b

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trum. Inectrum

Physics Letters B 605 (2005) 223–227

www.elsevier.com/locate/physlet

Making waves on CMB power spectrum and inflaton dynami

Masahiro Kawasaki, Fuminobu Takahashi, Tomo Takahashi

Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan

Received 2 August 2004; received in revised form 10 November 2004; accepted 11 November 2004

Available online 23 November 2004

Editor: T. Yanagida

Abstract

We discuss cosmic microwave background anisotropies in models with an unconventional primordial power specparticular, we consider an initial power spectrum with some “spiky” corrections. Interestingly, such a primordial power spgenerates “wavy” structure in the CMB angular power spectrum. 2004 Elsevier B.V. All rights reserved.

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Result of the first year Wilkinson microwave anistropy probe (WMAP)[1] is almost consistent with thfollowing assumption[2]: the present universe consists of 5% baryon, 23% cold dark matter (CDMand 72% dark energy. The primordial power spectris almost scale invariant with adiabatic fluctuatioThese assumptions form so-called the standardnario of cosmology.

Although the fit to the WMAP data with the stadard scenario seems to be excellent, there maysome deviations from the standard scenario in thegular power spectrum given by WMAP. One of theis that the angular power spectrum seems to bepressed at low multipoles compared to the predictioof the standard cosmological scenario.1

E-mail address: fumi@icrr.u-tokyo.ac.jp(F. Takahashi).1 In fact, this suppression was already observed by COBE

this issue was discussed in Ref.[3].

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.11.033

Since the cosmic variance error is very large at lmultipole region, we may say that the suppressiojust a statistical fluctuation. However, if we take thseriously, it may suggest some new physics. Thare many works proposing mechanisms to explainsuppression[4].

The other one is that, if you look at the WMAP agular power spectrum carefully, there is “wavy” struture around multipole region froml ∼O(10) up to thefirst peak. In particular, there seems to be a dip arothe first peak. Although this feature is not robust at tpoint, if it is an intrinsic one, it may suggest some uusual physics since this kind of structure cannot beplained by the conventional scenario. There have bsome works inspired by this feature. In Ref.[5], usinga cosmic inversion method, reconstruction of the pmordial power spectrum which can accommodatefeature is discussed. The authors of Refs.[6–8]consid-ered a primordial power spectrum with oscillatory c

.

224 M. Kawasaki et al. / Physics Letters B 605 (2005) 223–227

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rections which can be generated from trans-Plancphysics. They showed that, with such an initial powspectrum, we can improve the fit to the WAMP dacompared with the best-fit value of the standard snario.

Although we may not be at the position whereshould take this result seriously, it is interestingconsider some models which give the wavy structon the CMB angular power spectrum. In this Lettwe discuss a mechanism which generates such sture in the CMB power spectrum. Here we consida scenario where the primordial fluctuation is mofied from the standard form. If we assume an initpower spectrum with some “spiky” parts, wavy struture of the CMB power spectrum can appear. Thetial power spectrum we assume is written as

(1)Ps = Askn−1

(1+

∑i

αi exp

[− (k − kci)

2

2σ 2i

]),

whereAs andn are the amplitude of primordial fluctuation and the spectral index as usually assumed. Inspectrum, “spiky” corrections are added to the stdard power spectrum which can be parameterizedαi, kci andσi for theith spike. We assumeαi > −1 tokeep the initial power spectrumPs positive.kci andσi

represent the position of spike and its width, resptively. These parameters can be determined by moof the mechanism generating the primordial fluctua-tion. Here we consider a situation where the widththe Gaussian function is very narrow. This kind of pmordial spectrum may arise from a particular kindinflaton dynamics, which will be discussed later.

First we discuss effects of the spiky correctionsthe initial power spectrum. InFig. 1, we show the re-sultant CMB angular power spectrum from the initpower spectrum given by Eq.(1). In the upper paneof the figure, we assume a single “dip” which is prameterized asα = −0.9, kc = 2.2 × 10−2h Mpc−1,and σ = 8.7 × 10−5h Mpc−1. In the lower paneof the figure, we assumed a single spike and agle dip in the initial power spectrum with the prametersα1 = 3.0, kc1 = 4.3 × 10−3h Mpc−1, σ1 =7.6 × 10−5h Mpc−1 and α2 = −0.9, kc2 = 2.5 ×10−3h Mpc−1, σ2 = 1.4× 10−4h Mpc−1. The cosmo-logical parameters are chosen to beΩmh2 = 0.145,Ωbh

2 = 0.023, h = 0.69, ns = 0.97 andτ = 0.116which give the minimum value ofχ2 in the power-

-

Fig. 1. The CMB angular power spectrum with the primordial sptrum Eq.(1). The model parameters and cosmological parameare given in the text. The binned WMAP data is also plotted.

law CDM model.2 Here a flat universe and no tensmode are assumed. Very interestingly, with the pameter in the upper panel, you can see a dip aroundfirst peak, while wavy structure aroundl ∼ 50 appearsin the lower panel. Accordingly, the fit to the WMAdata becomes better. We calculatedχ2 using the codeprovided by WMAP[10] and obtained smallerχ2 byχ2 ∼ −10 compared to the value for the best-fit prameter of the power-lawCDM model in both panelsin Fig. 1.3

2 Although these cosmological parameters are a bit differenfrom those given by WMAP[2], this choice of cosmological parameters gives a better fit to the data withχ2 = 1428.6 [9].

3 For the case with the upper panel of the figure, assuming aintroduces three additional parameters thus the degrees of freedoincreases by 3. Similarly, in the case with the lower panel offigure, the degrees of freedom increases by 6 since we assumeand a spike.

