Adiabatic And Entropy Perturations In Inflationary Models Based On Non-linear Sigma Model

0.2 MB·English

by  N. A. Koshelev

ADIABATIC AND ENTROPY PERTURBATIONS IN INFLATIONARY
MODELS BASED ON NON-LINEAR SIGMA MODEL
N.A. Koshelev1
Ulyanovsk State University, 42 Leo Tolstoy St., Ulyanovsk 432700, Russia
Thescalarperturbationsininflationarymodels,basedonatwo-componentdiagonalnon-linearsigmamodel,
are considered. For inhomogeneities generated at an inflationary stage, the law of motion of the comoving
5
0 curvature is obtained (without using the slow roll approximations). A formal expressions for the power
R
0 spectrum and its spectral index are obtained, which are valid and after the slow-roll stage. As an example,
2 an inflationary model with a massive scalar field and quadratic curvature corrections is studied numerically.

n
a
J

7 1 Introduction.
general two-component diagonal sigma-model action has
2
the form:

1 A standard hypothesis of modern cosmology is the exis-
R 1
v tence of an accelerated expansion stage in the history of S = d4x g + T2(ϕ,χ)ϕ ϕ;µ
0 | | −2κ2 2 ;µ
the very early Universe, when the second time derivative Z (cid:26)
60 of the scale factor is positive. This expansion was termed p+ 1P2(ϕ,χ)χ;µχ;µ U(ϕ,χ) , (1)
1 inflationary. Presence of a long enough inflationary stage 2 − (cid:27)
0 can resolve such well-known difficulties of the Big Bang
wherethechiralmetriccomponentsT2(ϕ,χ),P2(ϕ,χ)are
5 theory,as the homogeneity,horizonandflatnessproblems
0 some functions of chiral fields (sigma model components)
[1]. Besides, the inflationary theory gives a root source of
/ ϕandχ. Thenon-linearsigmamodelsusuallyariseaftera
h primordial inhomogeneities, giving rise to the large-scale
conformal transformation in generalized Einstein theories
p structure ofthe Universe. Inthe simple models ofchaotic
- [6]. In Refs. [7], [8], inflationary models with an action of
inflationtheexistenceonlyonescalarfieldtermedinflaton
o
such view have received the name of chiral models. They
r is supposed. In a wide range of initial values of this field,
t are often call simply multi-field models [9, 10], however,
s the slow roll conditions is realized, and the scale factor
a grows quasi exponential. The primordial density inhomo- we do not use this term to stress the non-triviality of the
: scalar fields space.
v geneities arise from quantum vacuum fluctuations of the
i inflaton field. These quantum fluctuations, after horizon In two-fieldinflationarymodels, to study adiabatic and
X
entropy perturbations, Gordon at al. [11] constructedthe
crossing, can be considered as classical stochastic pertur-
r quantitiesδσ andδs. Thesevariablesareveryconvenient
a bations[2]. Atlargescales,theconservationlawoftheco-
for describing the time evolution of the comoving curva-
moving curvature perturbation allows to connect inho-
R ture , and for solving the perturbed field and Einstein
mogeneities generated on inflationary stage with primor- R
equations in slow roll. The perturbation equations in the
dial inhomogeneities in a radiation-dominated Universe.
new variables are also suitable for numerical simulations
The slow-roll models predict initial inhomogeneities with
[11, 12]. This approach was generalized to the case of
an approximately scale-invariant (Harrison - Zel’dovich)
a non-linear sigma model with chiral metric components
spectrum in good agreement with the astrophysical data
h =1, h =0,h =e2b(ϕ1) [13], and to the case ofthe
[3]. However, more realistic multi-field inflationary mod- 11 12 22
generalnon-linearsigma model under slow rollconditions
els also call the attention. In such models, even for long-
[10]. Thegoalofthepresentedworkistostudy theevolu-
wavelength inhomogeneities, the conservation law of the
tion of comoving curvature perturbations and to obtain a
quantity may not to hold (for example, strong ampli-
R power spectrum of initial inhomogeneities in inflationary
fication of long-wavelength inhomogeneities is possible at
models with the action (1).
