Scale dependences of local form non-Gaussianity parameters from a DBI isocurvature field
aa r X i v : . [ a s t r o - ph . C O ] A ug Preprint typeset in JHEP style - HYPER VERSION
Scale dependences of local form non-Gaussianity parametersfrom a DBI isocurvature field
Qing-Guo Huang ∗ and Chunshan Lin Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, Beijing 100190, China Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa,277-8583, Japan
Abstract:
We derive the spectral indices and their runnings of local form f NL and g NL from a DBIisocurvature field and we find that the indices are suppressed by the sound speed c s . This effect canbe interpreted by the Lorentz boost from the viewpoint in the frame where brane is moving. Keywords: inflation, non-Gaussianity. ∗ [email protected] ontents
1. Introduction 12. Dynamics and quantum fluctuation of a DBI isocurvature field 23. Scale dependence of local form non-Gaussianity parameters 5
4. Discussions 8
1. Introduction
Non-Gaussianity [1] has become a very important probe to the physics in the early universe. It ishelpful to figure out the mechanism for generating primordial curvature perturbation. A well-definednon-Gaussianity is the so-called local form non-Gaussianity which says that the curvature perturbationcan be expanded to the non-linear orders at the same spatial point ζ ( x ) = ζ g ( x ) + 35 f NL ζ g ( x ) + 925 g NL ζ g ( x ) + ... , (1.1)where ζ g is the Gaussian part of curvature perturbation, f NL and g NL are the non-Gaussianity param-eters which characterize the sizes of local form bispectrum and trispectrum respectively. Single-fieldinflation model predicts f NL ∼ O ( n s − ∼ O (10 − ) [2]. A convincing detection of a large local formnon-Gaussianity is the smoking gun for the multi-field inflation in which the quantum fluctuations ofthe isocurvature field contribute to the final curvature perturbation on the super-horizon scales. See,for example, [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] etc.In the literatures the non-Gaussianity parameters are taken as a constant. However, recentlythere are some hints at a possible scale dependence of f NL come from the numerous observations ofmassive high-redshift clusters [21, 22, 23, 24] which seems in greater abundances than expected froma Gaussian statistics [25, 26]. In fact, g NL may also be scale dependent. The scale dependences of f NL and g NL are measured by their spectral indices n NL and n g NL and their runnings α f NL and α g NL which are respectively defined by f NL ( k ) = f NL ( k p ) (cid:18) kk p (cid:19) n fNL + α fNL ln kkp , (1.2) g NL ( k ) = g NL ( k p ) (cid:18) kk p (cid:19) n gNL + α gNL ln kkp , (1.3)where k p is a pivot scale. The authors in [27] showed that Planck [28] and CMBPol [29] are able toprovide a 1- σ uncertainty on the spectral index of f NL as follows∆ n f NL ≃ . f NL p f sky for Planck , (1.4)– 1 –nd ∆ n f NL ≃ .
05 50 f NL p f sky for CMBPol , (1.5)where f sky is the sky fraction. The effects on the halo bias from the scale dependent f NL are discussedin [30, 31]. The studies on fingerprints of the scale-dependent g NL in CMB and large-scale structureare called for in the near future.Actually the scale-independent non-Gaussianity parameters are not generic predictions of inflationmodel. The large scale dependences of non-Gaussianity parameters can be obtained in the axion N-flation with different decay constants for different axion fields [32] and in the model with self-interactingcanonical isocurvature field [33, 34, 35, 36, 37, 38, 39]. It is interesting for us to ask how the resultsshould be modified for the non-canonical isocurvature field. In particular, we are curious that thescale dependence is enhanced or suppressed by the sound speed. For simplicity, we focus on the casewith a DBI isocurvature field.Our paper is organized as follows. In Sec. 2 we will discuss the dynamics and quantum fluctuationof a DBI isocurvature field. In Sec. 3 the spectral indices of f NL and g NL from a DBI isocurvaturefield will be calculated. Some discussions are contained in Sec. 4.
