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: christoffelhendriks@hotmail.com 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.
Abstract
The particle creation in two different inflationary spacetime models is computed. An introduction to quantum scalar fields is given and the relation between parti-cle 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 4
2.1 Quantum field theory . . . 4
2.2 Canonical quantization . . . 5
2.3 Defining vacuum and the number operator . . . 9
2.4 Bogoliubov transform . . . 10
2.5 Particle creation in curved spacetime . . . 12
3 Particle creation in Friedmann-Robertson-Walker space 14 3.1 The Friedmann-Robertson-Walker metric . . . 14
3.2 Particle creation in FRW . . . 16
4 De Sitter space 23 4.1 De Sitter hyperboloid . . . 23
4.2 Coordinate systems of de Sitter space . . . 24
4.2.1 Global coordinates . . . 24
4.2.2 Static coordinates . . . 27
4.2.3 Planar coordinates . . . 28
4.3 Transmission and reflection in global de Sitter space . . . 32
4.3.1 Direct solution of the Poschl-Teller wave equation . . . . 34
4.3.2 Approaching reflectionless potentials algebraically . . . . 40
4.4 Particle creation in de Sitter space . . . 43
4.5 De Sitter space in even dimensions . . . 45
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 uni-verse is known as de Sitter space. During the inflationary period the uniuni-verse 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 num-ber 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 produc-tion 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.
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.
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.
2.2
Canonical quantization
This section will quantize a free quantum field following [2]. For more informa-tion 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 ∂~xφa(~x, t). We define the
Lagrangian in terms of a so called Lagrangian density. For the 3+1-dimensional case we have:
L(t) =
Z
dx3L(φa, ∂µφa). (2.2)
This enables us to write down the action in terms of all dimensional variables:
S=
Z
dtL(t) =
Z
dx4L. (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 δS= Z dx4 ∂L ∂φa δφa+ ∂L ∂(∂µφa) δ(∂µφa) = Z dx4 " ∂L ∂φa − ∂µ ∂L ∂(∂µφa) δφa+∂µ ∂L ∂(∂µφa) δφa # =0. (2.5)
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:
∂µ ∂L ∂(∂µφa) − ∂L ∂φa =0. (2.6)
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]:
L=1 2η µν∂ µφa∂νφa− 1 2m 2φ2 a =1 2φ˙a 2 −1 2(∇φa) 2−1 2m 2φ2 a. (2.7)
Here is chosen for the signature(+1, −1, −1, −1)for the Minkowski metric ten-sor. The variable m is the total mass of the field. Now solving equation (2.6) we have:
∂µ(∂µφa) +m2φa=0. (2.8)
Denoting the Minkowski Laplacian ∂µ∂µ as we have arrived at the
Klein-Gordon 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
πa(~x, t)of the field:
πa(~x, t) = ∂L ∂ ˙φa
. (2.10)
Notice that πa 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 πa 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 πa 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), πa(~y, t0)] =iδ3(~x − ~y). (2.12)
We can assume further that φa and πa commutate with themselves just like x
and p:
[φa(~x, t0), φa(~x, t0)] = [πa(~y, t0), πa(~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
( +m2)φa= ∂2 ∂t2 − ∇ 2+m2 φa=0. (2.14)
We need a way to split up the infinite degrees of freedom of the field. This can be done using a Fourier transform[4]:
φ(~x, t) =
Z dk3
(2π)3/2e
i~k~xφ
a(~k, t). (2.15)
Putting this in the Klein-Gordon equation we get: Z dk3 (2π)3/2e i~k~x ∂2 ∂t2+k 2+m2 φa(~k, t) =0. (2.16)
Denoting the frequency ω~2
k=
~k2+m2 we can write the equation of motion for
φ(~k, t): (∂ 2 ∂t2 +ω 2 ~ k)φ( ~ k, t) =0. (2.17)
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 a~± k ≡ r ω~k 2 (φ(±~k, t)∓ iπ(±~k, t) ω~k ), (2.18)
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): φ(~k, t) = 1 p2ω~k (a+ −~ke iω~ kt+a− ~ke −iω~kt), (2.19) π(~k, t) =ir ω~k 2 (a − ~ ke −iω~kt− a+ −~ke iω~kt). (2.20)
Here we used the general relation a~±
k(t) =a
±
~ ke
±iω~k. Using the Fourier transform we get: (2.15) φ(~x, t) = Z dk3 (2π)3/2 1 p2ω~ k (a~+ ke iω~kt−i~k~x+a− ~ke −iω~kt+i~k~x), (2.21) with momentum π(~x, t) = Z dk3 (2π)3/2i r ω~ k 2 (−a + ~ ke iω~kt−i~k~x+a− ~ ke −iω~kt+i~k~x). (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:
[φ(~x, t), π(y, t)] = Z dp3dq3 (2π)3 i 2 rω q ωp
[a−p, a+q]ei(p~x−qy)−[a+p, a−q]e−i(p~x−qy)
The dirac delta function can be written:
δ3(~x − y) =
Z dp3
(2π)3/2e
ip(~x−y). (2.24)
Equations (2.23) and (2.24) can only be true when:
[a−p, a+q] = (2π)3/2δ3(p − q), (2.25)
and we had already assumed
[a−p, a−q ] = [a+p, a+q] =0. (2.26) By substituting: u~k= 1 (2π)3/2 1 p2ω~k eiω~kt+i~k~x, (2.27)
in equation (2.21) we can write down the field as a function of waves u~
k.
