Phantom Inflation with A Steplike Potential

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by  Zhi-Guo Liu

Phantom inflation with a steplike potential
Zhi-Guo Liu , Jun Zhang , and Yun-Song Piao
∗ † ‡
College of Physical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, China
The phantom inflation predicts a slightly blue spectrum of tensor perturbation, which might be
tested in coming observations. In normal inflation models, the introduction of step in its potential
generally results in an oscillation in the primordial power spectrum of curvatureperturbation. We
willcheckwhetherthereisthesimilarcaseinthephantominflationwithsteplikepotential. Wefind
that for same potentials, the oscillation of the spectrum of phantom inflation is nearly same with
that of normal inflation, thedifference between them is the tilt of power spectrum.
I. INTRODUCTION change in the potential will result in a universal os-
cillation in the spectrum of primordial perturbations
1
1 [30],[31],[32],[33],[34],[35],[36],[37], and also [38],[39]. A
The results of recent observations are consistent with
0 burst ofoscillations in the primordialspectrum seems to
an adiabatic and nearly scale invariant spectrum of pri-
2 providea better fit to the CMB angularpowerspectrum
mordialperturbations,aspredictedbythesimplestmod-
[40],[41]. In this Letter, we will check whether there is
n els of inflation. The inflation is supposed to have taken
a a similar feature in the phantom inflation with steplike
place at the earlier moments of the universe [1],[2],[3],
J potential. We will consider a quadratic potential with a
which superluminally stretched a tiny patch to become
1 stepandahybridpotentialwithastep,respectively,and
our observable universe today. During the inflation the
2 numericallycalculatethecorrespondingpowerspectrum,
quantumfluctuations inthe horizonwill be able to leave andthen compare them with that of normalinflationary
] the horizonandbecomethe primordialperturbationsre-
c model.
sponsible for the structure formation of observable uni-
q The plan of this work is as follows. In section II we
verse. In this sense, exploring different inflation models
- presentthe phantominflation scenario and give the sim-
r is still an interesting issue.
g ple analysis of the background evolution with steplike
[ Recently, the phantom field, for which the parame- potential. In section III we discuss the calculation of

ter of state equation ω < 1 and the weak energy the power spectrum and present numerical results for
2 −
condition is violated, has acquired increasely attention
v the primordial spectrum. Section IV contains discussion
[4], inspired by the wide use of such fields to describe
3 and conclusions. Note that we work in units such that
7 dark energy, e.g. [5],[6],[7],[8], [9],[10],[11]. The sim- ~=c=8πG=1.
6 plest realizationof phantom field is a normal scalar field
0 with reverse sign in its dynamical term. The quan-
2. tum theory of such a field may suffer from the causal- II. THE BACKGROUND OF PHANTOM
1 ity and stability problems [12, 13]. However, this does INFLATION WITH A STEP
0 not mean that the phantom field is unacceptable. Ac-
1 tionswithphantomlikeformmaybeariseinsupergravity
v: [14], scalar tensor gravity [15], higher derivative gravity The simplest realization of phantom field is a normal
i [16], braneworld [17], stringy [18], and other scenarios scalarfield with reversesignin its dynamicalterm. This
X [19, 20]. The phantom inflation, which is drived by the reversesignresultsinthat,differentfromtheevolutionof
r phantom field, has been proposed [21], and widely stud- normalscalarfieldduringthenormalinflation,thephan-
a
ied in [22],[23],[24],[25],[26],[27],[28]. In phantom infla- tom field during the phantom inflation will be driven by
tion, the power spectrum of curvature perturbation can its potential up along its potential, e.g.[9, 10]. Thus if
benearlyscaleinvariant. Thedualityofprimordialspec- initially the phantom field is in the bottom of poten-
trum to that of normal field cosmologyhas been studied tial, analogous to the slow rolling regime of the normal
in [22, 29]. In the meantime the spectrum of tensor per- scalar field, the phantom field will upclimbe and enter
turbationisslightlybluetilt,whichisdistinguishedfrom slow climbing regime. Hereafter, the phantom inflation
that of normal inflationary models [21]. begins, and after some time the phantom inflation ends
and the universe enters into a period dominated by the
The power spectrum of perturbations in the nor-
radiation. The exit from the phantom inflation can be
mal inflation depends on the inflaton potential. In
implemented by introducing an additional normal scalar
some inflationary models, there may be inflaton po-
field [21], in which the exit mechanism is similar to the
tentials with a large number of steps. The steplike
case of hybrid inflation [42, 43], the imposition of back-
reaction[26],the wormhole[23],thebrane/fluxannihila-
tion in string theory [27], or the nonminimally coupling
of the phantom to gravity [28].
