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We calculate the particle production rate in an expanding universe with a three-torus topology. We discuss also the complete evolution of the size of such a universe. The energy density of particles created through the nonzero modes is computed for selected masses. The unique contribution of the zero mode and its properties are also analyzed.
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CALT 68-2880
Particle creation in a toroidal universe
Bartosz Fornal
California Institute of Technology, Pasadena, CA 91125, USA
(Dated: July 27, 2012)
We calculate the particle production rate in an expanding universe with a three-torus topology. We discuss
also the complete evolution of the size of such a universe. The energy density of particles created through the
nonzero modes is computed for selected masses. The unique contribution of the zero mode and its properties
are also analyzed.
I. INTRODUCTION
Although current astrophysical observations provide pre-
cise information on the geometry of the universe [1], its topol-
ogy remains a mystery. We don’t even know whether the uni-
verse is compact or infinite. Nevertheless, lower bounds can
be put on its size for each compact topology (see, [2] and ref-
erences therein).
Amongst the possible topologies for the universe those with
some or all spatial dimensions compactified are especially in-
teresting, since then Casimir energies provide an additional
contribution to the energy density. This may lead to an in-
teresting vacuum structure for the standard model coupled to
gravity which is insensitive to quantum gravity effects [3–5].
The simplest flat topology with all dimensions compactified is
a three-torus, and that is the topology we will concentrate on
throughout this paper.
The scenario of our universe having a three-torus topol-
ogy was investigated by many authors (see, [6] and references
therein). Probably one of the most appealing features of such
a model is that the creation of a three-torus universe is much
more likely to occur than that of an infinite flat or closed uni-
verse [6, 7]. In addition, it has also been shown that a three-
torus topology can provide convenient initial conditions for
inflation [7].
Here we consider gravitational particle creation in an ex-
panding toroidal universe. The particle production formalism
was developed in [8–10] and investigated in great detail in
later works (see, [11–13] and references therein). Since then,
it has been thoroughly studied in the case of a FRW cosmol-
ogy, including its implications for dark matter creation around
the inflationary epoch [14]. However, particle production in a
toroidal universe hasn’t been extensively studied (see, [15] for
some work on the subject).
In this paper we provide a detailed numerical calculation
of the particle production in a three-torus universe. We start
with introducing the relevant formalism. We then discuss the
evolution of the size and energy density of the universe from
the Planck time to the present time. Next, we derive analyt-
ical formulae for the particle number and energy density at
early and late times. We then find full numerical solutions and
confirm that they agree with the analytical approximations in
the appropriate regions. The particle production through the
nonzero modes is somewhat similar as in the closed universe
case discussed in [12]. However, the three-torus particle cre-
ation includes an additional contribution from the zero mode,
which strongly depends on the choice of initial conditions.
II. THREE-TORUS METRIC
We start with the following spacetime interval,
ds2=dt2+tij dyidyj,(1)
where tij is the metric on the three-torus with i, j = 1,2,3,
and the compact coordinates are yi[0,2π). The 3×3matrix
with components tij is positive definite and has a determinant
equal to the volume modulus a3. A suitable parametrization
is given by,
tij =a2
(ρ3τ2)2/3
1τ1ρ1
τ1τ2
1+τ2
2ρ1τ1+ρ2τ2
ρ1ρ1τ1+ρ2τ2ρ2
1+ρ2
2+ρ2
3
,(2)
where (τ1, τ2, ρ1, ρ2, ρ3)are the shape moduli. We assume
that all the parameters in (2) are independent of the spatial co-
ordinates. Furthermore, we seek stable solutions to Einstein’s
equations for which the shape parameters are also constant in
time. It was shown in [5] that this is only possible for,
(τ1, τ2, ρ1, ρ2, ρ3) = 1
2,3
2,1
2,3
6,6
3!,(3)
which arises from the symmetries of the Casimir energies. For
a three-torus universe characterized by the parameters (3) only
the evolution of the volume modulus is nontrivial. The corre-
sponding metric takes the form,
tij =a2
3
4
211
121
112
,(4)
and this is the metric we will adopt in our further analysis.