How-ever, 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 values k. We can rescale the wave functions uk→(2πL)3/2uk so that:
u~
k=
1 p2Lω~k
eiω~kt+i~k~x (2.30)
This allows for the integral in equation (2.21) to be replaced by an infinite sum over k: φ(~x, t) = Z dk3[a−kuk+a+ku ∗ k] = X k [a−kuk+a+ku ∗ k]. (2.31)
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.
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 the amount of energy, equal to the excitations of the modes. Hence, a system withP nk particles has modes uk with excitation nk.The operators a+k and a−k are known as the creation and annihilation operators for the kthmode. Consider a state uk with nkparticles. Using the commutation
relation of the ladder operators(??) in bra-ket notation the following statements hold:
(a−ka+k − a+ka−k)|nki=|nki .
(2.32)
From this equation we can derive the eigenvalues of the ladder operators:
a−k |nki= p nk+D |nk− 1i , ak+|nki= p 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| a+ka − k |nki=hnk− 1| √ nk √ nk|nk− 1i=nk. (2.34)
The operator a+ka−k is called the number operator, from here on denoted ˆNk.
The expectation value of the summation all number operators will be equal to the total amount of particles in the system:
ˆ Nk≡ a+ka − k, Nˆ = X k ak+a−k, (2.35) hφ| ˆN |φi=X k nk. (2.36)
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 |0ki=0, (2.37)
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.
φ(x, t) =X
j
a−
juj(t, x) +a+ju∗j(t, x). (2.39)
Defining a different annihilation operator automatically defines a new vacuum state for this set:
a−j |0ji=0. (2.40)
We want to show that the expectation value of the new number operator ˆN
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:
uj=
X
k
αjkuk+βjku∗k. (2.41)
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:
uk=X
j
α∗
jkuj− βjku∗j. (2.43)
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 k a−kuk+a+ku ∗ k= X j a−j uj+a+ju∗j. (2.44)
Now using (2.41) to substitute uj and u∗j we get X k a−kuk+a+ku∗k= X j a−j X k αjkuk+βjku∗k +a+j X k α∗ jku∗k+βjk∗ uk (2.45) =X j X k (a−jαjk+a+jβjk∗ )uk+ (a−jβjk+a+jαjk∗ )u∗k, (2.46)
so that that for every kth mode:
a−kuk+a+ku∗k=X
j
(a−jαjk+a+j βjk∗ )uk+ (a−jβjk+aj+α
∗
jk)u∗k. (2.47)
Because the modes and its conjugates are orthogonal both parts uk and u∗k can
be computed separately a−k =X j (a−jαjk+a+jβ ∗ jk) (2.48) a+k =X j (a−jβjk+a+jα∗jk). (2.49)
In the same way the new annihilation and creation operators can be computed by using equation (2.43): a−j =X k (α∗jka−k − βjk∗ a+k), (2.50) a+j =X k (αjka+k − βjka−k). (2.51)
The operator a−k is defined as the annihilation operator for the old mode expan-sion 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 j (a−jαjk+aj+βjk∗ )|0i (2.52) =X j (αjka−j |0i+βjka+j |0i) (2.53) =X j βjk|1i 6=0. (2.54)
Therefore, the expectation value for the number operator will not necessarily be zero in the new vacuum state either:
h0| ˆNk|0i=h0| a+ka − k |0i= X j |βjk|2. (2.55)
Similarly, the new annihilation operator not necessarily annihilates the old vac-uum state: a−j |0i=X k −βjk∗ |1i , (2.56) h0| ˆNj|0i= X k |βjk|2. (2.57)
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 uink , and one with past infinity mode solutions, from here on the out-states uoutk . Following equation (2.41) the in-states can be represented in terms of the out-states:
uink =X
j
αkjuoutj +βkjuout∗j . (2.58)
When the difference between the in-states and out-states is only time depen-dent 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):
uink =αkkukout+βk−kuout∗−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 (2π)3/2 1 p2ω~ k eiω~kt+i~k~x. (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
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 −g∂µ(
√
−ggµν∂ν). (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
uink 6=uoutk (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 out-states 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-Robertson-Walker space.
