Download Particle Creation in Inflationary Spacetime

University of Amsterdam
BSc thesis
Particle Creation in Inflationary
Spacetime
Author:
Christoffel Hendriks
Supervisor:
dr. Alejandra Castro
Anich
July 14, 2014
Data
Title: Particle Creation in Inflationary Spacetime
Author: Christoffel Hendriks
E-mail: [email protected]
Studentnumber: 10218580
Study: Bachelor’s Physics and Astronomy
Supervisor: dr. Alejandra Castro Anich
Second reviewer: dr. Jan Pieter van der Schaar
Institute for Theoretical Physics
Universiteit van Amsterdam
Science Park 904, 1098 XH Amsterdam
http://iop.uva.nl/itfa/itfa.html
Back where I come from, we have universities, seats of
great learning, where men go to become great thinkers.
And when they come out, they think deep thoughts and
with no more brains than you have! But they have one
thing you haven’t got - a diploma.
The Wizard of Oz (1939)
1
Abstract
The particle creation in two different inflationary spacetime models is computed.
An introduction to quantum scalar fields is given and the relation between particle creation and the Bogoliubov coefficients is derived. The first spacetime model
used is the Friedmann-Robertson-Walker(FRW) space, representing a smoothly
expanding universe. The Bogoliubov coefficients for the transition between the
infinite past and future field modes are computed. It is concluded that particle
creation is a property of FRW space. The second spacetime model is de Sitter
space in global coordinates, representing an exponentially expanding universe.
The particle creation is computed by setting up a scattering problem for a scalar
field, propagated from past to future infinity. A reflectionless potential is found
for de Sitter space in odd dimensions, which is explored further algebraically.
Relating the scattering coefficients to the Bogoliubov coefficients revealed that
the infinite past vacuum state evolves into the infinite future vacuum state. It
is concluded that particle creation is a property of curved spacetime, although
not every curved spacetime model necessarily leads to particle creation.
Contents
1 introduction
2
2 Quantum field theory and Bogoliubov transformation
2.1 Quantum field theory . . . . . . . . . . . . . . . . . . .
2.2 Canonical quantization . . . . . . . . . . . . . . . . . . .
2.3 Defining vacuum and the number operator . . . . . . . .
2.4 Bogoliubov transform . . . . . . . . . . . . . . . . . . .
2.5 Particle creation in curved spacetime . . . . . . . . . . .
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3 Particle creation in Friedmann-Robertson-Walker space
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3.1 The Friedmann-Robertson-Walker metric . . . . . . . . . . . . . 14
3.2 Particle creation in FRW . . . . . . . . . . . . . . . . . . . . . . 16
4 De Sitter space
4.1 De Sitter hyperboloid . . . . . . . . . . . . . . . . . . . .
4.2 Coordinate systems of de Sitter space . . . . . . . . . . .
4.2.1 Global coordinates . . . . . . . . . . . . . . . . . .
4.2.2 Static coordinates . . . . . . . . . . . . . . . . . .
4.2.3 Planar coordinates . . . . . . . . . . . . . . . . . .
4.3 Transmission and reflection in global de Sitter space . . .
4.3.1 Direct solution of the Poschl-Teller wave equation
4.3.2 Approaching reflectionless potentials algebraically
4.4 Particle creation in de Sitter space . . . . . . . . . . . . .
4.5 De Sitter space in even dimensions . . . . . . . . . . . . .
5 Conclusion and discussion
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1
Chapter 1
introduction
In the first second after the Big Bang, the start of our universe, the universe
went through a period of exponential expansion[1]. This stage is known as the
inflationary period. Particle creation is believed to be a property of curved
spacetime. The curved spacetime model for an exponentially expanding universe is known as de Sitter space. During the inflationary period the universe is
assumed to be approximately de Sitter. Hence, the process of particle creation
could have greatly influenced the early develompent of the observable universe.
The purpose of this thesis is to explore the property of particle creation in curved
spacetime. This is done by embedding a quantum field theory in inflationary
curved spacetime.
The curvature of spacetime affects the excitation of the quantum scalar field.
As time elapses the curvature could excite the field, which is related to the number of particles in the system. This makes it impossible to define an absolute
vacuum state without particles for any observer in spacetime. Hence, curvature
of spacetime can induce a process of particle creation. In this thesis is tried to
confirm that particle production is a property of curved spacetime by exploring
two different expanding spacetime models.
The first model is Friedmann-Robertson-Walker space, representing a smoothly
expanding universe. There are different ways of obtaining the particle production in curved spacetime. With Friedmann-Robertson-Walker space it is done
by comparing the scalar field at the infinite past with the infinite future. The
second explored model is the spacetime representing an accelerated expanding
universe known as de Sitter space. Here the particle creation is obtained by
setting up a scattering problem for a normalized scalar field propagated from
past to future infinity.
2
The next chapter will give an introduction to quantum field theory and the
concept of particle creation. The third chapter will present the particle creation
in Friedmann-Robertson-Walker space. In the fourth chapter the same is done
for de Sitter space, and the last chapter will contain a discussion and conclusive
words about the subject.
3
Chapter 2
Quantum field theory and
Bogoliubov transformation
The start of this chapter will give an introduction to quantum field theory in
flat Minkowski space. Next will be shown that the properties of quantum fields
lead to particle creation in curved spacetime using Bogoliubov transformations.
Throughout this thesis we will use natural units:
c = } = 1.
2.1
(2.1)
Quantum field theory
Quantum mechanics combined with relativity violates the preservation of the
number of particles in a system[2]. At very small scales particle anti-particle
pairs can pop into existence. At any point in space, even empty space, these
particle pairs can appear and disappear. In quantum mechanics we had a finite
number of spatial degrees of freedom, equal to the amount of dimensions. We
now have a degree of freedom at any point in space, which is infinitely large.
To write down the Schrödinger equation for a single particle will therefore fail
as these particle pairs are neglected. Instead, we have to use a theory of fields.
A field is a function defined anywhere in space. Hence, the infinite amount
of degrees of freedom can be represented as a field φa (~x, t). The quantum
mechanical space operator is demoted to a variable of the field. The label ’a’
is the denotation of the dimensions in Lorentz invariant index notation. To
compute the particle creation in curved spacetime it is necessary to approach
every degree of freedom independently. We therefore need to write down φa (~x, t)
as a discrete summation of all these degrees of freedom. Why this is allowed for
a free field, a field without interactions, will be shown in the next section.
4
2.2
Canonical quantization
This section will quantize a free quantum field following [2]. For more information about quantum fields we also refer to [3]. Before this can be done we need
to look at the dynamics of the field φa (~x, t). Classically dynamics in space could
be determined with the Lagrangian, a function of the variables ~x and ~x˙ . We can
do the same thing for the field as a function of φa (~x, t) and φ˙a (~x, t). Because
we now have space as a variable it will also depend on ∂~x φa (~x, t). We define the
Lagrangian in terms of a so called Lagrangian density. For the 3+1-dimensional
case we have:
Z
L(t) = dx3 L(φa , ∂µ φa ).
(2.2)
This enables us to write down the action in terms of all dimensional variables:
Z
Z
S = dtL(t) = dx4 L.
(2.3)
From here on the Lagrangian density L will be called Lagrangian. An equation
of motion can be determined by the principle of least action:
δS = 0.
(2.4)
However, we are not looking a classical particle moving from point A to B. A
more correct way of viewing the action would be a field evolving from an initial
state to a final state. We should minimize this evolution. Using integration by
parts we have
Z
∂L
∂L
δ (∂µ φa )
δφa +
δS = dx4
∂φa
∂ (∂µ φa )
"
#
Z
∂L ∂L
∂L
4
= dx
− ∂µ
δφa = 0.
(2.5)
δφa + ∂µ
∂φa
∂ ( ∂µ φ a )
∂ (∂µ φa )
We can assume that the change in path of the field at spatial infinity is zero
δφa (~x, t) = 0, ~x → ∞. The second term is an integral of the derivative of this
path so will be equally zero. We arrive at the Euler-Lagrange equations of
motion for fields:
∂L
∂L
∂µ
−
= 0.
(2.6)
∂ (∂µ φa )
∂φa
Next we will derive the Klein-Gordon equation for fields. This is the equation
any free field should satisfy so it will be used frequently in this thesis. The
Lagrangian for a real scalar field is given[3]:
1
1
L = η µν ∂µ φa ∂ν φa − m2 φ2a
2
2
1 ˙ 2 1
1
2
= φa − (∇φa ) − m2 φ2a .
2
2
2
5
(2.7)
Here is chosen for the signature (+1, −1, −1, −1) for the Minkowski metric tensor. The variable m is the total mass of the field. Now solving equation (2.6)
we have:
∂µ (∂ µ φa ) + m2 φa = 0.
(2.8)
Denoting the Minkowski Laplacian ∂µ ∂ µ as we have arrived at the KleinGordon equation for fields. There is another property of fields that we can
determine using the Lagrangian. For classical space the momentum of a particle
can be computed:
p=
∂L
.
∂ ~x˙
(2.9)
We can do the same for the field Lagrangian to obtain the analogous momentum
π a (~x, t) of the field:
π a (~x, t) =
∂L
.
∂ φ˙a
(2.10)
Notice that π a is the conjugate momentum of φa by the properties of index
notation and derivatives. We might not know precisely what the momentum
of a field means but we do know it shares the same relation with the field as
the classical momentum and space variable. A well known relation between
momentum and space are the canonical commutation relations:
[xi , pj ] = iδij ,
[pi , pj ] = [xi , xj ] = 0.
(2.11)
This commutation relation required for ’x’ and ’p’ to be promoted to operators
in quantum mechanics. We want to set up analogous commutation relations
for fields. We therefore have to promote φa and π a to operators in the same
way. To write down the commutation relation we can’t use the Kronecker delta
δij . Different spatial points in the field φa have different degrees of freedom.
Therefore φa and π a should only be non-commutating when measured at the
same place and time. As space has become a variable we have to use the Dirac
delta to give an expression for ’the same place’:
[φa (~x, t0 ), π a (~y , t0 )] = iδ 3 (~x − ~y ).
(2.12)
We can assume further that φa and π a commutate with themselves just like x
and p:
[φa (~x, t0 ), φa (~x, t0 )] = [π a (~y , t0 ), π a (~y , t0 )] = 0.
(2.13)
These equations are known as the equal time commutation relations. We return
to the computed Klein-Gordon equation for fields(2.8). We will show that a free
field satisfying this equation allows the quantization of the field. We have
2
∂
2
2
( + m2 )φa =
−
∇
+
m
φa = 0.
(2.14)
∂t2
6
We need a way to split up the infinite degrees of freedom of the field. This can
be done using a Fourier transform[4]:
Z
dk 3
~
φ(~x, t) =
eik~x φa (~k, t).
(2.15)
(2π )3/2
Putting this in the Klein-Gordon equation we get:
2
Z
dk 3
i~k~
x ∂
2
2
+
k
e
+
m
φa (~k, t) = 0.
