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ABSTRACT
QUANTUM COSMOLOGY: SOLUTIONS TO THE MODIFIED
FRIEDMANN EQUATION
In this work, a detailed analysis of Standard Cosmological Inflation is
presented, which is then contrasted by Loop Quantum Cosmology (LQC), an
application to cosmology from Loop Quantum Gravity (LQG). Specifically,
the modified Friedmann equation of Loop Quantum Cosmology (LQC) is
solved, in order to obtain expressions used to assess an Inflationary era in the
early Universe. The expressions for the scale factor are derived when
considering two regions associated with the behavior of the modified
Friedmann equation, as well as the energy density and scalar field. The scale
factor expression will then be used to provide a solution to the horizon
problem that is related to the Big Bang model of the Universe, in contrast to
what has been presented in the literature.
James Anthony Rubio
December 2016
QUANTUM COSMOLOGY: SOLUTIONS TO THE MODIFIED
FRIEDMANN EQUATION
by
James Anthony Rubio
A thesis
submitted in partial
fulfillment of the requirements for the degree of
Master of Science in Physics
in the College of Science and Mathematics
California State University, Fresno
December 2016
APPROVED
For the Department of Physics:
We, the undersigned, certify that the thesis of the following
student meets the required standards of scholarship, format, and
style of the university and the student’s graduate degree program
for the awarding of the master’s degree.
James Anthony Rubio
Thesis Author
Gerardo Muñoz (Chair)
Physics
Douglas Singleton
Physics
Frederick A. Ringwald
Physics
For the University Graduate Committee:
Dean, Division of Graduate Studies
AUTHORIZATION FOR REPRODUCTION
OF MASTER’S THESIS
I grant permission for the reproduction of this thesis in part or
in its entirety without further authorization from me, on the
condition that the person or agency requesting reproduction
absorbs the cost and provides proper acknowledgment of
authorship.
X
Permission to reproduce this thesis in part or in its entirety
must be obtained from me.
Signature of thesis author:
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. Gerardo Muñoz, for his
continued guidance, wisdom, unwavering patience, and willingness to pursue
many topics that the research directed us towards. Most importantly, I would
like to thank him for his ability to help me understand difficult topics and
ideas in physics. I would also like to thank my thesis committee members, Dr.
Douglas Singleton and Dr. Frederick A. Ringwald, for all of their support and
enlightening discussions during the writing process. Finally, I would like to
thank the physics department faculty and staff for all of the knowledge
attained in my time as a student and Teaching Associate.
TABLE OF CONTENTS
Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
The Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
STANDARD COSMOLOGICAL INFLATION . . . . . . . . . . . . . . .
7
Friedmann–Lemaı̂tre–Robertson–Walker Universe . . . . . . . . . . .
7
Cosmic Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Issues of Cosmic Inflation . . . . . . . . . . . . . . . . . . . . . . . .
26
Loop Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . .
27
SOLUTIONS TO THE MODIFIED FRIEDMANN EQUATION FROM
LOOP QUANTUM COSMOLOGY . . . . . . . . . . . . . . . . . . .
30
Region I: t0 ≤ t ≤ tmax . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Region II: t > tmax . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
ANALYSIS OF SOLUTIONS IN REGIONS I AND II . . . . . . . . . .
43
Inflation in LQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
The Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
CONCLUSION AND SUMMARY . . . . . . . . . . . . . . . . . . . . .
47
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
LIST OF FIGURES
Page
Figure 1. Typical “flat” potential, in relation to an inflaton scalar field. 22
Figure 2. Graphical representation of two regions of solutions corresponding to H. Region I corresponds to t0 ≤ t ≤ tmax and Region
II corresponds to t > tmax . . . . . . . . . . . . . . . . . . . . . .
32
To my family, for their continued support of my education throughout the
years.
INTRODUCTION
Over many years, cosmology has seen several advances made as a
study of the Universe. Astrophysical observations have been made for
centuries and with time, they have increased knowledge of large objects in the
sky, as well as the nature of the light itself. In terms of the Universe as a
whole, a very significant observational study of galaxy redshifts was made in
the early twentieth century [1], which led to a new realization and
understanding among scientists. The results of the study made it clear that
the majority of the galaxies were redshifted from our location in the Universe
receding from our position. It was clear that the Universe was expanding with
time. This expansion rate was measured at the time, and presently, the value
is known within a small uncertainty. Another conclusion is that if the
Universe is expanding with time, it had a finite, hot, and dense beginning.
This early period of the Universe is referred to as the Big Bang. The Big
Bang model of the Universe has had both observations on and off Earth that
have strongly supported it as a scientific model of the Universe [2].
Measurements of light observed through astronomical telescopes have led to a
strongly supported theoretical era of nucleosynthesis in the early history of
the Universe, which serves as an explanation of the particle and chemical
history of elements. Along with nucleosynthesis is an understanding of the
thermal signature of the Cosmic Microwave Background observed in the sky,
as observed photons scattered off of electrons. However, there are
shortcomings of the Big Bang model. Representative shortcomings of the Big
2
Bang model are the flatness and horizon problems. The flatness problem
refers to the fact that all observations point to a spatially flat Universe, while
the Big Bang model allows for open, closed or flat solutions without a natural
mechanism that would select the flat solution from either initial conditions or
dynamical evolution. And the horizon problem arises from the observational
data showing that the temperature of widely separated regions of the
Universe is the same, despite the prediction of the Big Bang model that
causal contact between these regions is not possible.
The Cosmological Inflationary model is a large scope of this work, and
provides a solution to both the flatness and horizon problems. Inflation
precedes the Big Bang era, and is the subject of much research currently
around the world. Many aspects of Inflation will be discussed as well as an
alternative model that also provides a solution to the horizon problem. In this
work, solutions are obtained from the modified Friedmann equation. The
modified Friedmann equation is derived from Loop Quantum Cosmology, an
application of a proposed quantum gravity model, by which space and time
are quantized. Loop Quantum Cosmology (LQC), and the solutions obtained
from it, provides a new formalism to evaluate the accelerated expansion of the
Universe, and a reinterpretation of the horizon problem. The scope of this
work is limited to flat LQC models. The general case will be addressed in
future work.
3
The Big Bang
As mentioned previously, the Big Bang model came about mostly as a
result of astrophysical observations by both Vesto Slipher and Edwin Hubble
[1, 3, 4]. Redshift, z, is defined as the ratio of the difference of an observed
(λ0 ) and emitted (λ) wavelength of light, over the emitted wavelength,
z≡
λ0 − λ
λ
(1)
where if z > 0, then the galaxy is redshifted, and if z < 0, the galaxy is
blueshifted. What was evident after observations by Hubble was that the
majority of galaxies were redshifted, as opposed to blueshifted, especially for
galaxies at high redshift. Hubble plotted the redshifts, z, versus the estimated