M. Kawasaki et al. / Physics Letters B 605 (2005) 223–227 225

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avy

The reason why such structure arises is simple.angular power spectrum can be written as

(2)Cl = 2

π

∞∫0

dk k2Ps(k)∣∣Θl(k, η0)

∣∣2,

whereΘl(k, η) is the transfer function andPs is theprimordial power spectrum.η is the conformal timeand the subscript 0 represents a quantity defined apresent time. The transfer function is given by[11]

Θl(k, η0) =η0∫

0

dηg(η)(Θ0(k, η) + Ψ (k,η)

)

× jl

(k(η0 − η)

)

−η0∫

0

dηg(η)ivb(k, η)

k

d

dηjl

(k(η0 − η)

)

+η0∫

0

dη e−τ[Φ(k, η) − Ψ (k, η)

]

(3)× jl

(k(η0 − η)

),

where jl is a spherical Bessel function of degreel,and vb is the velocity of baryon.Ψ and Φ are thegravitational potential and the curvature perturbatirespectively.g(η) is the visibility function which has apeak aroundη∗ where∗ represents the epoch of the rcombination. Since we are considering the case whσi is very small (i.e., the width is very narrow), we creplace the second term of Eq.(1) with delta functions∑

i δ(k − kci). Approximating with the delta functioin the initial power spectrum Eq.(1), we can writeCl

as

Cl C(standard)l

+∑

i

αi

∣∣∣∣F (1)l (kci, η0, η∗)jl

(kci(η∗ − η0)

)

(4)+ F(2)l (kci, η0, η∗)

d

dηjl

(kci(η∗ − η0)

)∣∣∣∣2

,

where F(1,2)l (kci, η0, η∗) are functions which vary

slowly with l and C(standard)l is a CMB power spec

trum calculated by using the standard initial powspectrum. As you can see from Eq.(4), when thespiky correction term in the initial power spectru

Fig. 2. The difference inCl between with and without spiky correction terms. The model and cosmological parameters are the sain the upper panel ofFig. 1.

can give sizable contribution toCl , the form of Cl

exhibits the oscillatory structure ofjl . This is the rea-son why the wavy structure ofCl can be realizedassuming the initial power spectrum given by Eq.(1).To see the effects of a spherical Bessel function,plot Cl/C

(standard)l = (C

(standard)l − Cl)/C

(standard)l in

Fig. 2with the same model and cosmological paramters in the upper panel ofFig. 1. Whenαi is negative,the second term in Eq.(4) gives a negative contribution to Cl , thus, in this case, a dip can appear as inupper panel ofFig. 1. For a positiveαi case, the second term is added to give oscillatory power spectruWithout a spiky part in the initial power spectrum asis usually assumed, oscillatory structure of the sphical Bessel function is smeared out by contributiofrom superposition ofk-modes around the corresponing multipole region. Hence we do not usually ssuch structure. Notice also that if the width of tGaussian part in the initial power spectrum of Eq.(1)is not narrow, the oscillatory structure of a spheriBessel function is also smeared out, thus we cansee the wavy structure on the CMB power spectrThen how narrow should the width of the spiky struture be to give a distinctive feature toCl? Noting thatthe spherical Bessel functionjl(z) has the highest peawith the widthz ∼ l1/3 aroundz∗ ∼ l, the followinginequality needs to be satisfied in order to make wstructure around multipolel:

(5)σi z ∼ l−2/3.

kci z∗

226 M. Kawasaki et al. / Physics Letters B 605 (2005) 223–227

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This inequality is satisfied for the model parametwe have adopted inFig. 1.

Now we discuss possible models which generthe initial power spectrum of the form of Eq.(1). Usu-ally, the primordial density fluctuation is represenby the amplitude of the density perturbation athorizon crossingδH (k) which can be approximatelwritten in the spatially flat gauge as[12],

(6)δH (k) ∼ H 2

5πφ,

whereH is the Hubble parameter during inflation aφ is an inflaton field. The dot represents derivativwith respect to time. UsingδH , the initial power spectrum can be written asPs(k) ≡ δ2

H(k). From Eq.(6),we can see that, if the inflaton field almost stoinstantaneously,δH gets enhanced at correspondiscalekc, which may make some spiky structure in tpower spectrum.4 Inversely, if the inflaton field movevery fast instantaneously,δH gets suppressed at corrsponding scalekc. In this case, this makes some diin δH (k). Thus the primordial power spectrum in thform of Eq.(1) may be generated from an unusualflaton dynamics. However, this simple argument isenough to evaluate the primordial power spectrum.more detailed investigation of this issue, see Ref.[14].

In summary, we considered the primordial powspectrum with some spikes and/or spiky dips. Assuing this kind of initial power spectra, wavy structuappears in the CMB angular power spectrum, whmay be inferred from the first year WMAP observtion. Such wavy structure may originate to the oslatory structure of a spherical Bessel function. Whthere is a spike in the initial power spectrum andwidth is very narrow, for a fixedl, only the small num-ber ofk-modes is picked up to give contributions toCl

with oscillatory structure of the spherical Bessel funtion. If this mode can give sizable contribution toCl ,the oscillatory behavior of the spherical Bessel fution directly affects the form ofCl . When the widthof the spike is not so narrow, several modes can ctribute toCl . Thus oscillatory structure is smeared oby the projection effect as the conventional case.

4 Note that the power spectrum does not diverge even in theof φ → 0 [13].

Acknowledgement

We would like to thank K. Ichikawa for useful discussion. F.T. and T.T. would like to thank the JapSociety for Promotion of Science for financial suppoThis work is partially supported by the JSPS GrandAid for Scientific Research No. 14540245 (M.K.). Wacknowledge the use of CMBFAST[15] for numericalcalculations.

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