∗Email: [email protected]
†Email: [email protected] When the phantom field is minimally coupled to the
‡Email: [email protected] gravitational field, the Friedmann equation can be writ-
2
ten as
1.00040
1
3H2 = φ˙2+V(φ), (1)
−2 1.00035
where H2 is positive, which means in all cases for the 1.00030
H
phantomevolutionits dynamicalenergymustbesmaller 1.00025
than its potential energy, thus the phantom inflation is
not generally interrupted by the step in despite of the 1.00020
height of step. This can be compared with that in nor-
1.00015
mal inflation, in which it is possible that φ˙2 > V(φ)
for a high step and the inflation is interrupted for a 1.00010
-10 -5 0 5 10
short interval. The phantom field satisfies the equation
φ¨+3Hφ˙ V (φ)=0. Thephantomfieldisdriventoup- N
′
−
climbe alongits potentialisreflectedinthe minus before
FIG. 1: Evolution of the Hubble parameter H. H is in the
V term. Define the slow-climb parameters [21]
′ unit of H0 which is a model dependent parameter. H in-
creases and is different from normal inflation
H˙ φ¨
ǫ , δ (2)
pha ≡−H2 pha ≡−Hφ˙
normal inflation, since in the phantom inflation the en-
ergy density is increased. There is a very abrupt change
when the conditions ǫ << 1 and δ << 1 are
pha pha
| | | | due to the existence of the step.
satisfied,Eq(1)canbe solvedsemianalytically. Thenwe
have a eHt approximately, which means the universe Theslow-climbparametersintermofαcanbechanged
∼ as follows
entersinto the inflationaryphasedrivenbythe phantom
field. Hα φαα Hα
ǫ = , δ = . (6)
The numerical solutions of evolution equations are re- pha − H pha − φ − H
α
quired for accurately evaluating the perturbation spec-
The introduction of the step leads to a deviation from
trum. We shift the independent variable to α = lna,
slow-climbinflation, thoughduring this intervalitis still
whichwillfacilitatethe numericalintegration. Withthis
inflation. We haveplotted the evolutionof the twoslow-
replacementandusingtheenergyconservationequation,
climb parameters ǫ and δ around the time when
we have pha pha
the field crosses the step in Figure 2.
1
2
H = Hφ , (3)
α 2 α
III. THE PERTURBATION OF PHANTOM
INFLATION WITH A STEP
H 1
α
φαα+( H +3)φα− H2V′ =0. (4) We will investigate the power spectrum of curvature
perturbationsduringthephantominflation. Inthescalar
where the subscript α denotes differentiation for α and
caseitisadvantageoustodefineagaugeinvariantpoten-
the prime denotes differentiation with the scalar field φ.
tial [44],[45]
In general, the step in the potential can be modelled
by introducing the term proportional to tanh(φ−φstep). u= z (7)
δ − R
We consider a simple inflaton potential m2φ2, then this
where z a 2ǫ , and a denotes the scale factor, H
potential with a step can be given by ≡ | pha|
is the Hubblpe parameter, and the dot is the derivative
with respect to the physical time t.
1 φ φ
2 2 step
V(φ)= m φ 1+βtanh( − ) . (5) The equation of motion uk in the momentum space is
2 (cid:18) δ (cid:19)
z
2 ′′
This potential has a step at φ = φ with size and u′k′+ k uk =0, (8)
step (cid:18) − z (cid:19)
gradient governed by β and δ. We will focus on small
features in the potential, and thus will limit the param- where the prime denotes differentiation with respect to
eter β small. In this case, the phantom will upclimbe conformal time and k is the wave number. The solution
2
continuouslythroughthestepwhiletheinflationwillnot depends on the relative sizes of k and z′′/z. The z′′/z
be ceased. termcanbeexpressedas2a2H2plustermsthataresmall
The numerical results with the potential (5) can be during the phantom inflation. In general, on subhorizon
seen in Figure 1. We can see that H is nearly constant scales k2 z′′/z,the solution of perturbation is a plane
≫
in the inflationary era, but there are some differences wave
from the normal background. H increases slowly in the 1
phantom inflation compared with decrease slowly in the uk → √2k e−ikτ . (9)
3
0.00000 is constant.
