The Tachyon Inflationary Models with Exact Mode Functions

5 Pages·0.13 MB·English

by  Xin-zhou Li

The Tachyon Inflationary Models with Exact Mode Functions
Xin-zhou Li∗ and Xiang-hua Zhai
Shanghai United Center for Astrophysics,Shanghai Normal University, Shanghai 200234 , China
(Dated: February 1, 2008)
We show two analytical solutions of the tachyon inflation for which the spectrum of curvature
(density) perturbations can be calculated exactly to linear order, ignoring both gravity and the
self-interactionsofthetachyonfield. Themainfeatureofthesesolutionsisthatthespectralindices
are independentwith scale.
PACSnumbers: 04.20.Jb,98.80.Cq
I. INTRODUCTION ever, the accuracy of any approximation scheme should
3 be controlled which means that the errors should quan-
0 tified. This can only be done by comparing the approx-
0 The inflationary scenario[1]provides a mechanism to
imate power spectrum to an exact one[13]. Therefore,
2 produce the primordial fluctuations of spacetime and
it is very interesting to find the exact solutions of mode

n matter, which lead to the CMBR(cosmic microwave equation for the tachyon inflation.
a backgroundradiation)anisotropiesand to the large scale
J structure[2]. In standard inflationary models[3], the

0 physics lies in the inflation potential. The underlying II. THE EQUATIONS OF MOTION
1 dynamics is simply that of a single scalar field rolling

in the potential. The scenario is generically referred to
1 We consider spatially flat FRW line element given by:
as chaotic inflation in reference to its choice of initial
v
3 conditions. This picture is widely favored because of its
6 simplicity and has received by far the most attention to ds2 = dt2 a2(t) dx2+dy2+dz2
0 date. Some potentials that give the correct inflationary −
1 propertieshavebeenproposed[4]inthepasttwodecades. = a2(τ) dτ2 (cid:0) dx2+dy2+dz(cid:1)2 (1)
−
0
Recently, pioneered by Sen[5], the study of non-BPS ob-
3 where τ is the(cid:2)conform(cid:0) altime, with dt(cid:1)=(cid:3) adτ. As shown
jects such as non-BPS branes, brane-antibrane configu-
0 by Sen[7], a rolling tachyoncondensate in either bosonic
ration or space-like branes[6] has been attracting phys-
h/ ical interests in string theory. Sen showed that classi- or supersymmetric string theory can be described by a
p fluid which in the homogeneous limit has energy density
caldecayofunstableD-braneinstringtheoriesproduces
- and pressure as follows
p pressureless gas with non-zero energy density[7]. Gib-
e bons took into account coupling to gravitational field
h by adding an Einstein-Hilbert term to the effective ac- V(T)
v: tion of the tachyon on a brane, and initiated a study ρ =
1 T˙2
Xi of ”tachyon cosmology”[8]. The basic idea is that the −
usual open string vacuum is unstable but there exists a p = pV(T) 1 T˙2 (2)
r stable vacuum with zero energy density. There is evi- − −
a p
dence that this state is associatedwith the condensation where T and V(T) are the tachyon field and potential,
ofelectricflux tubes ofclosedstring[7]. These flux tubes and an overdot denote a derivative with respect to the
described successfully using an effective Born-Infeld ac- coordinatetimet. Totakethegravitationalfieldintoac-
tion[9]. Thestringtheorymotivatedtachyoninflation[10] counttheeffectivelagrangiandensityintheBorn-Infeld-
and quintessence[11]have been discussed. These investi- type form is described by:
gation are based on the slow-roll approximation so that
they are rather incomplete. It is obvious that tachyon
R
inflation might occur in a much broader context and we L=√ g V(T) 1+gµν∂ T∂ T
µ ν
should discriminate between generic predictions of infla- − 2κ −
(cid:18) (cid:19)
tion and predictions of a specific scenario. p (3)
where κ = 8πG = M−2. For a spatially homogeneous
p
For any model of inflation, the scalar power spectrum
tachyon field T, we have the equation of motion
canbe expressedas a series[12]. One oftenrely oneither
on approximations or on numerical mode-by mode inte-
grationfor predicting the coefficients of the series. How- V′
T¨+3HT˙ 1 T˙2 + 1 T˙2 =0
− V −
(cid:16) (cid:17) (cid:16) (cid:17) (4)
whichisequivalenttotheentropyconservationequation.
