Primordial fluctuations in the warm inflation scenario with a more realistic coarse - grained field

11 Pages·0.16 MB·English

by  Mauricio Bellini

Primordial Fluctuations in the Warm inflation scenario with a more realistic coarse -
grained field
Mauricio Bellini∗
Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales
Universidad Nacional de Mar del Plata,
Funes 3350, (7600) Mar del Plata, Buenos Aires, Argentina.
Istudyasemiclassical approachtowarminflationscenariointroducedinpreviousworks. Inthis
work, I define the fluctuations for the matter field by means of a new coarse - grained field with a
suppression factor G. This field describes the matter fluctuations on the now observable scale of
the universe. The power spectrum for the fluctuations of the matter field is analyzed in both, de
Sitter and power - law expansions for the universe. The constraint for the spectral index gives a
constraintforthemassofthematterfieldinthedeSitterexpansionandaconstraintforthefriction
parameter in thepower - law expansion for theuniverse.
0
0
0
2

n
a PACS number(s): 98.80.Cq, 05.40.+j, 98.70.Vc
J

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3
v
3
I. INTRODUCTION
6
0
8 The inflationary universe scenario asserts that, at some very early time, the universe went through a de Sitter
0 phaseexpansionwithscalefactora(t)growingasa eHo t. Inflationisneededbecauseitsolvesthehorizon,flatness,
9 ∼
and monopole problems of the very early universe and also provides a mechanism for the creation of primordial
9
density fluctuations [1]. Quantum fluctuations [2] and thermal fluctuations [3] of matter fields play a prominent role
/
c ininflationarycosmology. Theyleadtodensity perturbationsthatwouldbe responsibleforthe originofstructuresin
q
theuniverse[4]. Structureformationscenarios,inparticular,canreceiveimportantrestrictionsbasedonthemeasured
-
r δTr/ < Tr >= 1.1 10−5. According to the standard inflation model, the formation of large - scale structure in the
g
universe has its origin in the growth of primordial inhomogeneities in the matter distribution.
:
v The standard slow - roll inflation model separates expansion and reheating into two distinguished time periods. It
i is first assumed that exponential expansion from imflation places the universe in a supercooled phase. Subsequently
X
thereaftertheuniverseisreheated. Twooutcomesarisefromsuchasecenario. First,therequireddensityperturbations
r
in this cold universe are left to be created by the quantum fluctuations of the inflaton. Second, the temperature
a
cliff after expansion requires a temporally localized mechanism that rapidly distributes sufficient vacuum energy for
reheating.
The warm inflation scenario takes into account separately, the matter and radiation energy fluctuations. In this
scenariothefluctuationsofthematterfieldleadtoperturbationsformatterandradiationenergydensities[5,6],which
areresponsibleforthefluctuationsoftemperature. Thefieldϕinteractswiththeparticleswhichareinathermalbath
with a mean temperature smaller than the GUT critical temperature, < T > < T 1015 GeV. This scenario
r GUT
∼
was introduced by A. Berera [7], and developed in other work [8]. In the Berera’s formalism the temperature of the
universe is constant during inflation and the temporal evolution for the amplitude of the fluctuations of temperature
are not considered.
Inthewarminflationera,thekineticcomponentofenergydensityρ mustbesmallwithrespecttothevacuum
kinetic
energy, which is given by the potential V(ϕ)
ρ(ϕ) ρ V(ϕ) ρ .
m kinetic
∼ ∼ ≫
where
∗
E-mail address: [email protected]
1
1
ρ = ρ (ϕ)+ ϕ˙2,
kinetic r
2
and the radiation energy density is
τ(ϕ)
ρ (ϕ) = ϕ˙2.
r
8H(ϕ)
Here, ϕ is a scalar field of matter, τ(ϕ) is a friction parameter due to the interaction of the matter field ϕ with other
fields ofthe thermalbathwithtemperatureT . Furthermore,the dotdenotes the derivativewithrespecttothe time.
r
The conventional treatment of scalar field dynamics assumes that it is pure vacuum energy dominated. The various
kinematic outcomes are a result of specially chosen Lagrangians. In most cases the Lagrangian is unmotivated from
particle phenomenology. Clear exceptions are the Coleman - Weinberg potential with a untuned coupling constant,
which is motivated by grand unified theories and supersymmetric potentials [9]. Making an extension to the warm
inflationpicture,thebehaviorofthescalefactorcanalsobe alteredforanygivenpotentialwhenthe radiationenergy
is present.
