerically to obtain models of an evolving universe containing matter, radiation
and a scalar field. It can be seen that
z2=κ2ρr
3H2=8πGρr
3H2=ρr
ρc
= Ωr.(89)
Using this and the Friedmann constraint
H2=κ2
3 ρr+˙
φ2
2+V!,(90)
we can obtain an equation relating the density parameter of the scalar field to the variables xand y
Ωφ=κ2
3 ˙
φ2
2+V!=x2+y2,(91)
Page 21

Bradley Aldous
where xand yare the kinetic energy and potential energy terms of the scalar field respectively. Thus, if
we assume a flat universe, such that Ω = 1, we can see that the density parameter of matter is given by
Ωm= 1 −Ωφ−Ωr= 1 −x2−y2−z2.(92)
In this system of ODEs there is one free parameter in λas well as our choice of initial conditions in x0,
y0and z0.
Figure 9 shows the evolution of the density parameters of matter, radiation and the scalar field for a
variety of initial conditions with a fixed value for λ(= 1). It is seen that we need a large value for z0and
small values for x0and y0in order to obtain a plot resembling our universe. Figure 9d is the only plot in
which the domination sequence follows the order of radiation domination, matter domination and then
dark energy (scalar field) domination. For this reason we now fix the initial conditions at these values
and vary λ.
Figure 10 shows how the density parameters evolve for 1 <λ<4 with these fixed initial conditions.
From Figures 10a and 10b we can see that in late time dark energy domination is achieved in both cases,
however in Figures 10c and 10d it is shown that, in late time, dark energy domination is never achieved,
and that matter dominates for the rest of the universe. In fact there is a critical value of λ= 2.45, at
which the densities of matter and dark energy stay roughly equal for late time, and so any deviation from
this value causes matter or dark energy domination, dependent on which side of this critical value λis.
For λ < 2.45 we get dark energy domination for late time, and for λ > 2.45 we get matter domination.
5.2 Evolution of the scalar field equation of state parameter
Studying the evolution of the scalar field equation of state parameter allows us to understand the different
categories into which scalar field models can fall into. The equations needed to numerically solve in order
to obtain plots of the equation of state parameter evolution are stated in [99] (an excellent review paper
on quintessence by Tsujikawa) and are given by a system of coupled non-linear ODEs
w0= (w−1)[(3(1 + w)−λq3(1 + w)Ωφ],(93)
Ω0
φ=−3(w−wm)Ωφ(1 −Ωφ),(94)
λ0=−q3(1 + w)Ωφ(Γ −1)λ2,(95)
with
Γ≡V V,φφ
V,2
φ
,(96)
where the ,φφ denotes a second derivative with respect to φ.
Tsujikawa in [99] mentions three different cases of scalar field models characterised by their potentials.
The first case corresponds to the inverse power law potential
V(φ) = M5
φ,(97)
which corresponds to the case of a tracker field [100][101][9]. Tracker fields are described as quintessence
fields which closely track the radiation energy density until the point of matter-radiation equality,
whereby the quintessence field begins to act like dark energy, later leading to dark-energy domina-
tion. It has been argued [100] that this provides a partial solution to the cosmological constant problem.
This case is represented by Figures 11a and 11b.
The second case corresponds to the exponential potential
V(φ) = V1exp −λ1φ
Mpl +V2exp −λ2φ
Mpl ,(98)
which is representative of a scaling quintessence model [98][102]. Both of these cases belong to a class
of scalar field models known as freezing models which are thought to be vacuumless as the minimum
of the potential is not accessible within a finite range of φ. In these models the field decelerates down
Page 22

The early and late-time accelerated expansion of the universe
0.0 0.2 0.4 0.6 0.8 1.0
a
−0.76
−0.74
−0.72
−0.70
−0.68
w
(a)
0.0 0.5 1.0 1.5 2.0
z
−0.76
−0.74
−0.72
−0.70
−0.68
w
(b)
0.0 0.2 0.4 0.6 0.8 1.0
a
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
w
(c)
0 10 20 30 40
z
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
w
(d)
0.0 0.2 0.4 0.6 0.8 1.0
a
−1.0
−0.8
−0.6
−0.4
w
(e)
0.0 0.5 1.0 1.5 2.0
z
−1.0
−0.8
−0.6
−0.4
w
(f)
Figure 11: A plot showing the evolution of the equation of state parameter wfor the scalar field. Plot (a) shows this
evolution with the scale factor afor the tracking freezing model. Plot (b) shows this evolution with the redshift zfor the
tracking freezing model. This corresponds to the inverse power law potential (). Plot (c) shows this evolution with the
scale factor afor the scaling freezing model. Plot (d) shows this evolution with the redshift zfor the scaling freezing model.
