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The Cosmological Constant,Vacuum Energy and Dark Energy
Fatima Talhi
Master 2013
Supervisor: Badis Ydri
June 20, 2013
Abstract
It is well established by now that the universe is spatially flat and is composed of 4 per cent
ordinary mater, 23 per cent dark matter and 73 per cent dark energy. The dominant component,
dark energy, is believed to be the same thing as the cosomlogical constant introduced by Einstein in
1917 which in turn is believed to originate in the energy of the vacuum. Dark energy is characterized
mainly by a negative pressure and no dependence on the scale factor and its density behaves as
∼ H20 Λpl where H0 is the Hubble parameter and Λpl = 1/
√8πG is the Planck mass. The reality of
the energy of the vacuum is exhibited in a dramatic way in the Casimir force. In this dissertation
we present a discussion of various aspects of the cosmological constant, vacuum energy and dark
energy. We pay a particular attention to the calculation of vacuum energy in curved spacetimes
such as the FLRW universes and de Sitter spacetime which requires the use of quantum field theory
in the presence of a non zero gravitational background. de Sitter spacetime is of particular interest
since we know that both the early universe as well as its future evolution is dominated by vacuum,
i.e. FLRW universes may be understood as a deformation of de Sitter. We compute the vacuum
energy in an expanding de Sitter spacetime and show that it behaves in the right way as H2Λ20
where H is the de Sitter Hubble parameter and Λ0 is a comoving cutoff.
2
Acknowledgments
I would like to express my sincere gratitude to my thesis supervisor, Dr. Badis Ydri who has
consistently inspired me in this study and provided me with precious suggestions and advices. Without
his attentive guidance, this thesis would not have been possible to accomplish.
Special thanks to Dr. Adel Bouchareb for his time, patience, and understanding. He has provided
me with many constructive suggestions and comments not only on the structure and wording of the
thesis but also on the content.
My sincere thanks also go to Professor Reda Attallah, Dr. Rafik Chemam, and Dr. Mohamed Cherif
Talai for their endless patience, encouragement and advices with their extensive knowledge through the
two years of my master.
I am also grateful to all my teachers in the departement of physics.
Last but not least, I am deeply appreciative of my family, who have always supported me through the
difficult times. It is their love and support that have always encouraged me to stick on to the difficult
task through all my studies.
Dedication
To the two pillars of my life,
Mom and Dad,
you have given me so much, thanks for your faith in me, and for teaching me that I should never
surrender.
CONTENTS 3
Contents
1 Introduction 4
2 Special and General Relativity 4
2.1 Special Relativity and Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 4
2.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Einstein Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.4 Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The Hilbert-Einstein Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Cosmology 8
3.1 The Friedmann-Lemaıtre-Robertson-Walker Universe . . . . . . . . . . . . . . . . . . 8
3.1.1 Scale Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.2 The Friedmann-Lemaıtre Equations of Motion . . . . . . . . . . . . . . . . . 9
3.2 Concordance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Cosmological Constant, Vacuum Energy and Dark Energy 13
4.1 Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Einstein Static Universe and Vacuum Energy . . . . . . . . . . . . . . . . . . . . . . 14
5 Calculation of Vacuum Energy in Curved Backgrounds 19
5.1 Elements of QFT in curved spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Quantization in FLRW Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 Quantization in de Sitter Spacetime and Bunch-Davies Vacuum . . . . . . . . . . . . 24
6 Is Vacuum Energy Real? 28
6.1 The Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 The Dirichlet Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.3 Another Derivation Using The Energy-Momentum Tensor . . . . . . . . . . . . . . . 33
7 Conclusion 36
Bibliography 36
1 Introduction 4
1 Introduction
2 Special and General Relativity
2.1 Special Relativity and Lorentz Transformations
Special relativity depends upon two fundamental postulates:
• The laws of physics take the same form in all inertial reference frames.
• The speed of light in vacuum has the same value in all inertial reference frames.
The Lorentz transformation relating two observers, O and O′, where O′ moves with speed u in the
x direction, is given by the equations
x′ =x− ut√1− u2/c2
y′ = y
z′ = z
t′ =t− (u/c2)x√
1− u2/c2. (2.1)
By denoting
β =u
c, γ =
1√1− β2
,
the Lorentz transformations may be rewritten as
x0′ = γ(x0 − βx1)
x1′ = γ(x1 − βx0)
x2′ = x2
x3′ = x3. (2.2)
In the above equation : x0 = ct , x1 = x , x2 = y , x3 = z. Lorentz transformations can also be
rewritten as
xµ′
= Λµνxν (2.3)
Λ =
γ −γβ 0 0
−γβ γ 0 0
0 0 1 0
0 0 0 1
. (2.4)
In general a 4-vector is any set of numbers (a0, a1, a2, a3) wich transforms as (x0, x1, x2, x3) under
Lorentz transformations
aµ′
= Λµνaν
The numbers aµ are called the contravariant components of the 4-vector a. We define the covariant
components aµ by
a0 = −a0 , a1 = a1 , a2 = a2 , a3 = a3.
2.1 Special Relativity and Lorentz Transformations 5
The 4-dimentional scalar product must therefore be defined by the Lorentz invariant combination
ab = −a0b0 + a1b1 + a2b2 + a3b3
= aµbµ. (2.5)
We define the separation 4-vector ∆x between two events A and B occuring at the points (x0A, x
1A, x
2A, x
3A)
and (x0B , x
1B , x
2B , x
3B) by the components
∆xµ = xµA − xµB .
The distance squared between two events A and B (interval) is defined by
∆s2 = ∆xµ∆xµ = −c2∆t2 + ∆~x2.
This is a Lorentz invariant quantity. It could be positive, negative or zero, viz
• ∆s2 < 0 : the interval is called timelike.
• ∆s2 > 0 : the interval is called spacelike.
• ∆s2 = 0 : the interval is called lightlike.
At any event E in spacetime we can define a light cone. The past light cone contains all events
which can effect E, the future light cone contains all events that can be affected by E.
The interval ds2 between two infinitesimally close events A and B in spacetime is given by
ds2 = −c2(dt)2 + (d~x)2
The proper time dτ is defined by the equation
c2dτ2 = −ds2.
This is the time elapsed between the two events A and B as seen by an observer moving on a
straight line.
We can also write this interval as
ds2 = ηµνdxµdxν = ηµνdxµdxν . (2.6)
The 4×4 matrix η is called the metric tensor and it is given by
ηµν = ηµν =
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
. (2.7)
Clearly we can also write
ds2 = ηνµdxµdxν .
The metric η is used to lower and raise Lorentz indices, viz
xµ = ηµνxν .
The interval ds2 is invariant under Poincare transformations which combine translations a with
Lorentz transformations Λ, viz
xµ → x′µ = Λµνxν + aµ.
We compute
ds2 = ηµνdx′µdx′ν = ηµνdx
µdxν .
This leads to the condition
ηµνΛµρΛνσ = ηρσ ⇐⇒ ΛT ηΛ = η.
2.2 General Relativity 6
2.2 General Relativity
There are three essential ideas underlying general relativity (GR):
• Space-time is described by a curved, four-dimensional mathematical structure called a pseudo-
Riemannian manifold.
• At every spacetime point there exists locally inertial reference frames, corresponding to lo-
cally flat coordinates carried by freely falling observers, in which the physics of GR is locally
indistinguishable from that of special relativity. This is Einstein’s famous strong equivalence
principle and it makes general relativity an extension of special relativity to a curved space-
time.
• Dynamics is govern by Einstein’s equation.
2.2.1 Covariant Derivative
The covariant derivative is given by the expressions [1]
∇µV ν = ∂µVν + ΓνµλV
λ. (2.8)
∇µων = ∂µων − Γλµνωλ. (2.9)
Generally
∇σTµ1µ2...µkν1ν2...νl
= ∂σTµ1µ2...µk
ν1ν2...νl+ Γµ1
σλTλµ2...µk
ν1ν2...νl+ Γµ2
σλTµ1λ...µk
ν1ν2...νl+ ...
− Γλσν1Tµ1µ2...µk
λν2...νl− Γλσν2T
µ1µ2...µkν1λ...νl
− ... .(2.10)
Γνµλ is the Christoffel symbol. It is given by the expression
Γσµν =1
2gσρ(∂µgνρ + ∂νgρµ − ∂ρgµν). (2.11)
It is symmetric
Γσµν = Γσνµ. (2.12)
The covariant derivative of the metric and its inverse are always zero
∇σgµν = 0, ∇σgµν = 0. (2.13)
2.2.2 Riemann Curvature Tensor
The information about curvature is contained in a four-component tensor known as the Riemann
curvature tensor. It is given by the formula [1]
R σµαβ ≡ ∂αΓσµβ − ∂βΓσµα + ΓσαλΓλµβ − ΓσβλΓλµα. (2.14)
This tensor has the nice property that all of the components of R vanish if and only if the space
is flat. Operationally, flat means that there exists a global coordinate system in which the metric
components are everywhere constant.
The Riemann tensor obeys the properties [1]
• R σµνρ = −R σ
νµρ .
• R σ[µνρ] = 0.
• Rµνρσ = −Rµνσρ.
• The Bianchi identity ∇[µRλ
νρ]σ = 0.