In an alternative formalism of warm inflation [10] I demonstrated that, for a power - law inflation model, the
amplitude for both, thermalfluctuationsandmeantemperature,decreasewithtime fora sufficiently rapidexpansion
of the scale factor of the universe. Thus, at the end of inflation the spectrum of the coarse - grained matter field can
be calculated [5,6]. This is the most significant difference with the Berera’s formalism.
In this formalism a semiclassical expansion of the inflaton field was proposed [11]
ϕ(~x,t) = φ (t)+φ(~x,t).
c
Here the expectation value of the field operator ϕ, in some unknown state E >, is given by the classical field φ (t)
c
|
and the expectation value for the quantum fluctuations φ(~x,t) is zero
< E ϕ(~x,t)E >= φ (t); < E φ(~x,t)E >= 0.
c
| | | |
Thefieldφ isresponsiblefortheexpansionofthe universe,whilethe quantumfluctuations φ(~x,t)takesintoaccount
c
the local fluctuations of the matter field with respect to φ .
c
II. FORMALISM
The Lagrangianthat describes the warm inflation scenario is
R 1
(ϕ,ϕ )= √ g + gµνϕ ϕ +V(ϕ) + , (1)
,µ ,µ ,ν int
L − − 16π 2 L
(cid:20) (cid:21)
where R is the scalar curvature, gµν the metric tensor and g is the metric. The Lagrangian takes into account
int
the interaction of the field ϕ with other fields of the thermal bath with mean temperature <LT > < T 1015
r GUT
∼
GeV.Thislowertemperatureconditionimpliesthatmagneticmonopolesuppressionworkseffectively. Aswasrecently
showed, certain string inspired models solve all the cosmological puzzles and have a motivation from string theory
[12,13]. All particlelike matter which existed before inflation would have been dispersed by inflation.
We consider the metric
ds2 = dt2+a2(t)d~x2. (2)
−
The eq. (2) represents a flat Friedmann - Robertson - Walker (FRW) metric for a globally flat, isotropic and
homogeneous universe. The Hubble parameter H[ϕ(~x,t)] can be written as an expansion around H (φ )
c c
∞
1
H[ϕ(~x,t)]=H (φ )+ H(n)(φ ) φn(~x,t), (3)
c c c
n!
n=1
X
where H(n)(φ )= dnH(ϕ) H(n)(φ ). The equation of motion for the operator ϕ is
c dϕn |φc ≡ c
1
ϕ¨ 2ϕ+[3H(ϕ)+τ(ϕ)] ϕ˙ +V′(ϕ)=0. (4)
− a2∇
2
The semiclassical Friedmann equation for a globally flat universe is given by
8π
E H2(ϕ) E = E ρ +ρ E , (5)
3M2 h | m r| i
p
(cid:10) (cid:12) (cid:12) (cid:11)
where M is the Planckian mass. Here ρ(cid:12) and ρ(cid:12) are
p m r
ϕ˙2 1 2
ρ (ϕ)= + ~ϕ +V(ϕ), (6)
m 2 a2 ∇
τ(ϕ) (cid:16) (cid:17)
ρ (ϕ)= ϕ˙2. (7)
r
8H(ϕ)
The scalar potential V(ϕ) also can be written as an expansion around V(φ )
c
∞
1
V(ϕ)=V(φ )+ V(n)(φ ) φn(~x,t), (8)
c c
n!
n=1
X
where V(n)(φ ) dnV(ϕ) . Here the prime denotes the derivative with respect to the field ϕ. Furthermore V′(ϕ)
c ≡ dϕn φc
can be expanded as (cid:12)
(cid:12)
(cid:12)
∞
1
V′(ϕ)=V′(φ )+ V(n)(φ ) φn−1(~x,t). (9)
c c
n!
n=2
X
In the following I will consider H(ϕ) H (φ ) and V(ϕ) and V′(ϕ) as first ordered expansions in φ.