III. PARTICLE PRODUCTION
We first derive general formulae for the gravitational parti-
cle production rate in a three-torus universe. Let us consider
a complex scalar field Ψ = Ψ(x)of mass mwith the La-
grangian density given by,
L=ggµν µΨνΨm2+R
6|Ψ|2,(5)
arXiv:1207.6230v1 [hep-th] 26 Jul 2012
2
where Ris the Ricci scalar,
R=6
a2a¨a+ ˙a2.(6)
The dot denotes the derivative with respect to time tand the
factor ξ= 1/6was chosen to have conformal invariance in the
limit m0. We note, however, that the results in this paper
are not very sensitive to this choice and would be similar, for
example, in the case of a minimally coupled scalar field (for
which ξ= 0).
The stress-energy tensor is given by,
Tµν =µΨνΨ+νΨµΨgµν L
g
1
3(Rµν +µνgµν γγ)|Ψ|2.(7)
The equation of motion for the field Ψis,
¨
Ψ+3˙a
a˙
Ψ− ∇2Ψ + m2+¨a
a+˙a2
a2Ψ = 0 ,(8)
where 2is the Laplacian on the three-torus. We write the
solutions of equation (8) as,
Ψλ(x) = uλ(t)φλ(~y),(9)
with λ= (λ1, λ2, λ3). In our case,
2φλ=˜
tij
a2λiλjφλ,(10)
where,
˜
tij tij /a2,(11)
and there is an implicit sum over the repeated indices i, j =
1,2,3, with λi= 0,±1,±2, .... The formula for φλis given
by,
φλ=C ei˜
tklλkyl.(12)
Note that for λ= (0,0,0) we have φ0= const, which corre-
sponds to the zero mode. Equation (8) takes the form,
¨uλ+ 3 ˙a
a˙uλ+ω2
λ
a2+¨a
a+˙a2
a2uλ= 0 ,(13)
where,
ω2
λ= (ma)2+˜
tij λiλj.(14)
We can now quantize the field introducing standard canonical
equal-time commutation relations for the field and its general-
ized momentum. Those relations are satisfied if we write the
field Ψas,
ˆ
Ψ = 1
(2πa)3
/2
X
λ123=−∞ hφλu
λˆaλ+φ
λuλˆ
b
λi,(15)
where ˆa
λand ˆaλare the creation and annihilation operators of
a particle in the state λ= (λ1, λ2, λ3), and ˆ
b
λand ˆ
bλare those
for antiparticles, all obeying the usual commutation relations.
Adopting such a convention, the Hamiltonian can be written
as [12],
ˆ
H=
X
λ123=−∞
ωλAλˆaλˆa
λ+ˆ
b
¯
λˆ
b¯
λ
+Bλˆa
λˆ
b
¯
λ+B
λˆaλˆ
b¯
λ,(16)
with,
Aλ=a2|˙uλ|2
2ωλ
+1
2ωλ|uλ|2,
Bλ=a2˙u2
λ
2ωλ
+1
2ωλu2
λ,(17)
and ¯
λdefined through φ¯
λ=φ
λ.
In general, the Hamiltonian (16) is non-diagonal. The re-
quirement of it being diagonal at some initial time t0imposes
the following conditions,
uλ(t0) = 1
pωλ(t0),˙uλ(t0) = ipωλ(t0)
a(t0).(18)
Now, we can diagonalize the Hamiltonian through the follow-
ing Bogoliubov transformation,
ˆaλ=α
λ(t) ˆa0
λ(t) + βλ(t)ˆ
b0†
¯
λ(t),
ˆ
bλ=α
λ(t)ˆ
b0
λ(t) + βλ(t) ˆa0†
¯
λ(t),(19)
where |αλ|2− |βλ|2= 1. It can be shown [12], from the re-
quirement that there be no non-diagonal terms ˆa0†
λˆ
b0†
λor ˆa0
λˆ
b0
λ,
that the equations for αλ(t)and βλ(t)are,
˙
βλ=˙ωλ
2ωλ
αλexp 2iZt
t0
ωλ(t0)
a(t0)dt0,(20)
˙αλ=˙ωλ
2ωλ
βλexp 2iZt
t0
ωλ(t0)
a(t0)dt0,(21)
with the initial conditions βλ(t0) = 0 and αλ(t0) = 1. The
function uλis expressed in terms of αλand βλas [12],
uλ=1
ωλα
λexp iZt
t0
ωλ(t0)
a(t0)dt0
+βλexp iZt
t0
ωλ(t0)
a(t0)dt0.(22)
Relations (20) and (21) can be combined into one second order
differential equation for βλ(t),
¨
βλ+˙ωλ
ωλ¨ωλ
˙ωλ
+2λ
a˙
βλ˙ω2
λ
4ω2
λ
βλ= 0 ,(23)
with the initial conditions,
βλ(t0)=0,˙
βλ(t0) = ˙ωλ(t0)
2ωλ(t0).(24)
3
epoch
pre-inflationary era inflation radiation era matter era dark energy era