2.2 General Relativity 7
We define the Ricci tensor by
Rαβ = R λαλβ = gµνRµν . (2.15)
It is symmetric
Rµν = Rνµ. (2.16)
The trace of the Ricci tensor yields the Ricci scalar
R = R λλ = gµνRµν . (2.17)
2.2.3 Einstein Equation
In General Relativity, the equation of motion for the metric is the Einstein equation
Rµν −1
2Rgµν = 8πGTµν . (2.18)
G is Newton’s constant of gravitation, Tµν is a symmetric two-index tensor called the stress- energy-
momentum tensor. Thus the left hand side of this equation measures the curvature of spacetime
while the right hand side measures the energy and momentum contained in it.
If we take the trace of both sides of the above equation we obtain
−R = 8πGT.
We replace back to obtain Einstein’s equation in the form
Rµν = 8πG(Tµν −1
2Tgµν). (2.19)
In vacuum there is no energy or momentum then Tµν = 0. In this case Einstein’s equation is
Rµν = 0.
We define the Einstein tensor by
Gµν ≡ Rµν −1
2Rgµν . (2.20)
The divergence of this tensor vanishes
∇µGµν = 0. (2.21)
2.2.4 Perfect Fluid
A perfect fluid, defined to be a fluid which is isotropic (the same in all direction) in its rest frame,
is completely specified in terms of its rest-frame energy density ρ and its rest-frame pressure P [2].
If Uµ stands for the four-velocity of a fluid element, the stress-energy-momentum tensor takes the
form
Tµν = (ρ+ P )UµUν + Pgµν . (2.22)
If we raise one index and use the normalization gµνUµUν = −1, we get
T νµ =
−ρ 0 0 0
0 P 0 0
0 0 P 0
0 0 0 P
. (2.23)
The conservation of energy and momentum is
∇µTµν = 0. (2.24)
2.3 The Hilbert-Einstein Action 8
2.3 The Hilbert-Einstein Action
The Einstein’s equations for general relativity read
Rµν −1
2gµνR = 8πGTµν . (2.25)
They can be derived from the action [3]
S = SHE + SM . (2.26)
SHE is the Hilbert-Einstein action which is given by
SHE =1
16πG
∫d4x√−detg R. (2.27)
SM is the matter action which is given by
SM =
∫d4x√−detg LM . (2.28)
This is related to the stress-energy-momentum tensor by
Tµν = − 2√−detg
δSMδgµν
. (2.29)
We will be mostly interested in scalar field. The action of a scalar field in curved spacetime is given
by
Sφ =
∫dnx
√−detg
[− 1
2gµν∇µφ∇νφ− V (φ)
]. (2.30)
The corresponding stress-energy-momentum tensor is given by
Tµν = ∇µφ∇νφ−1
2gµνg
αβ∇αφ∇βφ− gµνV (φ). (2.31)
3 Cosmology
3.1 The Friedmann-Lemaıtre-Robertson-Walker Universe
3.1.1 Scale Factor
We assume that the universe is homogeneous (all points are the same, this is invariance under
translations) and isotropic (the universe looks the same in all directions, this is invariance un-
der rotations). Equivalently this means that there exists a foliation of spacetime consisting of
3-dimensional maximally symmetric spatial slices Σ. The only possible homogeneous and isotropic
metric describing an expanding universe is the Robertson-Walker metric which is given by [5]
ds2 = −dt2 +R2(t)
[dr2
1− kr2+ r2dΩ2
]. (3.1)
Where dΩ2 = dθ2 + sin2 θdφ2. The scale factor R(t) gives the volume of the spatial slice Σ at the
instant of time t and k is the spatial curvature parameter given by
k =
−1 : this is an open universe in which the hypersurfaces are three hyperboloides Σ = H3.
0 : this is a flat universe in which the hypersurfaces are flat space Σ = R3.
+1 : this is a closed universe in which the hypersurfaces are three spheres Σ = S3.The scale factor R(t) has units of distance and thus r is actually dimensionless. We reinstate a
dimensionful radius ρ by ρ = R0r. The scale factor becomes dimensionless given by
a(t) =R(t)
R0.
3.1 The Friedmann-Lemaıtre-Robertson-Walker Universe 9
The function a(t) known as the scale factor is a measure of the size of the spacelike hypersurface
Σ, whereas the curvature becomes dimensionful given by κ = k/R0. The Robertson-Walker metric
becomes
ds2 = −dt2 + a2(t)
[dρ2
1− κρ2+ ρ2dΩ2
]. (3.2)
The non-zero components of the Ricci tensor in the Robertson-Walker metric are
R00 = −3a
a. (3.3)
Rρρ =1
1− κρ2(aa+ 2a2 + 2κ). (3.4)
Rθθ = ρ2(aa+ 2a2 + 2κ). (3.5)
Rφφ = ρ2 sin2 θ(aa+ 2a2 + 2κ). (3.6)
Einstein’s equations are given by (we will work mostly with 8πG = 1)
Rµν = 8πG(Tµν −1
2gµνT ). (3.7)
3.1.2 The Friedmann-Lemaıtre Equations of Motion
We will assume that the matter and energy content of the universe is given by a perfect fluid, the
stress-energy-momentum tensor of a perfect fluid is
Tµν = (ρ+ P )UµUν + Pgµν .
The fluid is obviously at rest in comoving coordinates. In other words Uµ = (1, 0, 0, 0) and hence
Tµλ = diag(−ρ, P, P, P ). (3.8)
The trace is clearly T µµ = −ρ+ 3P
The µ = 0, ν = 0 component of Einstein’s equations is
R00 = 8πG(T00 +1
2T )⇒
−3a
a= 8πG(ρ+ P − P +
1
2(−ρ+ 3P ))
= 4πG(ρ+ 3P )
We obtain the equationa
a= −4πG
3(ρ+ 3P ). (3.9)
This is the second Friedmann equation.
From the other hand the µ = ρ, ν = ρ component of Einstein’s equations is
Rρρ = 8πG(Tρρ −1
2gρρT )⇒
aa+ 2a2 + 2κ = 4πG(ρ− P )a2
Using equation (3.9) we get
a(−a4πG
3(ρ+ 3P )) + 2a2 + 2κ = 4πG(ρ− P )a2(
a
a
)2
=8πG
3ρ− κ
a2. (3.10)
3.1 The Friedmann-Lemaıtre-Robertson-Walker Universe 10
This is the first Friedmann equation.
The expansion rate of the universe is measured by the Hubble parameter which has the unit of
inverse time, it is defined by
H ≡ a
a. (3.11)
Friedmann equations (3.9) and (3.10) can then be rewritten as
H2 =8πG
3ρ− κ
a2. (3.12)
H +H2 = −4πG
3(ρ+ 3P ). (3.13)
We introduce the critical density ρc and the density parameter Ω by
ρc =3H2
8πG. (3.14)
Ω =8πG
3H2ρ =
ρ
ρc. (3.15)
Using these two parameters in the first Friedmann equation we get
H2 =8πG
3ρ− κ
a2
=8πG
3(3H2
8πG)Ω− κ
a2⇒
(Ω− 1)H2 =κ
a2(3.16)
The first Friedmann equation becomes
Ω− 1 =κ
H2a2≡ ρ− ρc
ρc. (3.17)
We get immediately the behavior
The closed universe : κ > 0↔ Ω > 1↔ ρ > ρc. (3.18)
The flat universe : κ = 0↔ Ω = 1↔ ρ = ρc. (3.19)
The open universe : κ < 0↔ Ω < 1↔ ρ < ρc. (3.20)
The critical density is the density when the universe is precisely flat. The universe will be open if
the density is less than this critical value, closed if it is greater
We consider the conservation law ∇µTµ ν = ∂µTµν + Γµ µαT
αν − Γα µνT
µα = 0.
The ν = 0 component of this conservation law in comoving coordinates (Tµλ = diag(−ρ, P, P, P ))
is
ρ+3a
a(ρ+ P ) = 0.
In cosmology the pressure P and the rest mass density ρ are related by the equation of state
P = wρ.
The conservation of energy becomes
ρ
ρ= −3(1 + w)
a
a.
3.1 The Friedmann-Lemaıtre-Robertson-Walker Universe 11
For constant w the solution is of the form
ρ ∝ a−3(1+w). (3.21)
We set 3(1 + w) = n then ρ ∝ a−n.
The first Friedmann equation gives therefore a ∝ a1−n/2, the solution behaves as
a ∝ t 2n . (3.22)
For the matter content of the universe there are three cases of interest
• The matter-dominated universe (MD): Matter (also called dust) is a set of collision-less
non-relativistic particles which have zero pressure. For examples stars and galaxies may be
considered as dust since pressure can be neglected to a very good accuracy. Since PM = 0 we
have w = 0 and as a consequence
ρM ∝ a−3. (3.23)
It means that the energy density decreases as the volume increases.
We have also
a(t) ∝ t2/3 (3.24)
• The radiation-dominated universe (RD): Radiation consists of photons (obviously) but
also includes any particles with speeds close to the speed of light. For an electromagnetic field
the stress-energy-momentum-tensor satisfies Tµµ = 0. The stress-energy-momentum tensor
of a perfect fluid satisfies
Tµµ = (ρ+ P )UµU
ν + Pg νµ
= (ρ+ P )(−1) + P (4)
= −ρ+ 3P.
Thus for radiation we must have the equation of state PR = ρR/3 and as a consequence
w = 1/3 and hence
ρR ∝ a−4, a(t) ∝ t1/2. (3.25)
In a radiation dominated universe, the number of photons decreases as the volume increases,
and the energy of each photon redshifts and amount proportional to a(t)
• The vacuum-dominated universe (Λ): The vacuum energy (or the cosmological constant)
is a perfect fluid with equation of state PΛ = −ρΛ, i.e. w = −1 and hence
ρΛ ∝ a0, a(t) ∝ eHt. (3.26)
The vacuum dominated universe is also known as de Sitter space. In de Sitter space, the
energy density is constant, as is the Hubble parameter, and they are related by
H =
√8πGρΛ
3= constant.