c c
≡
A. Dynamics of the classical field
The classical equation of motion for the field φ is
c
φ¨ +[3H (φ )+τ (φ )]φ˙ +V′(φ )=0, (10)
c c c c c c c
where V′(φ ) dV(ϕ) and τ(ϕ) τ (φ ). Here, the laplacian term does not appears, since φ only depends on
c ≡ dϕ |φc ≡ c c c
time. The evolution of φ and H (φ ) are [10]
c c c
M2 τ −1
φ˙ = pH′ 1+ c , (11)
c − 4π c 3H
(cid:18) c(cid:19)
M2 τ −1
H˙ =H′φ˙ = p(H′)2 1+ c , (12)
c c c − 4π c 3H
(cid:18) c(cid:19)
where H decreases with time due to the fact that H˙ (t) < 0. Furthermore, from eq. (5), the classical Friedmann
c c
equation is
4π τ
H2(φ )= 1+ c φ˙2+2V(φ ) . (13)
c c 3M2 4H c c
p (cid:20)(cid:18) c(cid:19) (cid:21)
Replacing (11) in eq. (13), one obtains the classical scalar potential
3M2 M2 τ τ −2
V(φ )= p H2(φ ) p (H′)2 1+ c 1+ c . (14)
c 8π " c c − 12π c (cid:18) 4Hc(cid:19)(cid:18) 3Hc(cid:19) #
This expression relates the potential with the Hubble and friction parameters.
On the other hand, the radiation energy density is
τ M2 2 τ −2
ρ (φ )= c p (H′)2 1+ c . (15)
r c 8H 4π c 3H
c ! (cid:18) c(cid:19)
From eq. (15) one obtains the mean temperature of the thermal bath, in which there are particles that would be
dispersed during inflation
<T > [ρ (φ )]1/4. (16)
r r c
∝
The eq. (16) gives the temperature of the background,but does not describes the local fluctuations δT /<T >.
r r
3
B. Dynamics of the quantum perturbations
I consider the quantum fluctuations of the matter field for an universe with a FRW metric (2). I will consider the
Hubble and friction parameters as spatially homogeneous [i.e., H(ϕ) H (φ ) and τ(ϕ) τ (φ )]. The equation of
c c c c
≡ ≡
motion for the quantum fluctuations is
1
φ¨ 2φ+[3H +τ ]φ˙+V′′(φ )φ=0, (17)
− a2∇ c c c
which can be simplified with the map χ=e3/2 (Hc+τc/3)dt φ
R
k2
χ¨ a−2 2χ oχ=0. (18)
− ∇ − a2
Here,k (t)isthewavenumberthatseparatestheinfraredsector(withk <k )andtheshort-wavelengthsector(with
o o
k >k )
o
2
9 τ (t) 3 τ˙ (t)
k2(t)=a2(t) H (t)+ c V′′(t)+ H˙ (t)+ c . (19)
o "4(cid:18) c 3 (cid:19) − c 2(cid:18) c 3 (cid:19)#
Here, the parameters H (φ ) and τ (φ ) are evaluated in t, due to the fact that φ φ (t). The infrared sector
c c c c c c
≡
describes the macrophysics of the universe (i.e., describes the universe in a cosmological scale), while the short -
wavelength sector takes into account the microphysics of it. The eq. (17) is a Klein - Gordon equation with a time
dependent parameter of mass µ(t)= ko.
a
We can write the redefined field χ as a Fourier expansion — in terms of the modes ξ ei~k.~x — in the k - space
k
1
χ(~x,t)= d3k a ei~k.~xξ (t)+H.c , (20)
(2π)3/2 k k
Z h i
where ξ (t) are the time dependent modes with wavenumber k. The annihilation and creation operators a and a†,
k k k
satisfy the following commutation relations
ak,a†k′ =δ(3)(k−k′); [ak,ak′]= a†k,a†k′ =0. (21)
h i h i
Furthermore, the operators χ and χ˙ satisfy
[χ(~x,t),χ˙(~x,t)]=iδ(3)(~x x~′). (22)
−
The interpretationfor eq. (22)is just thatχ andχ˙ arecanonicallyconjugate variable if~x and~x′ are within the same
smeared point inside the light cone. Otherwise, χ and χ˙ can be measured independently.