5·1044 s5·1036 s1033 s7·104years 1010 years
tinit
'8·1020 GeV1'7·1012 GeV1'109GeV1'3·1036 GeV1'5·1041 GeV1
5·1036 s1033 s7·104years 1010 years 1.37 ·1010 years
tfinal
'7·1012 GeV1'109GeV1'3·1036 GeV1'5×1041 GeV1'7·1041 GeV1
ap0t ai0exp pρtot/(3M2
p)tar0t am0t2/3ad0exp pρtot/(3M2
p)t
a(t)
ap0'4·1010 1
GeV ai0'1015 GeV1ar0'6·1020 1
GeV am0'6·1014 1
3
GeV ad0'2·1042 GeV1
2·1035 m2·1031 m4 m 2·1023 m8·1026 m
a(tinit)
'1019 GeV1'1015 GeV1'2·1016 GeV1'1039 GeV1'4·1042 GeV1
2·1031 m4 m 2·1023 m8·1026 m1027 m
a(tfinal)
'1015 GeV1'2·1016 GeV1'1039 GeV1'4·1042 GeV1'5·1042 GeV1
ρtot(t) 3M2
p/(4t2) const 3M2
p/(4t2) 4M2
p/(3t2) const
ρtot(tinit )7·1074 GeV49·1058 GeV44·1054 GeV48·1037 GeV43·1047 GeV4
9·1058 GeV4
ρtot(tfinal )9·1058 GeV4after reheating: 5·1037 GeV43·1047 GeV43·1047 GeV4
4·1054 GeV4
TABLE I: Estimated timeline for the evolution of a three-torus universe from the Planck time until the present time tuniv 13.7 billion years, including the
size of the universe and total energy density during different epochs, assuming a pre-inflationary expansion a(t) = ap0tand the current size of the universe
a(tuniv)10 RH.
The normal ordered Hamiltonian now takes the diagonal form,
ˆ
H=
X
λ123=−∞
ωλˆa0†
λˆa0
λ+ˆ
b0†
λˆ
b0
λ,(25)
and the physical vacuum depends on time through,
ˆa0
λ(t)|0(t)i=ˆ
b0
λ(t)|0(t)i= 0 .(26)
It is now straightforward to arrive at the formula for the num-
ber density of particle pairs created after time t,
n(t) = 1
(2πa)3
X
λ123=−∞ |βλ|2.(27)
The energy density and pressure of the created particles are
calculated from the vacuum expectation values of the appro-
priate components of the stress-energy tensor (7). After nor-
mal ordering the result is (compare with [12]),
ρ(t) = 1
4π3a4
X
λ123=−∞
ωλ|βλ|2,(28)
pii(t) =
3
2
12π3a4
X
λ123=−∞
ωλ
×|βλ|2m2a2
2ωλ|uλ|21
ωλ,(29)
pij (t) = pii(t)
2for i6=j , (30)
where i, j = 1,2,3. Note that the pressure is not isotropic in
our case. It would be isotropic only for the simplest choice of
the three-torus metric diag(a2, a2, a2).
IV. EVOLUTION OF THE THREE-TORUS UNIVERSE
In order to calculate the particle production rate, we first
have to know how a(t)evolved in time. Our case is different
from the usual FRW cosmology, in which a(t)is the scale
factor and can be rescaled by an arbitrary number. For a three-
torus universe a(t)has a physical meaning – it describes the
size of the universe at a given time t. Recent analyzes (see,
[16] and references therein) set a lower bound on its present
value of a(tuniv)&6RH, where RHis the Hubble radius
today. Let’s therefore assume that the current size of the three-
torus universe is,
a(tuniv) = 10 RH1027 m.(31)
Table I shows the estimated timeline, the size of the uni-
verse and the total energy density during different epochs in
the evolution of the universe. For each epoch the total energy
density is uniquely determined by the expansion rate through
the Friedmann equation,
˙a
a2
=ρtot
3M2
p
,(32)
where the reduced Planck mass Mp'2.4·1018 GeV. Us-
ing the formulae for a(t)for different epochs, we can evolve
4
a(tuniv)back in time and find its value at any particular in-
stance in the past.
