Chaotic new inflation and primordial spectrum of adiabatic ... - iucaa

Chaotic new inflation and primordial spectrum of adiabatic fluctuationsJun’ichi YokoyamaYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JapanReceived 24 December 1998; published 23 April 1999In a number of scalar potentials with an unstable local maximum at the origin, chaotic inflation is followedby new inflation if model parameters of the potential are appropriately chosen. We calculate the spectrum ofadiabatic fluctuations in such a double inflation model and show that it is possible to realize an almostscale-invariant spectrum with a sharp cutoff on a superhorizon scale, which may be favored observationally,and a tilted spectrum with the spectral index n eff 0.95 on large scales. S0556-28219900112-5PACS numbers: 98.80.CqPHYSICAL REVIEW D, VOLUME 59, 107303It is now commonly believed that the large-scale homogeneityand isotropy observed in the Universe were realizedas a result of a stage of accelerated expansion or inflation inthe early universe 1. The original model of inflation 2 wasbased on a first-order phase transition of a Higgs field ingrand unified theories GUTs. This model, however, failedbecause it could not achieve a graceful exit to the Friedmannera and the resultant universe turned out to be too inhomogeneous.Soon after that, the new inflation model 3 was proposedin which an inflationary stage was realized as the Higgs fieldrolls down its Coleman-Weinberg 4 potential slowly towarda global minimum during a second-order phase transition.It was shown that long-wave primordial density fluctuationswere generated out of quantum fluctuations of thescalar field in this slow roll over de Sitter stage 5. Unfortunately,however, it was observed that new inflation drivenby a GUT Higgs field suffered from several problems such astoo large density fluctuations and too short duration of inflation.But the most serious problem was that the initial conditionrequired for new inflation, namely, localization of thescalar field around the origin could not be realized by thepresumed mechanism, or high-temperature symmetry restoration,because the universe could not have been in thermalequilibrium in its very early stage due to the rapid cosmicexpansion 6.The chaotic inflation model 7 was proposed to resolvethe above mentioned problems. This model assumes the existenceof an inflation-driving inflation field, , with a potential,V, which has a gentle slope up to well above thePlanck scale, M Pl . At the Planck time, t Pl , when classicaldescription of spacetime becomes possible, the universe isexpected to be in a chaotic state and the only constraint imposedon the initial distribution of the inflaton field is thatboth kinetic or gradient energy density and potential energydensity should be smaller than the Planck density, M 4 Pl .Hence it is natural to assume that has an amplitude muchlarger than M Pl in some domain. Then inflationary expansionsets in due to its large potential energy density as gradually rolls down to a minimum.The amplitude and the spectrum of density and curvaturefluctuations generated in this model are also calculated in thestandard way. Let us write the perturbed metric in the longitudinalgauge,ds 2 „12x,t…dt 2 a 2 t„12x,t…dx 2 ,where a(t) is the scale factor and we use the notation of 8for the perturbation variables. Decomposing them to Fouriermodes as k d3 x2 3/2 x,te ikx ,we can calculate the root-mean-square rms amplitude of k k at the time k mode reenters the Hubble radius inthe postinflationary universe as k 2 1/2 f H k˙ t k fH 2, 2k 3 ˙ t kwhere k is the Fourier component of long-wave fluctuationsof a massless scalar field in the de Sitter spacetime withthe Hubble parameter H, and f 3/5(2/3) in the matter-radiation- dominated stage 5. The right-hand side of theabove equality should be evaluated at tt k whena(t k )H(t k )k during inflation. By multiplying the phasespacefactor, the rms amplitude of curvature perturbation onthe comoving scale r2/k is given by r 2 H 2 f f tk 2˙ k , 4t kand the effective spectral index is defined by123n eff d ln k 2 4. 5d ln kSince both H and ˙ are slowly varying during inflation wefind n eff 1 in most cases, which is the Harrison-Zel’dovichspectrum 9. For example, in the chaotic inflation modeldriven by a quartic potential, V 4 /4, we find n eff0.95 on the comoving scales corresponding to the largescalestructures today.In comparing various observational data of the anisotropyin cosmic microwave background radiation CMB and ofthe large-scale structures LSS with theoretical models 10,a single power-law primordial fluctuation spectrum with a0556-2821/99/5910/1073034/$15.0059 107303-1©1999 The American Physical Society