The multicomponents of the universe:
In general, matter, radiation and vacuum can contribute simultaneously to the evolution of the
universe. The Friedmann equation takes in this case the form
H2 =8πG
3
∑i
ρi −κ
a2. (3.27)
3.2 Concordance Model 12
Let H0 be the value of the Hubble parameter at the present time t0. Define the critical density at
the present time t0 by
Ωi0 =8πG
3H20
ρi0. (3.28)
The scale factor is normalized such that a(t0) = 1, i.e. ρi = ρi0a−3(1+wi). The Friedmann equation
becomes
H2
H20
=∑i
Ωi0a−3(1+wi) − κ
H20a
2. (3.29)
Let H0 be the value of the Hubble parameter at the present time t0. Define the critical density at
the present time t0 by
Ωi0 =8πG
3H20
ρi0. (3.30)
The scale factor is normalized such that a(t0) = 1, i.e. ρi = ρi0a−3(1+wi). The Friedmann equation
becomes
H2
H20
=∑i
Ωi0a−3(1+wi) − κ
H20a
2. (3.31)
The spatial curvature will be thought of as giving another contribution to the mass density given
by
ρk = − 3
8πG
κ
a2. (3.32)
By analogy the density parameter of the spatial curvature will be given by
Ωk =8πGρk3H2
= − κ
H2a2. (3.33)
The Friedmann equation becomes
H2
H20
=∑i
Ωi0a−3(1+wi) + Ωk0a
−2. (3.34)
At the current epoch we must then have
1 =∑i
Ωi0 + Ωk0. (3.35)
The mass densities of matter and radiation are always positive whereas the mass densities corre-
sponding to vacuum and curvature can be either positive or negative.
3.2 Concordance Model
From a combination of cosmic microwave background (CMB) and large scale structure (LSS) ob-
servations we deduce that the universe is spatially flat and is composed of [4] 4% ordinary mater,
23% dark matter and 73% dark energy (vaccum energy or cosmological constant Λ), i.e.
Ωk ∼ 0. (3.36)
ΩM ∼ 0.04 , ΩDM ∼ 0.23 , ΩΛ ∼ 0.73. (3.37)
4 Cosmological Constant, Vacuum Energy and Dark Energy 13
4 Cosmological Constant, Vacuum Energy and Dark Energy
4.1 Dark Energy
It is generally accepted now that there is a positive dark energy in the universe which affects in
measurable ways the physics of the expansion. The characteristic feature of dark energy is that
it has a negative pressure (tension) smoothly distributed in spacetime so it was proposed that
a name like ”smooth tension” is more appropriate to describe it (see reference [8]). The most
dramatic consequence of a non zero value of ΩΛ is the observation that the universe appears to be
accelerating.
From an observational point of view astronomical evidence for dark energy comes from various
measurements. Here we concentrate, and only briefly, on the two measurements of CMB anisotropies
and type Ia supernovae.
CMB Anisotropies: The main point is as follows. The temperature anisotropies are given
by the power spectrum Cl. At intermediate scales (angular scales subtended by H−1CMB where
HCMB is the Hubble radius at the time of the formation of the cosmic microwave background
(decoupling, recombination, last scattering) we observe peaks in Cl due to acoustic oscillations in
the early universe. The first peak is tied directly to the geometry of the universe. In a negatively
curved universe photon paths diverge leading to a larger apparent angular size compared to flat
space whereas in a positively curved universe photon paths converge leading to a smaller apparent
angular size compared to flat space. The spatial curvature as measured by Ω is related to the first
peak in the CMB power spectrum by
lpeak ∼220√
Ω. (4.1)
The observation indicates that the first peak occurs around lpeak ∼ 200 which means that the
universe is spatially flat. The Boomerang experiment gives (at the 68 per cent confidence level) the
measurement.
0.85 ≤ Ω ≤ 1.25. (4.2)
Since Ω = ΩM + ΩΛ this is a constraint on the sum of ΩM and ΩΛ. The constraints from the CMB
in the ΩM − ΩΛ plane using models with different values of ΩM and ΩΛ is shown on figure 3 of
reference [11]. The best fit is a marginally closed model with
ΩCDM = 0.26 , ΩB = 0.05 , ΩΛ = 0.75. (4.3)
4.2 Einstein Static Universe and Vacuum Energy 14
Figure 1: power spectrum
Type Ia Supernovae: This relies on the measurement of the distance modulus m−M of type
Ia supernovae where m is the apparent magnitude of the source and M is the absolute magnitude
defined by
m−M = 5 log10[(1 + z)dM (Mpc)] + 25. (4.4)
z is the cosmological redshift. dM is the proper distance which is given between any two sources at
redshifts z1 and z2 by the formula
dM (z1, z2) =1
H0
√|Ωk0|
Sk
(H0
√|Ωk0|
∫ 1/(1+z2)
1/(1+z1)
da
a2H(a)
). (4.5)
Type Ia supernovae are rare events which thought of as standard candles. They are very bright
events with almost uniform intrinsic luminosity with absolute brightness comparable to the host
galaxies. They result from exploding white dwarfs when they cross the Chandrasekhar limit.
Constraints from type Ia supernovae in the ΩM − ΩΛ plane are consistent with the results
obtained from the CMB measurements although the data used is completely independent. In
particular these observations strongly favors a positive cosmological constant.
4.2 Einstein Static Universe and Vacuum Energy
The cosmological constant was introduced by Einstein in 1917 in order to produce a static universe.
To see this explicitly let us rewrite the Friedmann equations as
H2 ≡(a
a
)2
=8πGρ
3− κ
a2. (4.6)
a
a= −4πG
3(ρ+ 3P ). (4.7)
The scale factor a(t) measures the size of the universe, thus if the universe is static then a is a
constant.
The first Friedmann equation is compatible with a static universe: when we set a = 0, we obtain
ρ = 3κ/(8πGa2). For ordinary matter ρ > 0 then κ should be positive. The second equation is not
compatible with a static universe, when we set a = 0, we obtain (ρ+ 3P ) = 0 which is impossible
4.2 Einstein Static Universe and Vacuum Energy 15
for ordinary matter and ordinary energy (ρ > 0, P > 0).
Einstein solved this problem by modifying his equations as follows
Rµν −1
2gµνR+ Λgµν = 8πGTµν . (4.8)
The new free parameter Λ is precisely the cosmological constant. This new equations of motion
will entail a modification of the Friedmann equations. To find the modified Friedmann equation we
rewrite the modified Einstein’s equations as
Rµν −1
2gµνR = 8πGTµν − Λgµν
= 8πG(Tµν −Λ
8πGgµν)
We get
Rµν −1
2gµνR = 8πG(Tµν + TΛ
µν). (4.9)
Where
TΛµν = −ρΛgµν , ρΛ =
Λ
8πG. (4.10)
Λ = 8πGρΛ. (4.11)
We have
T νµ =
−ρ 0 0 0
0 P 0 0
0 0 P 0
0 0 0 P
, TΛ νµ =
−ρΛ 0 0 0
0 −ρΛ 0 0
0 0 −ρΛ 0
0 0 0 −ρΛ
. (4.12)
Then the modification of Einstein’s equations isρ→ ρ+ ρΛ.
P → P − ρΛ.
We insert this modification into Friedmann equations as follows
H2 =8πG
3(ρ+ ρΛ)− κ
a2
=8πGρ
3− κ
a2+
8πGρΛ
3
H2 =8πGρ
3− κ
a2+
Λ
3. (4.13)
a
a= −4πG
3((ρ+ ρΛ) + 3(P − ρΛ))
= −4πG
3(ρ+ 3P ) +
8πGρΛ
3
a
a= −4πG
3(ρ+ 3P ) +
Λ
3. (4.14)
The equations (4.13),(4.14) admit a static solution with positive spatial curvature and all the
parameters ρ, P , and Λ nonnegative. This solution is called the Einstein static universe.
• In static universe (a = 0, a = 0). From (4.13) we have
ρ =3κ
8πGa2− Λ
8πG.
4.2 Einstein Static Universe and Vacuum Energy 16
With ordinary matter ρ > 0 then Λ < 3κ/a2. From (4.14) we have
P =1
3
(Λ
4πG
)− ρ
3
=1
3
(Λ
4πG
)− 1
3(
3κ
8πGa2− Λ
8πG)
P =Λ
8πG− κ
8πGa2.
With ordinary matter P > 0 then Λ > κ/a2. The Einstein static universe corresponds to κ > 0
(Σ = S3) and Λ > 0 in the rangeκ
a2≤ Λ ≤ 3κ
a2,
with positive mass density and pressure given by
ρ =3κ
8πGa2− Λ
8πG> 0 , P =
Λ
8πG− κ
8πGa2> 0. (4.15)
The discovery by Hubble that the universe is expanding eliminated the empirical need for a static
universe model. The cosmological constant is however of fundamental importance to cosmology as
it might be relevent to dark energy.