Chapter 3
Particle creation in
Friedmann-Robertson-Walker
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
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
a(t)=dx. (3.3)
Integrating both side of the equation will get for the first light ray Z t2 t1 dt a(t)= Z xf 0 dx, (3.4)
and for the second light ray Z t2+δt2 t1+δt1 dt a(t)= Z xf 0 dx. (3.5)
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 a(t) =0= 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
0= Z t2+δt2 t2 dt a(t)− Z t1+δt1 t1 dt a(t). (3.7)
Assuming that both δt1, δt2 a/ ˙a the integrals can be approximated:
0= δt2 a(t2)
− δt1
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) a(t1) =δt2 δt1 =λ2 λ1 =1+z. (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
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)(dη2− dx2) =C(η)(dη2− 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 Klein-Gordon equation(2.8):
( +m2)φ=0, (3.14)
with the Laplacian for curved spacetime[4]:
≡√1 −g∂µ(
√
−ggµν∂ν). (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=eikxχk(η), (3.16)
with χk(η) the yet unknown time dependent part of the mode. with gµν 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≡ gµνdxµdxν, (3.17) so that: gµν=C(η) 1 0 0 −1 , gµν= 1 C(η) 1 0 0 −1 , (3.18) g=−C(η). (3.19)
The Laplacianbecomes
= 1
C(η)(∂
2
η− ∂x2). (3.20)
Using this in the Klein-Gordon equation equation (3.14) the following equation of motion for χk can be obtained:
∂η2χk(η) + (k2+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)1/2η
, 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≡(k2+Am2− Bm2)1/2, ωout≡(k2+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:
uink = (2Lωin)−1/2eikx−iωinη, (3.25)
uoutk = (2Lωout)−1/2eikx−iωoutη. (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
∂η=∂z dz dη= 1 2ρsech 2(ηρ)∂ z=2ρz(1 − z)∂z, ∂ηχ=2ρz(1 − z)∂zχ, ∂η2χ=4ρ2z2(1 − z)2∂2 z+2ρz(1 − z)2ρ(1 − 2z)∂zχ. (3.27)
The equation of motion can be written:
0=2ρz(1 − z)∂2z+2ρ(1 − 2z)∂z+ k2+Am2+Bm2(2z − 1) 2ρz(1 − z) χ =∂z2+ (1 z − 1 1 − z)∂z+ k2+Am2+Bm2(2z − 1) 4ρ2z2(1 − z)2 χ. (3.28)
With the substitutions ωin/out = (k2+Am2± Bm2)1/2 this can further be simplified to 0=∂2z+ (1 z− 1 1 − z)∂z+ ωin2 (1 − z) +ωout2 z 4ρ2z2(1 − z)2 χ =∂2z+ (1 z− 1 1 − z)∂z+ ωin2 4ρ2z2(1 − z) + ωout2 z 4ρ2z(1 − z)2χ. (3.29)
This equation is of the form of the Riemann’s differential equation[8]:
∂2zχ+ 1 − α − α 0 z − a + 1 − β − β0 z − b + 1 − γ − γ0 z − c ∂zχ + αα 0(a − b)(a − c) z − a + ββ0(b − c)(b − a) z − b + γγ0(c − a)(c − b) z − c χ (z − a)(z − b)(z − c)=0, (3.30) where α=−α0=iωin 2ρ , β=−β 0=iωout 2ρ , γ=1 − γ 0=1, (3.31) a=0, b=1, c=∞, (3.32) so that α+α0+β+β0+γ+γ0=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
χ1=C1( z − a z − c) α0(z − b z − c) β 2F1 α0+γ+β, α0+γ0+β; 1+α0− α;(b − c)(z − a) (z − c)(b − a) , (3.34) and χ2=C2( z − a z − c) α0(z − b z − c) β 2F1 α0+γ+β, α0+γ0+β; 1+β − β0;(a − c)(z − b) (z − c)(a − b) , (3.35) where C1and C2are 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
C1=C2= (−c)α
0
Putting in the variables of the Riemann differential and using that c z this becomes
χ1= (z)
−iωin
2ρ (1 − z)iωout2ρ
2F1(1+ (iω−/ρ), iω−/ρ; 1 −(iωin/ρ); z),
(3.37) χ2= (z) −iωin 2ρ (1 − z) iωout 2ρ
2F1(1+ (iω−/ρ), iω−/ρ; 1+ (iωout/ρ); 1 − z).