distance r, from the galaxies and obtained a linear relationship,
z=
H0
r
c
(2)
where H0 is the Hubble Constant, currently measured at H0 = 73.24 ± 1.74
km s−1 Mpc−1 [5], where a Megaparsec, Mpc, is equal to 3.09 × 1022 m , and c
is the speed of light, 3.00 × 108 m/s. The values measured by Hubble were
small, so the assumption of non-relativistic Doppler shifts applied, which
meant that z = vc . Therefore, the recessional speed of galaxies is proportional
to the estimated distance,
v = H0 r .
(3)
4
With (3), known as Hubble’s Law, inferred from the observational data, the
implication was that the Universe was expanding, and therefore has a finite
age. The best estimate currently is (13.799 ± 0.021) × 109 years [6]. The
concept of an expanding Universe with time had been considered theoretically
as a consequence of General Relativity by Albert Einstein [7], the details of
which will be covered in the next section. The idea of a once tiny, hot, and
dense Universe paved the way for research into the chemical nature and
composition of how the elements formed, specifically light elements. The idea
behind nucleosynthesis is that the light elements, such as hydrogen, were
formed when the Universe cooled from an initially hotter, denser radiation at
temperatures that would not allow the binding energies of atom formation. In
sum, as science has become knowledgeable of the energies associated with
particles, predictions as to the elemental abundances present within the
Universe support the notion of nucleosynthesis with current estimates from
observation [8]. This is called the era of Big Bang Nucleosynthesis (BBN) and
occurred when the Universe was in a hot, dense gaseous form [9]. Along with
nucleosynthesis, strong support for the Big Bang model comes from the
detection and properties of the Cosmic Microwave Background (CMB) of the
Universe. In 1964, two researchers from Bell Telephone Laboratory, Arno A.
Penzias and Robert W. Wilson, initially measured the CMB radiation which
from thermodynamics, resembled blackbody radiation [7, 10]. The observation
of a blackbody distribution immediately corresponded to a temperature that
is currently measured at 2.72548 ± 0.00057 K [11]. This CMB temperature
corresponds to photons scattered off of electrons when the Universe was about
5
380, 000 years old. Further observational data of the Cosmic Microwave
Background (CMB) were measured by the COBE satellite in 1992, which
showed an almost perfect blackbody spectrum of photons, with a temperature
of T0 = 2.725 ± 0.001 K [12]. Results of the COBE satellite measurements
were confirmed and improved upon by the Wilkinson Microwave Anisotropy
Probe (WMAP ) spacecraft [13]. Further observations of the CMB were made
by another spacecraft called Planck [6]. All of these experiments showed a
blackbody spectrum, but with fluctuations of order 10−5 . These fluctuations
are important outcomes of these observations and will be discussed in the
next section.
As mentioned previously, some aspects of the theoretical foundations
of the Big Bang model leave much to be desired. Among these shortcomings
are mainly two important problems: the flatness and the horizon problems.
The flatness problem has to do with the unique features of the measured
energy density of the Universe. Specifically, with current estimates of a
Cosmological Constant being very small, the ratio of energy density of the
Universe compared to a theoretically critical energy density differs from unity
by a strikingly small amount 0.0008+0.0040
−0.0039 [6, 9, 14]. The Big Bang model
does not specify the cause of why the ratio of energy densities is close to the
value unity, which due to the Friedmann equation, implies that the spatial
curvature of the Universe is flat. More interestingly, when extrapolations to
past epochs of time are made from our current time back to the Planck era,
the ratio of energy densities is projected to be even closer to unity. This
represents either a coincidence of nature, or the possibility that the Big Bang
6
model is incomplete. In addition, there is the horizon problem, which is linked
to the CMB. As described before, the CMB lends significant support to the
Big Bang model of the Universe. However, along with the measurement of the
CMB temperature came the realization that parts of the Universe, which are
not expected to be causally connected, have the same thermal features in
about one part in 100, 000 [4]. This means that these parts of the Universe
could not have possibly been in contact, as indicated by measurements of the
allowed distances that light could have traveled. Yet somehow, these parts of
the Universe reached thermal equilibrium.
How is it possible that the Universe has such features? These
problems, as well as the apparent scarcity of magnetic monopoles [15], led to
the proposal of Cosmological Inflation as a possible solution. At its core as a
theory, Cosmological Inflation is characteristic of a moment preceding the Big
Bang era, at a time when the scale factor of the Universe is accelerating. And
it is because of this idea of Cosmological Inflation that both the flatness and
horizon problems, as well as other issues of the Big Bang model, are solved.
The next section features more details and components of Cosmological
Inflation, issues researchers have with it, and an alternative theory which
takes a very different route to an era of an accelerating scale factor.
STANDARD COSMOLOGICAL INFLATION
Cosmology has been an active field of research for several decades, for
both theory and observation. In this section, the physical and mathematical
background of our current model of the Universe is presented. Beginning with
assumptions of homogeneity and isotropy, within the context of General
Relativity, one eventually reaches the Friedmann equations. The Friedmann
equations are most important to the Cosmic Inflation possibility, which is
simply an idea of an accelerating scale factor in the early Universe, postulated
in terms of negative pressures associated with scalar field dynamics. Along
with an overview of Cosmic Inflation, the problems of the Big Bang model are
presented along with solutions to these issues, provided that an Inflationary
era occurred early in the Universe. Also, a discussion of issues and limitations
of the Inflationary paradigm are discussed, which lead to an alternative theory
based on Loop Quantum Gravity. Loop Quantum Gravity, and its application
to the cosmological setting, is the basis of this work. Modified Friedmann
equations obtained from Loop Quantum Cosmology are presented, which are
comparable to the standard Friedmann equations. The modified Friedmann
equations will lead to solutions of the scale factor, which will be analyzed in
the next section.
Friedmann–Lemaı̂tre–Robertson–Walker Universe
A discussion of Cosmic Inflation usually begins with considering a
spacetime background metric. From General Relativity, a spacetime
8
background metric is defined by
ds2 =
X
gµν dxµ dxν
(4)
µ,ν=0
where µ, ν = 0, 1, 2, 3. gµν is referred to as the metric tensor. As mentioned
previously, on large scales, the Universe appears homogeneous and isotropic,
which simply means that it has invariance under spatial translations and
rotations. Thus when considering a spacetime background for an expanding
Universe, homogeneity and isotropy must be ensured. To represent an
expanding, homogeneous, and isotropic Universe, (4) takes the following form
(setting c = 1)
2
2
2
ds = −dt + a (t)
dr2
2
2
2
2
+ r dθ + sin θdφ
.
1 − kr2
(5)
The form of the spacetime metric of (5) is referred to as the
Friedmann–Lemaı̂tre–Robertson–Walker (FLRW) metric, named after
Alexander Friedmann, Georges Lemaı̂tre, Howard P. Robertson, and Arthur
G. Walker. These researchers popularized this metric in the early twentieth
century. Embedded in the spacetime metric is k, which represents the spatial
geometry of the hypersurface described by the spacetime, where
k=