Observational data suggest that we currently live in a
dark energy dominated universe, where the energy density is
roughly constant and the expansion is exponential. It was pre-
ceded (at t.10 billion years) by a matter dominated era with
the size of the universe increasing like t2/3. Before the matter
era (at t.70000 years), radiation was driving the expansion
like t. It is believed that before the radiation epoch there
was a brief period of inflation (1035 s.t.1033 s), an
exponential expansion resulting from a constant energy den-
sity. The inflationary energy density in table I was estimated
adopting 72 e-folds of inflation and assuming that it ended at
t1033 s.
The expansion rate in the pre-inflationary era is unknown.
However, in the case of a three-torus universe it might have
a natural explanation. As mentioned in the introduction, a
compact topology results in the appearance of Casimir ener-
gies of existing fields. The full formulae for the Casimir en-
ergies in a three-torus universe for an arbitrary set of shape
moduli was derived in [5]. For a real scalar field with mass
m1015 GeV the Casimir energy density before inflation
for our choice of metric is,
ρCas(a) = 2
(2π)4
1
a4(3
22/3
π
12 +21/3
33/2
ζ(3)
π+ 21/3π2
360
+211/6
33/4
X
n2,n3=1 n3
n23/2
cos (π n2n3)K3/23π n2n3
+
X
n2,n3=−∞
0
X
n1=1
cosh2
3πn1(n2+n3)iq25/3
3(n2
2n2n3+n2
3)
×1
n1
K142
3πn1q(n2
2n2n3+n2
3)),(33)
which can be written as,
ρCas(a) = κ
a4,(34)
where κ'5×103. In a general case, this formula in-
cludes also a factor corresponding to the number of degrees
of freedom for a given field and an additional minus sign for
fermions. Interestingly, the Casimir energies at a1/m
satisfy the same equation of state as radiation, i.e.,
pCas(a(t)) = 1
3ρCas(a(t)) .(35)
Therefore, if there were more fermionic degrees of freedom
(nf) than bosonic (nb) and Casimir energies dominated the
total energy density before inflation, the universe would ex-
pand according to a(t) = a0t, with the total energy density
given by,
ρtot
Cas(a) = (nfnb)κ
a4.(36)
Surprisingly, if we choose in our case (nfnb)20, we ob-
tain the total Casimir energy density at the time when inflation
-20
-10
0
10
20
30
40
-20
-10
0
10
20
30
40
Log10Ha@GeV-1DL
FIG. 1: Size of the three-torus universe as a function of time, assuming
a(tuniv) = 10 RH.
-20
-10
0
10
20
30
40
-40
-20
0
20
40
60
80
Log10HΡtot @GeV4DL
FIG. 2: Total energy density in the three-torus universe as a function of time.
started,
ρtot
Cas(a(tinf )) ρtot (tinf),(37)
where a(tinf )is the universe size at the beginning of inflation
and ρtot(tinf )is the total energy density of the universe during
inflation, both of which where estimated before independently
of the Casimir energies through the evolution back in time. Of
course, the required value of (nfnb)which satisfies condi-
tion (37) strongly depends on the inflationary parameters.
Another interesting observation concerning the three-torus
topology is that assuming the current size of the universe
a(tuniv)10 RHand a pre-inflationary expansion a(t) =
ap0t, the evolution back in time yields the size at Planck
time a(tp)lp, where lpis the Planck length. Note that an
evolution from the Planck size at Planck time to ten Hubble
radii at present time wouldn’t be possible if we assumed a
universe with a closed topology, since then the curvature term
contribution would entirely dominate the total energy density
during the early stages of the universe evolution.
The plot of the size of the universe a(t)corresponding to
the values from table I is given in figure 1. A similar plot for
the total energy density is shown in figure 2. We note that a
similar calculation to the one presented in this paper can be
done assuming a different pre-inflationary expansion.