(3.38) For notational convenience we have defined
ω±≡1
2(ωout± ωin). (3.39) After returning to the time parameter by putting
z=1 2(tanh(ρη) +1) = e2ρη e2ρη+1, (3.40) we get χ1= ( e2ρη e2ρη+1) −iωin/2ρ( 1 e2ρη+1) iωout/2ρ
2F1(1+ (iω−/ρ), iω−/ρ; 1 −(iωin/ρ);
1 2(tanh(ρη) +1), (3.41) χ2= ( e2ρη e2ρη+1) −iωin/2ρ( 1 e2ρη+1) iωout/2ρ
2F1(1+ (iω−/ρ), iω−/ρ; 1+ (iωout/ρ);
1
2(1 − tanh(ρη)). (3.42)
In the limit η → −∞ we can approximate 1+e2ρη≈ 1. Hence, the first solution can be approximated:
χ1→ e−ωinη. (3.43)
By taking the limit η → ∞ so that 1+e2ρη≈ e2ρη, the second solution can be
approximated:
χ2→ e−ωoutη. (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 2F1(a, b, c; z) = Γ (c)Γ(c − a − b) Γ(c − a)Γ(c − b)2F1(a, b, a+b+1 − c; 1 − z) +Γ(c)Γ(a+b − c) Γ(a)Γ(b) (1 − z) c−a−b 2F1(c − a, c − b, 1+c − a − b; 1 − z), (3.45)
so χ1 can be written
χ1=z
−iωin
2ρ (1 − z)iωout2ρ
Γ
(1 − iωin/ρ)Γ(−iωout/ρ)
Γ(−iω+/ρ)Γ(1 − iω+/ρ) 2
F1(1+ (iω−/ρ), iω−/ρ; 1+ (iωout/ρ); 1 − z)
+Γ(1 − iωin/ρ)Γ(iωout/ρ)
Γ(iω−/ρ)Γ(1+iω−/ρ)
(1 − z)−iωout/ρ
2F1(−iω+/ρ, 1 − iω+/ρ; 1 − iωout/ρ; 1 − z)
. (3.46)
Another property of hypergeometric functions is the equation
2F1(a, b; c; z) = (1 − z)c−a−b2F1(c − a, c − b; c; z). (3.47)
Using this we can derive
2F1(−iω+/ρ, 1 − iω+/ρ; 1 − iωout/ρ; 1 − z) =ziωin/ρ2F1(1 − iω−/ρ, −iω−; 1 − iωout/ρ; 1 − z).