The spectrum (k) is defined as
-0.00005 PR
-0.00010 2π2 3
hRk1R∗k2i= k3 PRδ (k1−k2), (12)
-0.00015
Ε and is given by
-0.00020
-0.00025 1/2 k3 uk
(k)= . (13)
PR r2π2 (cid:12) z (cid:12)
-0.00030 (cid:12) (cid:12)
(cid:12) (cid:12)
We will numerically calculate . However, before this
-3 -2 -1 0 1 2 3 it is interesting to show the PreRsult of in the slow
N climbing approximation, which is PR
2
1 H 2 k nR−1
1 = , (14)
PR 2ǫpha (cid:18)2π(cid:19) (cid:18)aH(cid:19)
| |
0
where the spectrum index is n 1 4ǫ +2δ .
pha pha
-1 This power spectrum may be eiRth−er bl≃ue−or red. The re-
∆
sultsaredependentontherelativemagnitudeofǫ and
-2 pha
δ , see [21] for the details. Eq.(14) is valid only when
pha
-3 the slow climb approximation is satisfied, i.e.|ǫpha| ≪ 1
and δ 1, however,when the potential has a sharp
pha
| |≪
-4 step,thedeviationofδislarge,seeFig2. Inthiscase,we
have to evolve the full mode equation numerically with-
-3 -2 -1 0 1 2 3
out any approximations.
N The perturbation equation (8), with the replacement
α=lna, can be written as
FIG.2: Evolutionofthetwoslow-climbparametersǫpha(top)
aolinfndet)hδ,epβhβa=(abn1od0tt−δo2mp,δa)rw=amit5het×tehr1se:0i−nβt4r=o(dgr1ue0ce−tni2o,ndδaos=fhteh1de0−lsit3neep()r.,eDdβidff=oetr5teen×dt uαα+(1+ HHα)uα+(cid:18)e2αkH2 2 − ez2α′′/Hz2(cid:19)u=0 (15)
10−3,δ = 10−3 (purple dashed line), β = 5×10−3,δ = 5×
10−4 (bluedotted line). with
z H H2 H φ V
′′ =a2H2 2 5 α 2 α 4 α αα + ′′ (16)
The 1 is obtained by the quantization of mode func- z (cid:18) − H − H2 − H φα H2(cid:19)
√2k
tion uk. Here a normal quantization condition has actu- Theevolutionofspectrumisgovernedbythecompetition
ally been applied, like that in normal field, which seems between the k2 and z /z terms. The overall normaliza-
′′
inconsequential for phantom field. However, it is gener- tion of is proportional to m2, φ determines the
step
ally thought that the phantom field might be only the wavelenPgtRhat which the feature appears. The dominant
approximativesimulationofafundamentaltheorybelow contribution to z /z is from the V term and is pro-
′′ ′′
certainphysicalcutoff,andthefulltheoryshouldbewell portional to β/δ2. Thus the range of k affected by the
quantized. In another viewpoint, we might assume that feature roughly depends on the square root of β/δ2.
initiallythereisnotphantomfield,thustheperturbation We plot the z /z term in Fig 3 and solve the equa-
′′
deep inside the horizonfollows normalquantizationcon- tion numerically and give the results of power spectrum
dition, then the evolution with w < 1 emerges for a pe- in Fig 4. In Fig 3, we can see that the z /z term is very
′′
riod,whichissimulatedphenomenologicallybythephan- different from 2a2H2. It has a sharp oscillation near the
tom field, as in island cosmology [46] or [27]. Thus the step. Fig 4 shows the power spectrum of phantom in-
primordial perturbation induced by the phantom fields flation with potential of Eq.(5). We plot it with three
has to have a normal quantization condition as its ini- group parameters and can see that the introduction of
tial condition, or it cannot be matched to that of initial step results in a large deviation from slow climbing in-
background. While on superhorizon scales k2 z′′/z flation and then a burst of oscillations superimposed on
≪
the dominated mode is the nearly scale invariant scalar power spectrum. The
magnitude and extent of oscillation is dependent on the
uk ∝z (10) amplitude and gradient of the step. Thus as in normal
inflation, these oscillations will inevitably leave interest-
which means that the curvature perturbation
ingimprintsinthe CMBangularpowerspectrum,which
k = uk/z , (11) might provide a better fit e.g.[40],[41].