+1 positive curvature, closed Universe




0
spatially flat Universe






−1 negative curvature, open Universe
9
Also included in (5) is a = a (t), which is the scale factor. If the scale factor
increases or decreases, the spacetime is either expanding or contracting. The
scale factor, a (t), has a very important role when considering the possibility
of Cosmic Inflation, as will be discussed. It is also important to note that
there is a direct relation of the scale factor corresponding to an emitted and
observed wavelength of a galaxy, following a null geodesic (ds2 = 0),
a (t0 )
λ0
=
a (t)
λ
where t is the time of emission, and t0 is the time at observation. Expressed
in terms of the redshift, z,
1+z =
1
a (t)
(6)
where a (t0 ) ≡ 1 by convention. Upon an expansion of a (t) in a power series,
the Hubble parameter can be written as
H=
ȧ (t)
.
a (t)
(7)
The relation Hubble obtained in (3) is therefore verified. More importantly,
the Hubble parameter, H, is directly related to the scale factor, a (t), and
with measurement of H, we have a measure of a (t). The Einstein equations,
which relate the spacetime geometry to the matter in the Universe, are then
used to derive the Friedmann equations, which relate the scale factor to
10
energy density and pressure of the Universe as follows:
1
Gµν = Rµν − gµν R + Λgµν = 8πGTµν
2
(8)
where Gµν is the Einstein tensor, Tµν is the energy–momentum tensor, Λ is
the Cosmological Constant, Rµν is the Ricci tensor, and R is the Ricci scalar
(R = g µν Rµν ). The FLRW metric tensor, gµν , has the following form

gµν

0
0
0
−1



−1
 0 a2 (t) {1 − kr2 }

0
0


=

0

2
2
0
a (t)r
0




2
2
2
0
0
0
a (t)r sin θ
(9)
In terms of the Christoffel symbol:
Rµν = Γαµν ,α − Γαµα ,ν + Γαβα Γβµν − Γαβµ Γβµα
(10)
and
Γµαβ
∂gαβ
g µν ∂gαν ∂gβν
+
−
.
=
2 ∂xβ
∂xα
∂xν
(11)
For the energy–momentum tensor, Tµν , we have

Tµν = gµν T α ν

−1
0
0
0

 −ρ


 0 a2 (t) {1 − kr2 }−1
 0
0
0


=

0
 0
2
2
0
a
(t)r
0




0
0
0
0
a2 (t)r2 sin2 θ

0

P 0 0


0 P 0


0 0 P
0
0
11


0
0
0
ρ



−1
0 P a2 (t) {1 − kr2 }

0
0


=

0

0
P a2 (t)r2
0




2
2
2
0
0
0
P a (t)r sin θ
(12)
where ρ is the energy density, and P is the pressure in the rest frame of a
perfect fluid, as seen by a comoving observer [16]. The covariant derivative of
Tµν gives a covariant generalization of the conservation equation of a perfect
fluid, such as the continuity and Euler equation,
∇µ T µν = ∂µ T µν + Γµµλ T λν + Γνµλ T µλ = 0 .
For the first term,
∂µ T µ0 = ∂0 T 00 =
∂ρ
.
∂t
For the second term, we get
Γµµλ T λν = Γµµ0 T 0ν .
This implies:
Γii0 T 00 = 3
ȧ(t)
ρ.
a(t)
For the third term, we get:
Γ0µλ T µλ = Γ0µλ T µλ
= Γ011 T 11 + Γ022 T 22 + Γ033 T 33 .
(13)
12
This implies:
Γ0µλ T µλ = 3
ȧ(t)
P.
a(t)
Thus from (13), with ν = 0, we have
∂ρ
ȧ(t)
+3
(ρ + P ) = 0 .
∂t
a(t)
Dividing (14) by ρ, and defining the equation of state as w =
(14)
P
,
ρ
dρ
da
+ 3 (1 + w)
=0.
ρ
a
This implies:
ρ ∝ a−3(1+w) .
(15)
Through the study of different eras of the Universe, w is categorized in terms
of the different theoretical contributions of energy density and pressure [8].
For nonrelativistic matter, w = 0, which includes baryons (electrons and
nuclei), and non–baryonic dark matter. For radiation, w = 1/3, which
includes photons, neutrinos, and possibly other relativistic particles. For
Cosmological Constant Λ, w = −1, which observational data seems to support
13
[8, 14]. In summary,
ρ∝