(3.48) The function χ1 can now be written
χ1=z
−iωin
2ρ (1 − z)iωout2ρ Γ(1 − iωin/ρ)Γ(−iωout/ρ)
Γ(−iω+/ρ)Γ(1 − iω+/ρ) 2
F1(1+ (iω−/ρ), iω−/ρ; 1+ (iωout/ρ); 1 − z)
+ziωin/2ρ(1 − z)−iωout2ρ Γ(1 − iωin/ρ)Γ(iωout/ρ)
Γ(iω−/ρ)Γ(1+iω−/ρ) 2
F1(1 − iω−/ρ, −iω−; 1 − iωout/ρ; 1 − z)
(3.49)
=Γ(1 − iωin/ρ)Γ(−iωout/ρ)
Γ(−iω+/ρ)Γ(1 − iω+/ρ)
χ2+Γ
(1 − iωin/ρ)Γ(iωout/ρ)
Γ(iω−/ρ)Γ(1+iω−/ρ)
χ∗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):
uink (η, x) =p2Lωineikxχ1(η), (3.51)
uoutk (η, x) =p2Lπωouteikxχ2(η). (3.52)
Hence, the full in-states uink can be written in terms of the full out-states uoutk by: uink =αkukout+βkuout∗−k , (3.53) where αk= r ωout ωin
Γ(1 − iωin/ρ)Γ(−iωout/ρ)
Γ(−iω+/ρ)Γ(1 − iω+/ρ) , (3.54) βk=
r ωout
ωin
Γ(1 − iωin/ρ)Γ(iωout/ρ)
Γ(iω−/ρ)Γ(1+iω−/ρ)
. (3.55)
This tells us that αk and βk are the Bogoliubov coefficients for transition from
producing the out-state number operator. For this we need the absolute squared values of the Bogoliubov coefficients:
|αk|2=|ωout
ωin
|Γ(1 − iωin/ρ)Γ(1+iωin/ρ)Γ(−iωout/ρ)Γ(iωout/ρ) Γ(−iω+/ρ)Γ(iω+/ρ)Γ(1 − iω+/ρ)Γ(1+iω+/ρ)
, (3.56)
|βk|2=|
ωout
ωin
|Γ(1 − iωin/ρ)Γ(1+iωin/ρ)Γ(−iωout/ρ)Γ(iωout/ρ) Γ(−iω−/ρ)Γ(iω−/ρ)Γ(1 − iω−/ρ)Γ(1+iω−/ρ)
. (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 Γ(1 − iωin/ρ)Γ(1+iωin/ρ) =− ωin ρ 2Γ (iωin/ρ)Γ(−iωin/ρ), (3.60) Γ(1 − iω±/ρ)Γ(1+iω±/ρ) =− ω± ρ 2 Γ(iω±/ρ)Γ(−iω±/ρ). (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
Γ(iωin/out)Γ(−iωin/out) =−
π
iωin/outsin(πiωin/out)
, (3.63)
Γ(iω±)Γ(−iω±) =−
π iω±sin(πiω±)
. (3.64)
By using sin iz=i sinh z the following functions will remain
|αk|2=
ωout
ωin
(ωinπω+)2sin2(iπω+/ρ)
(πω+)2ωinωoutsin(iπωin/ρ)sin(iπωout/ρ)
= sinh
2(πω +/ρ)
sinh(πωin/ρ)sinh(πωout/ρ)
, (3.65)
|βk|2=
ωout
ωin
(ωinπω−)2sin2(iπω−/ρ)
(πω−)2ωinωoutsin(iπωin/ρ)sin(iπωout/ρ)
= sinh
2(πω −/ρ)
sinh(πωin/ρ)sinh(πωout/ρ)
. (3.66)
The expectation value of the out-state number operator ˆN can be computed
from the absolute second Bogoliubov coefficient:
h ˆN i=X
k
Putting in the functions for the frequenties using equations (3.23) and (3.39) we get the particle creation:
h ˆN i=X
k
sinh2[π/ρ(√k2+Am2+Bm2−√k2+Am2− Bm2)]
sinh(π√k2+Am2+Bm2)sinh(π√k2+Am2− Bm2). (3.68)
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 space-time 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.
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 Bogoli-ubov 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 re-lation.
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
i=1
Xi2=R2, (4.1)
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.
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 min-imal 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 hyper-boloid. 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 ,
Xi=ωicosh τ , i=1, ..., n − 1,
(4.2) The variables ωiare the normalized parametrization of the hypersphere Sn−1
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 j=1 sin θj, i=2, ..., n − 2, ωn−1= n−1 Y j=1 sin θj (4.3)
where 0 ≤ θi < π and 0 ≤ θn−1 < 2π. The coordinates are normalized so that n−1
X
i=1
ω2i =1. (4.4)
The metric on the Sn−1hypersphere can be computed by taking the differentials:
dω1=sin2θ1dθ12, dωi= i−1 X k=1 cos2θicos2θk Y j6=k [sin2θj]dθ2k + i Y j=1 [sin2θj]dθ2i, dωn−1= n−1 X k=1 cos2θk Y j6=k sin2θjdθ2k. (4.5)
Then the metric becomes:
dΩ2n−1= n−1 X i=1 dωi2=dθ21+ n−1 X k=2 k−1 Y j=1 sin2θjdθ2k. (4.6)
The global coordinate system satisfies the coordinate equation for de Sitter(4.1):
−X2 0+ n−1 X i=1 Xi2= n−1 X i=1 ω2icosh2τ − sinh2τ=1. (4.7)
The metric with signature (-, +, ..., +) can be written in terms of the metric
dΩn−1 of the hypersphere Sn−1:
ds2=gµνdXµdXν=−(cosh2τ − n−1
X
i=1
ωi−1sinh2τ)dτ2+cosh2τ dΩ2n−1
=−dτ2+cosh2τ 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.