∗Electronicaddress: [email protected] Here, the Hubble parameter H is defined as H a˙/a,
≡
2
and V′ = dV/dT . If the stress-energy of the universe where we have introduced the new variable
is dominated by the tachyon field T, the Einstein field
equations for the evolution of the background metric, aT˙
G =κT , can be written as z (14)
µν µν
≡ H
The evolutionof the scalarperturbations is calculated
κ V(T)
H2 = (5) by the Einstein action. The first-order perturbation
3 1 T˙2 equations of motion are given by a second-order action.
−
Therefore, the gravitational and matter sectors are sep-
p
arated and each expanded to second-order in the per-
a¨ κ V(T) 3
=H2+H˙ = 1 T˙2 turbations. The action for the matter perturbations can
a 3 1 T˙2 (cid:18) − 2 (cid:19) be determined by expanding the Lagrangianas a Taylor
− (6) series about the background equations and integrating
p
From Eqs.(5) and (6) we deduce that by parts. Note that the inflationary requirement a¨ > 0
as long as T˙2 < 2. In the chaotic scenario, the infla-
3
dT 2 H′(T) tion will slowly roll down its potential, i.e., T˙2 ≪ 23 and
= (7) H2 κV(T). Hence, the tachyon equation of motion
dt −3H2(T) ≈ 3
(4) is approximated to the one with the Lagrangian of
and leading to the normal nearly quadratic form. Furthermore, we can
stateasmallparameter√ǫ= 2 H′ whichsuppressed
3 H2
V2(T)= 9 H4 4 H′2 (8) the non-linear contributions. qThe(cid:0)acti(cid:1)on of linear scalar
κ2 − κ2 perturbation of tachyon field is given by
1 z
a 3 T H3 S = 2 dτd3x (∂τu)2−δij∂iu∂ju+ zττu2
a0 =exp"−2ZT0 H′dT# (9) Z h i (15)
Eq.(15) is formally equivalent to the action for a
scalar field with the standard kinetic term and a time-
3 T H2 dependent effective mass m2 = zττ/z in flat space-time.
t= dT (10) Note that the Eq.(15) is formally the same as the case
−2 H′
ZT0 of ordinary scalar field, but it is new because the defi-
where a is the initial value of scale factor during infla- nition of variable z in Eq.(14) is different from the one
0
tion. in Ref.[15]. In fact z is dependent on the equation of
Bardeen et.al. [14] have shown that the general form tachyon motion(4).Quantizing u(τ,x), we have
ofthe metricforthe backgroundandscalarperturbation
id given by d3k
uˆ(τ,x) = (2π)3/2 uk(τ)aˆkeik·x+u∗k(τ)aˆ†ke−ik·x
Z h i
a−2(τ)ds2 = (1+2A)dτ2−2∂iBdxidτ aˆk,aˆ†l = δ3(k−l),aˆk|0i=0,etc. (16)
[(1 2R)δij +2∂i∂jE]dxidyj (11) h i
− − From Eq.(15), the mode functions u satisfy following
k
The intrinsic curvature perturbation of the comoving equation
R
hypersurfaces can be written as
d2u 1d2z
k + k2 u =0 (17)
H dτ2 − zdτ2 k
= R δT (12) (cid:18) (cid:19)
R − − T˙ Using Eqs.(4),(7) and (8), it is easy to find that
duringevolutionoftheuniverse,whereδT representsthe
fluctuation of the tachyon field and T˙ and H are deter- 1 1d2z
= 1+4ǫ(T) 3η(T)+9ǫ2(T)
minedbythebackgroundfieldequationsEqs.(7)-(8). On 2a2H2zdτ2 −
theanalogyofdiscussionfortheinflationdrivenordinary 1
scalarfieldwiththestandardkineticterm[15],weusethe 14ǫ(T)η(T)+2η2(T)+ ξ2(T)(18)
− 2
gauge-invariantpotential
where
T˙ 2 H′(T) 2
u=a δT + R z (13) ǫ(T)= (19)
" H #≡− R 3 H2(T)
(cid:20) (cid:21)
3
III. THE SOLUTION OF TYPE I
1H′′(T)
η(T)= (20) In this case, we should demand that H satisfies the
3H3(T)
differential equation
and
H4 2HH′′+2H′2 =0 (28)
− 3
1
2 H′H′′′(T) 2
ξ(T)= (21) which has the solution
3 H6(T)
(cid:18) (cid:19)
1
Using Eqs.(7)-(10), we can give the exact expression in
H(T)= (29)
whichthe modefunctionuk isrelatedto the fieldT. One AT +B 3T2
has the boundary conditions − 2
q
whereAandBarearbitraryintegrationconstants. How-
ever, we can chose A = 0 without loss of generality, as
1
u e−ikτ, aH k (22) it can be recovered by making a translation of the field,
k
→ 2k ≪ T T+A. FromEqs.(8)-(10),we havethe correspond-
→ 3
ing tachyon cosmology,
u z, aH k (23)
k
∝ ≫
3 B 5T2
V(T)= − 2 (30)
which guarantees that the perturbation behaves like a κv B 3T2 3
free field well inside the horizon and is fixed at super- uu − 2
horizon scales. Eqs.(14)-(17) make a difference between t(cid:0) (cid:1)
tachyon and ordinary scalar, because the curvature per-
a T
turbations couple to the stress-energy of tachyon field. a(T)= 0 0 (31)
T
On each scale is constant well outside the horizon.