∂t2
(2π )3/2
(2.16)
Denoting the frequency ω~2 = ~k 2 + m2 we can write the equation of motion for
k
φ(~k, t):
∂2
+ ω~k2 )φ(~k, t) = 0.
(2.17)
∂t2
This is exactly the equation of motion for the ordinary one dimensional harmonic
oscillator so we can treat the field φ(~k, t) the same. The harmonic oscillator is
exactly solvable for discrete k. The ladder operators of the harmonic oscillator
are defined
r
ω~k
iπ (±~k, t)
±
(φ(±~k, t) ∓
),
(2.18)
a~ ≡
k
2
ω~k
(
with φ(~k, t) the field and π (~k, t) its conjugate momentum. We can rewrite this
to substitute φ(~k, t) and π (~k, t):
1
φ(~k, t) = p
(a+ eiω~k t + a~− e−iω~k t ),
k
2ω~k −~k
r
ω~k − −iω t
π (~k, t) = i
(a e ~k − a+~ eiω~k t ).
−k
2 ~k
(2.19)
(2.20)
Here we used the general relation a~± (t) = a~± e±iω~k . Using the Fourier transform
k
k
we get: (2.15)
Z
dk 3
1
~
~
p
φ(~x, t) =
(2.21)
(a+ eiω~k t−ik~x + a~− e−iω~k t+ik~x ),
3/2
k
2ω~k ~k
(2π )
with momentum
Z
π (~x, t) =
r
ω~k
dk 3
~
~
i
(−a~+ eiω~k t−ik~x + a~− e−iω~k t+ik~x ).
3/2
k
k
2
(2π )
(2.22)
By using the commutation relations of φ (2.12) we can derive commutation
relations for the ladder operators. We assume that the ladder operators do not
commute with itself:
Z
r
dp3 dq 3 i ωq − + i(p~x−qy )
+ − −i(p~x−qy )
[
a
,
a
]
e
−
[
a
,
a
]
e
[φ(~x, t), π (y, t)] =
p
q
p
q
(2π )3 2 ωp
= iδ 3 (~x − y )
(2.23)
7
The dirac delta function can be written:
Z
dp3
3
δ (~x − y ) =
eip(~x−y ) .
(2π )3/2
(2.24)
Equations (2.23) and (2.24) can only be true when:
+
3/2 3
[a−
δ (p − q ),
p , aq ] = (2π )
(2.25)
and we had already assumed
−
+ +
[a−
p , aq ] = [ap , aq ] = 0.
(2.26)
By substituting:
u~k =
1
1
~
p
eiω~k t+ik~x ,
(2π )3/2 2ω~k
(2.27)
in equation (2.21) we can write down the field as a function of waves u~k . However, this k is still continuous value, prohibiting us of looking at the waves
independently. This problem can be solved by placing the field in a large but
finite box with sides L so that the volume becomes V = L3 . We can set up
boundary conditions for the field at the edges of the box:
φ(x = 0, y, z, t) = (x = L, y, z, t) = φ(x, y = 0, z, t)
= φ(x, y = L, z, t) = φ(x, y, z = 0, t) = φ(x, y, z = L, t).
(2.28)
This limits the frequencies of the wave functions and its index values k to discrete
values:
k=
2πn
,
L
n ∈ Z.
(2.29)
We have found a way of describing the field φ(~x, t) as a function of discrete
3/2 u so that:
values k. We can rescale the wave functions uk → ( 2π
k
L )
u~k = p
1
~
eiω~k t+ik~x
2Lω~k
(2.30)
This allows for the integral in equation (2.21) to be replaced by an infinite sum
over k:
Z
X
+ ∗
+ ∗
φ(~x, t) = dk 3 [a−
[ a−
(2.31)
k uk + ak uk ] =
k uk + ak uk ].
k
Now every degree of freedom can be approached independently as a harmonic
oscillator which was the goal of this section. As will be shown this property is
needed to compute the particle creation in curved spacetime.
8
2.3
Defining vacuum and the number operator
Before we can calculate the particle production we first need to establish a
definition for a state with particles and how the amount of particles is measured.
This will be done in this section following [5]. Quantum field theory prohibits
defining single particles for a system. The closest description of particles in a
free quantum field would be thePamount of energy, equal to the excitations of the
modes. Hence, a system with
nk particles has modes uk with excitation nk .
The operators ak+ and a−
k are known as the creation and annihilation operators
for the kth mode. Consider a state uk with nk particles. Using the commutation
relation of the ladder operators(??) in bra-ket notation the following statements
hold:
+
+ −
(a−
k ak − ak ak ) |nk i = |nk i .
(2.32)
From this equation we can derive the eigenvalues of the ladder operators:
p
nk + D |nk − 1i ,
a−
|n
i
=
k
k
p
+
ak |nk i = nk + 1 + D |nk + 1i ,
(2.33)
with D some constant we define equal to zero. The creation and annihilation
operators are each others complex conjugate. Therefore the expectation value
of the two operators combined will be equal to the amount of particles nk :
√ √
hnk | ak+ a−
k |nk i = hnk − 1| nk nk |nk − 1i = nk .
(2.34)
The operator ak+ a−
k is called the number operator, from here on denoted N̂k .
The expectation value of the summation all number operators will be equal to
the total amount of particles in the system:
X
N̂k ≡ ak+ a−
,
N̂
=
ak+ a−
(2.35)
k
k,
k
hφ| N̂ |φi =
X
nk .
(2.36)
k
The definition of a state with particles is therefore a state with a non zero
number operator. The lowest energy state is the state that becomes zero after
applying the annihilation operator. If every mode is in its lowest energy state the
expectation value of the number operator will be zero. The state of no particles,
the vacuum state, is therefore defined to be in this lowest energy state.
a−
k |0k i = 0,
(2.37)
hφ0 | N̂ |φ0 i = 0.
(2.38)
9
2.4
Bogoliubov transform
This section will link the mode expansion(2.31) to the particle production using
Bogoliubov transformations[5]. As stated in section 2.2 a field can be expanded
as a complete set of modes. As will be shown the particle production can be
expressed by using such mode expansions. However, following the basic principle
of Einstein’s theory of relativity there is no absolute reference frame. Hence,
defining a new reference frame allows us to expanded the same field as complete
set of different modes.
X
+ ∗
a−
(2.39)
φ(x, t) =
j uj (t, x) + aj uj (t, x) .
j
Defining a different annihilation operator automatically defines a new vacuum
state for this set:
a−
j |0j i = 0.
(2.40)
We want to show that the expectation value of the new number operator N̂
in the old vacuum state |0i will be non zero. This means that particles are
created between the transition from the old to the new mode expansion. Because
both sets form a complete basis for the field any mode can be expressed as an
expansion of the modes of the other set:
X
uj =
αjk uk + βjk u∗k .
(2.41)
k
Both the sets should satisfy the commutation relations for fields (2.12). This
can only be true when normalized[6]:
|αjk |2 − |βjk |2 = 1.
(2.42)
Using this normalization condition the inverse transformation will have the form:
X
∗
uj − βjk u∗j .
(2.43)
uk =
αjk
j
Equations (2.41) and (2.43) are called the Bogoliubov transformations and the
operators αjk and βjk are the Bogoliubov coefficients. As both of the different
mode expansions (2.31) and (2.39) represent the same field they can be equated.
This leads to an expression for the old annihilation and creation operators in
terms of the new annihilation and creation operators:
X
X
+ ∗
+ ∗
a−
(2.44)
a−
u
+
a
u
=
k
j uj + aj uj .
k k
k
j
k
10
Now using (2.41) to substitute uj and u∗j we get
X
+ ∗
a−
k uk + ak uk =
X
a−
j
X
j
k
X
∗
∗ ∗
uk
uk + βjk
αjk uk + βjk u∗k + aj+
αjk
k
k
(2.45)
=
XX
j
(a−
j αjk
∗
)uk
+ aj+ βjk
+ (a−
j βjk
∗
)u∗k ,
+ aj+ αjk
(2.46)
k
so that that for every kth mode:
X
−
+ ∗
+ ∗
+ ∗
∗
(a−
a−
j αjk + aj βjk )uk + (aj βjk + aj αjk )uk .
k uk + ak uk =
(2.47)
j
Because the modes and its conjugates are orthogonal both parts uk and u∗k can
be computed separately
X
+ ∗
a−
=
(a−
(2.48)
j αjk + aj βjk )
k
j
ak+
=
X
+ ∗
(a−
j βjk + aj αjk ).
(2.49)
j
In the same way the new annihilation and creation operators can be computed
by using equation (2.43):
X
∗ −
∗ +
(αjk
ak − βjk
ak ),
(2.50)
a−
j =
k
aj+
=
X
(αjk ak+ − βjk a−
k ).
(2.51)
k
The operator a−
k is defined as the annihilation operator for the old mode expansion of the field so for the vacuum state a−
k |0i = 0. However, the other vacuum
state for the new mode expansion will not necessarily be annihilated by a−
k:
a−
k |0i =
X
+ ∗
(a−
j αjk + aj βjk ) |0i
(2.52)
+
(αjk a−
j |0i + βjk aj |0i)
(2.53)
βjk |1i 6= 0.
(2.54)
j
=
X
j
=
X
j
Therefore, the expectation value for the number operator will not necessarily
be zero in the new vacuum state either:
X
h0| N̂k |0i = h0| ak+ a−
|βjk |2 .
(2.55)
k |0i =
j
11
Similarly, the new annihilation operator not necessarily annihilates the old vacuum state:
X
∗
−βjk
|1i ,
(2.56)
a−
j |0i =
k
h0| N̂ j |0i =
X
|βjk |2 .
(2.57)
k
Although starting out in a vacuum state without particles, an observer with a
different representation will encounter a non-vacuum particle state. Hence, by
changing the representation of the field particles are created.
2.5
Particle creation in curved spacetime
In this thesis the particle creation is explored between the infinite past and the
infinite future for different curved spacetime models. Because of the spacetime
curvature the modes will look different at past infinity than future infinity.
Hence, a mode expansion like (2.31) will also be different in the past and future
infinity limit. We therefore have two different mode expansions representing the
same field: One with mode solutions for the past infinity limit, from here on
called the in-states uin
k , and one with past infinity mode solutions, from here on
the out-states uout
.
Following
equation (2.41) the in-states can be represented
k
in terms of the out-states:
X
out∗
uin
αkj uout
.
(2.58)
j + βkj uj
k =
j
When the difference between the in-states and out-states is only time dependent the spatial part of the modes will not be effected. Therefore the in-state
expansion will have the same k modes as the out-state expansion. All modes are
defined to be orthogonal so only the two out-state modes with the same index
can contribute to the Bogoliubov transformation (2.58):
out
out∗
uin
k = αkk uk + βk−k u−k .