thepreheatingstage). Manyauthorsconsideredtheevolu-
tion foranumberofmulti-fieldmodelsbothanalytically
R
andnumerically. For double-fieldinflationary models, the 2 Background.
equation of motion of the comoving curvature perturba-
tion was written out in the pioneering papers [4], [5]. We consider the spatially flat Friedmann - Robertson -
Inthiswork,weconsiderinflationarymodelsbasedona Walker (FRW) universe, described by the action (1) with
non-linear sigma model with self-coupling potential. The the line element
1e-mail: [email protected] ds2 =dt2 a2(t)δ dxidxj. (2)
ij
−
1
The equations of motion for the homogeneous back- ofconstant-timehypersurfacesandspatialcoordinatesin-
ground chiral fields ϕ, χ and scale factor a(t) are side them. Here we operate without specified any partic-
ular gauge conditions.
H2 = κ2 T2ϕ˙2+ P2χ˙2+U , (3) Using perturbations of the energy-momentum tensor
3 2 2 δTµ andthose ofthe EinsteintensorδGµ[14], onecanob-
(cid:18) (cid:19) ν ν
κ2 tainthe perturbedEinsteinequationsforinhomogeneities
H˙ = T2ϕ˙2+P2χ˙2 , (4) with the comoving wave number k:
− 2
(cid:0) (cid:1)
ϕ¨+3Hϕ˙ +2TT˙ϕ˙ − TT,ϕϕ˙2− PTP2,ϕχ˙2+ UT,2ϕ =0, (5) 3H(Hφk+ψ˙k)+ ka22 ψk+H(a2E˙k−aBk)
h κ2 i
P˙ P TT U = δρk, (14)
χ¨+3Hχ˙ +2 χ˙ ,χχ˙2 ,χϕ˙2+ ,χ =0, (6) − 2
P − P − P2 P2
κ2
where H ≡ aa˙ is the Hubble rate. ψ˙k+Hφk =− 2 δqk, (15)
Following the papers [10, 11], we define the new ”adia-
batic” field σ such that
ψ¨k+3Hψ˙k+Hφ˙k+(2H˙ +3H2)φk
σ˙ =Tϕ˙cosΘ+Pχ˙sinΘ, (7) κ2
= (δρk 2U,ϕδϕk 2U,χδχk), (16)
2 − −
where
Tϕ˙ Pχ˙ φk ψk+H(aBk a2E˙k)+(aBk a2E˙k)·=0, (17)
=cosΘ, =sinΘ. (8) − − −
T2ϕ˙2+P2χ˙2 T2ϕ˙2+P2χ˙2 where the quantities δρk, δqk are
p p
Using definitions δρk =T2ϕ˙δϕ˙k+(U,ϕ+PP,ϕχ˙2+TT,ϕϕ˙2)δϕk
U U +P2χ˙δχ˙k+(U,χ+PP,χχ˙2+TT,χϕ˙2)δχk σ˙2φk, (18)
,ϕ ,χ −
U = cosΘ+ sinΘ,
σ
T P
U,χ U,ϕ δqk =−T2ϕ˙δϕk−P2χ˙δχk. (19)
U = cosΘ sinΘ, (9)
s
P − T In studying the evolution of perturbations, gauge-
invariant quantities are very convenient. The following
from the chiral field equations (5) and (6) we can write
gauge-invariantquantities are often used:
very useful relations:
σ¨+3Hσ˙ +U =0, (10) Φ=φ+ aB a2E˙ ·,
σ −
(cid:16) (cid:17)
U P T Ψ=ψ H aB a2E˙ . (20)
Θ˙ = s ,ϕχ˙ + ,χϕ˙. (11) − −
− σ˙ − T P (cid:16) (cid:17)
They were constructed by Bardeen [16] and are widely
3 Perturbations. usedintheinfluentialreport[14]. Thecomovingcurvature
perturbation k canbe expressedusingthequantitiesΦk
R
We investigate scalar linear perturbations about the spa- and Ψk as follows [17]:
tiallyflatFRWbackground. Thecorrespondingperturbed