The modified Einstein’s equations (4.8) can be derived from the action
S =1
16πG
∫d4x√−detg (R− 2Λ) +
∫d4x√−detg LM . (4.16)
Thus the cosmological constant Λ is just a constant term in the Lagrangian density. We call Λ the
bare cosmological constant. The effective cosmological constant Λeff will in general be different from
Λ due to possible contribution from matter. We consider a scalar field with Lagrangian density
LM = −1
2gµν∇µφ∇νφ− V (φ). (4.17)
The stress-energy-momentum tensor is given by
Tµν = ∇µφ∇νφ−1
2gµνg
αβ∇αφ∇βφ− gµνV (φ). (4.18)
The configuration φ0 with lowest energy density (the vacuum) is the configuration which minimizes
separately the kinetic and potential terms and as a consequence ∂µφ0 = 0 and V′(φ0) = 0. The
corresponding stress-energy-momentum tensor is therefore T(φ0)µν = −gµνV (φ0). In other words
the stress-energy-momentum tensor of the vacuum acts precisely like the stress-energy-momentum
tensor of a cosmological constant. We write (with T(φ0)µν ≡ T vac
µν , V (φ0) ≡ ρvac)
T (φ0)µν = −gµνV (φ0). (4.19)
The stress-energy-momentum tensor of the vacuum acts precisely like the stress-energy-momentum
tensor of a cosmological constant.
We write (with Tφ0µν ≡ T vac
µν , V (φ0) ≡ ρvac)
T vacµν = −ρvac gµν . (4.20)
The vacuum φ0 is therefore a perfect fluid with pressure given by
Pvac = −ρvac. (4.21)
4.2 Einstein Static Universe and Vacuum Energy 17
Thus the vacuum energy acts like a cosmological constant Λφ given by
Λφ = 8πGρvac. (4.22)
In other words the cosmological constant and the vacuum energy are completely equivalent. We
will use the two terms ”cosmological constant” and ”vacuum energy” interchangeably.
The effective cosmological constant Λeff is therefore given by
Λeff = Λ + Λφ (4.23)
= Λ + 8πGρvac. (4.24)
This calculation is purely classical.
Quantum mechanics will naturally modify this result. We follow a semi-classical approach in which
the gravitational field is treated classically and the scalar field (matter fields in general) are treated
quantum mechanically. Thus we need to quantize the scalar field in a background metric gµν which
is here the Robertson-Walker metric. In the quantum vacuum state of the scalar field (assuming
that it exists) the expectation value of the stress-energy-momentum tensor Tµν must be, by Lorentz
invariance, of the form [7]
< Tµν >vac= − < ρ >vac gµν . (4.25)
The Einstein’s equations in the vacuum state of the scalar field are
Rµν −1
2gµνR+ Λgµν = 8πG < Tµν >vac . (4.26)
The effective cosmological constant Λeff must therefore be given by
Λeff = Λ + 8πG < ρ >vac . (4.27)
The energy density of empty space < ρ >vac is the sum of zero-point energies associated with
vacuum fluctuations together with other contributions resulting from virtual particles (higher order
vacuum fluctuations) and vacuum condensates.
We will assume from simplicity that the bare cosmological constant Λ is zero. Thus the effective
cosmological constant is entirely given by vacuum energy, viz
Λeff = 8πG < ρ >vac . (4.28)
We drop now the subscript ”eff”without fear of confusion. The relation between the density ρΛ of
the cosmological constant and the density < ρ >vac of the vacuum is then simply
ρΛ =< ρ >vac . (4.29)
From the concordance model we know that the favorite estimate for the value of the density pa-
rameter of dark energy at this epoch is ΩΛ = 0.7. We recall G = 6.67 × 10−11m3kg−1s−2 and
H0 = 70kms−1Mpc−1 with Mpc = 3.09× 1024cm. We compute then the density
ρΛ =3H2
0
8πGΩΛ (4.30)
= 9.19× 10−27ΩΛkg/m3. (4.31)
We convert to natural units (1GeV = 1.8× 10−27kg, 1GeV−1 = 6.58× 10−25s) to obtain
ρΛ = 39ΩΛ(10−12GeV)4. (4.32)
4.2 Einstein Static Universe and Vacuum Energy 18
To get a theoretical order-of-magnitude estimate of < ρ >vac we use the flat space Hamiltonian
operator of a free scalar field given by
H =
∫d3p
(2π)3ω(~p)
[a(~p)+a(~p) +
1
2(2π)3δ3(0)
]. (4.33)
The vacuum state is defined in this case unambiguously by
a(~p)|0 >= 0
We get then in the vacuum state energy
Evac =< 0|H|0 >
Where
Evac =1
2(2π)3δ3(0)
∫d3p
(2π)3ω(~p). (4.34)
If we use box normalisation then (2π)3δ3(~p − ~q) will be replaced with V δ~p~q where V is spacetime
volume. The vacuum energy density is therefore given by (using also ω(~p) =√~p2 +m2)
< ρ >vac=1
2
∫d3p
(2π)3
√~p2 +m2. (4.35)
This is clearly divergent. We introduce a cutoff λ and compute
< ρ >vac =1
2
∫ λ
0
4πp2dp
(2π)3
√p2 +m2
=1
4π2
∫ λ
0
p2dp√p2 +m2
=1
4π2
[(1
4λ3 +
m2
8λ
)√λ2 +m2 − m4
8ln
(λ
m+
√1 +
λ2
m2
)]. (4.36)
In the massless limit (the mass is any case much smaller than the cutoff λ) we obtain the estimate
< ρ >vac=λ4
16π2. (4.37)
By assuming that quantum field theory calculations are valid up to the Planck scale Mpl =
1/√
8πG = 2.42× 1018GeV then we can take λ = Mpl and get the estimate
< ρ >vac= 0.22(1018GeV)4. (4.38)
By taking the ratio of the value (4.32) obtained from cosmological observations and theoretical
value (4.38) we get
(ρΛ
< ρ >vac)1/4 = 3.65× Ω
1/4Λ × 10−30. (4.39)
For the observed value ΩΛ = 0.7 we see that there is a discrepancy of 30 orders of magnitude
between the theoretical and observational mass scales of the vacuum energy which is the famous
cosmological constant problem. Let us note that in flat spacetime we can make the vacuum energy
vanishes by the usual normal ordering procedure which reflects the fact that only differences in
energy have experimental consequences in this case. In curved spacetime this is not however
possible since general relativity is sensitive to the absolute value of the vacuum energy. In other
words the gravitational effect of vacuum energy will curve spacetime and the above problem of the
cosmological constant is certainly genuine.
5 Calculation of Vacuum Energy in Curved Backgrounds 19
5 Calculation of Vacuum Energy in Curved Backgrounds
5.1 Elements of QFT in curved spacetime
We rewrite Friedmann equations with a cosmological constant which are given by
H2 =8πGρ
3− κ
a2+
Λ
3. (5.1)
a
a= −4πG
3(ρ+ 3P ) +
Λ
3. (5.2)
We will assume that ρ and P are those of a real scalar field coupled to the metric minimally with
action given by
SM =
∫d4x√−detg
(−1
2gµν∇µφ∇νφ− V (φ)
). (5.3)
If we are interested in an action which is at most quadratic in the scalar field then we must choose
V (φ) = m2φ2/2. In curved spacetime there is another term we can add which is quadratic in φ
namely Rφ2 where R is the Ricci scalar . The full action should then read (in arbitrary dimension
n)
SM =
∫dnx
√−detg
(−1
2gµν∇µφ∇νφ−
1
2m2φ2 − 1
2ξRφ2
). (5.4)
The choice ξ = (n−2)/(4(n−1)) is called conformal coupling. At this value the action with m2 = 0
is invariant under conformal transformations defined by [1]
gµν → gµν = Ω2(x)gµν(x), φ→ φ = Ω2−n2 (x)φ(x). (5.5)
The Lagrangian density is clearly
L = −1
2gµν∇µφ∇νφ−
1
2m2φ2 − 1
2ξRφ2. (5.6)
Euler-Lagrange equation of motion is
δLδφ− ∂µ
δLδ∂µφ
= 0. (5.7)
We get the equation of motion
(∇µ∇µ −m2 − ξR)φ = 0. (5.8)
Let φ1 and φ2 be two solutions of this equation of motion. We define their inner product by
(φ1, φ2) = −i∫
Σ
(φ1∂µφ∗2 − ∂µφ1.φ
∗2)dΣnµ. (5.9)
dΣ is the volume element in the spacelike hypersurface Σ and nµ is the timelike unit vector which
is normal to this hypersurface. This inner product is independent of the hypersurface Σ. Indeed
let Σ1 and Σ2 be two non intersecting hypersurfaces and let V be the four-volume bounded by Σ1,
Σ2 and (if necessary) timelike boundaries on which φ1 = φ2 = 0. We have from one hand
i
∫V
∇µ(φ1∂µφ∗2 − ∂µφ1.φ
∗2)dV = i
∮∂V
(φ1∂µφ∗2 − ∂µφ1.φ
∗2)dΣµ
= (φ1, φ2)Σ1− (φ1, φ2)Σ2
. (5.10)
From the other hand
5.1 Elements of QFT in curved spacetime 20
i
∫V
∇µ(φ1∂µφ∗2 − ∂µφ1.φ
∗2)dV = i
∫V
(φ1∇µ∂µφ∗2 −∇µ∂µφ1.φ∗2)dV
= i
∫V
(φ1(m2 + ξR)φ∗2 − (m2 + ξR)φ1.φ∗2)dV
= 0. (5.11)
Hence
(φ1, φ2)Σ1− (φ1, φ2)Σ2
= 0. (5.12)
There is always a complete set of solutions ui and u∗i of the equation of motion (5.8) which are
orthonormal in the above inner product (5.9), i.e. satisfying
(ui, uj) = δij , (u∗i , u∗j ) = −δij , (ui, u
∗j ) = 0. (5.13)
We can then expand the field as
φ =∑i
(aiui + a∗i u∗i ). (5.14)
We now canonically quantize this system.