(2.59)
We will discuss the concept of particle creation as a result of curved spacetime.
irst will be shown that there is no particle creation in flat Minkowski space. In
section 2.2 we solved the Klein-Gordon equation(2.8) assuming a flat Minkowski
metric tensor, independent of time or space. We found an exact solution for field
in terms of the modes uk :
u~k =
1
1
~
p
eiω~k t+ik~x .
3/2
2ω~k
(2π )
(2.60)
Varying the time or space variables would not change the Klein-Gordon equation
so these are the solution for any point in spacetime. We can therefore say that
out
uin
k = uk .
12
(2.61)
Comparing this with equation (2.59) we conclude that αk = 1 and βk = 0 for
flat space. The amount of particles created is related to the second Bogoliubov
coefficient(2.55). We can therefore conclude that there is no particle creation
in flat space. However, for curved spacetime we have a metric tensor that is
not independent of time or space. The curved spacetime models discussed in
this thesis have a time dependent metric tensor. The Laplacian in curved
spacetime is defined [4]:
√
1
≡ √ ∂µ ( −gg µν ∂ν ).
−g
(2.62)
When the metric tensor gµν depends on time, the Laplacian will change when
varying the time. Consequently, we have a different Klein-Gordon differential
equation for the field at the past and future infinity limit. We can conclude for
curved spacetime
out
uin
k 6= uk
(2.63)
We chose a space independent metric tensor so equation (2.59) still holds. It
should therefore be possible to write the difference between the in- and outstates in terms of the conjugate out-state. The result is a non-zero value for the
second Bogoliubov coefficient. We can conclude that particle creation should be
possible as a consequence of the spacetime curvature. For two different curved
spacetime models the particle creation will be computed in this thesis. The first
spacetime model to be explored will be the so called Friedmann-RobertsonWalker space.
13
Chapter 3
Particle creation in
Friedmann-RobertsonWalker
space
In this chapter the particle creation in Friedmann-Robertson-Walker space, from
here on FRW, will be explored[5]. The in- and out-states will be computed for
the past and future infinity limit. Finally the Bogoliubov coefficients and the
particle creation will be derived by trying to write the in-states in terms of the
out-states.
3.1
The Friedmann-Robertson-Walker metric
FRW is a spacetime model representing a universe that undergoes a smooth
expansion. It is a conformally flat spacetime, meaning that its metric tensor
can be represented as a function times the flat Minkowski metric tensor:
gµν = Ω2 (x)ηµν .
(3.1)
Here is gµν the FRW metric tensor, ηµν the flat Minkowsi metric tensor and
Ω2 (x) some function depending on time or space. For the calculation of the
particle production between the infinite past and future limit is chosen for a
1+1-dimensional FRW universe. Its line element can be represented by the
equation
ds2 = dt2 − a2 (t)dx2 .
(3.2)
The time dependent function a2 (t) is called the scale factor and the cause of the
expansion. It is related to the redshift. This can be shown conducting a thought
14
experiment with two light rays, one sent at time t1 and one at time t1 + δt1 [7].
At time t2 the first light ray will hit its target xf , for example another galaxy,
and the second at time t2 + δt2 . At lightspeed c = 1 the line element is lightlike
so ds2 = 0. We now have:
dt
= dx.
a(t)
(3.3)
Integrating both side of the equation will get for the first light ray
Z t2
Z xf
dt
=
dx,
t1 a ( t )
0
(3.4)
and for the second light ray
t2 +δt2
Z
t1 +δt1
dt
=
a(t)
xf
Z
dx.
(3.5)
0
As both equations have the same right hand side we can equate the left hand
sides. Subtracting the first integral we will get
Z
t2
t1
dt
−
a(t)
Z
t2
t1
dt
=0=
a(t)
Z
t2 +δt2
t1 +δt1
dt
−
a(t)
Z
t2
t1
dt
.
a(t)
(3.6)
The time interval between t1 + δt1 and t2 cancels out leaving
Z
t2 +δt2
0=
t2
dt
−
a(t)
Z
t1 +δt1
t1
dt
.
a(t)
(3.7)
Assuming that both δt1 , δt2 a/ȧ the integrals can be approximated:
0=
δt2
δt1
−
.
a ( t2 ) a ( t1 )
(3.8)
The time for a light ray to propagate to a certain distance is related to its
wavelength like λ = cδt. The increase of the wavelength of light is proportional
to the redshift z and therefore to the expansion of the universe:
a(t2 ) δt2
λ2
=
=
= 1 + z.
a(t1 ) δt1
λ1
(3.9)
If the initial scale factor is chosen a(t1 ) = 1, we get the scale factor used in the
line element(3.2):
a(t) =
1
.
1+z
15
(3.10)
3.2
Particle creation in FRW
Next we move on to the calculation of the particle creation following [5]. We need
to establish expressions for the in- and out-states which enables us to compute
the Bogoliubov coefficients and the particle creation. For this we parametrize the
time variable introducing conformal time dη = dt/a(t). Then the line element
can be written:
ds2 = a2 (η )(dη 2 − dx2 ) = C (η )(dη 2 − dx2 ).
(3.11)
The conformal scale factor C (η ) can be chosen:
C (η ) = A + B tanh ρη,
(3.12)
with constants A, B and ρ. This represents a universe with smooth expansion
but asymptotically flat at past and future infinity:
C (η ) → A ± B,
η → ±∞.
(3.13)
Because C (η ) is independent of x there is still translational symmetry. This
allows us to define states with separable spatial and time variables. The next
step will the computation of the time dependent part by solving the KleinGordon equation(2.8):
( + m2 )φ = 0,
(3.14)
with the Laplacian for curved spacetime[4]:
√
1
≡ √ ∂µ ( −gg µν ∂ν ).
−g
(3.15)
As discussed in section 2.2 the field can be expanded as independent modes uk
conjugates u∗k . For the calculation of particle production we are only going to
vary the time variable. Therefore we can take the spatial part of the states from
the mode solutions for flat space derived in the first section2.2. We could write
uk = eikx χk (η ),
(3.16)
with χk (η ) the yet unknown time dependent part of the mode. with g µν the
metric tensor, g the determinant of gµν and m the mass of the field. The metric
tensor can be calculated from the line element (3.11) by the definition
ds2 ≡ gµν dxµ dxν ,
(3.17)
so that:
gµν = C (η )
1
0
1
0
1
, g µν =
−1
C (η ) 0
16
0
,
−1
(3.18)
g = −C (η ).
(3.19)
The Laplacian becomes
=
1
(∂ 2 − ∂x2 ).
C (η ) η
(3.20)
Using this in the Klein-Gordon equation equation (3.14) the following equation
of motion for χk can be obtained:
∂η2 χk (η ) + (k 2 + C (η )m2 )χk (η ) = 0.
(3.21)
To get an idea what the time dependent part of the modes should look like we
approximate the Klein-Gordon equation for the far past and future limit. With
approximation (3.13) the wave equation can be solved:
χk (η ) = Dei(k
2 +Am2 ±Bm2 )1/2 η
,
η → ±∞.
(3.22)
Following section 2.4 we call the infinite past and future state respectively the
in- and out-states. We therefore can define the frequencies:
ωin ≡ (k 2 + Am2 − Bm2 )1/2 ,
ωout ≡ (k 2 + Am2 + Bm2 )1/2 .
(3.23)
From equation (2.30) in section 2.2 we derive that the factors Din and Dout
should be chosen:
Din ≡ (2Lωin )−1/2 ,
Dout ≡ (2Lωout )−1/2 .
(3.24)
An approximated function for the remote past and future can be obtained:
−1/2 ikx−iωin η
uin
e
,
k = (2Lωin )
uout
k
= (2Lωout )
−1/2 ikx−iωout η
e
.
(3.25)
(3.26)
These approximations of the in- and out-states will serve as a reference for
the real solutions to the Klein-Gordon equation. We need to solve equation
(3.21) without this approximation to get the exact wave functions. The exact
solution is a linear combination of hypergeometric functions. To derive this the
substitution z ≡ 12 (1 + tanh ρη ) is needed so that
1
dz
= ρsech2 (ηρ)∂z = 2ρz (1 − z )∂z ,
dη 2
∂η χ = 2ρz (1 − z )∂z χ,
2 2
2
2 2
∂η χ = 4ρ z (1 − z ) ∂z + 2ρz (1 − z )2ρ(1 − 2z )∂z χ.
∂η = ∂z
(3.27)
The equation of motion can be written:
k 2 + Am2 + Bm2 (2z − 1) 0 = 2ρz (1 − z )∂z2 + 2ρ(1 − 2z )∂z +
χ
2ρz (1 − z )
1
1
k 2 + Am2 + Bm2 (2z − 1) = ∂z2 + ( −
)∂z +
χ.
(3.28)
z 1−z
4ρ2 z 2 (1 − z )2
17
With the substitutions ωin/out = (k 2 + Am2 ± Bm2 )1/2 this can further be
simplified to
2 z
ω 2 (1 − z ) + ωout
1
1
0 = ∂z2 + ( −
)∂z + in 2 2
χ
z 1−z
4ρ z (1 − z )2
ω2
1
1
ω2 z
χ.
)∂z + 2 2 in
+ 2 out
= ∂z2 + ( −
2
z 1−z
4ρ z (1 − z ) 4ρ z (1 − z )
(3.29)
This equation is of the form of the Riemann’s differential equation[8]:
1 − α − α0
1 − β − β0
1 − γ − γ0
2
∂z χ +
+
+
∂z χ
z−a
z−b
z−c
0
αα (a − b)(a − c) ββ 0 (b − c)(b − a) γγ 0 (c − a)(c − b)
χ
+
+
+
= 0,
z−a
z−b
z−c
(z − a)(z − b)(z − c)
(3.30)
where
α = −α0 =
iωin
,
2ρ
β = −β 0 =
iωout
, γ = 1 − γ 0 = 1,
2ρ
a = 0, b = 1, c = ∞,
(3.31)
(3.32)
so that
α + α0 + β + β 0 + γ + γ 0 = 1.
(3.33)
The solution of the Riemann’s differential equation is a linear combination of
24 different functions. Two of them are of particular interest for this problem.
They consist of the hypergeometric series
z − a α0 z − b β
(b − c)(z − a)
0
0
0
0
χ1 = C 1 (
) (
) 2 F1 α + γ + β, α + γ + β; 1 + α − α;
,
z−c
z−c
(z − c)(b − a)
(3.34)
and
χ2 = C 2 (
z − a α0 z − b β
(a − c)(z − b)
) (
) 2 F1 α0 + γ + β, α0 + γ 0 + β; 1 + β − β 0 ;
,
z−c
z−c
(z − c)(a − b)
(3.35)
where C1 and C2 are some constant coefficients. We choose that the coefficients
for the other 22 solutions are equal zero: C3 , ..., C24 = 0. As will be shown
these functions represent the time dependent part of the in- and out-state we
are looking for. We choose the coefficients to be
0
C1 = C2 = (−c)α cβ .