This corresponds to the exponential potential (). Here each line corresponds to: λ1= 10, λ2= 0 (green line), λ1= 15,
λ2= 0 (orange line) and λ1= 30, λ2= 0 (blue line). Plot (e) shows this evolution with the scale factor afor the thawing
model. Plot (f ) shows this evolution with the redshift zfor the thawing model. This corresponds to the cosine potential
(). Here each line corresponds to: φ
fa= 0.5 (blue line), φ
fa= 0.25 (orange line) and φ
fa= 7.6×10−4(green line).
Page 23

Bradley Aldous
the potential, resulting in the equation of state parameter evolving towards wφ=−1. This case is
represented by Figures 11c and 11d.
The third case corresponds to the cosine potential
V(φ) = µ41 + cos φ
fa,(99)
which is representative of thawing quintessence models. Thawing models has an accessible minimum of
the potential and are characterised by an equation of state evolution which starts off at wφ=−1 and as
the field rolls down the potential, wφevolves towards 0. This case is represented by Figures 11e and 11f.
6 Conclusion
In this paper I have discussed, in reasonable detail, the standard model of modern cosmology, touching
on the necessary steps needed to derive the evolution equations. These evolution equations were used
to compare various models of the universe, paying particular attention to the general cases of flat, open
and closed universes. The sequence of species domination in our universe was also demonstrated through
the use of these equations.
I briefly discussed the history and the fundamental ideas behind dark energy and the cosmic acceler-
ation puzzle. I highlighted the key pieces of evidence for the existence of dark energy and the methods
through which these were obtained. Some of the main alternative theories for dark energy were discussed,
including scalar field models and modified gravity. These modified gravity theories sometimes tend to
suffer from instabilities and, occasionally, are only vaguely consistent with observations, whereas scalar
field models like quintessence tend to not suffer from instabilities are more promising candidates for dark
energy.
I have shown the derivations required to obtain a system of couple non-linear ODEs that can be
numerically solved to study how the density of the scalar field evolves with time. Here I presented a
number of different scenarios by varying the free parameters and initial conditions, finding that Figure
9d/10a best describes the history of our universe. From studying Figures 9 and 10, one finds that the
evolution of this model in late time is not affected by varying the initial conditions - the only parameter
that can affect the late time outcome of the model is λwhich has a critical value of 2.5, above which
matter continues to dominate and below which the scalar field dominates in late time.
The distinct models of a universe containing a scalar field have been discussed. These include thawing
models and freezing models. Thawing models describe a scenario where the equation of state parameter
is frozen at wφ=−1 by Hubble friction but as time evolves and the scalar field rolls down the potential,
this value evolves towards 0. In freezing models, the scalar field decelerates as it rolls down the potential
and evolves towards wφ=−1 as it ’freezes’. It has been shown that the trajectories of both of these
cases exist in well-defined regions on a dwφ/dln(a) plane (see Figure 1 in Ref. [103]).
Scalar field theories are incredibly interesting as a dynamical alternative to the cosmological constant
and are consistent with observations. As our observations of the CMB and other cosmological observables
become more precise in the near future, we will be able to obtain tighter constraints on the equation of
state parameter. At present, we have constraints which are completely consistent with a cosmological
constant as well as scalar field theories among others. The future of this field is a deeply intriguing one
- as we make more constraints on the cosmological parameters defining dark energy, we can understand
more about the nature of our universe, the constituents that comprise it and its ultimate fate.
Acknowledgements
I would like to thank Dr Karim Malik for his supervision and advice in researching for and writing this
report.