Replacing eq. (20) in eq. (22) one obtains the following condition for the time dependent modes ξ
k
ξ ξ˙∗ ξ˙ ξ∗ =i. (23)
k k − k k
When the time dependent modes are real one obtains
ξ ξ˙∗ ξ˙ ξ∗ =ξ ξ˙ ξ˙ ξ =0, (24)
k k − k k k k− k k
and the operators χ and χ˙ commute [5,6]. Thus, the field χ becomes classical when all the modes ξ (t) in eq. (20)
k
are real. Note that k depends on time and continuosly new time dependent modes ξ enters in the infrared sector
o k
(k k ) from the short- wavelengthsector (k k ). We are interestedin studying the sector of the universe where
o o
≪ ≫
theredefinedquantumfluctuationsχbecomeclassical. Ofcourse,thisconditionimposesrestrictionsoverthevacuum
[5,6] and thus the asymptotic vacuum must be real in the infrared sector. However,the modes ξ — for k >k (t) —
k o
are complex, and during inflation χ and χ˙ does not commute.
4
C. The data COBE coarse - grained field
In this work I define a coarse - grained field which is obtained from the experimental data [14]
1
χ = d3k G(k,t) a ei~k.~xξ (t)+H.c. , (25)
Ccg (2π)3/2 k k
Z h i
where the suppression factor is
1
G(k,t)= , (26)
N
v1+ ko(t)
u k
u
t (cid:16) (cid:17)
with N =m n. Causality places a strict constraint on the suppression index: N 4 n. A suppressionfactor like
− ≥ −
eq. (26) (with n 1) also has been found in a model with cosmic strings plus cold or hot dark matter [15,16].
∼
Furthermore, the squared fluctuations for the COBE coarse - grained field is [17]
∞ dk
E χ2 E = (k) (27)
Ccg k PχCcg
Z0
(cid:10) (cid:12)(cid:12) (cid:12)(cid:12) (cid:11) = 1 kodk k2 ξ (t)2G2(k,t), (28)
2π2 | k |
Z0
where the power spectrum (k) when the horizon exit is assumed as [16]
PχCcg
n
k
(k)=A(t ) f(k). (29)
PχCcg ∗ k (t )
(cid:18) o ∗ (cid:19)
Thepowerspectrum containsfourparameters: theamplitudeA,thespectralindexn,thesuppressionwavenum-
PχCcg
berk ,andthesuppressionindexm. Furthermore,t denotesthetimewhenthehorizonentry,forwhichk (t ) πH
o ∗ o ∗ o
in comoving scale. Replacing eq. (25) in (18), and neglecting the terms with 1 2χ , since they are very≃small
−a2∇ cg
with respect to the another terms, one obtains the quantum stochastic equation for the COBE coarse - grained field
2
k (t) N
o
χ¨ χ = [ξ (~x,t)+ξ (~x,t)]. (30)
Ccg Ccg 1 2
− a(t) k (t)
(cid:18) (cid:19) o
Here the noises ξ and ξ are given by
1 2
ξ (~x,t)= k˙okoN d3k k−N G3(k,t) a ei~k.~xξ˙ (t)+H.c. , (31)
1 −(2π)3/2 k k
Z h i
kN−1 k N
ξ (~x,t)= o d3k k−N G5(k,t) o 3k˙2 2k k¨ +2k˙2(1 N) 2k k¨
2 4(2π)3/2 Z "(cid:18) k (cid:19) (cid:16) o − o o(cid:17) o − − o o#
a ei~k.~xξ (t)+H.c. . (32)
k k
×
h i
The function G(k,t) determines the stochastic properties and spectrum of the noises ξ and ξ .
1 2
Replacing eq. (20) in eq. (18), one obtains the equation of motion for the modes ξ (t)
k
ξ¨ (t)+ω2 ξ (t)=0, (33)
k k k
whereω (t)=a−1[k2 k2(t)]1/2istheoscillationfrequencyofthemodes. InadeSitterexpansion[18]thesefrequencies
are consktant. Most ge−neorally, the frequencies depends on time. Note that for k < k the squared frequencies ω2 are
o k
positives and the solutions of (33) are real, but it does not occur for k > k where the solutions of (33) become
o
complex.