a−4




−3
a





a0

Radiaton Era
Matter Era
Cosmological Constant Era
In relation to the information obtained from the energy–momentum tensor,
the Einstein equations allow for further analysis of the scale factor when
considering the spacetime background. Using (10), the non–vanishing
components of the Riemann tensor are
ä
a
aä + 2ȧ + 2k
=
1 − kr2
R00 = −3
(16)
R11
(17)
R22 = r2 (aä + 2ȧ + 2k)
(18)
R33 = r2 (aä + 2ȧ + 2k) sin2 θ .
(19)
Computing the Ricci Scalar, R = g µν Rµν , yields
"
ä
R=6
+
a
#
2
ȧ
k
+ 2 .
a
a
(20)
Considering the space–space, and time–time components of (8),
1
G00 = R00 − g00 R + Λg00 = 8πGT00
2
1
Gij = Rij − gij R + Λgij = 8πGTij
2
(21)
(22)
14
the following equations are obtained
2
ȧ
8πG
k
Λ
ρ
+ 2− =
a
a
3
3
4πG
Λ
ä
=−
(ρ + 3P ) + .
a
3
3
(23)
(24)
Equations (23) and (24) are called the Friedmann equations, and they will
will be used in the analysis of Cosmic Inflation.
Cosmic Inflation
The earliest literature on Cosmological Inflation theory came from
Alan H. Guth [15], Alexei A. Starobinsky [17], Andreas J. Albrecht and Paul
J. Steinhardt [18], and Andrei D. Linde [19, 20]. As mentioned in the
introduction, Inflation solves both the horizon and flatness problems that the
Big Bang model is unable to address, primarily beginning with a discussion of
the scale factor, a (t). By use of (23) and (24), as well as introducing scalar
fields in the form of energy density and pressure, Cosmic Inflation becomes a
mechanism for addressing the otherwise unexplained origin of the initial
conditions of the early Universe. It also generates small perturbations
observed in the CMB, which lead to large–scale structure [12]. To begin with,
the horizon problem is the observation that causally disconnected regions of
the Universe, on large scales, share similarities of thermal characteristics,
courtesy of the observations made by COBE, WMAP, and Planck spacecraft
[6, 12, 13]. To illustrate, consider the null path of a light ray, ds2 = 0,
traveling in a homogeneous and isotropic Universe (dθ = dφ = 0). From (5),
15
(setting k = 0), the following is obtained
0 = −dt2 + a2 (t)dr2
Z t2
dt
dH =
t1 a(t)
(25)
where dH is the particle horizon in comoving coordinates. Observational data
suggests [9, 14] that
dHe dH0
where dHe is the particle horizon of a light ray from the time at the end of
Inflation, te , to the time of last scattering, tls , defined by
tls
Z
dHe =
te
dt
a(t)
as well as dH0 , the particle horizon of a light ray from the time of last
scattering to the current time, t0 , defined as
Z
t0
dH0 =
tls
dt
.
a(t)
Each time t, corresponding to the upper and lower limits of the integrals are
the following: the time at the end of inflation te , is approximately 10−32±6 s
[9], the time of last scattering of CMB photons tls , is about 380, 000 years
after the Big Bang [4], and the present age of the Universe t0 , is about 13.8
billion years after the Big Bang [6]. As a result of dH0 being much greater
than dHe , light could not have traveled far enough to establish thermal
16
contact between widely separated regions of the Universe, yet the observed
smallness of the temperature variations of the CMB strongly supports these
regions having nearly the same temperature of one part in 100,000 [4].
From the perspective of the Big Bang model, this is a major issue, but
it is solved by Cosmic Inflation. The flatness problem can be analyzed from
(23) as well, where
H2 +
Setting Ω =
ρ
,
ρcr
8πG
k
Λ
ρ
− =
2
a
3
3
(26)
where ρcr = 3MP2 l H 2 , currently measured at
ρcr,0 ' 1.88h2 × 10−29 g · cm−3 (h ' 0.70) [14], and MP l =
√1
8πG
is called the
reduced Planck mass, (26) takes the following form
Ωtotal − 1 =
where Ωtotal = Ω + ΩΛ and ΩΛ =
Λ
.
3H 2
k
a2 H 2
(27)
Expressed this way, the Friedmann
equation directly relates the ratio of energy density to the spatial flatness, and
Ωtotal , even when considering a non–zero Cosmological Constant Λ, is
currently measured to be very close to 1. It follows that the right–hand side of
(27) approaches zero, as Ωk = − a2kH 2 = 0.0008+0.0040
−0.0039 [6, 14] indicates from
observation. Again, the Big Bang model has no explanation as to why the
energy density of the Universe is close to a critical value, strongly implying a
spatially flat Universe. Even more perplexing is the realization that in the
early Universe, at Big Bang Nucleosynthesis, Ωtotal must have equaled 1 to
within 1 part in 1016 [14].
17
Cosmic Inflation also answers why the Ωtotal clearly tends to 1. The
standard definition of Cosmic Inflation is that
ä > 0 .
(28)
From (24), this implies (letting Λ → 0)
Ḣ + H 2 =
ä
4πG
=−
(ρ + 3P ) > 0 .
a
3
(29)
In order to satisfy (29),
ρ + 3P < 0 ,
which implies:
ρ
P <− .
3
(30)
The accelerating scale factor condition in (28), is equivalent to
d
(aH)−1 < 0
dt
(31)
where (aH)−1 is the comoving Hubble radius, and is decreasing during
Cosmic Inflation. The solution to the horizon problem provided by inflation is
is illustrated in [21] by considering the physical distance of light traveled by a
null geodesic, xH = a (t) dH , which from (25) gives
Z
t2
xH = a (t)
t1
dt
.
a(t)
(32)
18
Comparing the horizon distance of light traveled from the beginning of
Cosmic Inflation to the time of last scattering,
Z
tls
xls = a (tls )
ti
dt
a(t)
(33)
where during Inflation, the scale factor can be modeled as a (t) ∼ eHt , and the
horizon distance of light traveled since last scattering,
Z
t0
x0 = a (t0 )
tls
dt
a(t)
(34)
where using (15) and solving the Friedmann equation (23), the scale factor
has the form,
2
a (t) ∝ t 3(1+w)
(35)
2
and thus for the Matter era of the Universe (w = 0), a (t) ∼ t 3 . The comoving
horizons take the form from (33) and (34) of
Z
tls
a (tls )
ti
Z δt
dt
dt
= a (tls )
Ht
Ht
e
0 e
1 − e−Hδt
Hδt
=e
H
= H −1 eHδt − 1
19
where tls is set to δt, or the end of Inflation, and ti = 0 for the beginning of
Inflation, and
Z
t0
a (t0 )
tls
Z t0
dt
dt
= a (t0 )
2
a(t)
tls t 3
2 1
1
= t03 3t03 − 3tls3
' 3t0 .
The term Hδt is usually referred to as the number of e–foldings, N . In order
to yield a result that is greater than the horizon distance of 3t0 (or
equivalently
2
H0
2
with a (t0 ) ∼ t03 and H0 =
a˙0
),
a0
researchers constrain the
number of e–folds, N , to
N ≥ 60
and the horizon problem is avoided [8, 14, 22]. Specifically, before Inflation is
said to occur, the comoving Hubble distance is large, encompassing large
parts of the Universe. During Inflation, the scale factor, a, is accelerating, or
increasing exponentially. Thus the comoving Hubble distance shrinks. Hence,
before Inflation occurred, these parts of the Universe were in causal contact.
In order to solve the flatness problem, from (27), Ωtotal approaches 1 as
the comoving Hubble radius, (aH)−1 is decreasing during Inflation, and
increases for other eras in the Universe [9]. Cosmic Inflation is also largely
analyzed by the energy density and pressure relationship in accordance with
(30). The energy density is a positive quantity, implying that the pressure
must be a negative quantity. Ordinary matter and radiation do not feature
20
phenomena that possesses a negative pressure [8]. Therefore, scalar fields are
used, which provide a necessary negative pressure and are similar to electric
and magnetic fields, but without a direction [9]. A scalar field, ϕ, called an
inflaton, which is minimally coupled to gravity, leads to the form [8, 9, 14, 15]
T
α
β
=g
αν
∂ϕ ∂ϕ
1 µν ∂ϕ ∂ϕ
α
−g β g
+ V (ϕ)
∂xν ∂xβ
2
∂xµ ∂xν
(36)
of the energy–momentum tensor. Due to the FLRW metric, which preserves
homogeneity and isotropy of the spacetime, the momentum density is zero,