line element can in general be written as [14, 15] k =Ψk H(HΦk+Ψ˙k). (21)
R − H˙
ds2 =(1+2φ)dt2 2aB dxidt
,i
− Eq. (17) can be rewritten in the simple form
a2[(1 2ψ)δ +2E ]dxidxj. (12)
ij ,ij
− −
Φk =Ψk. (22)
The non-linear sigma model components can also be
decomposed into a homogeneous background fields ϕ(t), Taking a time derivative of Eq. (21), using the back-
χ(t) and small inhomogeneities δϕ, δχ: ground field equations and Eqs. (14) - (17), we find the
equation of motion of the comoving curvature perturba-
ϕ(x,t)=ϕ(t)+δϕ(x,t),
tion
χ(x,t)=χ(t)+δχ(x,t). (13)
˙k =2Hϕ˙χ˙T2ϕ˙U,χ−P2χ˙U,ϕ δϕk δχk
Gaugetransformationst˜=t+ξ0(t,x),x˜ =x+ξ,i(t,x), R (T2ϕ˙2+P2χ˙2)2 ϕ˙ − χ˙
(cid:18) (cid:19)
whereξ0(t,x),ξ(t,x)arethearbitraryscalarfunctions,al- H k2
low one to simplify the metric tensor by a suitable choice +H˙ a2Ψk. (23)
2
In the description of the evolution of the inhomo- k2 U,χχ U,ϕχ U,χ
+ + δχk+ δϕk+2 φk
geneities, we shall also use the new variables δs and δσ a2 P2 P2 P2
(cid:18) (cid:19)
[10]: =χ˙ φ˙k+3ψ˙k k2Bk +k2E˙k . (28)
− a
δs=P cosΘδχ T sinΘδϕ, (cid:18) (cid:19)
−
δσ =P sinΘδχ+TcosΘδϕ. (24) After some calculations, the following equations for the
inhomogeneities δσk and δsk can be derived:
Thequantities δσ andδs weretermedinRef. [11]”adi-
abatic”and”entropy”perturbations. Asindicated,bythe k2 U U P˙
name, perturbations with δs = 0 are only adiabatic. By δ¨σk+3Hδ˙σk+ a2+ σ˙sΘ˙+Uσσ− σ˙,χP2 sinΘ
definition, it is clear that δσ is associated with inhomo-
geneities of the field σ while δs corresponds to inhomo- U,ϕ T˙ Us · Uσ Us
cosΘ δσk+2 δsk 2 δsk
geneities of the entropy field s defined by − σ˙ T2 ! (cid:18)σ˙ (cid:19) − σ˙ σ˙
ds=PcosΘdχ−T sinΘdϕ. (25) = 2Uσφk+σ˙ φ˙k+3ψ˙k+k2E˙k k2Bk , (29)
− − a
(cid:18) (cid:19)
In some two-field inflationary models [18], there is a very
simple relationship between the generalized entropy per-
U
turbationsδsafterinflationandthespecificentropyatthe δ¨sk+3Hδ˙sk+ σ (P,ϕcosΘ+T,χsinΘ)
start of radiation dominated stage [11, 19]. PT
(cid:18)
Using variables δσ and δs, Eq. (23) can be reduce to P T P T P T σ˙2
,ϕ ,ϕ ,χ ,χ ,ϕϕ ,χχ
+ +
T2 P2 − T − P PT
H k2 U (cid:18) (cid:19)
R˙k = H˙ a2Ψk−2Hσ˙2sδsk, (26) +U,χ cosΘ P,ϕ sinΘ P,χ cosΘ
P2 T − P
(cid:18) (cid:19)
whichformallycoincideswiththecorrespondingequations U T T
,ϕ ,ϕ ,χ
sinΘ sinΘ cosΘ
of Refs. [11], [13]. For purely adiabatic perturbations −T2 T − P
(cid:18) (cid:19)
(δϕ/ϕ˙ = δχ/χ˙), we have δs = 0, and at large scales
k2 U 2 U
(k/a≪H), a conservation law is valid for comoving cur- +a2 +3 σ˙s +Uss δsk =2σ˙2sǫm, (30)
vature perturbation k. (cid:18) (cid:19) !