We choose a foliation of spacetime into spacelike hypersurfaces. Let Σ be a particular hypersurface
with unit normal vector nµ corresponding to a fixed value of the time coordinate x0 = t and with
induced metric hij which is given by the formula
hij = gij + ninj . (5.15)
We write the action as SM =∫dx0LM where LM =
∫dn−1x
√−detg LM
π =δLMδ(∂0φ)
= −√−detg gµ0∂µφ
= −√
deth nµ∂µφ. (5.16)
Such that [1] √−detg = N
√deth. (5.17)
gµ0 =nµ
N, (5.18)
where N is the norm of nµ
We promote φ and π to hermitian operators φ and π and then impose the equal time canonical
commutation relations
[φ(x0, xi), π(x0, yi)] = iδn−1(xi − yi). (5.19)
The delta function satisfies the property∫δn−1(xi − yi)dn−1y = 1. (5.20)
The coefficients ai and a∗i become annihilation and creation operators ai and a+i satisfying the
commutation relations
[ai, a+j ] = δij , [ai, aj ] = [a+
i , a+j ] = 0. (5.21)
5.1 Elements of QFT in curved spacetime 21
The vacuum state is given by a state |0u > defined by
ai|0u >= 0. (5.22)
The entire Fock basis of the Hilbert space can be constructed from the vacuum state by repeated
application of the creation operators a+i .
The solutions ui, u∗i are not unique and as a consequence the vacuum state |0u > is not unique.
Let us condider another complete set of solutions vi and v∗i of the equation of motion (5.8) which
are orthonormal in the inner product (5.9). We can then expand the field as
φ =∑i
(bivi + b∗i v∗i ). (5.23)
After canonical quantization the coefficients bi and b∗i become annihilation and creation operators
bi and b+i satisfying the standard commutation relations with a vacuum state given by |0v > defined
by
bi|0v >= 0. (5.24)
We introduce the so-called Bogolubov transformation as the transformation from the set ui, u∗i
(which are the set of modes seen by some observer) to the set vi, v∗i (which are the set of modes
seen by another observer) as
vi =∑j
(αijuj + βiju∗j ). (5.25)
By using orthonormality conditions we find that
αij = (vi, uj), βij = −(vi, u∗j ). (5.26)
We can also write
ui =∑j
(α∗jivj + βjiv∗j ). (5.27)
The Bogolubov coefficients α and β satisfy the normalization conditions∑k
(αikαjk − βikβjk) = δij ,∑k
(αikβ∗jk − βikα∗jk) = 0. (5.28)
The Bogolubov coefficients α and β transform also between the creation and annihilation operators
a, a+ and b, b+. We find
ak =∑i
(αik bi + β∗ik b+i ), bk =
∑i
(α∗kiai + β∗kia+i ). (5.29)
Let Nu be the number operator with respect to the u-observer, viz Nu =∑k a
+k ak. Clearly
< 0u|Nu|0u >= 0. (5.30)
We compute
< 0v|a+k ak|0v > = < 0v|
∑j
βjk bj∑i
β∗ik b+i |0v >
=∑j
∑i
βjkβ∗ik < 0v| bj b+i |0v >
=∑j
∑i
βjkβ∗ik < 0v|[bj , b+i ]|0v >
=∑j
∑i
βjkβ∗ik δij
=∑i
βikβ∗ik. (5.31)
5.2 Quantization in FLRW Universes 22
Thus
< 0v|Nu|0v >= trββ+. (5.32)
In other words with respect to the u-observer the vacuum state |0v > is not empty but filled with
particles. This opens the door to the possibility of particle creation by a gravitational field.
5.2 Quantization in FLRW Universes
We go back to the equation of motion (5.8), viz(∇µ∇µ −m2 − ξR
)φ = 0. (5.33)
The flat FLRW universes are given by
ds2 = −dt2 + a2(t)(dρ2 + ρ2dΩ2). (5.34)
The conformal time is denoted here by
η =
∫ t dt1a(t1)
. (5.35)
In terms of η the FLRW universes are manifestly conformally flat, viz
ds2 = a2(η)(−dη2 + dρ2 + ρ2dΩ2). (5.36)
The d’Alembertian in FLRW universes is
∇µ∇µφ =1√−detg
∂µ(√−detg ∂µφ)
= ∂µ∂µφ+
1
2gαβ∂µgαβ∂
µφ
= −φ+1
a2∂2i φ− 3
a
aφ. (5.37)
The Klein-Gordon equation of motion becomes
φ+ 3a
aφ− 1
a2∂2i φ+ (m2 + ξR)φ = 0. (5.38)
In terms of the conformal time 1 this reads (where d/dη is denoted by primes)
φ′′
+ 2a′
aφ′− ∂2
i φ+ a2(m2 + ξR)φ = 0. (5.39)
The positive norm solutions are given by
uk(η, xi) =ei~k~x
a(η)χk(η). (5.40)
Indeed we check that φ ≡ uk(η, xi) is a solution of the Klein-Gordon equation of motion provided
that χk is a solution of the equation of motion (using also R = 6(a/a+ a2/a2) = 6a′′/a3)
χ′′
k + ω2k(η)χk = 0. (5.41)
ω2k(η) = k2 +m2a2 − (1− 6ξ)
a′′
a. (5.42)
1To quantize this field system we reduce the field to a collection of independent degrees of freedom for which the
quantization is known. To this end we choose the conformal time coordinate η.
5.2 Quantization in FLRW Universes 23
In the case of conformal coupling m = 0 and ξ = 1/6 this reduces to a time independent harmonic
oscillator. This is similar to flat spacetime and all effects of the curvature are included in the
factor a(η) in equation (5.40). Thus calculation in a conformally invariant world is very easy.
The condition (uk, ul) = δkl becomes (with nµ = (1, 0, 0, 0), dΣ =√
deth d3x and using box
normalization (2π)3δ3(~k − ~p) −→ V δ~k,~p) the Wronskian condition
iV (χ∗kχ′
k − χ∗′
k χk) = 1. (5.43)
The negative norm solutions correspond obviously to u∗k. Indeed we can check that (u∗k, ul) = −δkland (u∗k, ul) = 0. The modes uk and u∗k provide a Fock space representation for field operators.
The quantum field operator φ can be expanded in terms of creation and annhiliation operators as
φ =∑k
(akuk + a+k u∗k). (5.44)
Alternatively the mode functions satisfy the differential equations (with χk = v∗k/√
2V )
v′′
k + ω2k(η)vk = 0 (5.45)
They must satisfy the normalization condition
1
2i(v′
kv∗k − vkv∗
′
k ) = 1. (5.46)
The scalar field operator is given by φ = χ/a(η) where (with [ak, a+k′
] = V δk,k′ , etc)
χ =1
V
∑k
1√2
(akv∗kei~k~x + a+
k vke−i~k~x
). (5.47)
The stress-energy-momentum tensor in minimal coupling ξ = 0 is given by
Tµν = ∇µφ∇νφ−1
2gµνg
ρσ∇ρφ∇σφ− gµνV (φ). (5.48)
We compute immediately in the conformal metric ds2 = a2(−dη2 + dxidxi) the component
T00 =1
2(∂ηφ)2 +
1
2(∂iφ)2 +
1
2a2m2φ2
=1
2a2
[χ′2 − 2
a′
aχχ′+a′2
a2χ2]
+1
2a2(∂iχ)2 +
1
2m2χ2. (5.49)
The conjugate momentum (5.16) in our case is π = a2∂ηφ. The Hamiltonian is therefore
H =
∫dn−1x π∂0φ− LM
=
∫dn−1x
√−detg
1
a2T00
= −∫dn−1x
√−detg T 0
0 . (5.50)
In the quantum theory the stress-energy-momentum tensor in minimal coupling ξ = 0 is given by
T00 =1
2a2
[χ′2 − a
′
a(χχ
′+ χ
′χ) +
a′2
a2χ2]
+1
2a2(∂iχ)2 +
1
2m2χ2. (5.51)
5.3 Quantization in de Sitter Spacetime and Bunch-Davies Vacuum 24
We assume the existence of a vacuum state |0 > with the properties a|0 >= 0, < 0|a+ = 0 and
< 0|0 >= 1. We compute
< χ′2 > =
1
2V 2
∑k
∑p
v∗′
k v′
pei~k~xe−i~p~x < 0|aka+
p |0 >
=1
2V
∑k
|v′
k|2. (5.52)
< χ2 > =1
2V 2
∑k
∑p
v∗kvpei~k~xe−i~p~x < 0|aka+
p |0 >
=1
2V
∑k
|vk|2. (5.53)
< (∂iχ)2 > =1
2V 2
∑k
∑p
v∗kvp(kipi)ei~k~xe−i~p~x < 0|aka+
p |0 >
=1
2V
∑k
k2|vk|2. (5.54)
We get then
< T00 > =1
2a2
1
2V
∑k
[|v′
k|2 −a′
a(v∗kv
′
k + v′∗k vk) +
a′2
a2|vk|2 + k2|vk|2 + a2m2|vk|2
]
=1
4a2
1
V
∑k
[|v′
k|2 + (k2 +a′′
a+ a2m2)|vk|2 − ∂η(
a′
a|vk|2)
]. (5.55)
The mass density is therefore given by
< ρ >vac=1
a2< T00 > =
1
4a4
∫d3k
(2π)3
[|v′
k|2 + (k2 +a′′
a+ a2m2)|vk|2 − ∂η(
a′
a|vk|2)
].(5.56)
5.3 Quantization in de Sitter Spacetime and Bunch-Davies Vacuum
In the limit a −→ ∞ (the future) it is believed that vacuum dominates and thus spacetime is
approximately de Sitter spacetime.