5
Knowing the shape of a(t), we can now numerically solve
equation (23) for the Bogoliubov coefficients and then use for-
mulae (27) and (28) to calculate the number density and en-
ergy density for the gravitationally created pairs of scalar par-
ticles of mass m. Our results can be easily generalized for
other types of particles by adopting the appropriate form of
the stress-energy tensor. We will perform the calculation for
three different masses: m1= 109GeV,m2= 103GeV, and
m3= 103GeV, without the zero mode first, and then com-
puting its contribution as well. In our calculation we will be
using natural units, i.e.,
2·1016 m1 GeV17·1025 s.(38)
We will also assume that there is no back-reaction of the cre-
ated particles on the background evolution.
V. ANALYTICAL RESULTS FOR NONZERO MODES
In order to fully understand the numerical solutions it is
very helpful to derive their analytical behavior in two regions:
t1/m and t1/m. In contrast to [11, 12], we choose
not to specify the initial conditions at the singularity, but at the
Planck time, since the physics at earlier times is unknown,
βλ(tp) = 0 ,˙
βλ(tp) = ˙ωλ(tp)
2ωλ(tp).(39)
However, we note that for the nonzero modes the solutions
are insensitive to the value of t0at which we impose those
conditions, as long as we look at the region tt0.
Let us first consider the case mt 1and focus only on the
pre-inflationary epoch, for which we adopted a(t) = ap0t.
With the additional assumption ma 1the largest contribu-
tion to the sum (27) comes from terms with small λi. Thus, we
can assume λita, in which case the solution to equation
(23) is given by,
βλ(t)ωλ
2(˜
tij λiλj)1
/4(˜
tij λiλj)1
/4
2ωλ
.(40)
After plugging (40) into (27), the particle number density is,
n(t)1
(2πa)3
X
λ123=−∞
0ωλ(˜
tij λiλj)1
/22
4ωλ(˜
tij λiλj)1
/2
m4a
128π3
X
λ123=−∞
01
(˜
tij λiλj)2'm4a
250 ,(41)
where the prime excludes the zero mode.
The calculation of the energy density and pressure in the re-
gion mt 1cannot be performed with the same assumption
λita, as now terms with large λicontribute significantly.
However, for ma 1we have |βλ|  1and αλ1. With
equation (20) we can approximate βλ(t)by,
βλm2
2˜
tij λiλjZt
0
a(t0) ˙a(t0)
×exp 2i(˜
tij λiλj)1
/2
a(t0)t0dt0.(42)
Using this result to rewrite (28), we arrive at the energy den-
sity for the created particles during the pre-inflationary era,
ρ(t)m4a4
p0
64π3a4
X
λ123=−∞
01
(˜
tij λiλj)3
/2Zt
0
dt1Zt
0
dt2
×exp 2i(˜
tij λiλj)1
/21
ap0t1t2.(43)
The biggest contribution in equation (43) comes from sum-
ming the terms with large λi. It turns out that we can ap-
proximate our three-torus by a three-sphere of radius 6
2a,
for which it is possible to perform the sum over λs. Equation
(43) takes the form,
ρ(t)m4a4
p0
64 3
2π3a4Zt
0
dt1Zt
0
dt2
×
X
λ=1
1
λexp 2iλ
ap0t1t2
=m4a4
p0
64 3
2π3a4Zt
0
dt1
Zt
0
dt2log
1
1exp h2i
ap0t1t2i
.
(44)
The double-integral above is real and positive. The resulting
energy density for mt 1is essentially constant in time.
Using a similar approximation one can derive the formula
for the pressure,
pij (t)˜
tij
ρ(t)
3˜
tij
m4a2
p0
96π3a2(m2a2
p0
4Zt
0
dt1Zt
0
dt2
×Li3exp 2i
ap0t1t2
2 Re Zt
0
dt1log 1exp 2i
ap0t1t).(45)
A numerical check shows that for mt 1the pressure for
the created particles is also almost constant and satisfies the
“quasi-vacuum-like” equation of state,
pij (t)≈ −˜
tij ρ(t).(46)
We will not go into the details of calculating the parti-
cle number or energy density in a three-torus universe for
mt 1, since this case is analogous to that of an infinite
flat or closed universe, which was explored in detail in [12].
Briefly summarizing, for mt 1all Bogoliubov coefficients
βλare roughly constant and particle creation doesn’t occur.
