and 3) and start to
oscillate rapidly. This is
shown in Fig. 4.8. After
a short time the
oscillations are damped
out and the fields
acquire a "fixed" value.
Then the fields do roll
down to their minimum
and inflation works. The
numerical solutions for
1 a
(5.31) A simple exercise
shows that the new
creation and
annihilation operators
b + k and b k can be
expressed in terms of
the "old" creation and
annihilation operators
as b + k = k a + k
+ka k b k = ka
k + k a + k .
(5.32) This is a
Bogoliubov
transform
0|T[(x)(x 0 )]|
0 = ( 0 )0|
(x)(x 0 )|0 +(
0 )0|(x 0 )
(x)|0, (5.89) where
is the Heaviside function. The Feynman
propagator is Lorentz
invariant, and hence
the physical propagator.
We are actually able to
calculate this
propagator explicitly if
we inser
CMBR. Also, quantum
fluctuations can
influence the dynamics
of fermions, which we
will discuss in chapter
6. For now, let us
continue with the
quantization of our
inflaton field. We could
also use the conformal
FLRW metric by making
the substitution dt =
and the coordinate z =
k are real, we find
that (2) k () =
(1) k (). Of course
this will be different
when for example is
complex, as we will see
when we quantize the
two-Higgs doublet
model. If we now look
at the original mode
fluctuations k, we
can wri
conclusion from our
numerical simulations
is that inflation also
works in a
nonminimally coupled
two-Higgs doublet
model. However, only
one particular
combination of the two
fields is the actual
inflaton. The other
linear combination is an
oscillating mod
fermion propagator.
However, as we will see
in the next sections,
quantizing a field
theory is
straightforward in
Minkowski space, but
quite subtle in an
expanding universe. For
good literature on
quantum field theory in
curved backgrounds,
see [30] and [
analytical results. The
large negative
nonminimal coupling
has the additional
advantage that the
constraint on the
smallness of the quartic
self-coupling in a
minimal model,
imposed by a
calculation of density
perturbations, can be
relaxed by many orders
Again we have derived
the field equations, and
numerical calculations
for the two real scalar
fields show that
inflation is still
successful in this
model. One linear
combination of the
fields serves as the
inflaton, whereas the
other is an oscillating
mo
However, the usual
vacuum that we define
from the annihilation
operators at this time is
not a good vacuum, in
the sense that the
vacuum is not the
lowest energy state at a
later time. We should
therefore construct a
better vacuum. If we
are in the adiaba
chapter 6. Chapter 5
Quantum field theory
in an expanding
universe In the previous
chapter we have given
an elaborate discussion
on inflation through a
nonminimally coupled
scalar field. We have
shown that nonminmal
inflation works for a
theory with a qua
sum, we change our
summation variable n
n+1, such that the
sum now runs from n =
1 to , and we split
off the n = 1 term. The
reason for this will
become clear in a
moment. The
propagator can then be
written as i(x, x 0 ) =
[(1) 2HH0 ] D 2 1
(4) D 2 ( D 2
(5.105). In the next
section we repeat the
quantization procedure
for the two-Higgs
doublet model. We will
see that there are only
some small changes
with respect to the
model for one real
scalar field, i.e. the
standard Higgs model.
5.3 Quantization of t
could do the same for
the out region, and
write (x,t) = Z d 3k
(2) 3 b k u out k +
b + k (u out k ) e
ikx . (5.37) Again, the
annihilation operators
in the out region also
define a vacuum by b
k |0out = 0, which is
the state of lowest
energy in the out
r
6lm)m " 2 X
l,m=1,2 (lm 6lm)
( l m) # = P k=1,2
ikk M2 p 2 P
k,l,m=1,2 k kl(lm
6lm)m " 6 X l,m=1,2
lm[ l V m +
V l m]4V # V
i (4.95) , where V
= V(1,2). We have
written the potential
terms on the righthand
side of the equation of
motion. In the slow-roll
+). (5.108) We stress
that in higher order
terms of n the 1 s 3 +
term does not appear
because we do not
have to expand the
digamma-function (z)
around z = 0 for n 1.