|R | | |
4
6 Thepowerspectrumoftensorperturbationisonlyde-
pendent onǫ . We set the parameterβ small,thus ac-
pha
4 tually ǫpha 1aroundthestep. Inthiscase,thetensor
| |≪
spectrumishardlyaffected,whichremainnearlyscalein-
L2
H^ 2 variant. However, due to ǫpha < 0, thus nT 2ǫpha is
^2 slightly blue tilt for the phantom inflation [2≃1],−which is
a
2
L(cid:144)Hz 0 distinguished from the normal inflation, in which nT is
(cid:144)z'' generally red tilt.
H
-2
1´10-8
-4
-3 -2 -1 0 1 2 3 5´10-9
N
2´10-9
FIG. 3: Evolution of z′′/z for β = 0.01 and δ = 0.001 with pR 1´10-9
the efolding number of inflation N and we have set N=0 at
thestep in the potential 5´10-10
2´10-10
1´10-8
0.1 0.5 1.0 5.0 10.0 50.0100.0
5´10-9 k(cid:144)aH
2´10-9 FIG. 5: Effects of a step in the potential on the power spec-
pR trumofcurvatureperturbationforthephantominflationand
1´10-9 the normal inflation, respectively. The blue solid line corre-
sponds to the phantom inflation with m = 7.5×10−6,β =
5´10-10 10−2,δ = 5×10−2 and the green dashed line corresponds
to the normal inflation with m = 7.5×10−6,β = 10−2,δ =
2´10-10 2×10−2
0.1 0.5 1.0 5.0 10.0 50.0100.0
k(cid:144)aH
FIG.4: Thepowerspectrumofcurvatureperturbationforthe IV. CONCLUSION AND DISCUSSION
phantom inflation with the potential (5), β = 5×10−3,δ =
4×10−2(true red line),β = 5×10−3,δ = 10−2(dotted blue
line),β =10−2,δ=4×10−2(dashed green line). Thephantomfieldcannaturallyappearineffectiveac-
tions of some theories, which might be only the approx-
imative simulation of a fundamental theory below cer-
tain physical cutoff. Thus the phantom cosmology have
Wecomparetheprimordialspectrumofphantominfla-
been widely studied. The phantom inflation predicts a
tionandnormalinflationinFig5,the topline showsthe
slightly blue spectrum of tensor perturbation, which is
primordial power spectrum of phantom inflation for the
potential with m = 7.5 10 6,β = 10 2,δ = 5 10 2. distinguished fromthat of the normalinflation, in which
− − −
× × the tensor perturbation is generally red tilt. This is
The normal inflation model with a same potential has a
a smokegun for the phantom inflation, which might be
red spectrum n < 1, however, the phantom inflation
background modRel has a blue spectrum n > 1. This tested in coming observations.
is because what we consider here is the pRotential with In normal inflation models, the introduction of step
2 2 in its potential generally results in an oscillation in the
m φ . In principle, we can have the models of phantom
primordialpowerspectrumofcurvatureperturbation. In
inflation with n < 1 by choosing a suitable potential
[21]. We can conRsider a model of phantom inflation with this Letter, we find that for same potentials with the
step,theoscillationofthespectrumofphantominflation
the potential
can be nearly same with that of normal inflation, and
1 φ φ the magnitude and extent of oscillation is dependent on
2 2 step
V(φ)=V0+ m φ 1+βtanh( − ) (17) the amplitude and gradient of the step. The difference
2 (cid:18) δ (cid:19)
between them is the tilt of power spectrum. However,
whereV0isconstantwhichdominatesthepotential. This the same tilt can be obtained by considering a different
potential is same with that of hybrid inflation with nor- potential of the phantom inflation.