For a universe expanding according to a(t) = a0tq, where
0< q 2/3, the number density and the energy density of
the created particles behave in the following way,
n(t)m33q
t3q1
a3,(47)
ρ(t)m43q
t3q1
a3,(48)
while the pressure pij (t)ρ(t).
6
-18
-16
-14
-12
-10
-8
-6
-100
-50
0
Log10Ht@GeV-1DL
Log10Hn@GeV3DL
FIG. 3: Number density of the created particles for m1= 109GeV (blue
solid line). The green dashed line corresponds to the mt 1and ma 1
approximation given by equation (41) and the orange dot-dashed line corre-
sponds to the mt 1approximation given by equation (47).
-18
-16
-14
-12
-10
-8
-6
-100
-50
0
50
Log10Ht@GeV-1DL
Log10HΡ@GeV4DL
FIG. 4: Energy density of the created particles for m1= 109GeV (blue
solid line). The green dashed line corresponds to the constant density for
mt 1given by equation (44) and the orange dot-dashed line corresponds
to the mt 1approximation given by equation (47). For comparison, the
total energy density of the universe is plotted in solid red.
VI. NUMERICAL RESULTS FOR NONZERO MODES
Having an analytical understanding of how the functions
n(t)and ρ(t)behave for mt 1and mt 1, in this section
we discuss the full numerical solutions.
We first solve numerically equation (23) and find βλ(t)for
different sets of λis, with ωλ(t)defined through equation
(14). The initial conditions are chosen according to (39). We
then plug the calculated βλ(t)to equations (27) and (28), and
in this way obtain the particle number and energy density.
Figure 3 shows the plot of the particle number density
(in blue) corresponding to a mass of the created particles of
m1= 109GeV. The green dashed line increasing like t
is the mt 1and ma 1approximation given by equa-
tion (41). Since the particle mass is large, this approxima-
tion breaks down relatively quickly and the number density
starts slowly decreasing. During the inflationary epoch, it falls
down rapidly by many orders of magnitude. Particles of mass
109GeV are no longer produced after inflation. This is con-
-15
-10
-5
0
-100
-50
0
Log10Ht@GeV-1DL
Log10Hn@GeV3DL
FIG. 5: Same as figure 3, but for m2= 103GeV.
-15
-10
-5
0
-100
-50
0
50
Log10Ht@GeV-1DL
Log10HΡ@GeV4DL
FIG. 6: Same as figure 4, but for m2= 103GeV.
firmed by fitting the line falling off like t3/2(orange dot-
dashed line) corresponding to the relation (47).
Figure 4 shows the time evolution of the energy density for
the created particles of mass m1= 109GeV. The energy den-
sity is nearly constant for mt 1and its value agrees with
that from equation (44), represented by the green dashed line.
The subsequent behavior of the energy density is similar as in
the number density case – following a mild decrease there is
a rapid fall during inflation, after which the density continues
to decrease like t3/2(orange dot-dashed line), since particle
creation no longer takes place. Note that the density of the
created particles is small compared to the total energy density
in the universe (denoted in figure 2 by the red solid line) at
any given time.
Figures 5–8 show the number density and energy density
for the created particles with masses m2= 103GeV (figures
5, 6) and m3= 103GeV (figures 7, 8). There are two qual-
itative differences compared to the previous plots. Because
the particles are lighter, the conditions mt 1and ma 1
hold also for later times and the plots match the approxima-
tions (41) and (44) (green dashed line) all the way until the
inflation epoch. Secondly, particle creation continues after in-
flation and stops only at t1/m, which is confirmed by
fitting the line (orange dot-dashed) going like t3/2.
7
-15
-10
-5
0
5
-100
-50
0
Log10Ht@GeV-1DL
Log10Hn@GeV3DL
FIG. 7: Same as figure 3, but for m3= 103GeV.
-15
-10
-5
0
5
-100
-50
0
50
Log10Ht@GeV-1DL
Log10HΡ@GeV4DL
FIG. 8: Same as figure 4, but for m3= 103GeV.
We emphasize that although particle creation lasts arbitrar-
ily long for light enough particles, their energy density is still
negligible compared to the total energy density. For instance,
if we wanted to have particles created today, we would need a
particle of mass mlight 1042 GeV. From formula (44) the
energy density is suppressed approximately by m4/a4. Tak-
ing into account also the huge decrease in energy density of
the created particles during inflation, there is no way it can
ever compete with the total energy density of the universe.