To summarize, we have
obtained a relatively
simple expression for
the scalar propagator
particle number, this
gives 0in|N out k |
0in = 0in|b + k b
k |0in = Z d 3k 0 (2) 3
|k0k| 2 . (5.43) Thus
we see that although
the particle number is
zero in the in region, it
becomes nonzero in
the out region. The
important requirement
is the time depen
some natural choice for
the parameters k and
k? In general we
cannot do this, because
the lowest energy state
at one instant of time
will not be the lowest
energy state at a later
time in an expanding
universe. However, in
the previous section we
showed t
fluctuations, which
allows us to write the
action for the quantum
fluctuations SQ = Z d 4 x
p g X i=1,2 g
(i) (i) X
i,j=1,2 i jR i j
X i,j=1,2 " i V
i j j + 1 2 i
V ij j + 1 2
i V i j
j # , where i(x)
= 0 i (t)+i(x,t), with
0 i (t) the classical
infl
inflation, see Eq. (4.19).
We can make an
expansion for s 1 in
the expression for .
This gives = 1 1
3 2 + s 3 . (5.103)
Now we also expand for
1, which is also true
during inflation. We get
= 3 2 + s 3 +O(s).
(5.104) When we now
look at the scalar
pro
k(t). (5.10) A
convenient way to
quantize the fields is
now to use the mode
expansion k(t) = 1 p
2k a k e ik t +
a + k e ik t , (5.11)
where a + k and a k
are the creation and
annihilation operators.
Basically, we have
replaced the k and k
in the general
1,2), (5.114) where the
canonical momentum is
by definition i = L
i = a 20 i , (i =
1,2). (5.115) We could
also write 1
2 = a 2 0 1
0 2 a 2 0 .
(5.116) The system of
field equations for the
complex fields 1 and
2 that we want to
solve is linear (Eq.
(5.1
section 5.2, this is
precisely what happens
in an expanding
universe. As we have
shown in this section, a
harmonic oscillator
with a constant
frequency allows us to
define a unique vacuum
state that is the state of
lowest energy at all
times. If the frequ
scalar and fermion
propagators in D
dimensions. To
calculate the
propagator in D
dimensions we need
the solutions for the
field fluctuations k in
D dimensions, and only
a few things will change
with respect to the four
dimensional case. We
can still write
quickly decaying mode.
However, there will
now be a third physical
quantity, which is the
phase between the two
fields. If this phase is
changing in time, it
could be a source for
CP violation and
baryogenesis, see e.g.
Cohen, Kaplan, Nelson
[29]. In futu
fact that we are
working in an infinite
space volume. If we
would work in a box,
this would be the
volume V of the box.
Dividing by this term
then gives us the
energy and particle
number densities. We
can recognize the
number operator N k
from Eq. (5.18).
operators must satisfy
a k ,a + k0 = (2) 3
3 (kk 0 ) (5.14) a +
k ,a + k0 = a k
,a k0 = 0. We now
define the vacuum
state that is annihilated
by all annihilation
operators, i.e. a k |
0 = 0. (5.15) Excited
states are made by
acting with the creation
ope
their minimum. During
the inflationary period
the fields are
proportional to each
other, which suggests
that we can find linear
combinations of the
two fields such that one
of the linear
combinations is the
actual inflaton, whereas
the other combination
i
Sitter space, with the
scale factor from Eq.
(2.23). The field
equation (5.62) then
becomes 00 k + k 2
+ 1 2 m2 + R( 1 6 )
H2(1) 2 ! k = 0.
(5.74) We now write
this equation as d 2
d(k) 2 +1+ 1 4 2
(k) 2 ! k = 0, (5.75)
with 2 = 1 4 m2 +
R( 1 6 ) H2(1) 2
approx k (0) = 1 p
2k(0) approx0 k
(0) = ik(0) 1 2
0 k (0) k(0) ! 1 p
2k(0) . (5.71) This
yields an energy in the
mode k with respect
to the adiabatic
vacuum of Ek(0) = 1 2
|0 k | 2 + 2 k |
k| 2 =0 = 1
2 k + 1 16 02 k 3
k 1 2 k. (5.72) So the
energy in