18
(3.36)
Putting in the variables of the Riemann differential and using that c z this
becomes
χ1 = ( z )
−iωin
2ρ
(1 − z )
iωout
2ρ
2 F1 (1 + (iω− /ρ), iω− /ρ; 1 − (iωin /ρ); z ),
(3.37)
χ2 = (z )
−iωin
2ρ
(1 − z )
iωout
2ρ
2 F1 (1 + (iω− /ρ), iω− /ρ; 1 + (iωout /ρ); 1 − z ).
(3.38)
For notational convenience we have defined
1
ω± ≡ (ωout ± ωin ).
2
After returning to the time parameter by putting
(3.39)
1
e2ρη
z = (tanh(ρη ) + 1) = 2ρη
,
2
e
+1
(3.40)
we get
χ1 = (
e2ρη −iωin /2ρ
1
)
( 2ρη
)iωout /2ρ
e2ρη + 1
e
+1
2 F1 (1 + (iω− /ρ), iω− /ρ; 1 − (iωin /ρ);
χ2 = (
1
(tanh(ρη ) + 1), (3.41)
2
1
1
e2ρη −iωin /2ρ
)
( 2ρη
)iωout /2ρ 2 F1 (1 + (iω− /ρ), iω− /ρ; 1 + (iωout /ρ); (1 − tanh(ρη )).
+1
e
+1
2
(3.42)
e2ρη
In the limit η → −∞ we can approximate 1 + e2ρη ≈ 1. Hence, the first solution
can be approximated:
χ1 → e−ωin η .
(3.43)
By taking the limit η → ∞ so that 1 + e2ρη ≈ e2ρη , the second solution can be
approximated:
χ2 → e−ωout η .
(3.44)
Comparing these solutions with the expected solutions for the in-states (3.25)
and out-states (3.26) will tell us that χ1 represents the time dependent part of
the in-state and χ2 the time dependent part of the out-state. Following section
(2.4) writing the in-states in terms of the out-states will lead to the Bogoliubov
coefficients. This can be done by using some basic properties of hypergeometric
functions[4]. For example, any hypergeometric function can be written as the
linear combination
Γ (c) Γ (c − a − b)
2 F1 (a, b, a + b + 1 − c; 1 − z )
Γ (c − a) Γ (c − b)
Γ (c) Γ (a + b − c)
+
(1 − z )c−a−b 2 F1 (c − a, c − b, 1 + c − a − b; 1 − z ),
Γ (a) Γ (b)
(3.45)
2 F1 (a, b, c; z ) =
19
so χ1 can be written
Γ(1 − iωin /ρ)Γ(−iωout /ρ)
2 F1 (1 + (iω− /ρ), iω− /ρ; 1 + (iωout /ρ); 1 − z )
Γ(−iω+ /ρ)Γ(1 − iω+ /ρ)
Γ(1 − iωin /ρ)Γ(iωout /ρ)
+
(1 − z )−iωout /ρ 2 F1 (−iω+ /ρ, 1 − iω+ /ρ; 1 − iωout /ρ; 1 − z ) .
Γ(iω− /ρ)Γ(1 + iω− /ρ)
(3.46)
χ1 = z
−iωin
2ρ
(1 − z )
iωout
2ρ
Another property of hypergeometric functions is the equation
2 F1 (a, b; c; z ) = (1 − z )
c−a−b
2 F1 (c − a, c − b; c; z ).
(3.47)
Using this we can derive
2 F1 (−iω+ /ρ, 1 − iω+ /ρ; 1 − iωout /ρ; 1 − z ) = z
iωin /ρ
2 F1 (1 − iω− /ρ, −iω− ; 1 − iωout /ρ; 1 − z ).
(3.48)
The function χ1 can now be written
Γ(1 − iωin /ρ)Γ(−iωout /ρ)
2 F1 (1 + (iω− /ρ), iω− /ρ; 1 + (iωout /ρ); 1 − z )
Γ(−iω+ /ρ)Γ(1 − iω+ /ρ)
−iωout Γ (1 − iω /ρ) Γ (iω
out /ρ)
in
+ z iωin /2ρ (1 − z ) 2ρ
2 F1 (1 − iω− /ρ, −iω− ; 1 − iωout /ρ; 1 − z )
Γ(iω− /ρ)Γ(1 + iω− /ρ)
(3.49)
χ1 = z
=
−iωin
2ρ
(1 − z )
iωout
2ρ
Γ(1 − iωin /ρ)Γ(iωout /ρ) ∗
Γ(1 − iωin /ρ)Γ(−iωout /ρ)
χ2 +
χ .
Γ(−iω+ /ρ)Γ(1 − iω+ /ρ)
Γ(iω− /ρ)Γ(1 + iω− /ρ) 2
(3.50)
We succeeded in relating the time dependent parts of the in- and out-states.
Next is to find these relations for the whole mode solutions. The full in- and
out-states are given (3.25) (3.26):
p
uin
2Lωin eikx χ1 (η ),
(3.51)
k (η, x) =
p
ikx
out
(3.52)
uk (η, x) = 2Lπωout e χ2 (η ).
out
Hence, the full in-states uin
k can be written in terms of the full out-states uk
by:
out
out∗
uin
k = αk uk + βk u−k ,
(3.53)
where
ωout Γ(1 − iωin /ρ)Γ(−iωout /ρ)
,
ωin Γ(−iω+ /ρ)Γ(1 − iω+ /ρ)
r
ωout Γ(1 − iωin /ρ)Γ(iωout /ρ)
βk =
.
ωin Γ(iω− /ρ)Γ(1 + iω− /ρ)
r
αk =
(3.54)
(3.55)
This tells us that αk and βk are the Bogoliubov coefficients for transition from
the out-state to the in-state. The last step towards the particle creation is
20
producing the out-state number operator. For this we need the absolute squared
values of the Bogoliubov coefficients:
ωout Γ(1 − iωin /ρ)Γ(1 + iωin /ρ)Γ(−iωout /ρ)Γ(iωout /ρ)
|
,
ωin Γ(−iω+ /ρ)Γ(iω+ /ρ)Γ(1 − iω+ /ρ)Γ(1 + iω+ /ρ)
ωout Γ(1 − iωin /ρ)Γ(1 + iωin /ρ)Γ(−iωout /ρ)Γ(iωout /ρ)
|
.
|βk |2 = |
ωin Γ(−iω− /ρ)Γ(iω− /ρ)Γ(1 − iω− /ρ)Γ(1 + iω− /ρ)
|αk |2 = |
(3.56)
(3.57)
To simplify this basic properties of gamma functions are needed[4]:
Γ(1 + z ) = zΓ(z ),
(3.58)
Γ(1 − z ) = −zΓ(−z ),
(3.59)
so that
ωin 2
Γ(iωin /ρ)Γ(−iωin /ρ),
ρ
ω± 2
Γ(1 − iω± /ρ)Γ(1 + iω± /ρ) = −
Γ(iω± /ρ)Γ(−iω± /ρ).
ρ
Γ(1 − iωin /ρ)Γ(1 + iωin /ρ) = −
(3.60)
(3.61)
Equations (3.56) and (3.57) can further be simplified by using the property
Γ(z )Γ(−z ) = −
π
,
z sin πz
(3.62)
so that
π
,
iωin/out sin(πiωin/out )
π
Γ(iω± )Γ(−iω± ) = −
.
iω± sin(πiω± )
Γ(iωin/out )Γ(−iωin/out ) = −
(3.63)
(3.64)
By using sin iz = i sinh z the following functions will remain
|αk |2 =
=
|βk |2 =
=
ωout
(ωin πω+ )2 sin2 (iπω+ /ρ)
ωin (πω+ )2 ωin ωout sin(iπωin /ρ) sin(iπωout /ρ)
sinh2 (πω+ /ρ)
,
sinh(πωin /ρ) sinh(πωout /ρ)
(3.65)
ωout
(ωin πω− )2 sin2 (iπω− /ρ)
2
ωin (πω− ) ωin ωout sin(iπωin /ρ) sin(iπωout /ρ)
sinh2 (πω− /ρ)
.
sinh(πωin /ρ) sinh(πωout /ρ)
(3.66)
The expectation value of the out-state number operator N̂ can be computed
from the absolute second Bogoliubov coefficient:
X
hN̂ i =
|βk |2 .
(3.67)
k
21
Putting in the functions for the frequenties using equations (3.23) and (3.39) we
get the particle creation:
√
√
X sinh2 [π/ρ( k 2 + Am2 + Bm2 − k 2 + Am2 − Bm2 )]
√
√
.
(3.68)
hN̂ i =
2 + Am2 + Bm2 ) sinh(π k 2 + Am2 − Bm2 )
sinh
(
π
k
k
As can be seen the particle creation will be equal to zero when B is equal to
zero. This confirms that particle creation is caused by the curvature of spacetime because the expansion scale factor is defined to increase with a factor B
(3.12). The other situation where the particle creation will decrease will be
when the mass of the field m is very small. This explaines something about the
nature of the particles created. The mass in the field creates a gravitational
field. The spacetime expansion causes for a change in the gravitational field,
which provides the energy needed for the particle creation. When the field is
massless there will be no gravitational field so there will be no particle creation.
With B 6= 0 and m 0 we have encountered a non zero expectation value of
the out-state number operator. Hence, the lowest energy vacuum in-state will
propagate to an excited out-state in Friedmann-Robertson-Walker spacetime.
This means that a smoothly expanding massive universe will have particles at
future infinity even if it started out without particles at the infinite past. This
shows that particle production is indeed a property of curved spacetime.
22
Chapter 4
De Sitter space
The second curved spacetime model explored in this thesis will be de Sitter
space. Different coordinate system can represent de Sitter. The next section
will analyze different coordinate systems to find one for the computation of
particle creation. A different approach to particle creation is used than with
the FRW spacetime. As will be shown, there is a relation between the Bogoliubov coefficients and the transmission and reflection coefficients of a propagated
scalar from past to future infinity. The Bogoliubov coefficients will be computed
by determining the transmission and reflection coefficients and exploring this relation.
4.1
De Sitter hyperboloid
De Sitter space is a model with a positive cosmological constant, so it represents
an accelerated expanding universe. The n-dimensional the Sitter space can be
described by the equation
−X02 +
n−1
X
Xi2 = R2 ,
(4.1)
i=1
where R is the so called de Sitter radius[9]. For computational convenience we
choose R = 1. In 2+1-dimensions it can be represented by the hyperboloid in
figure 4.1.
The minimum of the hypersphere Sn−1 radius is at time t = 0, where it
is equal to the de Sitter radius. The expansion rate of the radius increases
infinitely as time elapses to the infinite past or future. However, the absolute
limit to the speed of anything that contains mass is lightspeed. Just like a
Minkowski diagram lightspeed moves in 45 degree angles in the figure. Hence,
an observer residing in de Sitter space will experience a horizon at the point
where the expansion exceeds lightspeed. Any information beyond this horizon
can never get to the observer while in de Sitter spacetime.