and with the space–space and time–time components, T 0 0 and T i j , we obtain
expressions for the energy density and pressure in terms of the scalar field, ϕ,
and the scalar field potential, V (ϕ) as
ϕ̇2
+ V (ϕ)
2
ϕ̇2
Pϕ =
− V (ϕ) .
2
ρϕ =
(37)
(38)
Taking a time derivative of (23), and using (24), the scalar wave equation is
obtained, which is the following
ϕ̈ + 3H ϕ̇ + V 0 (ϕ) = 0
(39)
21
where V 0 (ϕ) =
dV (ϕ)
.
dϕ
By imposing the condition of ρϕ + 3Pϕ < 0, for the
scalar field and the potential,
2
ϕ̇2
ϕ̇
ρϕ + 3Pϕ =
+ V (ϕ) + 3
− V (ϕ) < 0
2
2
= 2 ϕ̇2 − V (ϕ) < 0 .
This implies:
ϕ̇2 < V (ϕ) .
(40)
The condition that arises between the scalar field and the potential, by (40),
is usually modeled as a scalar field rolling towards the minimum of the
potential [22]. In Figure 1, there is a region where the scalar potential is
considered flat, which corresponds to a “false” or temporary vacuum for the
minimum energy density, while the true minimum energy density corresponds
to the bottom of the curve [23]. The “false” vacuum also corresponds to the
negative pressure of the scalar field, which from (38) and (40) indicates the
scalar potential is dominant at that point. As the energy density at this
“false” vacuum location is nearly is at a minimum, the Inflationary expansion
takes place. The time it takes for the scalar field to “roll” down the potential
corresponds to the end of inflation, where the field resides at the true vacuum
or minimum energy density.
22
V(φ)
φ
Figure 1. Typical “flat” potential, in relation to an inflaton scalar field.
At this point a phase called Reheating is said to occur, which
researchers directly link to the eventual formation of a particle soup
corresponding to Big Bang Nucleosynthesis during the Radiation era
[9, 14, 22, 23]. There are many models of Cosmic Inflation, with varying
scalar potentials, which lead to different results and predictions, though
common among these models is the Slow–Roll approximation of the scalar
field as it moves from the “false” vacuum to the end of the scalar potential
23
dominance. Using the Friedmann equation, (26), (assuming k = 0)
H2 +
k
8πG
Λ
ρ
− =
2
a
3
3 8πG ϕ̇2
+ V (ϕ) .
=
3
2
This implies:
1
H =
3MP2 l
2
ϕ̇2
+ V (ϕ)
2
.
(41)
From (40), (41) takes the form
H2 '
V (ϕ)
3MP2 l
(42)
Also, applying (40) to (39),
ϕ̈ + 3H ϕ̇ = −V 0 (ϕ) .
This implies:
3H ϕ̇ ' −V 0 (ϕ) .
(43)
In order to satisfy (42) and (43), (ϕ) 1, and |η(ϕ)| 1, where and η are
Slow–Roll parameters, valid under the approximations given, which are
24
defined as
M2
(ϕ) = P l
2
η(ϕ) = MP2 l
V0
V
2
(44)
V 00
.
V
(45)
Under these parameters, the scalar potential V (ϕ), is restricted in form, but ϕ̇
may be chosen freely, which may violate (43). In relation to the first condition
for accelerated expansion, ä > 0 implies
ä
= Ḣ + H 2 > 0
a
(46)
which is satisfied if Ḣ > 0, otherwise
−
Ḣ
<1
H2
(47)
and the Slow–Roll parameter, , becomes
Ḣ
M2
− 2 = Pl
H
2
V0
V
2
=.
(48)
Under the Slow–Roll Approximation, as 1, Cosmic Inflation is able to
occur. As the scalar field reaches the minimum energy density at the true
vacuum, (ϕ) approaches 1, and Inflation will end. Also worth noting is that
the Slow–Roll parameters correspond to measured quantities from
observation, which constrains models of Inflation with observational data.
25
The number of e–folds of Inflationary expansion is given by
a(tend )
N ≡ ln
=
a(ti )
Z
tend
H dt
(49)
ti
where ti is the initial time of Inflation, and tend is the time at the end of
Inflation. For most models, typically N ≥ 60, in order for these models to
correlate well with observation [22]. In terms of the Slow–Roll approximation,
the number of e–folds can also take the following form
a(tend )
N (t) = ln
=
a(ti )
Z
tend
ti
1
H dt ' 2
MP l
Z
ϕi
ϕend
V
dϕ
V0
(50)
where ϕend is when (ϕend ) = 1 and Inflation ends.
The deviations from smoothness in the scalar fields, or quantum
fluctuations, are important to modern research into Cosmic Inflation. As
cosmology is now largely a precision area of research, quantum fluctuations of
the scalar field are considered in a statistical setting [9]. The resulting
mathematical formalism of Gaussian statistics of Fourier modes leads to very
important research and predictions that correspond to Cosmic Inflation, and
small anisotropies in the CMB, indicated in the COBE data [12], which later
were precisely measured to a fine degree in the WMAP data, as well as the
more recent Planck data [24, 25, 26]. The statistical nature of the quantum
fluctuations are then related to parameters associated with actual data from
the CMB, which then constrains theoretical models of Inflation, as to clearly
indicating which model is observationally valid. Thus, predictions of Cosmic
Inflation correspond to large–scale structure observed in the CMB. Also,
26
primordial gravitational waves are predicted in models of Inflation, and
research into the detection of B–mode polarization of the CMB anisotropies is
actively being pursued, with the general expectation that at some level it will
provide a strong indication of Inflation occurring early in the Universe
[9, 26, 27, 28].
Issues of Cosmic Inflation
In 2011, Paul J. Steinhardt, an early contributor to the formation of
the Inflationary idea of the Universe, publicly discussed in [29] the serious
issues that are present in Inflation. As stated previously, observational data
from Planck and other spacecraft [24, 25] lead to support for Inflation
parameters corresponding to many models of Inflation. The degree to which
these parameters agree with the data is very precise. It is this precision that
some models of Inflation must have that lead to the implication that even the
slightest imprecise value obtained by a model can lead to predictions that
vary drastically with observation. Deviations from the data are specifically
related to the way the scalar potential and the scalar field are arranged in
different models. This consequence of being slightly off is referred to as Bad
Inflation. As pointed out in [29], Roger Penrose has argued that the
probability of obtaining a Universe that began with an early period of
Inflation and is flat and uniform, is far smaller than the probability for a
non–inflationary Universe to become flat and uniform. Penrose’s result was
supported by other researchers in [30], under a similar analysis of
extrapolations backwards in time under the laws of physics. Thus Cosmic
27
Inflation is the least probable outcome, and in addition, models of Inflation
are extremely finely tuned in order to agree with observational data.
As discussed in [31], and [29], when inflation begins, it never stops,
which is due to the random quantum nature of the accelerated expansion.
This eventually leads to infinitely many Universes with properties like the
ones observed, and infinitely many without, with similar analysis supported in
[32]. As described in [29], the issue arises when the predictive component of
Inflation is discussed in terms of quantum effects, where eternal Inflation is
valid. This is disconcerting for standard Cosmological Inflation and although
detection of gravitational waves by the B–mode polarization of the CMB
would still support the theory, the questions raised in [29] and by other
researchers remain unanswered.
Loop Quantum Cosmology
The ability to have a description of physics near a singularity of
spacetime is important for the discussion of the early Universe and the
scenario provided by Inflationary Cosmology. As there are issues associated
with Cosmic Inflation, as described previously, there are also issues with the