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
cos2T(−dT
2+dΩ2
n−1). (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 confor-mal coordinate change. As norconfor-mal 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 Pen-rose 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
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:
X0= p l2− r2sinh τ Xi=rωi, i=1, ..., n − 2 Xn−1= p l2− r2cosh τ . (4.11)
Using equation (4.4) the static coordinates satisfy the de Sitter space equa-tion (4.1): −X02+ n−1 X i=1 Xi2= (1 − r2)(cosh2τ − sinh2τ) +r2 n−1 X i=1 ω2i−1=12. (4.12)
The metric can be obtained by computing the differentials:
dX02= (1 − r2)cosh2τ dτ2+ r 2 1 − r2sinh 2τ dr2, dX12= (1 − r2)sinh2τ dτ2+ r 2 1 − r2cosh 2τ dr2, n−1 X i=1 dXi= n−1 X i=1 ωidr2+r2 n−1 X i=1 dωi2. (4.13)
The line element for static de Sitter space can be obtained:
ds2=−(1 − r2)dτ2+ 1 1 − r2dr 2+r2 n−1 X i=1 dω2i. (4.14)
The metric in static coordinates is independent of τ so we say it has a time-like 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 dτ2 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 di-agram 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:
X0=sinh τ − 1 2xix ie−τ, Xi=xie−τ, i=1, ..., n − 2, Xn−1=cosh τ − 1 2xix ie−τ, (4.15) (4.16)
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 i=1 Xi2= (cosh2τ − sinh2τ)−xix ie−τ 2 2 +xix ie−τ 2 2 + (sinh τ − cosh τ+e−τ)xixie−τ=1. (4.17) Like the other coordinate systems we compute the differentials:
dX02= cosh2τ+xix ie−τ 2 2 − cosh τ xixie−τ dτ2− xixie−2τdxidxi, dXi=e−2τdxidxi− xixie−2τdτ2, dXn−12 = sinh2τ+xix ie−τ 2 2 − sinh τ xixie−τ dτ2− xixie−2τdxidxi. (4.18) With the same metric signature we get the line element for planar de Sitter space:
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:
X0=sinh τ − 1 2xix ieτ, Xi=xieτ, i=1, ..., n − 2, Xn−1=cosh τ − 1 2xix ieτ, (4.20) ds2=−dτ2+e2τdxidxi. (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 co-ordinates 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 transla-tional 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].
4.3
Transmission and reflection in global de
Sit-ter 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 transmis-sion 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=−dτ2+cosh2τ dΩ2
n−1, (4.22)
with dΩn−1the metric of the hypersphere Sn−1as in (4.5). For example in 2+1
dimensions we have
dΩ2n−1=dθ2+sin2θdφ2. (4.23) From the definition of the line element ds2=gµνdxµdxν the metric tensor gµν
for de Sitter space can be obtained. For example the metric tensor in 2+1-dimensions becomes: gµν= −1 0 0 0 cosh2τ 0 0 0 sin2θ cosh2τ , (4.24) with determinant
det(g) =− sin2θ(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):
(τ , θ, φ)≡√1 −g∂µ √ −ggµν∂ν, (4.27) = 1 r2sin θcosh −2τ ∂ µr2sin θ cosh2τ gµν∂ν,
=− cosh−2τ ∂τ[cosh2τ ∂τ] +cosh−2(θ, φ), (4.28)
with(θ, φ)the spherical Laplacian of the 2 dimensional hypersphere. Extrap-olating 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
should then be written with powers of n − 1. We get the de Sitter Laplacian in
n dimensions:
=− cosh1−nτ ∂τ[coshn−1τ ∂τ] +cosh−2τSn−1. (4.29) This enables us two write down the equation of motion in de Sitter space:
(− cosh1−nτ ∂
τ[coshn−1τ ∂τ] +cosh−2τSn−1− m2)φ=0. (4.30) Like the computation for FRW space, we expand the field φ into modes:
φ=X
l
[a−l φl+a+l φ
∗
l]. (4.31)
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)
where the term cosh(τ)−n−12 is chosen for convenience of further computation.
