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

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