An interesting solution of the Friedmann equations (4.13) and (4.14) is precisley the maximally
symmetric de Sitter space with positive curvature κ > 0 and positive cosmological constant Λ > 0
and no matter content ρ = P = 0 given by the scale factor
a(t) =α
R0cosh
t
α. (5.57)
α =
√3
Λ, R0 =
1√κ. (5.58)
At large times the Hubble parameter becomes a constant
H ' 1
α=
√Λ
3. (5.59)
The behavior of the scale factor at large times becomes thus
a(t) ' a0eHt, a0 =
α
2R0. (5.60)
5.3 Quantization in de Sitter Spacetime and Bunch-Davies Vacuum 25
Thus the scale factor on de Sitter space can be given by
a(t) ' a0 exp(Ht).
In this case the curvature is computed to be zero and thus the coordinates t, x, y and z are
incomplete in the past. The metric is given explicitly by
ds2 = −dt2 + a20e
2Htdxidxi. (5.61)
In this flat patch (lower half of) de Sitter space is asymptotically static with respect to conformal
time η in the past. This can be seen as follows. First we can compute in closed form that
η = −e−Ht/(a0H) , a(t) = a(η) = −1/(Hη),
and thus η is in the interval ] − ∞, 0] (and hence the coordinates t, x, y and z are incomplete).
We then observe that Hη = a′/a = −1/η −→ 0 when η −→ −∞ which means that de Sitter is
asymptotically static.
de Sitter space is characterized by the existence of horizons. As usual null radial geodesics are
characterized by a2(t)r2 = 1. The solution is explicitly given by
r(t)− r(t0) =1
a0H(e−Ht0 − e−Ht). (5.62)
Thus photons emitted at the origin r(t0) = 0 at time t0 will reach the sphere rh = e−Ht0/(a0H) at
time t −→∞ (asymptotically). This sphere is precisely the horizon for the observer at the origin in
the sense that signal emitted at the origin can not reach any point beyond the horizon and similarly
any signal emitted at time t0 at a point r > rh can not reach the observer at the origin.
The horizon scale at time t0 is defined as the proper distance of the horizon from the observer
at the origin, viz a2(t0)rh = 1/H. This is clearly the same at all times.
The effective frequencies of oscillation in de Sitter space are
ω2k(η) = k2 +m2a2 − (1− 6ξ)
a′′
a
= k2 +[m2
H2− 2(1− 6ξ)
] 1
η2. (5.63)
These may become imaginary. For example ω20(η) < 0 if m2 < 2(1−6ξ)H2. We will take ξ = 0 and
assume that m << H. From the previous section we know that the mode functions must satisfy
the differential equations (with χk = v∗k/√
2V )
v′′
k +
(k2 +
[m2
H2− 2] 1
η2
)vk = 0 (5.64)
The solution of this equation is given in terms of Bessel functions Jn and Yn by
vk =√k|η|
[AJn(k|η|) +BYn(k|η|)
], n =
√9
4− m2
H2. (5.65)
The normalization condition (5.46) becomes (with s = k|η|)
ks(A∗B −AB∗)( ddsJn(s).Yn(s)− d
dsYn(s).Jn(s)) = 2i. (5.66)
We use the result
d
dsJn(s).Yn(s)− d
dsYn(s).Jn(s) = − 2
πs. (5.67)
We obtain the constraint
AB∗ −A∗B =iπ
k. (5.68)
We consider now two limits of interest.
5.3 Quantization in de Sitter Spacetime and Bunch-Davies Vacuum 26
The early time regime η −→ −∞: This corresponds to ω2k −→ k2 or equivalently
k2 >> (2− m2
H2)
1
η2. (5.69)
This is a high energy (short distance) limit. The effect of gravity on the modes vk is therefore
negligible and we obtain the Minkowski solutions
vk =1√keikη , k|η| >> 1. (5.70)
The normalization is chosen in accordance with (5.46).
The late time regime η −→ 0: In this limit ω2k −→ (m2/H2 − 2)1/η2 < 0 or equivalently
k2 << (2− m2
H2)
1
η2. (5.71)
The differential equation becomes
v′′
k − (2− m2
H2)
1
η2vk = 0. (5.72)
The solution is immediately given by vk = A|η|n1 + B|η|n2 with n1,2 = ±n + 1/2. In the limit
η −→ 0 the dominant solution is obviously associated with the exponent −n+ 1/2. We have then
vk ∼ |η|12−n , k|η| << 1. (5.73)
Any mode with momentum k is a wave with a comoving wave length L ∼ 1/k and a physical wave
length Lp = a(η)L and hence
k|η| = H−1
Lp. (5.74)
Thus modes with k|η| >> 1 corresponds to modes with Lp << H−1. These are the sub-horizon
modes with physical wave lengths much shorter than the horizon scale and which are unaffected
by gravity. Similarly the modes with k|η| << 1 or equivalently Lp >> H−1 are the super-horizon
modes with physical wave lengths much larger than the horizon scale. These are the modes which
are affected by gravity.
A mode with momentum k which is [10] sub-horizon at early times will become super-horizon at
a later time ηk defined by the requirement that Lp = H−1 or equivalently k|ηk| = 1. The time ηk is
called the time of horizon crossing of the mode with momentum k. The behavior a(η) −→ 0 when
η −→ −∞ allows us to pick a particular vacuum state known as the Bunch-Davies or the Euclidean
vacuum. The Bunch-Davies vacuum is a de Sitter invariant state and is the initial state used in
cosmology. In the limit η −→ −∞ the frequency approaches the flat space result, i.e. ωk(η) −→ k
and hence we can choose the vacuum state to be given by the Minkowski vacuum. More precisely
the frequency ωk(η) is a slowly-varying function for some range of the conformal time η in the limit
η −→ −∞. This is called the adiabatic regime of ωk(η) where it is also assumed that ωk(η) > 0.
By applying the Minkowski vacuum prescription in the limit η −→ −∞ we must have
vk =N√keikη , η −→ −∞. (5.75)
From the other hand by using Jn(s) =√
2/(πs) cosλ, Yn(s) =√
2/(πs) sinλ with λ = s− nπ/2−π/4 we can compute the asymptotic behavior
vk =
√2
π[A cosλ+B sinλ] , η −→ −∞. (5.76)
5.3 Quantization in de Sitter Spacetime and Bunch-Davies Vacuum 27
By choosing B = −iA and employing the normalization condition (5.68) we obtain
B = −iA , A =
√π
2k. (5.77)
Thus we have the solution
vk =1√kei(kη+nπ
2 +π4 ) , η −→ −∞. (5.78)
The Bunch-Davies vacuum corresponds to the choice N = exp(inπ2 + iπ4 ). The full solution
using this choice becomes
vk =
√π|η|
2
[Jn(k|η|)− iYn(k|η|)
], n =
√9
4− m2
H2. (5.79)
The mass density in FLRW spacetime was already computed in equation (5.56). We have
ρ =1
4a4
∫d3k
(2π)3
[|v′
k|2 + (k2 +a′′
a+ a2m2)|vk|2 − ∂η(
a′
a|vk|2)
]. (5.80)
For de Sitter space we have a = −1/(ηH) and thus
ρ =η4H4
4
∫d3k
(2π)3
[|v′
k|2 + (k2 +2
η2+
m2
H2η2)|vk|2 + ∂η(
1
η|vk|2)
]. (5.81)
For m = 0 we have the solutions
vk =
√π|η|
2
[J 3
2(k|η|)− iY 3
2(k|η|)
]. (5.82)
We use the results (x = k|η|)
J3/2(x) =
√2
πx
(sinx
x− cosx
), Y3/2(x) =
√2
πx
(− cosx
x− sinx
). (5.83)
We obtain then
vk = − i
k32
eikη
η− 1
k12
eikη. (5.84)
In other words
|vk|2 =1
k3
1
η2+
1
k, |v
′
k|2 = −1
k
1
η2+
1
k3
1
η4+ k. (5.85)
We obtain then (using also a hard cutoff Λ)
ρ =η4H4
4
∫d3k
(2π)3
[2k +
1
kη2
]=
η4H4
16π2(Λ4 +
Λ2
η2). (5.86)
This goes to zero in the limit η −→ 0. However if we take Λ = Λ0a where Λ0 is a proper momentum
cutoff then the energy density becomes independent of time and we are back to the same problem.
We get
ρ =1
16π2(Λ4
0 +H2Λ20). (5.87)
6 Is Vacuum Energy Real? 28
We observe that
ρdeSitter − ρMinkowski =H2
Λ20
Λ40
16π2
=H2
Λ20
ρMinkowski. (5.88)
We take the value of the Hubble parameter at the current epoch as the value of the Hubble parameter
of de Sitter space, viz
H = H0 =7× 6.58
3.0910−43GeV. (5.89)
We get then
ρdeSitter − ρMinkowski = 0.38(10−30)4.0.22(1018GeV)4
= 0.084(10−12GeV)4. (5.90)
6 Is Vacuum Energy Real?
6.1 The Casimir Force
The Casimir effect is the attractive force between two uncharged, conducting parallel plates that
is caused by quantum vacuum energy . It is also described as the force between two polarizable
atoms and between an atom and an uncharged conducting plate. The Casimir effect has now
been detected experimentally and its existence is commonly derived as due to the influence of the
conducting plates on the quantum vacuum energy.