5
D. Classicality conditions for the COBE coarse - grained field
Asinpreviousworks[5,6,10,11,18]weareinterestedinstudyingtheclassicalityconditionsforthequantumstochastic
equation (30). Observe that all the modes ξ (t) on the infrared sector are real. If we write the modes as a complex
k
function with components u (t) and v (t)
k k
ξ (t)=u (t)+i v (t), (34)
k k k
the condition for that the modes to be real is
v (t)
k
α (t)= 1. (35)
k
u (t) ≪
(cid:12) k (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The condition for the COBE coarse - grained field χCc(cid:12)g to b(cid:12)e classical during inflation becomes [5,6]
1 k≃ko(t)
α (t) 1, (36)
k
M(t) ≪
k=0
X
where M(t) is the time dependent number ofdegreesof freedominthe infrared(large- wavelength)sector. Whenall
the modes of the infrared sector are real (i.e., when α 1).
k≃ko ≪
E. Heisenberg representation for the COBE coarse - grained field
During inflation the field χ obeys the quantum stochastic equation
Ccg
2
k (t) N
o
χ¨ χ = [ξ (~x,t)+ξ (~x,t)], (37)
Ccg Ccg c1 c2
− a(t) k (t)
(cid:18) (cid:19) o
which can be written as
2
k (t)
o
χ¨ χ +ξ (~x,t)=0, (38)
Ccg Ccg c
− a(t)
(cid:18) (cid:19)
where ξ (~x,t)= N [ξ (~x,t)+ξ (~x,t)]. The new noises ξ and ξ are
c −ko(t) c1 c2 c1 c2
ξ (~x,t)= k˙okoN d3k k−N G3(k,t) a ei~k.~xξ˙ (t)+a†e−i~k.~xξ˙ (t) , (39)
c1 −(2π)3/2 k k k k
Z h i
kN−1 k N
ξ (~x,t)= o d3k k−N G5(k,t) o 3k˙2 2k k¨ +2k˙2(1 N) 2k k¨
c2 4(2π)3/2 Z "(cid:18) k (cid:19) (cid:16) o − o o(cid:17) o − − o o#
a ei~k.~xξ (t)+a†e−i~k.~xξ (t) . (40)
× k k k k
h i
These noises become from the increasing number of degrees of freedom of the infrared sector, as a consequence of
the modes which enters in the infrared sector from the short - wavelength sector. In general, ξ is a colored noise,
c1
while ξ gives non - local dissipation. Under special circumstances (i.e., for N ), the noise ξ is nearly delta
c2 c1
→ ∞
correlated and describes a nearly white and gaussian noise. Furthermore, in this case ξ generates local dissipation.
c2
The effective Hamiltonian associated with eq. (38) is
1 1
H (χ ,t)= P2 + µ2(t) χ2 +ξ χ , (41)
eff Ccg 2 Ccg 2 Ccg c Ccg
where P χ˙ and µ2(t) = ko2. Observe that ξ plays the role of an external classical stochastic force in the
Ccg ≡ Ccg a2 c
effective Hamiltonian (41). Thus, one can write the Scho¨dinger equation
∂ 1 ∂2 1
i ψ(χ ,t)= ψ(χ ,t)+ µ2(t)χ2 +ξ χ ψ(χ ,t). (42)
∂t Ccg −2∂χ2 Ccg 2 Ccg c Ccg Ccg
Ccg (cid:20) (cid:21)
6
Here, ψ(χ ,t) is the wave function that characterizethe fluctuations of the matter field in the observable universe.
Ccg
Since the time dependence ofµ(t), the Hamiltonianisnon- conservative,alsointhe caseinwhichone wouldneglects
the stochastic force. The only case where µ does not present a time dependence is a de Sitter expansion of the
universe. In this case µ is constant [18] and eq. (41) represents a harmonic oscillator with a stochastic external force
ξ . In this case we have a forced linear harmonic oscillatorand the solution is a coherentstate with the displacement
c
due to the action of the external force [19].
The effective Hamiltonian (41) represents an oscillator that experiences both, the squeezing, due to the time
dependent frequency, and the external force, due to the inflow of the short modes in the infrared sector.
