PHYSICAL REVIEW D 78, 043515 (2008)
Spectral index in curvaton scenario
School of physics, Korea Institute for Advanced Study, 207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, Korea (Received 28 May 2008; published 7 August 2008) A red-tilted primordial power spectrum is preferred by WMAP ﬁve-year data and a large positive localtype non-Gaussianity fNL might beobserved as well. In this short paper we ﬁnd that a red-tilted and large non-Gaussian primordial power spectrum cannot be naturally obtained in the curvaton model, because fNL is related to the initial condition of inﬂation.
DOI: 10.1103/PhysRevD.78.043515 PACS numbers: 98.80.Cq
Inﬂation  provides an elegant mechanism to solve many puzzles in the hot big bang model. The wrinkles in the cosmicmicrowave background (CMB) radiation and the large-scale structure of the Universe are seeded by the quantum ﬂuctuations generated during inﬂation . The shape of the primordial quantum ﬂuctuations is characterized by its amplitude P and tilt ns which can be measured by experiments. Wilkinson Microwave Anisotropy Probe (WMAP) ﬁve-year data  combined with the distance measurements from the TypeIa supernovae (SN) and the baryon acoustic oscillations (BAO) in the distribution of galaxies indicate P;obs ¼ 2:457þ0:092 Â 10À9 ; À0:093 ns ¼ 0:960þ0:014 : À0:013 (1) (2)
equil local À9 < fNL < 111 and À 151 < fNL < 253 ð95%CLÞ;
(4) À1 < 0:0037 ð95%CLÞ; (5)
where ‘‘local’’ and ‘‘equil’’ denote the shapes of the nonGaussianity. In  the authors reported that a positivelarge non-Gaussianity
local 27 < fNL < 147
Gravitational wave perturbations are also generated during inﬂation and its amplitude PT is only determined by the inﬂation scale. For convenience, we deﬁne a new quantity, named tensor-scalar ratio r ¼ PT =P , to measure the amplitude of gravitational wave perturbations. The primordial gravitational wave perturbation has not been detected. Thepresent limit on the tensor-scalar ratio is r < 0:20 (95% CL). The blue tilted primordial power spectrum ðns > 1Þ is disfavored even when gravitational waves are included. The non-Gaussianity is characterized by the nonGaussianity parameter fNL which is deﬁned as follows ÈðxÞ ¼ ÈL ðxÞ þ fNL ½È2 ðxÞ À hÈ2 ðxÞi; L L (3) where ÈL ðxÞ denotes the linear Gaussian part of the perturbation in realspace. The simplest model of inﬂation predicts a closely Gaussian distribution of primordial ﬂuctuations [4,5], namely jfNL j < Oð1Þ. The most general density perturbation is a superposition of an isocurvature density perturbation and an adiabatic density perturbation. 2 2 We introduce a new parameter À1 ¼ fiso =ð1 þ fiso Þ to measure the isocurvature density perturbation, where fiso is the ratio ofthe isocurvature and adiabatic amplitudes at the pivot scale. A Gaussian and adiabatic power spectrum of primordial perturbation is still consistent with WMAP ﬁveemail@example.com
is detected at 95% C.L.. A large non-Gaussianity is not a conclusive result from experiments, but it is still worthy for us to discuss the theoretical probabilities of the large nonGaussianity. If it is conﬁrmed bythe forthcoming cosmological experiments, it strongly shows up some very important new physics of the early Universe. An attractive model for a large positive local-type nonGaussianity is the curvaton model [7,8] in which the primordial power spectrum is generated by a light scalar ﬁeld, called curvaton , but not the inﬂaton , even though the dynamics of inﬂation is governed by the inﬂaton.Recently many issues about the curvaton model were discussed in [9–12]. In  we considered the case in which the Hubble parameter is roughly a constant during inﬂation and found that fNL is bounded by the tensor-scalar ratio r from above. Or equivalently a large non-Gaussianity gives a lower bound on the amplitude of the tensor perturbation in curvaton scenario. In  the authors pointed out that...
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