23
Figure 4.1: The 2+1-dimensional hyperboloid representing the whole of de Sitter
space. At time t = 0 the universe, represented by the dotted circle, has its minimal radius. As time elapses backward and forward the universe exponentially
expands.
4.2
Coordinate systems of de Sitter space
Different coordinate systems are available satisfying the equation for de Sitter
space (4.1). However, not every coordinate system describes the whole hyperboloid. This section will explore three different kinds of coordinate systems. We
refer to [9] for a more extensive disscussion on de Sitter coordinates systems.
4.2.1
Global coordinates
The whole hyperboloid of de Sitter space can be described with the so called
global coordinates. Here we have the substitutions:
X0 = sinh τ ,
i = 1, ..., n − 1,
Xi = ωi cosh τ ,
(4.2)
The variables ωi are the normalized parametrization of the hypersphere Sn−1
and τ is the time variable. This parametrization of a hypersphere will also be
24
used for other coordinate systems. The coordinates ωi can be written in terms
of the angles:
ω1 = cos θ1 ,
ωi = cos θi
i−1
Y
i = 2, ..., n − 2,
sin θj ,
j =1
ωn−1 =
n−1
Y
sin θj
(4.3)
j =1
where 0 ≤ θi < π and 0 ≤ θn−1 < 2π. The coordinates are normalized so that
n−1
X
ωi2 = 1.
(4.4)
i=1
The metric on the Sn−1 hypersphere can be computed by taking the differentials:
dω1 = sin2 θ1 dθ12 ,
Y
i−1 i
X
Y
2
2
2
2
dωi =
cos θi cos θk
[sin θj ]dθk +
[sin2 θj ]dθi2 ,
k =1
dωn−1 =
n−1
X
j =1
j6=k
cos2 θk
k =1
Y
sin2 θj dθk2 .
(4.5)
j6=k
Then the metric becomes:
dΩ2n−1 =
n−1
X
dωi2 = dθ12 +
i=1
n−1
X k−1
Y
sin2 θj dθk2 .
(4.6)
k =2 j =1
The global coordinate system satisfies the coordinate equation for de Sitter(4.1):
−X02 +
n−1
X
Xi2 =
i=1
n−1
X
ωi2 cosh2 τ − sinh2 τ = 1.
(4.7)
i=1
The metric with signature (-, +, ..., +) can be written in terms of the metric
dΩn−1 of the hypersphere Sn−1 :
ds2 = gµν dX µ dX ν = −(cosh2 τ −
2
n−1
X
i=1
2
ωi−1 sinh2 τ )dτ 2 + cosh2 τ dΩ2n−1
= −dτ + cosh τ dΩ2n−1 .
(4.8)
The hyperbolic cosine increases exponentially as τ approaches the infinite past
and future limit. This describes the total hyperboloid with an exponential
expansion rate as time elapses to past and future infinity.
25
A visual representation of the causal structure of global de Sitter can be
obtained by changing to a conformal time T :
cosh τ =
1
.
cos T
(4.9)
As normal time elapses −∞ < τ < ∞ the newly defined conformal time elapses
−π/2 < T < π/2. The metric now becomes:
1
(−dT 2 + dΩ2n−1 ).
cos2 T
(4.10)
The figure of the hyperboloid becomes a cylinder under the conformal time
change, as can be seen in figure 4.2. Because the coordinate change is conformal
the angles are preserved so light rays still move at 45 degrees in the figure.
Figure 4.2: The 2+1-dimensional hyperboloid of de Sitter space under a conformal coordinate change. As normal time elapses −∞ < τ < ∞ the newly defined
conformal time elapses −π/2 < T < π/2. Light rays move with 45 degrees in
the figure.
Figure 4.2 mapped onto the 1+1-dimensional spacetime is known as the Penrose diagram[10] for de Sitter space, drawn in figure4.3. The spatial coordinate
r is the distance to edge of the universe and the middle of the diagram represents t = 0, elapsing backward and forward vertically. The top and bottom of
26
the diagram represent the future and past lightlike infinity limit. Light rays are
represented by 45 degree lines.
An observer in the first region can never go to the third and fourth region
because then he would have to exceed light speed. The first region represents our
observable universe while the third represents a hypothetic unreachable inverse
universe. An observer that entered region four can never go back to either of
these universes. There are other coordinate systems that satisfy the coordinate
equation for de Sitter space(4.1). The next coordinate system explored is the
so called static coordinate system.
Figure 4.3: The Penrose diagram of de Sitter space. The spatial coordinate
r is the distance to the edge of the universe and the middle of the diagram
represents t = 0, elapsing backward and forward vertically. The top and bottom
of the diagram represent the future and past lightlike infinity limit. Light rays
are represented by 45 degree lines.
4.2.2
Static coordinates
Another coordinate system for de Sitter space is the static coordinate system.
We have the substitutions:
p
X0 = l2 − r2 sinh τ
Xi = rωi , i = 1, ..., n − 2
p
Xn−1 = l2 − r2 cosh τ .
27
(4.11)
Using equation (4.4) the static coordinates satisfy the de Sitter space equation (4.1):
−X02 +
n−1
X
Xi2 = (1 − r2 )(cosh2 τ − sinh2 τ ) + r2
i=1
n−1
X
2
ωi−1
= 12 .
(4.12)
i=1
The metric can be obtained by computing the differentials:
r2
sinh2 τ dr2 ,
1 − r2
r2
dX12 = (1 − r2 ) sinh2 τ dτ 2 +
cosh2 τ dr2 ,
1 − r2
n−1
n−1
n−1
X
X
X
dXi =
ωi dr2 + r2
dωi2 .
dX02 = (1 − r2 ) cosh2 τ dτ 2 +
i=1
i=1
(4.13)
i=1
The line element for static de Sitter space can be obtained:
n−1
ds2 = −(1 − r2 )dτ 2 +
X
1
2
2
dr
+
r
dωi2 .
1 − r2
(4.14)
i=1
The metric in static coordinates is independent of τ so we say it has a timelike killing vector ∂∂τ . This means that the metric remains invariant under time
translation τ → τ + a. This is the coordinate system as seen from a static
observer. Hence, it is given the signature ’static’ coordinates. At the point
r = 1 the time differential dτ 2 vanishes while the radial differential dr2 blows
up, creating a singularity. This singularity restrains the static coordinates of
accounting for anything beyond the point r = 1.
Hence, static coordinates only represent the first region of the Penrose diagram of de Sitter space, as shown in figure 4.4. This first region is therefore
called the static patch of de Sitter space.
4.2.3
Planar coordinates
Another coordinate system that satisfies the coordinate equation for de Sitter
space(4.1) is called planar coordinates:
1
X0 = sinh τ − xi xi e−τ ,
2
Xi = xi e−τ , i = 1, ..., n − 2,
1
Xn−1 = cosh τ − xi xi e−τ ,
2
(4.15)
(4.16)
28
Figure 4.4: The Penrose diagram of the static patch. With static coordinates
there is a singularity at the point r = 1. Static coordinates therefore only cover
the region where r < 1, colored gray in the figure.
so that
−X02 +
n−1
X
Xi2 = (cosh2 τ − sinh2 τ ) −
i=1
x xi e−τ 2 x xi e−τ 2
i
+ i
2
2
+ (sinh τ − cosh τ + e−τ )xi xi e−τ = 1.
(4.17)
Like the other coordinate systems we compute the differentials:
x xi e−τ 2
i
2
2
i −τ
dX0 = cosh τ +
− cosh τ xi x e
dτ 2 − xi xi e−2τ dxi dxi ,
2
dXi = e−2τ dxi dxi − xi xi e−2τ dτ 2 ,
x xi e−τ 2
2
dXn−1
= sinh2 τ + i
− sinh τ xi xi e−τ dτ 2 − xi xi e−2τ dxi dxi .
2
(4.18)
With the same metric signature we get the line element for planar de Sitter
space:
ds2 = −dτ 2 + e−2τ dxi dxi .
29
(4.19)
Instead of a timelike killing vector planar de Sitter space has a spacelike killing
vector. As time elapses backwards to past infinity the spatial coordinates expand
exponentially. The planar coordinates represent an expanding universe from
future to past infinity. It can be seen as the hyperboloid of de Sitter space, but
sliced up at a 45 degree angle, shown in figure 4.5. Constructing the conformal
Penrose diagram reveal that only the first and fourth region are covered by
planar coordinates.
Figure 4.5: The hyperboloid of de sitter space in planar coordinates. In planar
coordinates the universe expands exponentially from future to past infinity.
There only the gray area is covered by planar coordinates
It is also possible to set up planar coordinates for an expansion forward in
time covering the second and first region:
1
X0 = sinh τ − xi xi eτ ,
2
Xi = xi eτ , i = 1, ..., n − 2,
1
Xn−1 = cosh τ − xi xi eτ ,
2
2
2
2τ
ds = −dτ + e dxi dxi .
30
(4.20)
(4.21)
In figure 4.6 the diagram is shown for both future and past expanding planar
coordinates of de Sitter space.
Figure 4.6: The Penrose diagrams of the planar coordinates. The left diagram
shows the planar coordinates representing a universe, exponentially expanding
as time elapses backwards. Only the first and fourth region are covered by these
coordinates, colored gray. The right diagram shows the planar coordinates for
a future expanding universe. These coordinates only cover the first and second
region of the Penrose diagram.
The goal of this chapter is to compare the in- and out-states of the total de
Sitter space, so the global coordinate system seems the right coordinate system
to work with.
However, there is one problem with this coordinate system. The spatial coordinates in this system increase with time. This is not the case for an earthly
observer, in our perception length scales remain the same over time. This is
therefore the system of some meta observer living in a system with only translational symmetry. Nevertheless, the purpose of this thesis was to explore particle
creation in curved spacetime, not necessarily our spacetime. We will therefore
use the global coordinate system for the computation of particle creation. More
information about different coordinate systems of de Sitter space with reference
to the particle creation can be found in [11].
31
4.3
Transmission and reflection in global de Sitter space
In this section a start of the computation of the particle creation in de Sitter
space will be made following [12]. This will be done by computing the transmission and reflection of a propagated scalar field from past to future infinity. The
start of the problem will the same as the FRW computation. In n-dimensional
global coordinates the line element for de Sitter space is:
ds2 = −dτ 2 + cosh2 τ dΩ2n−1 ,
(4.22)
with dΩn−1 the metric of the hypersphere Sn−1 as in (4.5). For example in 2+1
dimensions we have
dΩ2n−1 = dθ2 + sin2 θdφ2 .
(4.23)
From the definition of the line element ds2 = gµν dxµ dxν the metric tensor g µν
for de Sitter space can be obtained. For example the metric tensor in 2+1dimensions becomes:


−1
0
0
,
0
gµν =  0 cosh2 τ
(4.24)
2
2
0
0
sin θ cosh τ
with determinant
det(g ) = − sin2 θ (cosh2 )2 .
(4.25)
To compute the transmission and reflection coefficients, we need the modes of a
scalar field, propagated from the infinite past. The scalar field φ should satisfy
the Klein-Gordon equation:
( − m2 )φ = 0, .