R
The perturbed chiral field equations also can be pre-
sented in the terms of the new variables. Directly per- where
turbingthe chiralfieldequations[20]forsigmamodel(1),
U U U
we find: Uss = P,χ2χ cos2Θ− P,ϕTχ sin2Θ+ T,ϕ2ϕ sin2Θ, (31)
T˙ T PP
δ¨ϕk+ 3H+2T!δ˙ϕk+2(cid:18)T,χϕ˙− T2,ϕχ˙(cid:19)δ˙χk Uσσ = UP,χ2χ sin2Θ+ UP,ϕTχ sin2Θ+ UT,ϕ2ϕ cos2Θ. (32)
+ 2(a3aT3TT,2ϕϕ˙)˙−(T22T),2ϕϕϕ˙2−(P22T),2ϕϕχ˙2!δϕk 3HHerδeqkw[e16u]s,ethtehecogmaouvgien-gindveanrisaitnytpqeuratnutribtaytiǫomn: ≡ δρk −
+ 2(a3TT,χϕ˙)˙ (T2),ϕχϕ˙2 (P2),ϕχχ˙2 δχk ǫm =σ˙ δσ˙k+2Usδsk σ˙φk σ¨δσk . (33)
a3T2 − 2T2 − 2T2 ! (cid:20) σ˙ − − σ˙ (cid:21)
k2 U U U
,ϕϕ ,ϕχ ,ϕ Since from the Eqs. (14) and (15) we have
+ + δϕk+ δχk+2 φk
a2 T2 T2 T2
(cid:18) (cid:19)
=ϕ˙ φ˙k+3ψ˙k− k2aBk +k2E˙k , (27) ka22Ψk =−κ22ǫm, (34)
(cid:18) (cid:19)
at large scales (k/a H) the right hand side of Eq. (30)
≪
becomes negligible. As follows from Eq. (30), the large-
P˙ P TT
δ¨χk+ 3H+2 δ˙χk+2 ,ϕχ˙ ,χϕ˙ δ˙ϕk scale entropy perturbations are decoupled from adiabatic
P! (cid:18)P − P2 (cid:19) andmetricperturbations,whichgeneralizestheconclusion
(a3PP χ˙)˙ (P2) (T2) of Ref. [11] on inflationary models with the action (1).
+ 2 ,χ ,χχχ˙2 ,χχϕ˙2 δχk Eq. (29) can be rewritten as an equation for Sasaki-
a3P2 − 2P2 − 2P2 ! Mukhanov gauge-invariantvariable
(a3PP χ˙)˙ (P2) (T2)
+ 2 ,ϕ ,ϕχχ˙2 ,ϕχϕ˙2 δϕk σ˙
a3P2 − 2P2 − 2P2 ! Q=δσ+ Hψ. (35)
3
Using this quantity, one can obtain: This allows us to disregard δ¨σ and δ¨σ in the perturbed
field equations (29), (30). Thus, Eq. (30) may be written
k2 U U P˙ as
Q¨k+3HQ˙k+ a2 + σ˙sΘ˙ − σ˙,χP2 sinΘ δ˙sk =A(t)δsk, (44)
U,ϕ T˙ κ2 a3σ˙2 · where
− σ˙ T2 cosΘ− a3 (cid:18) H (cid:19) +Uσσ!Qk A(t)= 1 Uσ (P cosΘ+T sinΘ)
,ϕ ,χ
−3H PT
Us · Uσ H˙ Us (cid:18)
= 2 δsk +2 + δsk. (36) U 2 U P P
− (cid:18) σ˙ (cid:19) σ˙ H! σ˙ +3 s + ,χ cosΘ ,ϕ sinΘ ,χ cosΘ
σ˙ P2 T − P
(cid:18) (cid:19) (cid:18) (cid:19)
Eqs. (30) and (36) in specific case T2 =1, P2 =P2(ϕ) U,ϕ T,ϕ T,χ
+U sinΘ sinΘ cosΘ
coincide with the corresponding equations of Ref. [13], ss− T2 T − P
(cid:18) (cid:19)
andin the slowrollconditions reduce to equationsof Ref. P T P T P T σ˙2
,ϕ ,ϕ ,χ ,χ ,ϕϕ ,χχ
[10]. + + . (45)
T2 P2 − T − P PT
(cid:18) (cid:19) (cid:19)
Taking into account the inequality (42), equations (15)
4 Application to slow roll infla-
and (21) can be simplified:
tionary models.