VII. ZERO MODE
Let us now consider the contribution of the zero mode. In
this case all the formulae simplify significantly. Relation (14)
becomes,
ω0(t) = m a(t).(49)
Equation (23) for the Bogoliubov coefficient reduces to,
¨
β0+˙a
a¨a
˙a+ 2im˙
β0˙a2
4a2β0= 0 ,(50)
with the initial conditions,
β0(t0) = 0 ,˙
β0(t0) = ˙a(t0)
2a(t0).(51)
The equation for u0(t)is,
¨u0+ 3 ˙a
a˙u0+m2+¨a
a+˙a2
a2u0= 0 ,(52)
where,
u0(t0) = 1
pm a(t0),˙u0(t0) = irm
a(t0).(53)
Once equations (50) and (52) are solved, the particle num-
ber density, energy density and pressure can be calculated us-
ing the simple relations,
n(t) = |β0|2
(2πa)3,(54)
ρ(t) = m|β0|2
4π3a3,(55)
pij (t) = ˜
tij
m
12π3a3|β0|21
2ma|u0|2+1
2.(56)
It turns out that equation (50) with the initial conditions (51)
can be solved exactly for simple choices of a(t). For instance,
in the pre-inflationary era with a(t) = a0t, the solution for
a particle of mass mtakes the form,
β0(t) = i
2mt1
/4t
5
/4
0e2im(t0t)U5
4,3
2,2imt0
×L1/2
5/4(2imt)U5
4,3
2,2imtL1/2
5/4(2imt0)
×4U5
4,3
2,2imt0L3/2
9/4(2imt0)
5U9
4,5
2,2imt0L1/2
5/4(2imt0)1
,(57)
where U(x, y, z)is the confluent hypergeometric function of
the second kind, and Lk
n(x)is the associated Laguerre poly-
nomial.
In the limit mt 1the dependence of the Bogoliubov
coefficients in formula (57) on tand t0is β0(t/t0)1
/4,
which translates to,
n(t, t0)ρ(t, t0)1
tt0
.(58)
Thus, setting initial conditions at a small enough t0, one can
have an arbitrarily large zero mode particle production at any
given instance of time. As mentioned earlier, this situation
doesn’t occur for the nonzero modes, for which ˙
βλ(t0)is in-
dependent of t0for ma 1. However, since we don’t know
the physics before the Planck time, choosing t0< tpis not
justified. In the subsequent calculation we assume t0=tp,
just as in the case of the nonzero modes.
8
-15
-10
-5
0
-20
0
20
40
60
Log10Ht@GeV-1DL
Log10Hn@GeV3DL
FIG. 9: Number density of the particles created through the zero mode for
m= 103GeV (blue solid line), assuming t0=tp. The green dashed
line corresponds to the mt 1approximation given by equation (58). The
orange dot-dashed line denotes the mt 1approximation given by equation
(60).
-15
-10
-5
0
-20
0
20
40
60
80
Log10Ht@GeV-1DL
Log10HΡ@GeV4DL
FIG. 10: Energy density of the particles created through the zero mode for
m= 103GeV (blue solid line), assuming t0=tp. The green dashed
line corresponds to the mt 1approximation given by equation (58). The
orange dot-dashed line corresponds to the mt 1approximation given
by equation (60). The red solid line denotes the total energy density of the
universe.
It turns out that also equation (52) possesses an analytical
solution for a(t) = a0t, given by,
u0(t) = π
4(ma0t)1/2(Y1/4(mt)2mt0J3/4(mt0)
+ (1 + 2imt0)J1/4(mt0)+J1/4(mt)
×2mt0Y3/4(mt0)(1 + 2imt0)Y1/4(mt0)),
(59)
where Jα(x)is the Bessel function of the first kind and Yα(x)
is the Neumann function.