BRIEF REPORTS PHYSICAL REVIEW D 59 107303constant power index is usually assumed. In particular, modelswith n1 are preferred as predicted from inflation. Infact, however, a number of mechanisms has been proposedto generate more complicated non-scale-invariant fluctuationswithin inflationary cosmology. For example, Hodgesand Blumenthal 11 argued possibility to generate arbitraryspectrum of adiabatic fluctuations by making use of weak dependence of asM PlV3/2 3 V , 6in the slow roll over regime. While they depicted variousshapes of nontrivial spectra, few of the corresponding potentialshave a plausible form in particle physics. Other possiblemechanisms for generation of non-scale-invariant spectra includemixture of adiabatic and isocurvature fluctuations,which requires multiple fluctuating fields 12,13, and multiplestages of inflation with multiple inflatons or combinationof an inflaton and higher-order gravity 14.Recently, we proposed a novel model of double inflationwhich contains only one source of inflation 15. That is, weobserved that in a number of scalar potentials with an unstablelocal maximum at the origin chaotic inflation is followedby new inflation if model parameters are appropriatelychosen. The primary concern of the previous paper was togenerate a spectrum of density fluctuations which has a sharppeak on a specific scale in order to allow significant formationof primordial black holes PBHs on the correspondingmass scale. We have found a successful model to explainmassive compact halo objects MACHOs in terms of thesePBHs by specifying some of the model parameters with theaccuracy of eight digits, because the abundance of PBHs isexponentially sensitive to the peak amplitude of fluctuations16.In the present article we investigate a more generic featureof the spectrum of adiabatic fluctuations generated inthis model and discuss its relevance to CMB anisotropy andLSS.We adopt the Coleman-Weinberg potential,V 4 4 ln v 1 4 16 v4 .This potential can drive chaotic inflation for emax(M Pl /8,4v), and new inflation around 0. Kungand Brandenberger 17 studied the initial distribution of ascalar field with a double-well potential and concluded thatchaotic inflation is much more likely than new inflation.Thus we also start with chaotic inflation with M Pl . Afterchaotic inflation, the scalar field starts oscillation around v if v is too large, or it overshoots the symmetric state0 and approaches the other minimum v if v is toosmall. If v is appropriately chosen at an intermediate value,on the other hand, it can spend a long enough time near theorigin and then slow roll over new inflation can set in, whichends up with either v or v depending on the signof when classical slow roll over regime begins.7We numerically solve equations of motion of the homogeneouspart of the fields,H aȧ 2¨ 3H˙ V0,2 823M Pl8˙ 2V, 92with the above initial condition. As a result we find that newinflation can be realized after chaotic inflation if we take v0.22M Pl . More precisely, the number of e-folds of newinflationary expansion, N new , with ḢH 2 is larger than 10for v0.2201M Pl 0.2259M Pl , and it satisfies N new 60for v0.2223M Pl 0.2239M Pl . In the latter case, the comovingscales which leave the Hubble radius during chaoticinflation are stretched beyond the current Hubble horizonand hence they are unobservable today.After new inflation settles to the positive potential minimumv if v0.2232M Pl and to the negative potentialminimum v if v0.2231M Pl . If we take even smallervalues of v, orv0.1066M Pl , will return to the positivepotential minimum after one oscillation around the negativepotential minimum. New inflation is possible when approachesthe origin after crossing v if v0.1064M Pl0.1066M Pl . Here we have assumed that dissipates itsenergy density only through cosmic expansion. Below weconcentrate on the cases where new inflation takes placewhen moves to the negative direction and relaxes to v without returning to v.The above results are independent of the initial conditionas long as has large enough initial value so that the systemrelaxes to the slow