κ2
Φk(t)= σ˙δσk, (46)
Let us consider anapplicationofthe above expressionsto 2H
slow roll inflationary models. Quantum vacuum fluctua-
H2
tionsofchiralfieldsbecomeclassicalinhomogeneitiesafter k = Φk. (47)
they leave the horizon. The Fourier component of chiral R − H˙
fields perturbations at horizon crossing are given by [5] Integrating Eq. (26), one can obtain the following ex-
pression for the long-wavelengthcomoving curvature per-
δϕk(tk)= 1 H(tk)eϕ(k), (37) turbation Rk :
T(tk) √2k3
δχk(tk)= 1 H(tk)eχ(k). (38) Rk(t)=Rk(tk)− H√(2tkk3)J(t,tk)es(k), (48)
P(tk) √2k3
where
Heretkistheinstantofhorizoncrossing(a(tk)H(tk)=k),
ande , e are realGaussianstochastic variables with the t2HU δs t2HU
ϕ χ s s
J(t,tk)= dt= αk(t)dt. (49)
following properties: Ztk σ˙2 δs(tk) Ztk σ˙2
eϕ(k)eχ(k′) =δϕχδ(3)(k k′), (39) Here the function αk(t) is a solution of Eq. (44), nor-
h i −
e (k) = e (k) =0. (40) malized by the condition αk(tk) = 1. Using first of re-
φ χ
h i h i lations (41), background equation (4) and also equations
From(24)and(37)onecanderivethe followingexpres- (46), (47) one can therefore write:
sions for δσk and δsk:
δσk(tk)= H(tk)eσ(k), δsk(tk)= H(tk)es(k), (41) Rk(t)= H√(2tkk3)(cid:18)Hσ˙((ttkk))eσ(k)−J(t,tk)es(k)(cid:19). (50)
√2k3 √2k3
The power spectrum [15] cannow be written in the fol-
wheree ,e areindependentGaussianrandomquantities lowing form:
σ s
with zero average and unit dispersion.
Being restricted to the most important non-decreasing R(t)= H2(tk) H2(tk) +J2(t,tk) . (51)
modes of long-wavelength inhomogeneities, as well as in P 4π2 σ˙2(tk)
(cid:18) (cid:19)
case of single-field inflationary models, it is possible to
This formula is valid also after the slow-roll stage if,
write the inequality:
for finding the function αk(t) a long-wavelength limit of
Φ˙k HΦk. (42) Eq. (30)isusedinsteadofEq. (44). If,fromcertaintime,
≪ theentropyperturbationδsissosmall,thatthecomoving
Another standard assumption is that, for non- curvature perturbation remains actually constant, the
R
decreasing modes in the slow-roll regime δ¨ϕ/δ˙ϕ 3H, expression obtained above describe the adiabatic power
δ¨χ/δ˙χ 3H and therefore [11], [10] | | ≪ spectrum.
| |≪
A commonly used characteristic of a power spectrum
δ¨σ 3H δ˙σ , δ¨s 3H δ˙s. (43) is its spectral index n [15]. Using (51) and background
s
| |≪ | | | |≪ | |
4
4
P(t)/P(t)
k
3.5 0
n − 1
s
−0.05
3
−0.1
2.5
−0.15
2 −0.2
0.5
0.4 5
0.3 4
1.5 0.2 3
2
ln a(t)/a(tk) m/M 0.1 0.01 0 1 φk/χk
1
0 10 20 30 40 50 60 70
Figure 2: Numerical simulations of the spectral index af-
Figure 1: The time dependence of (t)/ (tk) with ter inflation. Here it is assumed that the perturbations
P P
M/m=5, ϕ(tk)=14/κ, χ(tk)=4/κ. leavethehorizon60e-foldsbeforetheendofinflation;the
inflation ends when H˙/3H2 =1.