Figure 9 shows the number density and figure 10 presents
the energy density for particles of mass m= 103GeV cre-
-15
-10
-5
0
-0.4
-0.2
0.0
0.2
0.4
Log10Ht@GeV-1DL
pii HtLΡHtL
pii Ρ=-2133
pii Ρ = 2133
FIG. 11: Ratio of the pressure pii (t)and the energy density ρ(t)of the
particles created through the zero mode for m= 103GeV assuming the
initial condition at t0=tp.
ated through the zero mode, assuming a full evolution of a(t)
given in table I, with the initial conditions set at t0=tp. In
the region mt 1both the number density and energy den-
sity decrease according to (58), which is confirmed by fitting
the line (green dashed) going like 1/t. Because of choosing
a small mass of the created particles, the fit matches the plot
until the beginning of inflation, during which both the num-
ber density and energy density fall rapidly by many orders of
magnitude. The orange dot-dashed line corresponds to,
n(t)ρ(t)1
t3
/21
a3,(60)
i.e., no particle creation. It fits well the plot for t&1/m,
which implies that for the zero mode particle creation also
stops at t1/m. The energy density of the created parti-
cles is again small compared to the total energy density in the
universe.1
It is interesting to analyze the equation of state for the mat-
ter created through the zero mode, as it has significantly dif-
ferent properties than in the nonzero mode case. Let us for a
moment assume that the entire evolution proceeds according
to the pre-inflationary expansion a(t) = ap0t. An analysis
of formulae (57) and (59) shows that for such a scenario,
pij (t) = (˜
tij 1
3ρ(t) for mt 1,
˜
tij 1
3ρ(t) for mt 1.(61)
Since ˜
tii =3
2and ˜
tij = 1/3
4for i6=j, the created matter
would have a “quasi-accelerating” equation of state for times
t.1/m. This picture, however, is significantly modified
when we take into account inflation.
1We note, however, that if ρ(t)M4
pfor a range of t < tpand the equa-
tions in this paper are still valid in that region, by assuming a small enough
t0the energy density of the particles created through the zero mode might
in general be large enough to compete with ordinary matter and radiation
at some instance of time in the evolution of the universe.
9
Figure 11 shows the the ratio pii(t)(t)for the zero mode
production of particles with mass m= 103GeV including
the full evolution of the three-torus universe from table I. As
expected from equation (61), for the pre-inflationary epoch it
has the form pii(t)(t) = 3
2/3. However, it doesn’t stay
like this until t1/m – inflation itself changes the equation
of state to pii(t) = ρ(t)3
2/3, and it remains this way for
the rest of the evolution. This behavior seems generic for all
masses of the created particles. An interesting fact is that for
much shorter inflation times one could end up with an equa-
tion of state with pii(t)(t)anywhere between 3
2/3and
3
2/3at the end of inflation. However, such short inflation
periods are not realistic.
VIII. CONCLUSIONS
In the present work we calculated the rate for gravitational
particle creation in a three-torus universe. We performed the
calculation assuming the metric on the three-torus for which
the shape moduli are stable. We adopted the current size of the
universe of ten Hubble radii and estimated its full evolution in
time. We argued that the Casimir energies might dominate the
total energy density before inflation, resulting in the expansion
of the universe like in the radiation era. We then showed that
there are two types of contributions to the particle production
rate – one from the nonzero modes on a torus, and the other
coming from the zero mode.
The nonzero mode contribution is interesting because at
early times it is characterized by a “quasi-vacuum-like” equa-
tion of state for the produced matter. The particle production
itself continues until t1/m, after which it essentially stops.
It turns out that the energy density of particles created through
the nonzero modes is tiny compared to the total energy density
in the universe at all stages of its evolution.
The production through the zero mode is more unique. The
corresponding energy density of the created particles is very
sensitive to initial conditions. If we set those at the Planck
time, the energy density of the particles is again small with
respect to the total energy density of the universe. We showed
that although the particles produced through the zero mode
have a “quasi-accelerating” equation of state before inflation,
they are described by a “quasi-radiative” equation afterwards.
Acknowledgment
The author is extremely grateful to Mark Wise for stimu-
lating discussions and helpful comments. The work was sup-
ported in part by the U.S. Department of Energy under con-
tract No. DE-FG02-92ER40701.
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... Rather, our result suggests that there must be non-inflationary phase before the inflationary one, if one believes that no singularity exists in the real Universe. For example, it is pointed out that the Casimir energy behaves like radiation in the torus universe model and hence there would be the radiation dominant era before inflation [17,18], though in this case we will hit the big bang singularity. Quantum creation of compact universe [1][2][3] is also a possible scenario to avoid our mild singularity because semi-classical description is no longer valid near a = 0. Incompleteness of our torus universe is also interesting in the perspective of quantum gravity in de Sitter, especially in the context of holography [19,20]. ...
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