We consider two large and perfectly conducting plates of surface area A at a distance L apart
with√A >> L so that we can ignore edge contributions. The plates are in the xy plane at x = 0
and x = L. In the volume AL the electromagnetic standing waves take the form
ψn(t, x, y, z) = e−iωnteikxx+ikyy sin knz. (6.1)
They satisfy the Dirichlet boundary conditions
ψn|z=0 = ψn|z=L = 0. (6.2)
Thus we must have
kn =nπ
L, n = 1, 2, .... (6.3)
ωn =
√k2x + k2
y +n2π2
L2. (6.4)
These modes are transverse and thus each value of n is associated with two degrees of freedom.
There is also the possibility of
kn = 0. (6.5)
In this case there is a corresponding single degree of freedom.
The zero point energy of the electromagnetic field between the plates is
E =1
2
∑n
ωn
=1
2A
∫d2k
(2π)2
[k + 2
∞∑n=1
(k2 +n2π2
L2)1/2
]. (6.6)
6.1 The Casimir Force 29
The zero point energy of the electromagnetic field in the same volume in the absence of the plates
is
E0 =1
2
∑n
ωn
=1
2A
∫d2k
(2π)2
[2L
∫dkn2π
(k2 + k2n)1/2
]. (6.7)
After the change of variable k = nπ/L we obtain
E0 =1
2A
∫d2k
(2π)2
[2
∫ ∞0
dn(k2 +n2π2
L2)1/2
]. (6.8)
Casimir energy is the shift of the energy of the vacuum due to the plates.
We have then
E =E − E0
A=
∫d2k
(2π)2
[1
2k +
∞∑n=1
(k2 +n2π2
L2)1/2 −
∫ ∞0
dn(k2 +n2π2
L2)1/2
]. (6.9)
This is obvioulsy a UV divergent quantity. We regularize this energy density by introducing a cutoff
function fΛ(k) which is equal to 1 for k << Λ and 0 for k >> Λ. We have then (with the change
of variables k = πx/L and x2 = t)
EΛ =
∫d2k
(2π)2
[1
2fΛ(k)k +
∞∑n=1
fΛ(
√k2 +
n2π2
L2)(k2 +
n2π2
L2)1/2 −
∫ ∞0
dnfΛ(
√k2 +
n2π2
L2)(k2 +
n2π2
L2)1/2
]
=π2
4L3
∫dt
[1
2fΛ(
π
L
√t)t1/2 +
∞∑n=1
fΛ(π
L
√t+ n2)(t+ n2)1/2 −
∫ ∞0
dnfΛ(π
L
√t+ n2)(t+ n2)1/2
].(6.10)
This is an absolutely convergent quantity and thus we can exchange the sums and the integrals.
We obtain
EΛ =π2
4L3
[1
2F (0) + F (1) + F (2)....−
∫ ∞0
dnF (n)
]. (6.11)
The function F (n) is defined by
F (n) =
∫ ∞0
dtfΛ(π
L
√t+ n2)(t+ n2)1/2. (6.12)
Since f(k) −→ 0 when k −→∞ we have F (n) −→ 0 when n −→∞. We use the Euler-MacLaurin
formula
1
2F (0) + F (1) + F (2)....−
∫ ∞0
dnF (n) = − 1
2!B2F
′(0)− 1
4!B4F
′′′(0) + .... (6.13)
The Bernoulli numbers Bi are defined by
y
ey − 1=
∞∑i=0
Biyi
i!. (6.14)
For example
B2 =1
6, B4 = − 1
30, etc. (6.15)
6.2 The Dirichlet Propagator 30
Thus
EΛ =π2
4L3
[− 1
12F′(0) +
1
720F′′′
(0) + ....
]. (6.16)
We can write
F (n) =
∫ ∞n2
dtfΛ(π
L
√t)(t)1/2. (6.17)
We assume that f(0) = 1 while all its derivatives are zero at n = 0. Thus
F′(n) = −
∫ n2+2nδn
n2
dtfΛ(π
L
√t)(t)1/2 = −2n2fΛ(
π
Ln)⇒ F
′(0) = 0. (6.18)
F′′(n) = −4nfΛ(
π
Ln)− 2π
Ln2f
′
Λ(π
Ln)⇒ F
′′(0) = 0. (6.19)
F′′′
(n) = −4fΛ(π
Ln)− 8π
Lnf′
Λ(π
Ln)− 2π2
L2n2f
′′
Λ(π
Ln)⇒ F
′′′(0) = −4. (6.20)
We can check that all higher derivatives of F are actually 0. Hence
EΛ =π2
4L3
[− 4
720
]= − π2
720L3. (6.21)
This is the Casimir energy. It corresponds to an attractive force which is the famous Casimir force.
6.2 The Dirichlet Propagator
We define the propagator by
DF (x, x′) =< 0|T φ(x)φ(x
′)|0 > . (6.22)
It satisfies the inhomogeneous Klein-Gordon equation
(∂2t − ∂2
i )DF (x, x′) = iδ4(x− x
′). (6.23)
We introduce Fourier transform in the time direction by
DF (ω, ~x, ~x′) =
∫dte−iω(t−t
′)DF (x, x
′) , DF (x, x
′) =
∫dω
2πeiω(t−t
′)DF (ω, ~x, ~x
′).
(6.24)
We have
(∂2i + ω2)DF (ω, ~x, ~x
′) = −iδ3(~x− ~x
′). (6.25)
We expand the reduced Green’s function DF (ω, ~x, ~x′) as
DF (ω, ~x, ~x′) = −i
∑n
φn(~x)φ∗n(~x′)
ω2 − k2n
. (6.26)
The eigenfunctions φn(~x) satisfy
∂2i φn(~x) = −k2
nφn(~x)
δ3(~x− ~x′) =
∑n
φn(~x)φ∗n(~x′). (6.27)
6.2 The Dirichlet Propagator 31
In infinite space we have
φi(~x) −→ φ~k(~x) = e−i~k~x ,
∑i
−→∫
d3k
(2π)3. (6.28)
Thus
DF (ω, ~x, ~x′) = i
∫d3k
(2π)3
e−i~k(~x−~x
′)
~k2 − ω2. (6.29)
We can compute the closed form
DF (ω, ~x, ~x′) =
i
4π
eiω|~x−~x′|
|~x− ~x′ |. (6.30)
Equivalently we have
DF (x, x′) = i
∫d4k
(2π)4
e−ik(x−x′)
k2. (6.31)
Let us remind ourselves with few more results. We have (with ωk = |~k|)
DF (x, x′) =
∫d3k
(2π)3
1
2ωke−ik(x−x
′). (6.32)
Recall that k(x− x′) = −k0(x0 − x0′) + ~k(~x− ~x′). After Wick rotation in which x0 −→ −ix4 and
k0 −→ −ik4 we obtain k(x− x′) = k4(x4 − x′
4) + ~k(~x− ~x′). The above integral becomes then
DF (x, x′) =
∫d3k
(2π)3
1
2ωke−i(k4(x4−x
′4)−~k(~x−~x
′))
=1
4π2
1
(x− x′)2. (6.33)
We consider now the case of parallel plates separated by a distance L. The plates are in the xy
plane. We impose now different boundary conditions on the field by assuming that φ is confined in
the z direction between the two plates at z = 0 and z = L. Thus the field must vanishes at these
two plates, viz
φ|z=0 = φ|z=L = 0. (6.34)
As a consequence the plane wave eik3z will be replaced with the standing wave sin k3z where the
momentum in the z direction is quantized as
k3 =nπ
L, n ∈ Z+. (6.35)
Thus the frequency ωk becomes
ωn =
√k2
1 + k22 + (
nπ
L)2. (6.36)
We will think of the propagator (6.33) as the electrostatic potential (in 4 dimensions) generated at
point y from a unit charge at point x, viz
V ≡ DF (x, x′) =
1
4π2
1
(x− x′)2. (6.37)
6.2 The Dirichlet Propagator 32
We will find the propagator between parallel plates starting from this potential using the method
of images. It is obvious that this propagator must satisfy
DF (x, x′) = 0 , z = 0, L and z
′= 0, L. (6.38)
Instead of the two plates at x = 0 and x = L we consider image charges (always with respect to the
two plates) placed such that the two plates remain grounded. First we place an image charge −1 at
(x, y,−z) which makes the potential at the plate z = 0 zero. The image of the charge at (x, y,−z)with respect to the plane at z = L is a charge +1 at (x, y, z + 2L). This last charge has an image
with respect to z = 0 equal −1 at (x, y,−z− 2L) which in turn has an image with respect to z = L
equal +1 at (x, y, z + 4L). This process is to be continued indefinitely. We have then added the
following image charges
q = +1 , (x, y, z + 2nL) , n = 0, 1, 2, ... (6.39)
q = −1 , (x, y,−z − 2nL) , n = 0, 1, 2, ... (6.40)
The way we did this we are guaranteed that the total potential at z = 0 is 0. The contribution of
the added image charges to the plate z = L is also zero but this plate is still not balanced properly
precisely because of the original charge at (x, y, z).