R
Its spectrum is defined by
1 1
3 2 3 2
t(T) = B T2 B T2 (32)
= d3k k(τ)eik·x, (24) (cid:18) − 2 0(cid:19) −(cid:18) − 2 (cid:19)
R (2π)3/2R 1 1
Z √B √B− B− 32T02 2 √B+ B− 32T2 2
+ ln
2 h√B (cid:0)B 3T2(cid:1)21ih√B+(cid:0)B 3T2(cid:1)21i
2π − − 2 − 2 0
hRkR∗li= k3PRδ3(k−l), (25) The conformal thime is (cid:0) (cid:1) ih (cid:0) (cid:1) i
where (k) is the power spectrum. From eq.(25), we
R
have P 1 3 12 T 3 12
τ(T) = B T2 B T2 (33)
2a − 2 0 − 2a T − 2
0 (cid:18) (cid:19) 0 0 (cid:18) (cid:19)
PR12(k)=r2kπ32 uzk (26) + r23a0BT0 arcsinr23BT0−r23a0BT0 arcsinr23BT
(cid:12) (cid:12)
(cid:12) (cid:12) It is easy to find that τ tends to a constant value at late
Furthermore, we find(cid:12)a s(cid:12)imple relation as follows
time, or as T goes to zero.
Forthissolution,thecosmologicalpropertiesareeasily
1 derived. As the tachyon field T goes to zero or infinite,
dz 2 2
= a2 ǫ (1 2η+3ǫ) (27) the potential V(T) tends to non-zero value 3 or zero.
dτ − 3 − κB
(cid:18) (cid:19) The motion is not inflationary at all time. From Eq.(6),
we find that the period of accelerated expansion corre-
Theexactinflationarysolutionsforthemodeequationsof sponds to T˙2 < 2 and decelerate otherwise. Thus infla-
curvature perturbations might be found from two cases. 3
TypeI:westartfromthe quantityz isaconstant,which tion occurs only when T < B. If this model was to
| | 3
is equivalent to requiring 1 2η+3ǫ = 0. Type II: we produce all the 50 e-foldings oqf inflation needed to solve
−
also might start from the quantities ǫ(T),η(T)and ξ(T) the initial conditions problems in the standard model of
are constants. cosmology,tachyon field must evolve to be close to zero.
4
Next, we discuss the spectrum of curvature perturba- and
tions produced by this model, which is similar to one of
the inflation model driven by ordinary scalar field[16].