equations one can conclude, that the spectral index for
inflationary models under consideration is given by:
where the new field χ 3/2κ−1ln 1 R˜2/3M2 is
≡ −
∂ln R(t) 1 ∂ln R(t) H˙(tk) introduced. Note that whepn χ κ−1(cid:16), then U(ϕ,χ(cid:17))
ns−1≡ ∂Plnk = H ∂Ptk =2H2(tk) M2χ2/2+m2ϕ2/2. Someinflation≪arymodelsmoregener≈al
than(53)wereconsideredearlierinthepreheatingcontext
κ2+ 2H(tk)σ¨(tk) +2 2Us(tk)J + A(tk)J2
σ˙3(tk) σ˙2(tk) H(tk) in[21,22](Thesepaperstookintoaccountnotonlyhigher
. (52)
− H2(tk(cid:16)) +J2 (cid:17) curvatureterms,butalsoapossiblenon-minimalcoupling
σ˙2(tk) ξϕ2R˜).
The spectral index is time-dependent due the time- Weareinterestinghereinthecasewhenthechiralfields
dependence of the quantity J(t,tk). ϕ and χ are comparable in magnitude and are both in a
Eqs. (51),(26),(30)and(36)arethemainresultsofthis slow-rollregimeatthebeginningoftheinflationarystage.
work. These equations are very well suitable for numer- In the analysis of the background and inhomogeneities
ical calculations of perturbation evolution in inflationary evolution, we shall be restricted to the parameter area
models based on non-linear sigma models. M > m. Since the field χ is heavier, it breaks up faster
than field ϕ, and soon after the inflation influence of the
4.1 Numerical example. field χ on the scale factor dynamic and evolution of inho-
mogeneities is vanishingly small. Besides, after inflation,
As an example, we shall describe an inflationary model Ω2 = 1, therefore actions in the original frame (53) and
∼
with a single massive scalar field and additional higher in the Einstein frame (54) almost do not differ. Accord-
ordercurvature terms due to curvature squared. The cor- ingly, the inhomogeneities calculated in both frames after
responding action looks like: inflationasymptoticallycoincide. Itisclearthattheinho-
mogeneities in this model after the inflationary stage are
S = d4x g˜ R˜ + R˜2 to a great extent adiabatic.
Z | |(−2κ2 2κ2M2 The backgroundandperturbed equationsfor the sigma
p
1 1 model (54) can be solved by numerical simulations. For
+ ϕ ϕ g˜µν m2ϕ2 . (53)
;µ ;ν findingthelong-wavelengthcomovingcurvatureperturba-
2 − 2
(cid:27) tion generated on inflationary stage, it is very conve-
R
Making the conformal transformation g = Ω2g˜ with nient to use Eqs. (26) and (30). Eq. (30) is much easier
ij ij
Ω2 =1 R˜2 , one gets the action in the Einstein frame: tosolvethantheinitialsetofperturbedEinsteinandfield
− 3M2 equations. It can be made by using standard packages of
R 1 e−√23κχ numerical computations. The results of the calculations
S = d4x g − + χ χ;µ+ ϕ ϕ;µ are presented in Figs. 1 and 2. These pictures show the
Z | |(2κ2 2 ;µ 2 ;µ time dependence (t)/ (tk) for a specific choice of pa-
p P P
3M2 1 e−√23κχ 2 1e−2√23κχm2ϕ2 , (54) rameters and the dependence of the spectral index on the
−4 κ2 − −2 model parameters.
h i (cid:27)
5
4.2 Acknowledgement.
I thank S.V. Chervon and V.M. Zhuravlev for attention
to the work and useful discussions.
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