2. Dynamics and quantum fluctuation of a DBI isocurvature field
Brane inflation [40] is considered to be a popular inflation model embedded into string theory. A morerealistic setup is that a D3-brane moves in the internal six-dimensional Calabi-Yau manifold whichcontains one or more throats in Type IIB string theory [41]. For simplicity, we consider a D3-branewhich is mobile in a throat whose metric is given by ds = h − / g µν dx µ dx ν + h / (cid:2) dr + b ( r ) dθ + ... (cid:3) , (2.1)Here θ is an angular coordinate which is transverse to the radius direction r and b ( r ) is the radius ofthroat at r . In this paper, we choose signature ( − , + , + , +). The action for the mobile D3-brane isgiven by S = − T Z d x (cid:20)q − det (cid:2) h − / ( g µν + h∂ µ r∂ ν r + hb ∂ µ θ∂ ν θ ) (cid:3) − h − (cid:21) − Z d x √− gV ( r, θ ) . (2.2)It is convenient to define two canonical fields in the slow-rolling limit as follows φ = p T r, (2.3) σ = p T bθ. (2.4)Here we consider ˙ σ ≪ ˙ φ , and then φ and σ are taken to be the adiabatic and entropic direction duringinflation respectively.In the limit of f ˙ φ ≪ f ˙ σ ≪
1, these two-field inflation is reduced to the canonical case,where f = h/T . In this paper we focus on another limit in which c s ≃ q − f ˙ φ ≪ , (2.5)and ˙ σ / ˙ φ ≪ c s . (2.6)– 2 –rom the above action the equations of motion for φ and σ are roughly given by¨ φ + 3 H (1 − κ/
3) ˙ φ ≃ f ∂f∂φ − c s ∂V ( φ, σ ) ∂φ , (2.7)3 H ˙ σ ≃ c s (cid:18) η b c s H σ − ∂V ( φ, σ ) ∂σ (cid:19) , (2.8)where κ ≡ ˙ c s Hc s , (2.9) η b ≡ ˙ bHb , (2.10)and both κ and η b are assumed to be much smaller than unity. For simplicity, the cross-couplingbetween φ and σ is assumed to be negligibly small, and Eq. (2.8) is valid when (cid:12)(cid:12)(cid:12)(cid:12) V ′′ ( σ )3 H (cid:12)(cid:12)(cid:12)(cid:12) ≪ c − s . (2.11)This is the slow-roll condition for the isocurvature field σ . Now the dynamics of σ becomes3 H ˙ σ ≃ − c s (cid:2) ˜ m σ + V ′ ( σ ) (cid:3) , (2.12)where ˜ m ≡ − η b H /c s . (2.13)The slow variation of the radius of throat induces an effective mass for the isocurvature field alongthe angular direction.The quantum fluctuations of φ and σ have been well studied in [42]. See also [18, 43, 44]. Here wedon’t want to repeat the computations. We will only briefly recall the results in [42]. The canonicallynormalized quantum fluctuations of φ and σ are respectively given by v φ = ac / s δφ, v σ = a √ c s δσ, (2.14)whose Fourier modes corresponding to the Minkowski-like vacuum on very small scales are given by v φ,k ≃ v σ,k ≃ √ kc s e − ikc s χ (cid:18) − ikc s χ (cid:19) , (2.15)where χ = Z dta ( t ) (2.16)is the conformal time. The perturbation mode of k exits horizon at the time of c s k = aH . Thereforethe power spectra for δφ and δσ are P δφ ≃ (cid:18) H π (cid:19) , (2.17) P δσ ≃ (cid:18) H πc s (cid:19) . (2.18)– 3 –he amplitude of δσ is amplified by a factor of 1 /c s compared to the canonical one.Before closing this section, we want to estimate the typical value of σ during inflation. Thequantity of h σ i coming from its quantum fluctuations is h σ i = 1(2 π ) Z | δσ k | d k = 1(2 π ) Z (cid:18) a k + H k c s (cid:19) d k. (2.19)Considering that the physical momentum p is related to k by p = k/a = e − Ht k , we have h σ i = 1(2 π ) Z d pp (cid:18)