spacetime background near the small, hot, and infinitely dense region of
spacetime required by the Big Bang model. From General Relativity, matter
is directly related to the geometry of spacetime, but there is an implicit
assumption of a smooth, continuous spacetime background, a background
which is known to break down at singularities [33]. On a small scale,
presumably when the Universe is in a small, hot, and dense state, quantum
28
theory has a role, not only in the fluctuations of the scalar fields discussed
previously, but perhaps in the gravitational field itself. Loop Quantum
Gravity (LQG) is the quantum gravity model in which all facets of General
Relativity are quantized, for both the geometry of the spacetime, as well as
the corresponding matter [34, 35, 36, 37]. The Riemannian geometry used in
(8) is replaced by a quantum Riemannian geometry, developed in [38, 39, 40].
A very important property of LQG is the violation of the Stone–von
Neumann theorem in quantum theory, arising from the non–existence of a
local quantum operator corresponding to the classical connection [41]. This
leads to a quantum theory that is different from a Schrödinger representation,
along with new commutation relations. Loop Quantum Cosmology (LQC) is a
symmetry–reduced version of LQG, which closely follows the same methods of
derivation and eventually results in a quantum Hamiltonian constraint
equation from the canonical formulation of General relativity. Computing the
Hamiltonian constraint equation in [41] leads to the modified Friedmann
equation (k, Λ = 0), of the form
ρ
8πG
ρ 1−
H =
3
ρm
2
which features quantum corrections to (23), where the maximum energy
density, ρm ,
ρm =
3
8πGγ 2 λ2
(51)
29
where
√
λ2 = 4 3πγlP2 l .
The Planck length, lP l , is defined by lP2 l = G~ (c = 1), and γ is the
Barbero–Immirzi parameter of LQG [41], which corresponds to a value
usually inferred from entropy calculations of black holes [36]. The second
modified Friedmann equation (Raychaudhuri equation) takes the form of
ä
4πG
ρ
ρ
=−
ρ 1−4
− 4πGP 1 − 2
.
a
3
ρm
ρm
(52)
In the next section, these equations will be used to obtain complete solutions
for the scale factor, energy density, and the scalar field.
SOLUTIONS TO THE MODIFIED FRIEDMANN EQUATION FROM
LOOP QUANTUM COSMOLOGY
In this section we obtain solutions to the modified Friedmann and
Raychaudhuri equations, which are derived directly from Loop Quantum
Cosmology [41]. Specifically, the scale factor, a (t), is obtained, as well as its
derivatives, which correspond to two regions of the modified Friedmann
equation as the energy density, ρ, decreases from its maximum value. The
expression obtained will specify the scale factor in terms of the equation of
state parameter, w, which leads to further analysis of behavior under different
values. Finally, equations for the scalar field and energy density are presented
under a fast roll assumption, ϕ̇2 V (ϕ). Similar work of obtaining solutions
has been presented [42], though not to the extent of the calculations shown
here.
Beginning with the modified Friedmann equation, the initial attempt at
a solution for the scale factor resides with the analysis of an expression for Ḣ:
8πG
ρ
ρ 1−
.
H =
3
ρm
2
Solving (51) for the energy density term, ρ, upon which further analysis of
two corresponding regions will be made, we obtain
ρm
ρ=
2
q
1± 1−
3H 2
2πGρm
.
(53)
31
To obtain an expression for Ḣ, in terms of H, the Raychaudhuri equation,
(52), is used
ä
4πG
ρ
ρ
=−
ρ 1−4
− 4πGP 1 − 2
a
3
ρm
ρm
remembering that
Ḣ + H 2 =
ä
.
a
This implies:
Ḣ =
ä
− H2 .
a
(54)
Using (51), (52), and (54), Ḣ takes the form,
ä
4πG
ρ
ρ
ρ
8πG
Ḣ = = −
ρ 1−4
ρ 1−
− 4πGP 1 − 2
−
.
a
3
ρm
ρm
3
ρm
After simplification, the expression takes the form of
ρ
Ḣ = −4πG (1 + w) ρ 1 − 2
ρm
where w =
P
,
ρ
(55)
the equation of state. Using the expression obtained for the
energy density, (53), we obtain an equation for Ḣ, which serves as the basis
for determining solutions for the scale factor, a (t):
Ḣ = ±2πG (1 + w) ρm 1 ±
q
1−
3H 2
2πGρm
q
1−
3H 2
2πGρm
.
(56)
32
Figure 2 is a graphical representation of two regions of the modified
Friedmann equation, which arises when considering the positive (+) and
negative (−) values of (56), from which the scale factor is obtained. The
solutions are obtained from the two regions, as follows.
2 π G ρm
H2
3
II
I
ρm
ρm
2
Figure 2. Graphical representation of two regions of solutions corresponding
to H. Region I corresponds to t0 ≤ t ≤ tmax and Region II corresponds to
t > tmax .
Region I: t0 ≤ t ≤ tmax
In Region I, we consider first the positive value of (56), which is
Ḣ = +2πG (1 + w) ρm
q
1+ 1−
3H 2
2πGρm
q
1−
3H 2
2πGρm
(57)
ρ
33
q
letting x =
3
H,
2πGρm
which implies that
r
ẋ =
3
Ḣ
2πGρm
or
r
Ḣ =
2πGρm
ẋ
3
(58)
leads to the following form for (57)
r
n
op
p
2πGρm
2
ẋ = 2πG (1 + w) ρm 1 + 1 − x
1 − x2
3
(59)
upon separation of variables, the following integral is obtained
Z
x
xi
p
dx0
n
o√
= 6πGρm (1 + w) (t − ti ) .
√
1 + 1 − x02
1 − x02
(60)
The integral on the LHS results in
Z
x
xi
x
√
02 1− 1−x dx
o√
n
=
.
√
x0
1 + 1 − x02
1 − x02
xi
0
(61)
Thus, we obtain the following expression
1−
√
p
p
1 − x2 1 − 1 − x2i
−
= 6πGρm (1 + w) (t − ti ) .
x
xi
(62)
34
Letting Ci =
q
2
1− 1−xi
,
xi
then A = Ci +
√
6πGρm (1 + w) (t − ti ), the equation
simplifies to the form,
x=
2A
1 + A2
(63)
or
r
x=
√
2 Ci + 6πGρm (1 + w) (t − ti )
3
H=
2
√
2πGρm
1 + Ci + 6πGρm (1 + w) (t − ti )
from which a form of the Hubble parameter in Region I, HI , is obtained as
r
HI =
√
8πGρm
Ci + 6πGρm (1 + w) (t − ti )
.
√
3 1 + Ci + 6πGρm (1 + w) (t − ti ) 2
(64)
An expression for the scale factor a (t), can be obtained upon integration of
(64). From the definition of the Hubble parameter, H =
ȧ
a
=
d
dt
ln a, we obtain
the integral
r
ln a|aaIi
=
8πGρm
3
Z
√
6πGρm (1 + w) (t − ti )
2 dt .
√
1 + Ci + 6πGρm (1 + w) (t − ti )
Ci +
(65)
From the substitution of A, the integral becomes
aI
2
1
ln =
ai
3 (1 + w)
Z
A
Ai
A
dA .
1 + A2
(66)
35
The integration results in the following
2
1
3 (1 + w)
Z
A
A
A
1
2
0
ln
A
+
1
dA
=
Ai
1 + A2
2
2
1
A +1
aI
ln
ln =
ai
3 (1 + w)
A2i + 1
2
1
aI
A + 1 3(1+w)
=
ai
A2i + 1
(68)
) 1
2
√
6πGρm (1 + w) (t − ti ) + 1 3(1+w)
.
Ci2 + 1
(70)
Ai
(67)
(69)
or in the following terms
aI
=
ai
As xi =
q
(
3
H,
2πGρm i
Ci +
with the following assumption of xi 1, Ci takes the
form
1 − 1 − 21 x2i
Ci ≈
.
x0
This implies
1
Ci ≈ xi approaches 0 .
2
Thus for the scale factor, aI , the following expression results
aI = ai 6πGρm (1 + w)2 (t − ti )2 + 1
1
3(1+w)
.
(71)
36
A time derivative of (71) results in the following expression for ȧI ,
1 −1
ȧI
= 4πGρm (1 + w) (t − ti ) 6πGρm (1 + w)2 (t − ti )2 + 1 3(1+w) .
ai
(72)
An expression of äI is also obtained upon another time derivative, resulting in
the following,
1 −2
äI
= 6πGρm (1 + w)2 (t − ti )2 + 1 3(1+w) {4πGρm (1 + w) (1
ai
−2πGρm (1 + w) (1 + 3w) (t − ti )2 .
(73)
And the Hubble parameter, H, takes the form of
HI =
ȧI
4πGρm (1 + w) (t − ti )
=
.
aI
6πGρm (1 + w)2 (t − ti )2 + 1
(74)
It is also important to determine the value of t = tmax , which corresponds to
the maximum value of H, Hmax , and from (51),
r
Hmax =
2πGρm
=
3
ȧI
a
.
(75)
max
Thus, using (74), we obtain the expression for tmax as the following,
tmax = ti + √
1
6πGρm (1 + w)
.
(76)
37
Similarly, the maximum value of the second derivative of the scale factor,
äI = äI,max , can be obtained using (73) and (76),
1
äI,max
1
= 8πGρm 2 3(1+w) −1
ai
6
(77)
1
1
ämax = äi 2 3(1+w) −1 .
6
(78)
letting äi = ai 8πGρm
The energy density, ρ, can also be expressed in a similar form as the scale
factor, aI , which by using (53) and (74) becomes
v