References
[1] W. de Sitter, “The expanding universe,” Scientia, vol. 49, pp. 1–10, 1931.
[2] G. Lemaˆıtre, “The expanding universe,” Monthly Notices of the Royal Astronomical Society, vol. 91,
pp. 490–501, March 1931.
Page 24

The early and late-time accelerated expansion of the universe
[3] A. S. Eddington, “The expansion of the universe,” The Observatory, vol. 54, pp. 112–114, April
1931.
[4] E. A. Milne, “The expanding universe,” Nature, vol. 130, p. 508, October 1932.
[5] A. G. Doroshkevich, Y. B. Zel’dovich, and I. D. Novikov, “The origin of galaxies in an expanding
universe,” Soviet Astronomy, vol. 11, pp. 233–239, October 1967.
[6] H. Nariai and T. Kimura, “On the canonical formalism of the gravitational field in the expanding
universe,” Progress of Theoretical Physics, vol. 28, pp. 529–545, September 1962.
[7] E. V. Linder, “Exploring the expansion history of the universe,” Physical Review Letters, vol. 90,
no. 9, 2003.
[8] S. W. Hawking, Properties of expanding universes. PhD thesis, University of Cambridge, United
Kingdom. Supervisor: Dennis Sciama, http://adsabs.harvard.edu/abs/1966PhDT.......101H, 1966.
[9] R. R. Caldwell and M. Kamionkowski, “The physics of cosmic acceleration,” Annual Review of
Nuclear and Particle Science, vol. 59, pp. 397–429, March 2009.
[10] A. V. Filippenko and A. G. Riess, “Results from the high-z supernova search team,” Physics
Reports, vol. 307, pp. 31–44, July 1998.
[11] G. Goldhaber, “The acceleration of the expansion of the universe: A brief early history of the
supernova cosmology project (scp),” AIP Conference Proceedings, July 2009.
[12] T. Digges, A perfit description of the caelestial orbes. 1576.
[13] E. Hubble, “A relation between distance and radial velocity among extra-galactic nebulae,” Pro-
ceedings of the National Academy of Science, vol. 15, pp. 168–173, March 1929.
[14] L. Santos, W. Zhao, E. G. M. Ferreira, and J. Quintin, “Constraining interacting dark energy with
cmb and bao future surveys,” Physical Review D, vol. 96, p. 103529, November 2017.
[15] S. Cao and N. Liang, “Interaction between dark energy and dark matter: observational constraints
from ohd, bao, cmb and sne ia,” arXiv:1105.6274, May 2011.
[16] V. Mukhanov, Physical Foundations of Cosmology. Cambridge University Press, November 2005.
[17] A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, and A. Hamilton, “Significance of the largest
scale cmb fluctuations in wmap,” Physical Review D, vol. 69, p. 063516, March 2004.
[18] Y. Ali-Haimoud and M. Kamionkowski, “Cosmic microwave background limits on accreting pri-
mordial black holes,” Physical Review D, vol. 95, p. 043534, February 2017.
[19] A. M. Wolfe, D. A. Turnshek, H. E. Smith, and R. D. Cohen, “Damped lyman-alpha absorption
by disk galaxies with large redshifts. i - the lick survey,” Astrophysical Journal Supplement Series,
vol. 61, pp. 249–304, June 1986.
[20] H. Seo and D. J. Einsenstein, “Probing dark energy with baryon acoustic oscillations from future
large galaxy redshift surveys,” The Astrophysical Journal, vol. 598, pp. 720–740, December 2003.
[21] M. Davis and P. J. E. Peebles, “A survey of galaxy redshifts. v - the two-point position and velocity
correlations,” Astrophysical Journal, vol. 267, pp. 465–482, April 1983.
[22] M. S. Vogeley, C. Park, M. J. Geller, and J. P. Huchra, “Large-scale clustering of galaxies in the
cfa redshift survey,” Astrophysical Journal, Part 2 - Letters, vol. 391, pp. 5–8, May 1992.
[23] L. Hernquist, N. Katz, D. H. Weinberg, and J. Miralde-Escude, “The lyman-alpha forest in the
cold dark matter model,” Astrophysical Journal Letters, vol. 457, p. 51, February 1996.