roll over chaotic inflation solution 7.They are also independent of the coupling parameter . Infact, defining Ṽ 1 V and a new time variable 1/2 t, the equations of motion 8 and 9 are rewritten as 3H˜ Ṽ0,10a2 823M Pl 2.211H˜ 2 a Obviously, the system is now independent of . The amplitudeof fluctuation, on the other hand, is written in terms ofthe rescaled variables asH2 H˜ 2 1/22˙ 2 ,12so it is proportional to 1/2 .Figure 1 depicts the amplitude of fluctuation (r2/k) as a function of the natural logarithm of the comovingwave number, N(k)ln(k/k *), for various values of vwith 10 14 . Here k *is the comoving wave number leavingthe Hubble radius at the end of chaotic inflation. In thisfigure fluctuations on the left of the dip are due to chaoticinflation and those on the right are from new inflation. Thevalue of N(k) at the right end of each curve, N f , where 107303-2

BRIEF REPORTS PHYSICAL REVIEW D 59 107303FIG. 1. Spectrum of perturbation,(r), generated in the chaoticnew inflation model for 10 14 . Fluctuations generatedduring chaotic inflation are practicallydegenerate for all v’s consideredhere.sharply declines, is practically equal to N new for each v. Theamplitude of fluctuation (r) on the current Hubble radiuscan be obtained from the value at N(k)N f 60 and that onthe comoving scale corresponding to 1h 1 Mpc today fromN(k)N f 52 for each v. Here we have used the followingrelation between current length scale r and N(k):r310 3 h 1 Mpc 141/4expN10 f 60Nk,13where efficient reheating is assumed after new inflation withv0.22M Pl and h0.7.Let us summarize our main result shown in Fig. 1. Firstfor v0.20M Pl , the number of e-folds during new inflationis less than unity and the density fluctuation is entirely due tochaotic inflation. For 0.2223M Pl v(0.2239M Pl ), on thecontrary, all the currently observable scales have left theHubble radius during new inflation regime. The case withv0.22235M Pl is particularly interesting because there is asharp decline in the spectrum of fluctuation just above thecurrent Hubble radius. In fact, it has been argued in the literature18 that such a spectrum fits the observation of theanisotropy of CMB by the Cosmic Background ExplorerCOBE better than the conventional scale-invariant spectrum.For 0.20M Pl v0.2223M Pl , both chaotic inflation andnew inflation contribute to density fluctuations within thecurrent Hubble radius. In most cases, the spectrum of fluctuationson scales probed by LSS and CMB, namely, thosecorresponding to the comoving scales between N(k)N f52 and N f 60 in the unit of the horizontal axis in Fig. 1,can be fit by a power law. But models with v0.2222M Pl0.2223M Pl should be ruled out, because they produce alarge dip in the spectrum of fluctuations on scales probed byLSS and CMB. Table I shows effective spectral index obtainedon these scales. As is seen there we can realize a tiltedspectrum of adiabatic fluctuations with the spectral indexn eff 0.95. Previously a power-law spectrum with a fixed indexn1 has been known to be realized with a scalar fieldwith an exponential potential 19,Vexp 1n 16, 142n M Plwhere inflation is of power law, a(t) t (2n)/(1n) . Ourmodel provides another mechanism to produce a tilted spectrumwithin a renormalizable theory in the Einstein gravity.In addition, this model also predicts a peak of densityfluctuation on a smaller scale corresponding to the comovinghorizon scale at the onset of slow-roll new inflation. In TableI we have also shown the comoving mass scales of nonrelativisticmatter corresponding to the peak, taking the currentdensity parameter 0 0.3 and h0.7. For realistic valuesof the spectral index n eff 0.8 on large scales 20, thesemass scales are too small to leave any cosmological effects.So far we have assumed that the homogeneous part of theinflaton traces the classical trajectory and calculated thespectrum of adiabatic fluctuations based on the formula 4TABLE I. Effective spectral index, n eff , on comoving scales r310 3 h 1 Mpc and 1h 1 Mpc and the comoving mass scale atthe peak of fluctuations as a function of v.v n eff (310 3 h 1 Mpc) n eff (1h 1 Mpc) Peak mass0.20M Pl 0.95 0.940.2200M Pl 0.94 0.92 210 53 M 0.2210M Pl 0.93 0.91 110 45 M 0.2215M Pl 0.91 0.89 410 35 M 0.2220M Pl 0.85 0.78 110 15 M 0.2221M Pl 0.80 0.69 410 9 M 0.2222M Pl 0.65 6M 0.2223M Pl 0.94 310 12 M 107303-3