The probability to find the universe with a given χ in a given time t is
Ccg
P(χ ,t)=ψ(χ ,t)ψ∗(χ ,t), (43)
Ccg Ccg Ccg
where the asterisk denotes the complex conjugate. The problem with the Heisenberg’s representation is that is very
complicatedtofindsolutionsforψ(χ ,t). However,itgivesaclearconceptualnotionofthephysicalproblemunder
Ccg
consideration.
III. EXAMPLES
A. de Sitter expansion: supercooled scenario
I consider a de Sitter expansion for which H (φ ) H = constant. In this case H′ = 0 and thus, from eq. (15)
c c o
≡
one obtains a supercooledexpansionfor the universe[18](ρ =0). This means that the dissipationparameteris zero
r
(τ =0) during the expansion. From eq. (14), one obtains a constant potential
c
3M2
V = p H2. (44)
o 8π o
For a massive inflaton with mass m the square parameter µ2(t) = ko2(t) (where a eHot ) also becomes time
a2 ∝
independent
9
µ2 = H2 m2. (45)
4 o −
The equation for the time dependent modes ξ (t) is
k
9
ξ¨ (t)+ k2e−2Hot H2+m2 ξ (t)=0. (46)
k − 4 o k
(cid:20) (cid:21)
The general solution for ξ is
k
k k
ξ (t)=A (1) e−Hot +A (2) e−Hot , (47)
k 1Hν H 2Hν H
(cid:20) o (cid:21) (cid:20) o (cid:21)
where ν = 9 m2 < 3/2. Here (1) and (2) are the first and second species Hankel functions. For sufficiently
4 − Ho2 Hν Hν
large t, i.e.,qfor t H−1 one obtains the expression for (1) and (2)
≫ o Hν Hν
ν −ν
k 1 k i k
(1,2) e−Hot e−Hot Γ(ν) e−Hot . (48)
Hν H ≃ Γ(1+ν) 2H ± π 2H
(cid:20) o (cid:21) (cid:18) o (cid:19) (cid:18) o (cid:19)
Chosing A =0, and from the relation ξ ξ˙∗ ξ˙ ξ∗ =i, one obtains A = i πΓ(1+ν), and thus
1 k k− k k 2 2 νΓ(ν)Ho
q
ν −ν
i π k 1 Γ(ν+1)Γ(ν) k
ξ (t) e−Hot + e−Hot . (49)
k
≃ 2rνHoΓ(ν)Γ(ν+1)(cid:20)2Ho (cid:21) 2s νHoπ (cid:20)2Ho (cid:21)
For very large t, one obtains the asymptotic modes
7
−ν
1 Γ(ν)Γ(ν+1) k
ξ (t) e−Hot . (50)
k |t≫1 ≃ 2s νHoπ (cid:20)2Ho (cid:21)
Thus, the squared fluctuations on the observable scale, for very large t, are
22ν−3 Γ(ν+1)Γ(ν) kp
E φ2 E H2ν−1 e2(ν−3/2)Hot dk k2(1−ν) G2(k), (51)
Ccg ≃ π3 ν o
(cid:20) (cid:21) Z0
(cid:10) (cid:12) (cid:12) (cid:11)
where φCcg = e−3/2(cid:12)HotχC(cid:12)cg and kp is de wavenumber for the Planckian wavelength. Comparing (51) with (27) and
(29), one obtains the following relation for the spectral index
n 1=2(1 ν). (52)
− −
The standard choice n = 1 was first advocated by Harrison [20] and Zel’dovich [21] on the ground that it is scale
invariant at the epoch of horizon entry. The constraint
n 1 <0.3, (53)
| − |
gives the following values for the mass of the inflaton field
1.36 H <m<1.46 H . (54)
o o
Thus, the parameter ν only can take the values
0.351 < ν < 0.65. (55)
The power spectrum for the COBE coarse - grained field φ , is
Ccg
(k) k2(1−ν) G2(k). (56)
PφCcg ∝
Furthermore, the amplitude A(t) decreases with time during the inflation era
22ν−3 Γ(ν+1)Γ(ν)
A(t)= H2ν−1 e2(ν−3/2)Hot, (57)
π3 ν o
(cid:20) (cid:21)
for m=0. After the horizon entry (i.e., for t>t ) the amplitude A(t ) becomes
∗ ∗
6
22ν−3 Γ(ν+1)Γ(ν)
A(t )= H2ν−1 e2(ν−3/2)Hot∗. (58)
∗ π3 ν o
(cid:20) (cid:21)