The solution of mode equation (17) is simple, 1 1 √6
ǫ(T)= , η(T)= . and ξ(T)=
n n n
(40)
1 From Eqs.(8)-(10), we have the corresponding tachyon
u (τ)= e−ikτ (34)
k √2k cosmology,
faorrytchoengdriotwioinnsg. mSoindcee, atfhteercwonefhoramveailmtipmoseedτ ttheendbsoutnoda- V(T)= 2 n2 n 12 (T T )−2
0
κ − 3 −
constant, mode function u is essentially fixed at super-
k (cid:16) (cid:17) (41)
horizon scales. The spectral index n is
R
a(T)=a (T T )n (42)
dln R 0 − 0
n 1 P =2 (35)
R
− ≡ dlnk
Notethatthisresultisexactandindependentoftachyon 1
3 2
fieldT. Bynow,theinflationaryuniverseisgenerallyrec- t(T)= n (T T0) (43)
2 −
ognized to be the most likely scenario that explains the (cid:18) (cid:19)
origin of the Big Bang. So far, its predictions of the
The conformal time is
flatness of the universe and the almost scale-invariant
powerspectrumofthe curvatureperturbationthatseeds
structureformationsareingoodagreementwiththecos- 3 21
micmicrowavebackground(CMB)observations. Thekey τ(T)=a−01(1−n) 2n (T−T0)1−n
data of CMB are the curvature perturbation magnitude (cid:18) (cid:19)
(44)
measured by COBE[17] and its power spectrum index
Itiseasyto findthatthe conditionforinflationisn>1,
nR[18] whichcorrespondstoT˙2 < 2. Thismodelactuallyinflate
3
forever. The model equation (17) is solved in terms of
1
δ 1.9 10−5 (36) Besselfunctions,µk(τ)=(kτ)2 [C1Jµ(kτ)+C2J−µ(kτ)]
H ≃ × where C1 and C2 are constants fixed from Eq.(22). The
spectrum may be calculated exactly to read
and
κ
(k)= 22µ−1Γ2(µ)k−2µ
n 1 <0.1 (37) PR π2A2
| R− | (45)
where A is a constant
Therefore,thespectrumofthismodelis”blue”asitpos-
sesses more power at large values of k, or small scales,
which is ruled out by the observable universe. However, µ−1
this modelcanbe used to probe the accuracyof the first A=a23−µ(n 1)µ−21 3n 2 4
0 − 2
andsecondorderapproximationsfor the mode equation. (cid:18) (cid:19)
(46)
The corresponding spectral index is
IV. THE SOLUTION OF TYPE II
4
n =1 (47)
R
− n 1
In this case, we should demand that H satisfies the
−
differential equations
In particular, we have n 1 for n 1. In this limit,
R
the solution and slow roll in≈flation wi≫th ǫ=η = 1 agree
n
at the leading order in ǫ. Only for n > 201, i.e. for
ǫ(T)=const., η(T)=const. and ξ(T)=const.
1 > n > 0.98, the error of slow roll approximation is
R
(38)
less than 1 percent.
which have the solution We briefly conclude the main result of this paper. We
find two family of exact solutions for the mode equa-
tion (17) on curvature perturbations of tachyon infla-
1
2n 2 tion. This calculation is done to linear order, ignoring
H(T)= 3 (39)
T T both gravity and self-interactions of the tachyon field.
(cid:0) −(cid:1)0
5
Since the observed anisotropies are small, this approxi- ture perturbation can match the exact result to within
mation is considerably more accurate than the slow-roll 10 percent over most of the inflationary epoch. Finally,
approximation, and we need not attempt to go beyond we point out that the slow-roll approximation is the as-
it,thoughitispossibletoextendcalculationsbeyondlin- sumption that the field evolution is dominated by drag
ear perturbation theory[19]. These models can be probe from the expansion is small parameters ǫ=0. We have
the accuracy of the first and second order expressions found the solutions of mode equation are again Hankel
for the curvature perturbation spectra. In fact, almost functionofthefirstkindH(1)( kτ)withµ= 3+4ǫ 2η.
all analytical predictions for perturbation spectra from µ − 2 −
inflation rely on the slow roll approximation. Further-
more,the parameter η becomes largenear the topof the
acknowledgments
tachyon potential in type I model, indicating a break-
down of the slow roll assumption. In the cases that n This work was partially supported by National Na-
are not large enough, the slow roll conditions are badly ture Science Foundation of China under grant19875016,
violated by type II model. Therefore, the first order ex- Foundation of Shanghai Development for Science and
pressiondoesnotgivegoodagreementwiththeexactfor Technology under grant 01JC14035, Foundation of
all solutions. But we have confidence that we use slow ShanghaiEducationalMinistryundergrant01QN86and
roll approximation in more realistic situations, since the FoundationofShanghaiScienceandTechnologyMinistry
second order expression for the spectral index of curva- under grant 02QA14033.
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