(4.26)
Here m is the mass of the total field. We choose m to be large because there
will be no particle creation without mass if the case is similar to FRW space.
The next step is to determine the de Sitter Laplacian . This can be done with
the calculated metric tensor(4.24):
√
1
(τ , θ, φ) ≡ √ ∂µ −gg µν ∂ν ,
−g
1
= 2
cosh−2 τ ∂µ r2 sin θ cosh2 τ g µν ∂ν ,
r sin θ
= − cosh−2 τ ∂τ [cosh2 τ ∂τ ] + cosh−2 (θ, φ),
(4.27)
(4.28)
with (θ, φ) the spherical Laplacian of the 2 dimensional hypersphere. Extrapolating this to n dimensions means we have to change (θ, φ) to the spherical
Laplacian of the n-1-dimensional hypersphere Sn−1 . The hyperbolic cosines
32
should then be written with powers of n − 1. We get the de Sitter Laplacian in
n dimensions:
= − cosh1−n τ ∂τ [coshn−1 τ ∂τ ] + cosh−2 τ S n−1 .
(4.29)
This enables us two write down the equation of motion in de Sitter space:
(− cosh1−n τ ∂τ [coshn−1 τ ∂τ ] + cosh−2 τ S n−1 − m2 )φ = 0.
(4.30)
Like the computation for FRW space, we expand the field φ into modes:
X
+ ∗
(4.31)
φ=
[a−
l φl + al φl ].
l
We are going to solve the equation of motion for each mode separately. It is
convenient to separate the spatial and time dependent part of the modes. The
spatial coordinates can be written as the normalized spherical harmonics on the
hypersphere Sn−1 . We can write:
φl = ul (τ )(cosh τ )−
n−1
2
Yl (Ωn−1 ).
(4.32)
n−1
where the term cosh(τ )− 2 is chosen for convenience of further computation.
The spherical hamonics Yl (Ωn−1 ) are the eigenfunctions of the spherical Laplacian Sn−1 . The corresponding eigenvalues are[13]:
∆S n−1 Yl (Ωn−1 ) = −l (l + n − 2)Yl (Ωn−1 ).
(4.33)
The equation of motion (4.26) can be written:
− cosh1−n τ ∂τ [cosh(τ )n−1 ∂τ ]u(τ ) cosh(τ )
1−n
2
− (l (l + n − 2) cosh(τ )−2 + m2 )u(τ ) cosh(τ )
1−n
2
= 0.
(4.34)
By computing the first partial derivative with for notational convenience ∂τ u(τ ) =
u̇(τ ) and a = 1 − n we have:
a
a
+ u(τ ) sinh(τ ) cosh(τ )− 2 −1 ]
2
a
−(l (l + n − 2) cosh(τ )−2 + m2 )u(τ ) cosh(τ ) 2 = 0.
− cosha τ ∂τ [u̇(τ ) cosh(τ )
−a
2
(4.35)
After evaluating the second partial derivative it looks like:
a
a(a + 2) a(a + 2)
ü(τ ) + u(τ ) − u(τ )
+
+ l (l + n − 2) u(τ ) cosh(τ )−2 + m2 u(τ ) = 0.
2
4
4
(4.36)
By filling in a = 1 − n the equation of motion for de Sitter space is obtained:
2l + n − 3 2l + n − 1 1
(n − 1)2
2
∂τ2 +
+
(
m
−
)
u(τ ) = 0.
(4.37)
2
2
4
cosh2 τ
33
To create an idea what kind of equation we have encountered we make the
substitutions:
r
(n − 1)2
2l + n − 1
L≡
, ω ≡ m2 −
,
(4.38)
2
4
so that:
h
− ∂τ2 −
L(L − 1) i
u(τ ) = ω 2 u(τ ).
cosh2 τ
(4.39)
If the variable τ is viewed as a spatial variable, this equation is the ordinary one
dimensional time-independent Schrödinger equation. The potential corresponding to this Schrödinger equation is known as the Poschl-Teller potential[14]:
V =−
L(L − 1)
.
cosh2 τ
(4.40)
Notice that for odd dimensional de Sitter space the variable L becomes an
integer, and half integer for even dimensions. The difference between half integer
and integer L has a drastic influence on the calculation of particle creation. For
this thesis is chosen to stay restricted to the odd dimensional de Sitter space
with integer L.
4.3.1
Direct solution of the Poschl-Teller wave equation
In this subsection the transmission and reflection coefficients for the solutions
of the Poschl-Teller Schrödinger equation will be computed. We refer to [15]
for more information on solving the Poschl-Teller equation and other similar
equation. The equation of motion with the Posch-Teller potential is:
h
L(L − 1) i
u = 0.
∂τ2 + ω 2 +
cosh2 τ
(4.41)
At past and future infinity we can approximate
cosh τ → ∞,
τ → ±∞,
(4.42)
so the equation of motion becomes
[∂τ2 + ω 2 ]u = 0.
(4.43)
The solutions to this differential equation are
u = Ce±iω ,
τ → ±∞ C ∈ C.
(4.44)
Therefore the modes should be of the following form to compute the reflection
and transmission of a from the infinite past propagated wave:
u=
{e
iωτ
+ Re−iωτ , τ → −∞
.
T eiωτ , τ → +∞
34
(4.45)
First we are going to define some substitution to simplify the equation of motion.
After that the equation of motion, without the approximation, will be solved to
determine the exact modes. With the substitution y ≡ cosh2 τ we have:
q
dy
(4.46)
∂τ = ∂y
= 2 sinh τ cosh τ ∂y = −4(1 − y )y∂y ,
dτ
q
∂τ u = 2i y (1 − y )∂y u,
(4.47)
∂τ2 u = −4y (1 − y )∂y2 u − 2(1 − 2y )∂y u.
(4.48)
The differential equation becomes
[y (1 − y )∂y2 + (1/2 − y )∂y −
ω 2 L(L − 1)
−
]u = 0.
4
4y
(4.49)
By choosing u ≡ y L/2 v (y ) the differentials can be rewritten
L
v + ∂y v )y L/2 ,
2y
(4.50)
∂y2 = (L/2(L/2 − 1)y −2 v + Ly −1 ∂y v + ∂y2 v )y L/2 ,
(4.51)
u ≡ y L/2 v (y ), ∂y u = (
leading to the differential equation:
y (1 − y )∂y2 v + ((L + 1/2) − (L + 1)y )∂y v − 1/4(L2 + ω 2 )v = 0.
(4.52)
This is a hypergeometric equation[4] of the form
x(1 − x)∂x2 z + (c − (a + b + 1)x)∂x z − abz = 0,
(4.53)
where
x = 1 − y,
a = 1/2(L + iω ),
z = v,
(4.54)
c = 1/2,
(4.55)
ab = 1/4(L2 + ω 2 ).
(4.56)
b = 1/2(L − iω ),
The solution to equation (4.52) is a linear combination of the hypergeometric
functions:
p
1
1 3
1
(4.57)
v = A2 F1 (a, b, ; 1 − y ) + B 1 − y 2 F1 (a + , b + , ; 1 − y ),
2
2
2 2
where A and B are some constants. The constants A and B can be restricted
by setting up the scattering problem(4.45). First we change back to the original
time variable cosh2 τ = y. Using the proposed substitution for u (4.50) we get:
u = v coshL τ ,
1
= A coshL τ 2 F1 (a, b, ; − sinh2 τ ),
2
1
1 3
L
− B cosh τ sinh τ 2 F1 (a + , b + , ; − sinh2 τ ).
2
2 2
35
(4.58)
This function has an even and an odd part because of the respectively even and
odd properties of the hyperbolic cosine and sine. The modes u can be rewritten
u = Aue(ven) + Buo(dd) .
(4.59)
For the calculation of the reflection and transmission between the far past en future only the infinitely large negative and positive values of τ matter. Therefore
the hyperbolic functions can be approximated:
1
sinha τ = ( (eτ − e−τ ))a → (±1)a 2−a ea|τ | , for τ → ±, ∞
2
1
cosha τ = ( (eτ + e−τ ))a → 2−a ea|τ | , for τ → ±.∞
2
(4.60)
(4.61)
The hypergeometric functions written in terms of gamma functions [15] become:
Γ (b − a)
1
22a e−2a|τ |
ue (τ ) → 2−L eL|τ | Γ( )
2 Γ(b)Γ( 21 − a)
!
Γ (a − b)
2b −2b|τ |
2 e
,
+
Γ(a)Γ( 12 − b)
(4.62)
3
Γ (b − a)
uo (τ ) → ± 2−(L+1) e(L+1)|τ | Γ( )
22a+1 e−(2a+1)|τ |
2 Γ(b + 12 )Γ(1 − a)
!
Γ (a − b)
22b+1 e−(2b+1)|τ | , τ → ±∞.
(4.63)
+
Γ(a + 12 )Γ(1 − b)
The ± sign in the odd solution is used respectively for τ being in the infinite
past and future limit. By filling in the variables a and b (4.55) we get:
1
ue (τ ) → Γ( )(Ee−iω|τ | + E ∗ eiω|τ | ),
2
3
uo (τ ) → ±Γ( )(Oe−iω|τ | + O∗ eiω|τ | ),
2
(4.64)
(4.65)
where
E≡
O≡
Γ(−iω )
Γ( L2
−
iw
1−L
2 )Γ( 2
−
iw
2 )
−
iw
2 )
Γ(−iω )
1
Γ ( L+
2 −
iw
L
2 ) Γ (1 − 2
eiω log 2 ,
(4.66)
eiω log 2 .
(4.67)
We have found the mode solutions for the time dependent part of the de
Sitter equation of motion. We are now going to proceed different from the FRW
approach by setting up the scattering problem. The total solution is the linear
combination of the odd and even solutions:
u=
{
A(Ee−iωτ + E ∗ eiωτ ) + B (Oe−iωτ + O∗ eiωτ ), τ → ∞
A(Eeiωτ + E ∗ e−iωτ ) − B (Oeiωτ + O∗ e−iωτ ), τ → −∞
36
.
(4.68)
As mentioned before the past and future infinity limit of the modes should be
of the form:
u=
{
eiωτ + Re−iωτ , τ → −∞,
.
T eiωτ , τ → +∞.
(4.69)
The purpose of the following computation will be to determine the reflection
and transmission coefficients R and T . The scattering set up defines restriction
on the constants A and B by
AE + BO = 0,
(4.70)
AE − BO = 1,
(4.71)
so that the transmission and reflection coefficients become:
T = AE ∗ + BO∗ ,
(4.72)
∗
(4.73)
∗
R = AE − BE .
It can be assumed that E and O are functions of some arguments φe/o . These
arguments can be defined to be real by allocating real factors Ce and Co to the
solutions:
E = Ce eiφe ,
O = Co eiφo ,
(4.74)
where
φe , φo , Ce , Co ∈ R.
(4.75)
By equating (4.70) and (4.71) we will get
1
BCo = e−iφo .