Due to δ 2 = (k) [17],the spectral density is δ =k1−νG(k). From the condition(55), one obtains a positive
| k| PφCcg | k|
exponent for k in the spectral density. To obtain a negative exponent in the spectral density δ , the mass of the
k
| |
inflaton field must be very small
√5
m< H . (59)
o
2
B. Power - law inflation
In this example I consider a power - law expansion for the universe. In this case the scale factor and the Hubble
parameterarerespectivelya(t) (t/to)p,andHc(t)=p/t. Thetemporalevolutionoftheclassicalfieldise−φc(t)/m =
∝
(H /p)t. Furthermore, the temporal evolution for the radiation energy density is [see eq. (15)]
o
M2 2 p2 τ t −2
ρ (t)= p 1+ c t−2. (60)
r 4π m2 3p
! (cid:18) (cid:19)
The potential is given by
8
3M2 M2 p2 τ t −1 τ t
V [φ (t)]= pt−2 p2 p 1+ c 1+ c . (61)
c c 8π " − 12πm2 (cid:18) 3p(cid:19) (cid:18) 4p(cid:19)#
The equation of motion for the modes is
t2pk2
ξ¨ (t)+ o µ2(t) ξ (t)=0, (62)
k t2p − k
(cid:20) (cid:21)
where
9 τ 2 3 τ˙
µ2(t)= pt−1+ c V′′(t) pt−2 c , (63)
4 3 − c − 2 − 3
(cid:16) (cid:17) (cid:18) (cid:19)
and
3M2p5 1
V′′(t)= p t−2
c 32H2 [3H p3+τ2m2H3t3]
o o c o
4H3p2τ2t3(4τ H )+H3t2p−2 m2τ τ′′ 2m2(τ′)2+144pτ2+6mτ τ′ 45τ2
× o c c− o o c c − c c c c− c
+(cid:2)Ho3p−1t 3m2τc′′+432pτc+6mτc′−(cid:2) 141τc +2Ho3(216p−77) . (cid:3) (64)
Here the prime denotes the d(cid:2)erivative with respect to φ (i.(cid:3)e., τ′ dτc(t), etc(cid:3).). For simplicity I will consider
c c ≡ dφc
τ =pγt−1, where γ is a dimensionless constant. In this case the general solution for the eq. (62) is
c
k(t/t )1−p k(t/t )1−p
ξ (t)=A t/t (1) o +A t/t (2) o , (65)
k 1 oHν H (p 1) 2 oHν H (p 1)
(cid:20) o − (cid:21) (cid:20) o − (cid:21)
p p
where ν = 1 √1+4K2 and
2(p−1)
9 γ 2 3 γ 3p2M2 M2 γ γ −2
K2 = p2 1+ p 1+ + p 1 p 1+ 1+ . (66)
"4 3 − 2 3 2πm2 − 48πm2 4 3 !#
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
For A =0, the modes (65) for k(t/to)1−p 1 are
1 Ho(p−1) ≪
t/t k(t/t )1−p −ν
o o
ξ (t) Γ(ν) , (67)
k
≃ 2√π 2(p 1)H
p (cid:20) − o(cid:21)
where Γ(ν) is the gamma function with argument ν.
The square fluctuations on the observable scale are
Γ2(ν)22ν(p 1)2ν ko
E φ2 E − (t/t )2ν(p−1)+1H2ν dk k2(1−ν)G2(k,t), (68)
| Ccg| ≃ 8π3(p 1) o o
− Z0
(cid:10) (cid:11)
where φ =t−3/2p(1+γ/3)χ , and
Ccg Ccg
1
G2(k,t)= . (69)
N
1+ Ktp−1
ktpo
h i
From the condition n 1=2(1 ν) one obtains
− −
1
n 1=2 1 1+4K2 , (70)
− − 2(p 1)
(cid:20) − (cid:21)
p
where K2 is given by eq. (66). The constraint n 1 <0.3 in eq. (70) gives the following conditions
| − |
[1.3(p 1)]2 1
K2 < − − , (71)
4
[0.702(p 1)]2 1
K2 > − − . (72)
4
9
For example, taking p=4, with the scale m=1, one obtains the condition
7.1065 < γ < 7.1286, (73)
which implies that
7.1065 H (t)< τ < 7.1286 H (t). (74)
c c c
When the horizon entry, the condition (74) becomes
7.1065 H (t )< τ (t ) < 7.1286 H (t ). (75)
c ∗ c ∗ c ∗
IV. CONCLUSIONS
In this thermal scenario the rapid cooling followed by rapid heating of the standard inflation is replaced by a
smootheneddissipativemechanism. Theclassicalfieldφ (t)generatestheexpansionofthescalefactoroftheuniverse.