The spherical hamonics Yl(Ωn−1)are the eigenfunctions of the spherical
Lapla-cianSn−1. The corresponding eigenvalues are[13]:
∆Sn−1Yl(Ωn−1) =−l(l+n − 2)Yl(Ωn−1). (4.33)
The equation of motion (4.26) can be written:
− cosh1−nτ ∂
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−n2 =0. (4.34)
By computing the first partial derivative with for notational convenience ∂τu(τ) =
˙u(τ)and a=1 − n we have:
− coshaτ ∂τ[˙u(τ)cosh(τ)
−a
2 +u(τ)sinh(τ)a
2cosh(τ)
−a2−1]
−(l(l+n − 2)cosh(τ)−2+m2)u(τ)cosh(τ)a2 =0. (4.35)
After evaluating the second partial derivative it looks like:
¨u(τ) +u(τ)a 2 − u(τ) a(a+2) 4 + a(a+2) 4 +l(l+n − 2)u(τ)cosh(τ) −2+m2u(τ) =0. (4.36) By filling in a=1 − n the equation of motion for de Sitter space is obtained:
∂2τ+2l+n − 3 2 2l+n − 1 2 1 cosh2τ + (m 2−(n − 1)2 4 ) u(τ) =0. (4.37)
To create an idea what kind of equation we have encountered we make the substitutions: L ≡2l+n − 1 2 , ω ≡ r m2−(n − 1)2 4 , (4.38) so that: h − ∂2τ−L(L − 1) cosh2τ i u(τ) =ω2u(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 correspond-ing to this Schrödcorrespond-inger equation is known as the Poschl-Teller potential[14]:
V =−L(L − 1)
cosh2τ . (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
∂τ2+ω2+L(L − 1)
cosh2τ
i
u=0. (4.41)
At past and future infinity we can approximate
cosh τ → ∞, τ → ±∞, (4.42) so the equation of motion becomes
[∂τ2+ω2]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=
{
T eiωτ, τ →+∞
eiωτ+Re−iωτ, τ → −∞
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 ≡ cosh2τ we have:
∂τ =∂y dy dτ =2 sinh τ cosh τ ∂y= q −4(1 − y)y∂y, (4.46) ∂τu=2i q y(1 − y)∂yu, (4.47) ∂τ2u=−4y(1 − y)∂y2u − 2(1 − 2y)∂yu. (4.48)
The differential equation becomes
[y(1 − y)∂y2+ (1/2 − y)∂y−
ω2
4 −
L(L − 1)
4y ]u=0. (4.49) By choosing u ≡ yL/2v(y)the differentials can be rewritten
u ≡ yL/2v(y), ∂yu= (
L
2yv+∂yv)y
L/2, (4.50)
∂2y= (L/2(L/2 − 1)y−2v+Ly−1∂yv+∂y2v)yL/2, (4.51)
leading to the differential equation:
y(1 − y)∂y2v+ ((L+1/2)−(L+1)y)∂yv − 1/4(L2+ω2)v=0. (4.52)
This is a hypergeometric equation[4] of the form
x(1 − x)∂2xz+ (c −(a+b+1)x)∂xz − abz=0, (4.53)
where
x=1 − y, z=v, (4.54)
a=1/2(L+iω), b=1/2(L − iω), c=1/2, (4.55)
ab=1/4(L2+ω2). (4.56) The solution to equation (4.52) is a linear combination of the hypergeometric functions: v=A2F1(a, b, 1 2; 1 − y) +B p 1 − y2F1(a+ 1 2, b+ 1 2, 3 2; 1 − y), (4.57) 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 cosh2τ=y. Using the proposed substitution for u (4.50) we get:
u=v coshLτ , =A coshLτ 2F1(a, b, 1 2; − sinh 2τ), − B coshLτ sinh τ 2F1(a+ 1 2, b+ 1 2, 3 2; − sinh 2τ). (4.58)