mal field. Fig 6 shows that n < 1 which is similar to In general, φ˙2 for the phantom must be smaller than
that of the normal inflation wiRth m2φ2. its potential energy in all time. Thus the phantom in-
5
mal inflation, in which it is possible that for a high step
the inflation is interrupted for a short interval. Here, we
8.´10-9 have limited the parameter β small, however, it is inter-
7.´10-9 estingtoconsiderthephenomenaofβ 1,i.e. thereisa
≫
large step, by which the density of dark energy observed
6.´10-9
might be linked to that of inflation, as in the eternal ex-
R
p panding cyclic scenario, e.g.[47],[48]. This might lead to
5.´10-9
a lower CMB quadrupole in observable universe if the
stepis justin the positionofpotential,inwhichthe per-
4.´10-9 turbation with Hubble scale leaves the horizon during
the phantom inflation, as in the bounce inflation model
[38],[39]. It is possible that the phantom inflation with
0.5 1.0 5.0 10.0 50.0 100.0
steplike potential can be effectively implemented in cer-
k(cid:144)aH
tain warped compactifications with the brane/flux anni-
hilation,e.g.[27]. Weexpecttobacktotherelevantissues
FIG. 6: The power spectrum of curvature perturbation for
the phantom inflation with the potential (17), V0 = 3.7× in the coming works.
10−14,m=3.2×10−8,φstep =0.0125,β =5×10−4,δ =10−5. Acknowledgments
Corresponding to thespectra index ns <1.
This work is supported in part by NSFC under Grant
No:10775180,11075205,inpartbytheScientificResearch
flation is not interrupted by the step in despite of the Fund of GUCAS(NO:055101BM03),in part by National
height of step. This can be compared with that in nor- Basic Research Programof China, No:2010CB832804
[1] A.H.Guth,Phys.Rev.D 23,347(1981). [21] Y. S. Piao and Y. Z. Zhang, Phys. Rev. D 70,
[2] A.D.Linde, Phys. Lett. B 108 ,389(1982); A.J.Albrecht 063513(2004); Y.S. Piao and E Zhou, Phys. Rev. D68,
and P. J. Steinhardt,Phys. Rev.Lett. 48, 1220(1982). 083515 (2003).
[3] A.A.Starobinsky,Phys. Lett.B91, 99 (1980). [22] J. E. Lidsey, Phys.Rev.D 70, 041302 (2004).
[4] R.R.Caldwell, Phys.Lett. B545, 23 (2002). [23] P. F. Gonzalez-Diaz and J. A. Jimenez-Madrid, Phys.
[5] R.R. Caldwell, M. Kamionkowski and N.N. Weinberg, Lett. B 596, 16 (2004) [arXiv:hep-th/0406261]; P.F.
Phys.Rev.Lett. 91, 071301 (2003). Gonzalez-Diaz, Phys.Rev.D68, 084016 (2003).
[6] J.G. Hao, X.Z. Li, Phys. Rev.D67, 107303 (2003). [24] S. Nojiri and S. D. Odintsov, Gen. Rel. Grav. 38, 1285
[7] S. Nojiri and S. D. Odintsov, Phys. Lett. B 562, 147 (2006); S. Capozziello, S. Nojiri and S. D. Odintsov,
(2003) [arXiv:hep-th/0303117]. E. Elizalde, S. Nojiri Phys. Lett. B 632, 597 (2006); E. Elizalde, S. Nojiri,
and S. D. Odintsov, Phys. Rev. D 70, 043539 (2004) S. D. Odintsov, D. Saez-Gomez and V. Faraoni, Phys.
[arXiv:hep-th/0405034]. Rev. D 77, 106005 (2008).
[8] P. Singh, M. Sami and N. Dadhich, Phys. Rev. D 68, [25] M. Baldi, F. Finelli and S. Matarrese, Phys. Rev. D72,
023522 (2003). 083504 (2005).
[9] M.SamiandA.Toporensky,Mod.Phys.Lett.A19,1509 [26] P. Wu and H. W. Yu,JCAP 0605, 008 (2006).
(2004). . [27] Y. S. Piao, Phys.Rev. D 78, 023518 (2008).