2 
u
2
2

u
6πGρm (1 + w) (t − ti ) − 1 
ρm
t
1+
ρ=
2
2 
6πGρm (1 + w)2 (t − ti )2 + 1 
(79)
letting f 2 = 6πGρm (1 + w)2 (t − ti )2 , the expression ρI becomes
ρm
ρI =
2
2
2
(f + 1)
(80)
or
ρI =
ρm
.
6πGρm (1 + w)2 (t − ti )2 + 1
(81)
In addition to the expressions obtained, when considering scalar fields and
potentials, (37) allows for the ability to obtain an equation for both ϕ̇I (t) and
38
ϕ̇2
2
ϕI (t), with the assumption of
V (ϕ), which corresponds to stiff matter,
or w = 1. Using (37),
ϕ̇2I ' 2ρ .
(82)
Using (81), ϕ̇ becomes
ϕ̇I '
p
2ρm q
1
.
2
(83)
2
6πGρm (1 + w) (t − ti ) + 1
This implies:
Z
ϕ
dϕI '
p
Z
ϕi
Letting
√
t
2ρm
ti
1
q
dt .
6πGρm (1 + w)2 (t − ti )2 + 1
(84)
6πGρm (1 + w) (t − ti ) = sinh ξ, the integration takes the form,
√
Z ξ
2ρm
cosh ξ dξ
√
p
ϕI − ϕi '
6πGρm (1 + w) ξi
sinh2 ξ + 1
(85)
and the integration leads to
√
t
p
1
1
sinh−1
6πGρm (1 + w) (t − ti ) .
(1
+
w)
ti
3πG
Using the identity sinh−1 x = ln x +
1
1
ϕI ' ϕi + √
3πG (1 + w)
p
√
(86)
x2 + 1 , ϕ̇I (t) takes the form
q
2
2
6πGρm (1 + w) (t − ti ) + 6πGρm (1 + w) (t − ti ) + 1 .
(87)
39
Region II: t > tmax
Completing the derivations of expressions, in Region II, considering the
(−) value of (56),
Ḣ = −2πG (1 + w) ρm 1 −
q
1−
3H 2
2πGρm
q
1−
3H 2
2πGρm
.
(88)
Similar to the process of Region I, to obtain an expression for the Hubble
q
3
H,
parameter in Region II, HII , letting y = 2πGρ
m
r
ẏ =
3
Ḣ .
2πGρm
(89)
And as before, we obtain the following integration,
Z
y
ymax
p
dy
op
n
= − 6πGρm (1 + w) (t − tmax ) .
p
1 − 1 − y2
1 − y2
q
2
1+ 1−ymax
,
ymax
As done in Region I, and letting Cmax =
√
B = Cmax + 6πGρm (1 + w) (t − tmax ), we obtain
r
(90)
and then
3
2B
HII =
.
2πGρm
1 + B2
This implies:
r
HII =
√
8πGρm
Cmax + 6πGρm (1 + w) (t − tmax )
.
√
3 1 + Cmax + 6πGρm (1 + w) (t − tmax ) 2
(91)
40
q
2
1+ 1−ymax
,
ymax
Since Cmax =
q
m
Hmax = 2πGρ
=
3
ymax =
q
3
H ,
2πGρm max
and
ȧ
a max
ymax = 1
⇒ Cmax = 1 .
Thus, in Region II, we have:
r
HII =
√
1 + 6πGρm (1 + w) (t − tmax )
8πGρm
.
√
3 1 + 1 + 6πGρm (1 + w) (t − tmax ) 2
(92)
The derivation of the scale factor of Region II, aII (t), is equivalent to Region
I, thus we obtain
aII
=
amax
where g = 1 +
g2 + 1
2
gmax
+1
1
3(1+w)
(93)
√
6πGρm (1 + w) (t − tmax ). The value of g (tmax ) = gmax = 1,
and for Region II, the scale factor becomes
aII
=
amax
g2 + 1
2
1
3(1+w)
.
(94)
The time derivative of the scale factor , ȧII takes the form
ȧII
=
amax
r
8πGρm
g
3
g2 + 1
2
1
3(1+w)
−1
.
(95)
41
And thus äII becomes
äII
= −4πGρm
amax
g2 + 1
2
1
3(1+w)
−2
1
(1 + 3w) g 2 − 3 (1 + w) .
3
(96)
The value of a (tmax ) = amax is obtained using the expressions in Region I,
using a (t) from (76) and tmax from (71),
1
amax = ai 2 3(1+w)
(97)
or rewritten in a familiar form,
1
amax = ai · e 3(1+w) ln 2 .
(98)
A calculation of td , the time of deceleration, is obtained when considering
when äII = 0, which gives
td = tmax + √
s
1
6πGρm (1 + w)
!
3 (1 + w)
−1
(1 + 3w)
.
(99)
The derivation of energy density ρII , ϕ̇II , and ϕII are equivalent to that of
Region I, thus, considering the negative value of (53), and using (92), we
obtain
ρII =
ρm
.
g2 + 1
(100)
42
And for ϕ̇II and ϕII , we obtain
p
1
2ρm p
2
g +1
n p
√ o
1
1
ln g + g 2 + 1 − ln 1 + 2
ϕII ' ϕmax + √
.
3πG (1 + w)
ϕ̇II '
(101)
(102)
ANALYSIS OF SOLUTIONS IN REGIONS I AND II
Through simple substitution of the modified Friedmann equation, as
well as the Raychaudhuri equation, it was possible to obtain expressions for
the scale factor, a, the Hubble parameter, H, energy density, ρ, and a scalar
field, ϕ. Minimal assumptions were used in the derivations of the expression,
which have a dependence on the equation of state, w, as well as time t. In this
section, the expressions will be discussed in terms of their qualitative features,
comparison to other formalism present within the literature, as well as scale
factor leading to a solution to the horizon problem.
Inflation in LQC
As stated previously, the basic definition of an Inflationary Universe is
ä > 0 .
(103)
The expressions in both regions for the second time derivative for the scale
factor, ä, does meet the requirement, provided that the equation of state,
w > −1. The modified Friedmann equation of (51) assumes homogeneity and
isotropy, as well as a spatially flat geometry and a Cosmological Constant
equal to zero. The requirement on the equation of state does not appear to be
surprising as the effects of Vacuum energy or Cosmological Constant are not
incorporated into these equations. Another aspect of Inflation in the standard
cosmological setting is the amount of Inflation, or the number of e–folds of the
44
Inflationary expansion, which from (49) gives
N = ln
a(tend )
.
a(ti )
(104)
From (94) and (98), in Region II (t > tmax ), at the end of Inflation, where
t = td , the scale factor takes the form
aII (td ) = ai
2 (2 + 3w)
1 + 3w
1
3(1+w)
.
(105)
And (105) is general, yet in standard Cosmic Inflation, equations of state for
radiation and matter are never considered for an Inflationary era. Assuming
that w = 1 for stiff matter, it is clear that the number of e–folds is too small,
in respect to what is acceptable in classical cosmology [14]. Therefore, the
duration of accelerated expansion would seem to be short lived. Results of
independent investigations reinforce the conclusion of minimal inflation
[41, 43]. Also, in [44], the form of the scale factor is assumed to have power
law behavior, which we obtained in the general case. Inflation, by basic
definition, will occur for ρ >
ρm
.
2
The accelerated expansion ends in Region II
just past the top, as seen in Figure 2. Expressions for the Hubble parameter,
H, in both regions, are positive for t > ti if w > −1. Also, from the
derivations with the stiff matter assumption, the scalar field, ϕ, has a form
that is dependent on the equation of state, as well as time. Similar forms of
the scale factor, its derivatives, the Hubble parameter, and scalar fields have
also been obtained in [42]. The authors in [42] also used the assumption of