The image charge of the original charge with respect to the plate at z = L is a charge −1 at
(x, y, 2L−z) which has an image with respect to z = 0 equal +1 at (x, y,−2L+z). This last image
has an image with respect to z = L equal −1 at (x, y, 4L − z). This process is to be continued
indefinitely with added charges given by
q = +1 , (x, y, z + 2nL) , n = −1,−2, ... (6.41)
q = −1 , (x, y,−z − 2nL) , n = −1,−2, ... (6.42)
By the superposition principle the total potential is the sum of the individual potentials.
For q = +1 we have
(x− x′)new = (x, y, z + 2nl)− (x
′, y′, z′)
= x− x′+ (0, 0, 0, 2nl)
= x− x′+ 2nLe3 (6.43)
For q = −1 we have
(x− x′)new = (x, y,−z − 2nL)− (x
′, y′, z′)
= x− x′− 2(z + nL)e3 (6.44)
We get immediately
V ≡ DF (x, x′) =
1
4π2
+∞∑n=−∞
[1
(x− x′ − 2nLe3)2− 1
(x− x′ − 2(nL+ z)e3)2
].
(6.45)
This satisfies the boundary conditions (6.38). By the uniqueness theorem this solution must
therefore be the desired propagator. At this point we can undo the Wick rotation and return
to Minkowski spacetime.
6.3 Another Derivation Using The Energy-Momentum Tensor 33
6.3 Another Derivation Using The Energy-Momentum Tensor
The stress-energy-momentum tensor in flat space with minimal coupling ξ = 0 and m = 0 is given
by
Tµν = ∂µφ∂νφ−1
2ηµν∂αφ∂
αφ. (6.46)
The stress-energy-momentum tensor in flat space with conformal coupling ξ = 1/6 and m = 0 is
given by
Tµν =2
3∂µφ∂νφ+
1
6ηµν∂αφ∂
αφ− 1
3φ∂µ∂νφ. (6.47)
This tensor is traceless, i.e. Tµµ = 0 which reflects the fact that the theory is conformal. This
tensor is known as the new improved stress-energy-momentum tensor.
In the quantum theory Tµν becomes an operator Tµν and we are interested in the expectation
value of Tµν in the vacuum state < 0|Tµν |0 >. We are of course interested in the energy density
which is equal to < 0|T00|0 > in flat spacetime. We compute (using the Klein-Gordon equation
∂µ∂µφ = 0)
< 0|T00|0 >ξ= 16
=2
3< 0|∂0φ∂0φ|0 > −
1
6< 0|∂αφ∂αφ|0 > −
1
3< 0|φ∂µ∂ν φ|0 >
=2
3< 0|∂0φ∂0φ|0 > +
1
6< 0|∂0φ∂0φ|0 > −
1
6< 0|∂iφ∂iφ|0 > −
1
3< 0|φ∂2
0 φ|0 >
=5
6< 0|∂0φ∂0φ|0 > −
1
6< 0|∂iφ∂iφ|0 > −
1
3< 0|φ∂2
0 φ|0 >
(6.48)
From Klein -Gordon equation we have
∂µ∂µφ = 0
∂0∂0φ+ ∂i∂iφ = 0
−∂0∂0φ+ ∂i∂iφ = 0⇒ ∂20 φ = ∂2
i φ (6.49)
Thus we get
< 0|T00|0 >ξ= 16=
5
6< 0|∂0φ∂0φ|0 > −
1
6< 0|∂iφ∂iφ|0 > −
1
3< 0|φ∂2
i φ|0 >
We have (after partial integration)
∂i(φ∂iφ) = ∂iφ∂iφ+ φ∂i∂iφ⇒ φ∂i∂iφ = −∂iφ∂iφ (6.50)
And we obtain
< 0|T00|0 >ξ= 16=
5
6< 0|∂0φ∂0φ|0 > +
1
6< 0|∂iφ∂iφ|0 > . (6.51)
We regularize this object by putting the two fields at different points x and y as follows
< 0|T00|0 >ξ= 16
=5
6< 0|∂0φ(x)∂0φ(y)|0 > +
1
6< 0|∂iφ(x)∂iφ(y)|0 >
=
[5
6∂x0 ∂
y0 +
1
6∂xi ∂
yi
]< 0|φ(x)φ(y)|0 > . (6.52)
Similarly we obtain with minimal coupling the result
6.3 Another Derivation Using The Energy-Momentum Tensor 34
< 0|T00|0 >ξ=0 =
[1
2∂x0 ∂
y0 +
1
2∂xi ∂
yi
]< 0|φ(x)φ(y)|0 > . (6.53)
We use the result
DF (x− y) = < 0|T φ(x)φ(y)|0 >
=1
4π2
+∞∑n=−∞
[1
(x− y − 2nLe3)2− 1
(x− y − 2(nL+ x3)e3)2
]. (6.54)
We introduce (with a = −nL,−(nL+ x3))
Da = (x− y + 2ae3)2 = −(x0 − y0)2 + (x1 − y1)2 + (x2 − y2)2 + (x3 − y3 + 2a)2. (6.55)
We then compute
∂x0 ∂y0
1
Da= − 2
D2a
− 8(x0 − y0)2 1
D3a
. (6.56)
∂xi ∂yi
1
Da=
2
D2a
− 8(xi − yi)2 1
D3a
, i = 1, 2. (6.57)
∂x3 ∂y3
1
D−nL=
2
D2−nL
− 8(x3 − y3 + 2nL)2 1
D3−nL
. (6.58)
∂x3 ∂y3
1
D−(nL+x3)= − 2
D2−(nL+x3)
+ 8(x3 + y3 + 2nL)2 1
D3−(nL+x3)
. (6.59)
We can immediately compute
< 0|T00|0 >Lξ=0 =1
4π2
+∞∑n=−∞
[2
D2−nL
− 4(x3 − y3 + 2nL)2 1
D3−nL
− 4(x3 + y3 + 2nL)2 1
D3−(nL+x3)
]
−→ − 1
32π2
+∞∑n=−∞
1
(nL)4− 1
16π2
+∞∑n=−∞
1
(nL+ x3)4. (6.60)
This is still divergent. The divergence comes from the original charge corresponding to n = 0 in
the first two terms in the limit x −→ y. All other terms coming from image charges are finite. The
same quantity evaluated in infinite space is
< 0|T00|0 >∞ξ=0 =
∫d3k
(2π)3
ωk2e−ik(x−y). (6.61)
This is divergent and the divergence must be the same divergence as in the case of parallel plates
in the limit L −→∞, viz
< 0|T00|0 >∞ξ=0 = − 1
32π2
1
(nL)4|n=0. (6.62)
Hence the normal ordered vacuum expectation value of the energy-momentum-tensor is given by
< 0|T00|0 >Lξ=0 − < 0|T00|0 >∞ξ=0 = − 1
32π2
∑n 6=0
1
(nL)4− 1
16π2
+∞∑n=−∞
1
(nL+ x3)4. (6.63)
6.3 Another Derivation Using The Energy-Momentum Tensor 35
This is still divergent at the boundaries x3 −→ 0, L.
In the conformal case we compute in a similar way the vacuum expectation value of the energy-
momentum-tensor
< 0|T00|0 >Lξ= 16
=1
12π2
+∞∑n=−∞
[− 2
D2−nL
+4
D2−(nL+x3)
− 4(x3 − y3 + 2nL)2 1
D3−nL
− 4(x3 + y3 + 2nL)2 1
D3−(nL+x3)
]
−→ − 1
32π2
+∞∑n=−∞
1
(nL)4. (6.64)
The normal ordered expression is
< 0|T00|0 >Lξ= 16− < 0|T00|0 >∞ξ= 1
6= − 1
32π2
∑n 6=0
1
(nL)4
= − 1
16π2L4
∞∑n=1
1
n4
= − 1
16π2L4ζ(4). (6.65)
The zeta function is given by
ζ(4) =
∞∑n=1
1
n4=π4
90. (6.66)
Thus
< 0|T00|0 >Lξ= 16− < 0|T00|0 >∞ξ= 1
6= − π2
1440L4. (6.67)
This is precisely the vacuum energy density of the conformal scalar field. The electromagnetic field
is also a conformal field with two degrees of freedom and thus the corresponding vacuum energy
density is
ρem = − π2
720L4. (6.68)
This corresponds to the attractive Casimir force. The energy between the two plates (where A is
the surface area of the plates) is
Eem = − π2
720L4AL. (6.69)
The force is defined by
Fem = −dEem
dL
= − π2
240L4A. (6.70)
The Casimir force is the force per unit area given by
Fem
A= − π2
240L4. (6.71)
7 Conclusion 36
7 Conclusion
In this work we studied the cosmological constant Λ, the vacuum energy and their relation to dark
energy. Dark energy viewed as the energy of the vacuum is a perfect fluid with equation of state
Pvac = −ρvac and thus plays the same role of the cosmological constant. By comparing the value of
the energy density obtained from cosmological observations and the theoretical value we see that
there is a discrepancy of 30 orders of magnitude. It is hopped that a solution of this problem can
be found by a proper calculation of the vacuum energy in curved and expanding spacetimes. We
quantize a scalar field in FLRW universes since they are the spacetimes which describe more closely
the real world. In the future evolution of the universe it is believed that vacuum will dominate
and thus spacetime becomes approximately de Sitter spacetime. We quantize therefore in de Sitter
spacetime where a reasonable physically well founded vacuum known as the Bunch-Davies vacuum
can be used. An estimation of the vacuum energy which has the same form as dark energy is
obtained in de Sitter spacetime. FLRW universes may be thought of as small perturbation of de
Sitter. We also discuss the reality of the energy of the quantum fluctuations in the vacuum in the
context of the experimentally verified Casimir force between parallel plates.
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