[10] Z.K.Guo,Y.S.Piao,andY.Z.Zhang,Phys.Lett.B594, [28] C. J. Feng, X. Z. Li and E. N. Saridakis, Phys. Rev. D
247 (2004). 82, 023526 (2010) [arXiv:1004.1874 [astro-ph.CO]].
[11] J.M. Aguirregabiria, L.P. Chimento, R. Lazkoz, Phys. [29] Y.S.Piao,Phys.Lett.B606,245(2005).Y.S.Piao,Y.Z.
Rev.D70, 023509 (2004). Zhang, Phys. Rev.D70, 043516 (2004).
[12] S.M. Carroll, M. Hoffman and M. Trodden, Phys. Rev. [30] A.A. Starobinsky,JETP Lett. 55, 489 (1992).
D68023509 (2003). [31] J. A. Adams, G. G. Ross, and S. Sarkar, Nucl. Phys. B
[13] J.M. Cline, S. Jeon and G.D. Moore, Phys. Rev. D70, 503, 405 (1997) [arXiv:hep-ph/9704286].
043543 (2004). [32] S. M. Leach and A. R. Liddle, Phys. Rev. D 63, 043508
[14] H.P.Nilles, Phys. Rept.110 1 (1984). (2001).
[15] B. Boisscau, G. Esposito-Farese, D. Polarski and A.A. [33] S. M. Leach, M. Sasaki, D. Wands and A. R. Liddle,
Starobinsky,Phys.Rev.lett. 85 2236(2000). Phys. Rev.D 64, 023512 (2001).
[16] M.D. Pollock, Phys. Lett.B215, 635 (1988). [34] J. A. Adams, B. Cresswell, and R. Easther, Phys. Rev.
[17] V. Sahni and Y. Shtanov, JCAP 0311, (2003) 014; D 64, 123514 (2001).
D.J. Liu and X.Z.Li, Phys.Rev.D 68, 067301 (2003). [35] J. Hamann, L. Covi, A. Melchiorri and A. Slosar, Phys.
[18] P.Frampton, Phys. Lett.B555, 139 (2003). Rev. D 76, 023503 (2007).
[19] L.P. Chimento and R. Lazkoz, Phys. Rev. Lett. 91 [36] M. Joy, V. Sahni and A. A. Starobinsky, Phys. Rev. D
211301 (2003); J. G. Hao and X. Z. Li, Phys. Lett. B 77, 023514 (2008)
606, 7 (2005). [37] R.N.LernerandJ.McDonald,Phys.Rev.D79,023511
[20] H.Stefancic, Eur. Phys.J. C36, 523 (2004). (2009) [arXiv:0811.1933 [astro-ph]].
6
[38] Y.S. Piao, B. Feng, and X.M. Zhang, Phys. Rev. D69, [44] V.F. Mukhanov, JETP lett. 41, 493 (1985); Sov. Phys.
103520 (2004); Y.S. Piao, Phys. Rev. D71, 087301 JETP. 68, 1297 (1988).
(2005). [45] H.KodamaandM.Sasaki,Prog.Theor.Phys.Suppl.78,
[39] Y.F. Cai, X.M. Zhang, JCAP 0906 003 (2009). J. Liu, 1 (1984); M.Sasaki, Prog. Theor. Phys.76, 1036 (1986).
Y.F. Cai, H.Li, arXiv:1009.3372. [46] Y.S. Piao, Phys. Rev. D72, 103513 (2005); Phys. Lett.
[40] M. Joy, A. Shafieloo, V. Sahni and A. A. Starobinsky, B659, 839 (2008); Phys.Rev. D79, 083512 (2009).
JCAP 0906, 028 (2009). [47] B. Feng, M.Z. Li, Y.S. Piao, X.M. Zhang, Phys. Lett.
[41] D. K. Hazra, M. Aich, R. K. Jain, L. Sriramkumar and B634, 101 (2006)
T. Souradeep, JCAP 1010, 008 (2010). [48] C. Ilie, T. Biswas, K. Freese, Phys. Rev. D80, 103521
[42] A.D.Linde, Phys.Rev. D49, 748 (1994). (2009).
[43] E.J.Copeland,A.R.Liddle,D.H.Lyth,E.D.Stewartand
D.Wands, Phys.Rev.D49 6410 (1994).