2
1
ACe = e−iφe ,
2
(4.76)
Hence, the equations for the reflection and transmission coefficients can be
rewritten:
1
T = (e−2iφe − e−2iφo ),
2
1 −2iφe
R = (e
+ e−2iφo ).
2
(4.77)
(4.78)
For the calculation of the transmission and reflection the exponentials e−iφe and
e−iφo need to be calculated. From equations (4.66) and (4.67) one obtains:
Ce e−iφe = E ∗ =
Ce e−iφo = O∗ =
Γ(iω )
Γ( L2 +
iw
1−L
2 )Γ( 2
+
iw
2 )
+
iw
2 )
Γ(iω )
1
Γ ( L+
2
+
iw
L
2 ) Γ (1 − 2
37
e−iω log 2 ,
(4.79)
e−iω log 2 .
(4.80)
Now using the fact that |E| = Ce the even exponential can be written:
e−2iφe =
E ∗2
E ∗2
E∗
= ∗2=
2
Ce
|E |
E
= ei2φe = e2iω log 2
Γ(iω )Γ( L2 −
iω
1−L
iω
2 )Γ( 2 − 2 )
.
1−L
iω
Γ(−iω )Γ( L2 + iω
2 )Γ( 2 + 2 )
(4.81)
The same applies to the odd solution:
e−2iφo =
O∗2
O∗2
O∗
= ∗2=
2
Co
|O |
O
= e−2iω log 2
1
Γ(iω )Γ( L+
2 −
iω
L
iω
2 ) Γ (1 − 2 − 2 )
.
1
iω
L
iω
Γ(−iω )Γ( L+
2 + 2 ) Γ (1 − 2 + 2 )
(4.82)
Starting with the odd solution the arguments of the Gamma functions are substituted
Z1 ≡
L + 1 iω
− ,
2
2
(4.83)
so that
e−2iφo = e−2iω log 2
Γ(iω )Γ(Z1 )Γ( 23 − Z1∗ )
Γ(−iω )Γ(Z1∗ )Γ( 32 − Z1 )
.
(4.84)
By multiplying with
Γ(1 − Z1∗ )Γ(1 − Z1 )
,
Γ(1 − Z1∗ )Γ(1 − Z1 )
(4.85)
the gamma functions can be simplified. We can use the properties of gamma
functions[4]:
π
,
sin(πZ )
√
1
Γ(Z )Γ(Z + ) = 21−2Z πΓ(2Z ).
2
Γ (Z ) Γ (1 − Z ) =
(4.86)
(4.87)
The exponential takes the form
e−2iφo = e−2iω log 2
We can put back Z1 =
e−2iφo = e−2iω log 2
√
∗
Γ(iω )π sin(πZ1∗ )22Z1 −1 πΓ(2 − 2Z1∗ )
√
.
Γ(−iω )π sin(πZ1 )22Z1 −1 πΓ(2 − 2Z1 )
L+1
2
−
(4.88)
iω
2 :
1
iω L+iω
Γ(iω ) sin(π L+
Γ(1 − L − iω )
2 + π 2 )2
1
iω L−iω
Γ(−iω ) sin(π L+
Γ(1 − L + iω )
2 − π 2 )2
38
.
(4.89)
Using the fact that L is an integer, the sines can be simplified to
1
iω
sin(π L+
2 +π 2 )
1
sin(π L+
2
− π iω
2 )
=
L
iω
cos(π L2 ) cosh(π iω
2 ) − i sin(π 2 ) sinh(π 2 )
L
iω
cos(π L2 ) cosh(π iω
2 ) + i sin(π 2 ) sinh(π 2 )
= (−1)L .
(4.90)
Because L is also a positive integer, the gamma functions can be simplified using
the formula
Γ (z − n) =
n
Y
(z − l )−1 Γ(z ).
(4.91)
l =1
Both equations (4.90) and (4.91) do not hold for half integer values of L. This is
the reason why the particle creation is differently in even-dimensional de Sitter
space. Now the exponential will be maximally simplified:
L−1
Q
e
−2iφo
(iω − n)
n=1
= (−1)L L−1
Q
(−iω − n)
n=1
L−1
Q
(−iω + n)
L−1
Y ω + in
=1
.
= − nL−1
=−
ω − in
Q
n
=
1
(−iω − n)
(4.92)
n=1
The same can be done for the even solution:
Z2 ≡
L iω
− ,
2
2
e−i2φe
(4.93)
√
πΓ(1 − 2Z2∗ )
Γ(iω )π sin(πZ2∗ )2
√
= e−2iω log 2
Γ(−iω )π sin(πZ2 )22Z2 πΓ(1 − 2Z2 )
L−1
Q
(iω − n)
n=1
= (−1)L−1 L−1
Q
(−iω − n)
2Z2∗
(4.94)
(4.95)
n=1
L−1
Q
=
(−iω + n)
n=1
L−1
Q
=
(−iω − n)
L−1
Y
n=1
ω + in
.
ω − in
(4.96)
n=1
Following equation (4.77) the transmission and reflection coefficients can be
39
computed with these exponentials. The transmission becomes:
1
T = (e−2iφe − e−2iφo )
2
L−1
L−1
1 Y ω + in Y ω + in
+
)
= (
2
ω − in
ω − in
=
n=1
L−1
Y
n=1
(4.97)
n=1
ω + in
,
ω − in
(4.98)
and the reflection:
1
R = (e2iφe + e2iφo )
2
L−1
L−1
1 Y ω + in Y ω + in
= (
−
) = 0.
2
ω − in
ω − in
n=1
(4.99)
n=1
Thus, we have shown that odd dimensional de Sitter space has a reflectionless
potential. Before relating this to the Bogoliubov coefficients the phenomenon of
reflectionless potentials will be further explored. This will be done in the next
subsection with algebraic means.
4.3.2
Approaching reflectionless potentials algebraically
In the previous subsection we have encountered the phenomenon of reflectionless potentials. This is a property that normally would only be expected for an
equation of motion with a constant potential. This is obviously not the case for
the wave function in the de Sitter space(4.37). In this subsection the basic concept of reflectionless potentials is explored following[12]. An algebraic method
is used to construct arbitrary reflectionless potentials. As will be shown this
leads quite straightforward to the potential found for the de Sitter space. We
start with an arbitrary Hamiltonian of the form:
H ( τ ) = p2 + V ( τ ) ,
(4.100)
where p = −i∂τ and V (τ ) approaches a constant in the past and future infinity
limit: τ → ±∞. The Hamiltonian can be rewritten in terms of ladder operators
A+ and A− . To prevent confusion, these are ladder operators arbitrarily chosen
and have nothing to do with the ladder operators of the mode solutions. There
are two ways of combining these ladder operators so we can create two different
Hamiltonians defined H+ and H− :
H + ≡ A + A− ,
H− ≡ A− A+ .
(4.101)
The ladder operators are defined:
A± ≡ p ± iW (τ ),
40
(4.102)
with W (τ ) some potential that becomes a constant in the past and future infinity
limit. Now substituting the ladder operators we get for the hamiltonians:
H± = (p ± iW (τ )(p ∓ iW (τ ))
= p2 + W (τ )2 ∓ piW (τ ) ± iW (τ )
= p2 + W (τ )2 ∓ ∂τ [W (τ )].
(4.103)
As both H+ and H− should be of the form (4.100) we can define a pair for the
potentials in the same way:
V± ≡ W 2 ∓ ∂τ [W (τ )].
(4.104)
Just like W (τ ) will V± approach a constant in the past and future infinity limit.
We can write:
H± |ψ± i = E± |ψ± i .
(4.105)
where |ψ± i and E± are an eigenstate and corresponding energy of the Hamiltonian H± . The eigenstate |ψ± i can be converted to an eigenstate of the Hamiltonian H∓ by letting the operator A∓ act on it:
H∓ [A∓ |ψ± i] = A∓ A± A∓ |ψ± i = A∓ H± |ψ± i = A∓ E± |ψ± i = E± [A∓ |ψ± i].
(4.106)
We can set up the problem for the computation of the reflection and transmission
the same way as before but now for the + and − states independently:
ψ± =
eiωτ + R± e−iωτ , τ → −∞
{
T± eiωτ , τ → +∞
.
(4.107)
The two Hamiltonians will have different transmission and reflection coefficients
because their eigenstates differ. As stated before the eigenstates of H+ can be
converted to the eigenstates of H− so that:
aψ− = A− ψ+ ,
(4.108)
with a the eigenvalue of A− acting on ψ. Filling in A− and ψ± we get for the
far past limit:
a(eiωτ + R− e−iωτ ) = (p − iW (−∞))(eiωτ + R+ e−iωτ )
= ωeiωτ − iW (−∞)eiωτ − R+ ωe−iωτ − iW (−∞)R+ e−iωτ
= [ω − iW (−∞)]eiωτ − [ω + iW (−∞)]R+ e−iωτ . (4.109)
No limitations have been given on the eigenvalue a so we can choose a = ω −
iW (−∞). Now the terms with eiωτ cancel, leaving
R− =
ω + iW (−∞)
R+ .
−ω + iW (−∞)
41
(4.110)
The same can be done in the far future limit:
aψ− =(ω − iW (−∞))T− eiωτ = (p − iW (∞)T+ eiωτ ,
⇒ T− =
ω − iW (∞)
T+ .
ω − iW (−∞)
(4.111)
(4.112)
From equation (4.110) we can conclude that if the reflection coefficient R− is
equal to zero, the reflection coefficient R+ must be zero too. If we can construct
a potential V− that has no reflection we know that the partner potential V+ will
also be reflectionless. A wavefunction with a constant potential at any time τ
will have no reflection so we can choose the reflectionless potential:
V− = W (τ )2 + ∂τ (W (τ )) = 1.
(4.113)
This differential equation has the solution
W (τ ) = tanh τ .
(4.114)
From here the partner potential V+ can be computed:
V+ = W (τ )2 − ∂τ (W (τ ) = 1 −
2
.
cosh2 τ
(4.115)
A shift of the potential will not affect the reflectionless property so the constant
term can be neglected. Now we can choose V+ to be the lower potential of some
new potential, say V2 , in the exact same way as V− is the lower potential of V+ .
We will get a new differential equation for a new W2 (τ ):
W2 (τ ) + ∂ (W2 (τ )) =
2
.
cosh2 τ
(4.116)
By adding a constant that will not affect the reflectionless property this differential equation has a straightforward solution:
2
,
cosh2 τ
⇒ W2 (τ ) = 2 tanh τ .
W2 (τ ) + ∂ (W2 (τ )) = 4 −
(4.117)
(4.118)
The new potential V2 can be calculated which by equation (4.110) will also be
reflectionless:
W22 (τ ) − ∂ (W2 (τ )) = 4 −
6
.
cosh2 τ
(4.119)
This procedure can be repeated an infinitely many times and every new found
potential will be reflectionless. Take notice that V2 , neglecting the constant, is
the potential corresponding to the second repetition of this procedure and can
be written:
V2 = −
M (M + 1)
, M = 2.
cosh2 τ
42
(4.120)
Assuming this is the case for any arbitrary M ∈ Z we have
VM = −
M (M + 1)
cosh2 τ
(4.121)
M (M + 1)
cosh2 τ
⇒ WM +1 (τ ) = (M + 1) tanh τ .