[24] R. Cen, J. Miralde-Escude, J. P. Ostriker, and M. Rauch, “Gravitational collapse of small-scale
structure as the origin of the lyman-alpha forest,” Astrophysical Journal, Part 2 - Letters, vol. 437,
pp. 9–12, December 1994.
Page 25

Bradley Aldous
[25] D. J. Einsenstein, “Dark energy and cosmic sound,” New Astronomy Reviews, vol. 49, pp. 360–365,
November 2005.
[26] R. Lazkoz, J. Alcaniz, C. Escamilla-Rivera, V. Salzano, and I. Sendra
[27] N. Padmanabhan, X. Xu, D. J. Einsenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta, and E. Kazin,
“A 2 per cent distance to z = 0.35 by reconstructing the baryon acoustic oscillations - i. methods
and application to the sloan digital sky survey,” Monthly Notices of the Royal Astronomical Society,
vol. 427, pp. 2132–2145, December 2012.
[28] D. J. E. et al., “Detection of the baryon acoustic peak in the large-scale correlation function of the
sdss luminous red galaxies,” The Astrophysical Journal, vol. 633, pp. 560–574, November 2005.
[29] G. H. et al., “Nine-year wilkinson microwave anisotropy probe (wmap) observations: Cosmological
parameter results,” The Astrophysical Journal Supplement Series, vol. 208, p. 19, June 2013.
[30] P. C. et al., “Planck 2015 results. xiii. cosmological parameters,” Astronomy and Astrophysics,
vol. 594, p. A13, September 2016.
[31] C. Wetterich, “Cosmology and the fate of dilatation symmetry,” Nuclear Physics B, vol. 302, no. 4,
pp. 668–696, 1988.
[32] C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt, “Essentials of k-essence,” Physical
Review D, vol. 63, p. 103510, May 2001.
[33] E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,” International Journal of
Modern Physics D, vol. 15, pp. 1753–1935, June 2006.
[34] L. A. Boyle, R. R. Caldwell, and M. Kamionkowski, “Spintessence! new models for dark matter
and dark energy,” Physics Letters B, vol. 545, pp. 17–22, October 2002.
[35] N. Banerjee and S. Das, “Spintessence: a possible candidate as a driver of the late time cosmic
acceleration,” Astrophysics and Space Science, vol. 305, pp. 25–27, September 2006.
[36] J. Bloomfield, E. E. Flanagan, M. Park, and S. Watson, “Dark energy or modified gravity? an effec-
tive field theory approach,” Journal of Cosmology and Astroparticle Physics, vol. 2013, September
2013.
[37] A. Joyce, L. Lombriser, and F. Schmidt, “Dark energy vs. modified gravity,” Annual Review of
Nuclear and Particle Science, vol. 66, no. 1, pp. 95–122, 2016.
[38] A. D. Dolgov and M. Kawasaki, “Can modified gravity explain accelerated cosmic expansion?,”
Physics Letters B, vol. 573, pp. 1–4, August 2003.
[39] K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, “Dark energy cosmology: the equiva-
lent description via different theoretical models and cosmography tests,” Astrophysics and Space
Science, vol. 342, pp. 155–228, July 2012.
[40] A. Friedman, “ ¨
Uber die kr¨ummung des raumes,” Zeitschrift f¨ur Physik, vol. 10, no. 1, pp. 377–386,
1922.
[41] A. Einstein, “Die grundlage der allgemeinen relativit¨atstheorie,” Annalen der Physik, vol. 354,
no. 7, pp. 769–822, 1916.
[42] B. J. Carr and M. J. Rees, “The anthropic principle and the structure of the physical world,”
Nature, vol. 278, pp. 605–612, April 1979.
[43] J. Mosterin, “Anthropic explanations in cosmology,” March 2004.
[44] A. H. Guth, “Inflationary universe: A possible solution to the horizon and flatness problems,”
Physical Review D, vol. 23, pp. 347–356, January 1981.
[45] A. Einstein and W. de Sitter, “On the relation between the expansion and the mean density of the
universe,” Proceedings of the National Academy of Science, vol. 18, no. 3, pp. 213–214, 1932.