BRIEF REPORTS PHYSICAL REVIEW D 59 107303derived with the slow-roll approximation. Here we discussplausibility of the above treatment.First quantum fluctuations generated during the precedingchaotic inflation stage may affect the duration of new inflation.One can estimate the fluctuation of the number ofe-folds of slow-roll new inflation, N new ,as13N newN new2 s s.15Here s is the field amplitude when slow-roll approximationbecomes valid, which is nothing but the value of at thepeak of . s is the amplitude of fluctuation at this epochwithin the comoving scale corresponding to the current horizon.It is typically significantly smaller than H(4M Pl )20 1/2 M Pl . For 10 14 we numerically findthat s ranges between s 10 2 M Pl and310 3 M Pl for v0.22M Pl 0.2224M Pl . Thus we findN new 1 for the cases of our interest.Next we evaluate the correction factor, C, to be multipliedto Eq. 4, which accounts for the deviation from the slowrollapproximation. This factor could be important since theinflaton is not necessarily slowly rolling in the beginning ofnew inflation and in the final stage of both chaotic and newinflation. It has been given by Stewart and Lyth 21 asC12ln 2b2, Ḣ ¨, 2H H˙ ,16where b is the Euler-Mascheroni constant and so 2ln 2b0.7296. As a result of numerical calculation, we find0C10.03 until the end of chaotic inflation. In the beginningof new inflation C deviates from unity considerablyand the slow-roll formula 4 tends to overestimate the amplitudeof fluctuation. However, at the peak of fluctuation,C1 is already as small as 0.03. After that Eq. 4 is fairlyaccurate until (r) starts to decline sharply at the end of newinflation. Thus for all our practical purposes the slow-rollformula 4 is sufficient.In summary, we have calculated the spectrum of adiabaticfluctuation generated in the chaotic new inflation model inducedby a Coleman-Weinberg potential. As a result we haveshown that, depending on the choice of the model parameterv, this model can realize an almost scale-invariant spectrumwith a sharp cutoff on a super horizon scale, which may befavored by COBE observation of CMB anisotropy, and atilted spectrum with the spectral index n eff 0.95 which maybe desirable for successful structure formation.This work was partially supported by the Japanese Grantin Aid for Science Research Fund of the Monbusho No.09740334 and ‘‘Priority Area: Supersymmetry and UnifiedTheory of Elementary Particles #707.’’1 For a review of inflation see, e.g., A. D. Linde, Particle Physicsand Inflationary Cosmology Harwood, Chur, Switzerland,1990; K. A. Olive, Phys. Rep. 190, 181 1990; D. H. Lythand A. Riotto, hep-ph/9807278.2 A. H. Guth, Phys. Rev. D 23, 347 1981; K. Sato, Mon. Not.R. Astron. Soc. 195, 467 1981.3 A. D. Linde, Phys. Lett. 108B, 389 1982; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48, 1220 1982.4 S. Coleman and E. Weinberg, Phys. Rev. D 7, 788 1973.5 S. W. Hawking, Phys. Lett. 115B, 295 1982; A. A. Starobinsky,ibid. 117B, 175 1982; A. H. Guth and S-Y. Pi, Phys.Rev. Lett. 49, 1110 1982.6 J. Ellis and G. Steigman, Phys. Lett. 89B, 186 1980.7 A. D. Linde, Phys. Lett. 129B, 177 1983.8 H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 11984.9 E. R. Harrison, Phys. Rev. D 1, 2726 1970; Ya. B.Zel’dovich, Mon. Not. R. Astron. Soc. 160, 1p1972.10 See, e.g., E. Gawiser and J. Silk, Science 280, 1405 1998.11 H. M. Hodges and G. R. Blumenthal, Phys. Rev. D 42, 33291990.12 A. D. Linde, Phys. Lett. 158B, 375 1985; L. A. Kofman andA. D. Linde, Nucl. Phys. B282, 555 1987.13 J. Yokoyama, Astron. Astrophys. 318, 673 1997.14 L. F. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Lett.157B, 361 1985; J. Silk and M. S. Turner, Phys. Rev. D 35,419 1987; R. Holman, E. W. Kolb, S. L. Vadas, and Y.Wang, Phys. Lett. B 269, 252 1991; D. Polarski and A. A.Starobinsky, Nucl. Phys. B385, 623 1992; Phys. Rev. D 50,6123 1994; S. Gottlöber, J. P. Muecket, and A. A. Starobinsky,Astrophys. J. 434, 417 1994.15 J. Yokoyama, Phys. Rev. D 58, 083510 1998.16 B. J. Carr, Astrophys. J. 201, 11975.17 J. H. Kung and R. H. Brandenberger, Phys. Rev. D 42, 10081990.18 Y.-P. Jing and L.-Z. Fang, Phys. Rev. Lett. 73, 1882 1994;L.-Z. Fang and Y.-P. Jing, Mod. Phys. Lett. A 11, 15311996; A. Berera, L.-Z. Fang, and G. Hinshaw, Phys. Rev. D57, 2207 1998.19 F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316 1985.20 C. L. Bennet et al., Astrophys. J. Lett. 464, L11996.21 E. D. Stewart and D. H. Lyth, Phys. Lett. B 302, 171 1993.107303-4