WM +1 (τ )+∂ (WM +1 (τ )) = (M + 1)2 −
(4.122)
(4.123)
The new potential can be computed:
WM +1 (τ ) − ∂τ (WM +1 (τ )) = (M + 1)2 −
(M + 1)2 + (M + 1)
.
cosh2 τ
(4.124)
M (M +1)
This proofs that for any integer M ∈ Z the potential VM = − cosh2 τ is reflectionless. Even the defined lowest order potential V− can be written in terms of
this potential by choosing M = 0. The transmission coeffient can be computed
from (4.112) together with (4.123):
ω − iM tanh(−∞)
ω − iWM (−∞)
TM − 1 =
TM
ω − iWM (∞)
ω − iM tanh(∞)
ω + iM
=
.
ω − iM
TM =
(4.125)
T0 is defined to be equal to one so the transmission becomes:
T=
M
Y
ω + in
.
ω − in
(4.126)
n=1
Choosing M to be L − 1 we get the potential found for the de Sitter space with
the same transmission coefficient computed before.
L(L − 1)
,
cosh2 τ
L−1
Y ω + in
T=
.
ω − in
V =−
(4.127)
(4.128)
n=1
This confirms that odd dimensional de Sitter space has a reflectionless potential.
4.4
Particle creation in de Sitter space
In this section the final step towards particle creation in de Sitter space will be
made. The reflectionless property of the Poschl-Teller potential has an interesting consequence for the particle creation. To determine the particle creation
we first need to establish the relation between the transmission and reflection
and the Bogoliubov coefficients. More on this relation is found in [16].With
43
the reflection and transmission coefficients we can write exact solutions to the
Klein-Gordon equation at the infinite past and future limit. The first thing to
do is changing back to the original modes(4.32):
φl = u(τ )Yl (Ωn−1 ) cosh−
n−1
2
τ.
(4.129)
Following the problem for the reflection and transmission coefficients (4.45)we
can write:
φl =
{
Yl (Ωn−1 ) cosh−
n−1
2
τ (eiωτ + Re−iωτ ) τ → −∞,
Yl (Ωn−1 ) cosh−
n−1
2
τ T eiωτ , τ → +∞
.
(4.130)
Just like with the FRW spacetime we can obtain the Bogoliubov coefficients
by writing the in-state, where τ → −∞, as a linear combination of out-states,
where τ → ∞:
out
φin
+ βl−l uout∗
l = αll ul
−l .
(4.131)
By noticing that the hyperbolic cosine is real and an even function so that
cosh τ = cosh −τ this can be done in the current form:
−
φin
l = Yl ( Ωn−1 ) cosh
n−1
2
τ (eiωτ + Re−iωτ )
n−1
1
R
= Yl (Ωn−1 ) cosh− 2 τ ( T eiωτ + T e−iωτ
T
T
1 out R out∗
= φl + φl .
T
T
(4.132)
Hence, the Bogoliubov coefficients in global de Sitter space are given:
|R|2
,
|T |2
1
|αk |2 =
.
|T |2
|βk |2 =
(4.133)
(4.134)
In this case the reflection coefficient is equal to zero, therefore the second Bogoliubov coefficient is also zero. The expectation value of the number operator
is related to the absolute squared value of the second Bogoliubov coefficient:
N̂k = |βk |2 .
(4.135)
The de Sitter universe in odd dimensions with no particles at the infinite past,
the vacuum in-state, will have no particles at the infinite future. However, this
computation does not give information about particle creation in between the
infinities. It could be possible that particles are created and then annihilated
somewhere between the past and future infinity limit[11]. As we are not living in
the past or future infinity limit, this is another reason why our universe cannot
be described by this computation. Up till now we have been restricted to de
Sitter space in odd dimension. In the next section will be discussed in which way
the computation of the particle creation for de Sitter space in even dimensions
space differs from odd dimensions.
44
4.5
De Sitter space in even dimensions
This section will discuss the particle creation for de Sitter space in even dimensions. It is not necessary to set up the whole problem from the start to
compute the particle creation. Most of the computation done in section 4.3.1
is the same for the even-dimensional case. Instead of redoing the computation
only the differences will be discussed and the influence on the particle creation.
For odd dimensions the variable L in the equation of motion is an integer
(4.38) while for even dimensions L becomes a half integer. For half integer
values of L the algebraic method used in section 4.3.2 will get no results. The
reflectionless potential derived in that section looked like
VM = C −
M (M + 1)
,
cosh2 τ
(4.136)
with C some constant that did not affect the reflectionless property. The lowest
order potential for integer M was defined to be a reflectionless potential by
choosing M = 0:
V− = (0 + 1)2 −
0(0 + 1)
= 1.
cosh2 τ
(4.137)
However, now we have half integers for M so the lowest value can not be chosen
equal to zero. The lowest positive value will now be M = 1/2 so that
V− = (1/2)2 −
1/2(1/2 + 1)
3
= (1/4) −
.
2
cosh τ
4 cosh2 τ
(4.138)
For any half integer value of M there will a term with cosh−2 τ , leading to a
comparable differential equation as the equation we started with (4.41). Hence,
the only way to find out if the potential is reflectionless for even dimensions is
by computing the equation of motion directly. For odd dimensions this is done
in section 4.3.1.
We will now discuss where the direct solutions to the equation of motion for
de Sitter space differ in even and odd dimensions. In section 4.3.1 there are two
equations that exploit the odd-dimensional property. The first equation is the
simplification of the sines (4.90). This equation cannot be used anymore when
L is a half integer. The second equation is the simplification of the gamma
functions (4.91). When L is a half integer the gamma functions can only be
partially simplified using
L− 12
Γ(1 − L ∓ iω ) = Γ(∓iω + 1/2)
Y
n=1
45
(z − n)−1 .
(4.139)
Now using the same algebra as with the odd solution we will get the transmission
and reflection coefficients:
1
L+1
1 L−
iω Y2 ω + in
sin
(
π
Γ
(
iω
)
Γ
(
−iω
−
)
+
π
)
)
1 sin(π L2 + π iω
2
2
2
2
−
T=
1
iω
1
2 sin(π L2 − π iω
ω − in
sin(π L+
2 )
2 − π 2 ) Γ (−iω ) Γ (iω − 2 ) n=1
L− 1
= (−1)
2L+1
Γ(iω )Γ(−iω − 12 ) Y2 ω + in
i tanh(πω )
,
ω − in
Γ(−iω )Γ(iω − 12 )
(4.140)
n=1
1
iω 1
1 L−
Y2 ω + in
sin(π L+
1 sin(π L2 + π iω
2 )
2 + π 2 ) Γ (iω ) Γ (−iω − 2 )
R=
+
1
1
iω
2 sin(π L2 − π iω
ω − in
sin(π L+
2 )
2 − π 2 ) Γ (−iω ) Γ (iω − 2 ) n=1
L− 1
Γ(iω )Γ(−iω − 21 ) Y2 ω + in
1
=
.
cosh πω Γ(−iω )Γ(iω − 21 )
ω − in
n=1
(4.141)
The scattering coefficients found for even dimensions cannot be as easily simplified as the coefficients for odd dimensions. However, following (4.133) we need
the absolute squared values of the coefficients to compute the particle creation.
We can write:
|T |2 = T ∗ T = tanh2 (πω ),
1
|R|2 = R∗ R =
.
cosh πω
(4.142)
(4.143)
Now using equation (4.133) and (4.135) we will get the particle creation for de
Sitter space in even dimensions:
N̂k = |βk |2 =
|R|2
1
=
|T |2
sinh2 πω
(4.144)
We put back the frequencys in terms of the field mass (4.38):
1
N̂k =
sinh2 π
q
m2 −
(n−1)2
4
.
(4.145)
Notice that this expression becomes complicated when the mass m of the field
(n−1)2
is low. When m2 < 4
the particle creation becomes imaginary and the
periodic sine would blow up. These complications can be avoided by keeping
restricted to a massive field.
In summary, de Sitter space in global coordinates with even dimensions has
no particle creation between past and future infinity. However, when changing
to even dimensions this is not the case for a massive field and particles are
created.
46
Chapter 5
Conclusion and discussion
It has been shown that particle production is a property of curved spacetime.
The particle production is directly connected to the coefficients of the Bogoliubov transform. With this we tried to compute the particle creation in
Friedmann-Robertson-Walker space and de Sitter space. This is done by comparing the infinite past with the infinite future limit.
We started with Friedmann-Robertson-Walker space, the conformally flat
spacetime model describing a smoothly expanding universe. The result was a
non-zero value for the Bogoliubov coefficients for a massive field. The conclusion
could therefore be made that curved spacetime indeed leads to particle production. This is a very interesting property, meaning that a smoothly expanding
massive universe starting out as empty vacuum in the far past will have particles
at the far future.
The next explored spacetime model was odd dimensional de Sitter space, describing an accelerated expanding universe. There are many ways of putting up
coordinate systems satisfying the definition of de Sitter space. In this thesis we
have restricted us to the coordinate system covering the whole of de Sitter space,
so called global coordinates. The whole of de Sitter space can be described by a
hyperboloid: Infinite expansion at the far past and future limit and its minimum
at the present. It became clear that odd and even dimensions made a great difference to the computation of particle creation. The computation was done by
setting up a scattering problem of a propagated wave from past to future infinity.
It was found that there was no reflection of the propagated wave in odddimensional global de Sitter. This has been explored further by approaching
arbitrary Hamiltonians with reflectionless potentials algebraically. This led to
the the same differential equation found for de Sitter space. We explored the direct relation between the scattering coefficients and the Bogoliubov coefficients.
The conclusion was made that a vacuum state at past infinity will still be in the
vacuum state at future infinity in odd dimensional global de Sitter space. After
47
that the differences between de Sitter space in even and odd dimensions were explored. This led to the conclusion that, in contrast to odd-dimensional de Sitter
space, there is particle creation in massive de Sitter space with even dimensions.
In summary, there has been shown that curved spacetime can lead to particle
creation. However, not every curved spacetime model will necessarily lead to
particle creation. There is particle creation in massive Friedmann-RobertsonWalker space but in odd dimensional de Sitter space, when only looking at past
and future infinity, there is not.
This is not the last thing to be said about particle creation. There are many
other curved spacetime models that cause particle creation. The direct next step
would be to explore the other coordinate systems of de Sitter space. For example the static coordinates could be described as the system of a static earthly
observer. It then might be possible to try and describe astronomical observations with the theory of particle production. As mentioned in the introduction
the inflationary period was approximately de Sitter, so particle creation should
have greatly influenced the early development of our observable universe.
48
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