c
Furthermore the fluctuations of the matter field, with respect to the homogeneous field φ (t), are described by the
c
field φ(~x,t). In the framework of a more realistic treatment, the fluctuations on the sub - Hubble scale are described
bytheCOBEcoarse-grainedfieldχ ,whichisdefinedbyasuppressionfactorG(k),whichtendstozerofork 0.
Ccg
→
This field describes the observableuniverseafter the horizonentry. The quantumto classicaltransitionofthe COBE
coarse - grained field χ , is due to the complex to real transition of the modes ξ , during inflation. The infrared
Ccg k
sector takes into account only the modes with wavelength much bigger than the size of the horizon k−1 >k−1(t). In
o
particular, during inflation χ (~x,t) does not commutes with χ˙ (~x′,t) for ~x ~x′ <k−1. This is due to χ and χ˙
Ccg Ccg | − | o
arecanonicallyconjugatevariablesifare“measured”insideacausallyconectedregionofthespacetime. Otherwise,χ
and χ˙ can be “measured” independently. The now observable universe is composed by causally desconeted domains
during inflation.
During inflation, the number of degrees of freedom in the infrared sector M(t) is constantly increasing,since short
- wavelengths modes cross the horizon from the short - wavelength sector. This effect is seen in the second order
stochastic equation as a noise ξ and produces quantum decoherence [22] and non - local dissipative effects in the
c
infraredsector. Thestochasticpropertiesofξ aregivenbythesuppressionfactorG. Furthermore,thenoiseξ canbe
c k
seen as a stochastic force in the framework of the Heisenberg’s representationfor the redefined quantum fluctuations
χ . TheeffectiveHamiltonianintheHeisenberg’srepresentationdescribesadampedoscillatorwithavariableµ(t).
Ccg
This oscillator experiences both, squeezing, due to the time dependent µ(t), and a stochastic external force, due to
the inflow of the short - wavelength modes in the infrared sector.
Finally, two examples were considered. In the example for a de Sitter expansion of the universe, the radiation
component for the energy density becomes zero. In this case the parameter of mass µ = ko does not depends on
a
time. Furthermore, constraint n 1 <0.3 for the spectrum index gives restrictions for the mass of the inflaton field
| − |
(1.35H <m<1.46H )andtheparameterν (0.351 < ν < 0.65). Here,thespectraldensitybeing givenby δ =
o o k
k1−νG(k) andthe amplitude A(t) decreaseswith time as A(t) e2(ν−3/2)Hot for m=0. When the horizonentr|y (|for
∼ 6
t=t ),thisamplitudebecomesfreezenwithvalueA(t ). Intheexampleforapower-lawexpansionfortheuniverse,
∗ ∗
the constraint n 1 < 0.3 for the spectrum index gives restrictions for the friction parameter, which is now time
dependent. For|th−e p|articular case τ =pγt−1 and p=4, one obtains that 7.1065 H (t ) <τ (t )<7.1286 H (t ).
c c ∗ c ∗ c ∗
Acknowledgements
I thank Adnan Bashir for your careful reading of the manuscript and CIC of Universidad Michoacana for financial
support in the form of a research grant. Also, I thank the referee for comments and suggestions.
[1] T. Padmanabhan. Structure Formation in the Universe. Cambridge University Press (1993).
A.D. linde, Particles Physics and Inflationary Cosmology, (Princeton University Press, Princeton, NJ, 1993).
E. W. Kolb and M. S. Turner,The Early Universe, (Addison - Wesley, Redwood City,CA, 1990).
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