stiff matter (w = 1), or a fast rolling scalar field,
ϕ̇2
2
V (ϕ). However,
45
neither [42], nor the authors in [45] distinguish evolution of the energy density
of the modified Friedmann equation in the two regions, as was done in the
previous section. Thus, the solutions of the modified Friedmann equation
have provided a new path of research into cosmology that addresses the
quantum nature of the Universe. And we have also addressed the horizon
problem, as discussed in previous sections.
The Horizon Problem
As a final discussion, LQC presents an opportunity to address the
horizon problem, at least in some manner. In [46, 47], the authors claim that
the horizon problem is solved in accordance with the LQC contribution given
by the modified Friedmann equation (51). Their solutions to the horizon
problem either includes an infinite number of e–folds, or a large number of
e–folds, provided during the quantum bounce, or very near ρ ' ρm , where the
Inflationary phase occurs. With respect to the number of e–folds, (105)
implies a very small number, as referenced in [47], which is referred to as Bad
inflation. Also, the calculations in [46, 47] yield a particle horizon near the
start of Region I in Figure 2, which they claim is equal to infinity, thus solving
the horizon problem, as the particle horizon from the time of decoupling to
now will be large but finite. The expressions obtained in the previous section
were used to investigate the claim made by these authors, and in the form of
(25), it was shown that
dHRM →dec dHdec→now .
(106)
46
Here dHRM →dec is the particle horizon from the beginning of Region I, setting
ti = 0, to decoupling, tdec , which is divided into the following integrals,
Z
tRM
dHRM →dec =
0
dt
+
aI (t)
Z
tdec
tRM
dt
aI (t)
(107)
where tRM is the time at Radiation–Matter equality. Also, dHdec→now has the
form of
Z
tnow
dHdec→now =
tdec
dt
aI (t)
(108)
Upon substitution of tmax in (76) in aII (t) in (94), and using (97),
aII (t) = aI (t), and the scale factor is explicitly continuous in both regions.
Thus, the scale factor in Region I, aI (t), is used for the calculations. For the
first integral in (107), w = 1/3. For the second integral in (107), as well as the
integral in (108), w = 0, based on the different eras of evolution of the
Universe. A computation of (107) and (108) make it clear that (106) is valid,
without the integration in (107) being infinite. Further analysis is needed to
provide more detail as to whether the calculation should remain plausible.
The flatness problem in cosmology is also addressed in LQC, but it requires a
more general quantum Hamiltonian constraint than referenced in [41], where
spatial flatness was assumed before the modified Friedmann equations were
obtained. Nevertheless, the flatness problem should also be addressed in LQC
from a rigorous analysis based on first principles.
CONCLUSION AND SUMMARY
In summary, cosmology has been an area of research with many
developments, with major insights arriving in the early twentieth century.
From the observations made by Edwin Hubble of an expanding Universe [1],
to the discovery of the Cosmic Microwave Background, by Penzias and Wilson
[10], the foundation and successes of the Big Bang model has been reviewed in
this work. Along with the successes were the shortcomings of the model, as
more knowledge was obtained from spacecraft observations. The ideas and
proposal of Cosmic Inflation have also been discussed in detail, as a solution
to the problems in the Big Bang model. From the assumption of homogeneity
and isotropy, as well as the use of General Relativity, the Friedmann
equations were obtained. The Friedmann equations serve as the basis of
Cosmic Inflation, which strongly depends on the behavior of a scale factor of
an expanding Universe. Also important is the use of scalar fields and
potentials, used to describe the energy density and negative pressure relations
implied during Inflation, noted in Figure 1. The precision–based cosmology
and observational analysis was also discussed, as the current state of Inflation
is now described in terms of predictions made about large scale structure
observed in the CMB anisotropy. Also discussed was the implication of a
detection of B–mode polarization, indicating a primordial gravitational wave
background. Along with the forefront of Cosmic Inflation research, it was also
important to discuss the issues some researchers have with the Inflationary
era, which seem to be well founded.
48
In conclusion, the main analysis of this work came along with the
discussion of a modified Friedmann equation, which comes from LQC [41], an
application of LQG, a theory of quantum gravity [34, 35, 36, 37]. Solutions to
the modified Friedmann equation were presented, as well as their qualitative
features in comparison to what appears in Standard Cosmological Inflation.
Also, as discussed in the previous section, there are some articles in the
literature that have similar results to those derived in this work, although the
mathematical techniques are different. Finally, an attempt at a solution to
the horizon problem was introduced, which differs from work presented
previously [46, 47].
In future work, the flatness problem should also be addressed without
assumption, as observational data support a spatially flat geometry of the
Universe. Quantum gravity seems like the logical background in which the
smallest scale of the Universe should be studied, not solely to assess the
problems of the Big Bang model, but to verify if the theory is a valid
theoretical framework of physics. Further research and analysis will indicate if
the LQC–inspired formalism of an expanding Universe is appropriate, and
valid as a description of observed phenomena in the Universe. As LQC is a
recently proposed theory of cosmology, future results will decide what is the
correct quantitative description of the early Universe, as science will to seek
answers to Nature’s largest questions.
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