Page 26

The early and late-time accelerated expansion of the universe
[46] R. R. Caldwell, M. Kamionkowski, and N. N. Weinberg, “Phantom energy and cosmic doomsday,”
Physical Review Letters, vol. 91, no. 7, p. 071301, 2003.
[47] P. F. Schewe, “The big rip: a new cosmic doomsday,” Physics Today, vol. 56, p. 9, October 2003.
[48] Y. Wang, J. M. Kratochvil, and A. Linde, “Current observational constraints on cosmic doomsday,”
Journal of Cosmology and Astroparticle Physics, vol. 2004, no. 12, p. 6, 2004.
[49] D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer, and E. L. Wright, “The cosmic
microwave background spectrum from the full cobe firas data set,” Astrophysical Journal, vol. 473,
p. 576, December 1996.
[50] M. K. et al., “Improved cosmological constraints from new, old and combined supernova data sets,”
The Astrophysical Journal, vol. 686, pp. 749–778, October 2008.
[51] C. B. et al., “The wigglez dark energy survey: mapping the distance-redshift relation with baryon
acoustic oscillations,” Monthly Notices of the Royal Astronomical Society, vol. 418, pp. 1707–1724,
December 2011.
[52] C. B. et al., “The wigglez dark energy survey: testing the cosmological model with baryon acoustic
oscillations at z = 0.6,” Monthly Notices of the Royal Astronomical Society, vol. 415, pp. 2892–2909,
August 2011.
[53] W. J. P. et al., “Measuring the baryon acoustic oscillation scale using the sloan digital sky survey
and 2df galaxy redshift survey,” Monthly Notices of the Royal Astronomical Society, vol. 381,
pp. 1053–1066, November 2007.
[54] W. J. P. et al., “Baryon acoustic oscillations in the sloan digital sky survey data release 7 galaxy
sample,” Monthly Notices of the Royal Astronomical Society, vol. 401, pp. 2148–2168, February
2010.
[55] L. A. et al., “The clustering of galaxies in the sdss-iii baryon acoustic oscillation spectroscopic
survey: baryon acoustic oscillations in the data release 9 spectroscopic galaxy sample,” Monthly
Notices of the Royal Astronomical Society, vol. 427, pp. 3435–3467, December 2012.
[56] S. C. et al., “The 2df galaxy redshift survey: power-spectrum analysis of the final data set and
cosmological implications,” Monthly Notices of the Royal Astronomical Society, vol. 362, pp. 505–
534, September 2005.
[57] R. J. Adler, B. Casey, and O. C. Jacob, “Vacuum catastrophe: an elementary exposition of the
cosmological constant problem,” American Journal of Physics, vol. 63, pp. 620–626, July 1995.
[58] S. Weinberg, “The cosmological constant problem,” Reviews of Modern Physics, vol. 61, pp. 1–23,
January 1989.
[59] R. R. Caldwell, “A phantom menace? cosmological consequences of a dark energy component with
super-negative equation of state,” Physics Letters B, vol. 545, pp. 23–29, May 2002.
[60] A. H. Guth, The inflationary universe: the quest for a new theory of cosmic origins. Vintage, 1997.
[61] B. Ratra and P. J. E. Peebles, “Cosmological consequences of a rolling homogeneous scalar field,”
Physical Review D, vol. 37, pp. 3406–3427, June 1988.
[62] R. R. Caldwell, R. Dave, and P. J. Steinhardt, “Cosmological imprint of an energy component with
general equation of state,” Physical Review Letters, vol. 80, pp. 1582–1585, February 1998.
[63] T. Chiba, “Quintessence, the gravitational constant, and gravity,” Physical Review D, vol. 60,
August 1999.
[64] P. Brax and J. Martin, “Quintessence and supergravity,” Physics Letters B, vol. 468, pp. 40–45,
October 1999.
[65] A. Leithes, K. A. Malik, D. J. Mulryne, and N. J. Nunes, “Linear density perturbations in multifield
coupled quintessence,” Physical Review D, vol. 95, p. 123519, June 2017.
Page 27

Bradley Aldous
[66] L. Wang, R. R. Caldwell, J. P. Ostriker, and P. J. Steinhardt, “Cosmic concordance and