12 + H p c s (cid:19) . (2.20)The first term is contributed from vacuum fluctuations in Minkowski space and it can be eliminatedby renormalization. In addition, the physical momenta for the modes of quantum fluctuations weconcern in the time interval between 0 and t are those from c − s H to c − s He − Ht . Therefore h σ i = H π c s Z c − s Hc − s He − Ht dpp = H π c s t. (2.21)It indicates that the quantum fluctuation of the field σ in the inflation epoch can be modeled by arandom walk with step size H/ πc s per Hubble time. However the vacuum expectation value of σ cannot go like t for t → ∞ if σ has a potential. For example, let’s consider the potential of σ is V ( σ ) = 12 m σ σ + λσ n . (2.22)From Eq. (2.12), the dynamics of σ is described by3 H ˙ σ = − c s m σ (1 + nλσ n − /m ) , (2.23)where m = m σ + ˜ m . (2.24)Similar to [45], combining the contributions from quantum fluctuation and classical equation of motion,we have d h σ i dt = H π c s − c s m H h σ i (cid:20) nλm h σ i n/ − (cid:21) . (2.25)The above differential equation approaches a constant equilibrium value, namely h σ i = 3 H π m c s
11 + n s , (2.26)where s ≡ λ h σ i n/ − /m . (2.27)The typical value of σ during inflation is σ ∗ = p h σ i ∼ c − / s which is enhanced for c s ≪ . Scale dependence of local form non-Gaussianity parameters In this paper we consider that the curvature perturbation is generated by the isocurvature field σ atthe end of inflation or deep in the radiation dominated era, and the curvature perturbation can beexpanded to the non-linear orders by using the δN formalism [46]: ζ ( t f , x ) = N ,σ ( t f , t i ) δσ ( t i , x ) + 12 N ,σσ δσ ( t i , x ) + 16 N ,σσσ δσ ( t i , x ) + ... , (3.1)where N ,σ , N ,σσ and N ,σσσ are the first, second and third order derivatives of the number of e-foldswith respect to σ respectively. Here t f denotes a final uniform energy density hypersurface and t i labels any spatially flat hypersurface after the horizon exit of a given mode. Similar to [36, 38], t i isset to be t ∗ ( k ) which is determined by k = a ( t ∗ ) H ∗ for a given mode with comoving wavenumber k . H ∗ denotes the Hubble parameter during inflation from now on.From Eq. (3.1), the amplitude of curvature perturbation is given by∆ R = N ,σ ( t ∗ ) (cid:18) H ∗ πc s (cid:19) . (3.2)The amplitude of Gravitational waves perturbation only depends on the energy scale of inflation asfollows ∆ T = H ∗ π / . (3.3)Here we work on the unit of M p = 1. The scale dependence of gravitational wave perturbation ismeasured by n T which is defined by n T ≡ d ∆ T d ln k = − ǫ H , (3.4)where ǫ H ≡ − ˙ H ∗ H ∗ . (3.5)For convenience, the tensor-scalar ratio r T is introduced to measure the amplitude of gravitationalwaves: r T ≡ ∆ T / ∆ R = 8 c s N ,σ ( t ∗ ) . (3.6)Comparing Eq. (3.1) to (1.1), the non-Gaussianity parameters are given by f NL = 56 N ,σσ ( t ∗ ) N ,σ ( t ∗ ) . (3.7)and g NL = 2554 N ,σσσ ( t ∗ ) N ,σ ( t ∗ ) . (3.8)Following the method in [36, 38], we introduce a new time t r ( > t ∗ ) which is chosen as a time soonafter all the modes of interest exit the horizon during inflation and keep it fixed. The value of σ at t r is related to that at time t ∗ and the time t ∗ by Z σ r σ ∗ dσV ′ eff ( σ ) = − Z t r t ∗ c s ( t )3 H ( t ) dt, (3.9)– 5 –here V eff = 12 ˜ m σ + V ( σ ) . (3.10)The equation of motion in Eq. (2.12) can be rewritten by3 H ˙ σ = − c s V ′ eff . (3.11)Therefore we have ∂σ r ∂σ ∗ (cid:12)(cid:12)(cid:12)(cid:12) t ∗ = V ′ eff ( σ r ) V ′ eff ( σ ∗ ) , (3.12) ∂σ r ∂t ∗ (cid:12)(cid:12)(cid:12)(cid:12) σ ∗ = c s ( t ∗ ) V ′ eff ( σ r )3 H ( t ∗ ) . (3.13)Considering ddt ∗ F ( σ r ) = ∂F ( σ r ) ∂σ r ( ˙ σ ∗ ∂σ r ∂σ ∗ + ∂σ r ∂t ∗ ) (3.14)and Eq. (3.11), one finds dd ln k F ( σ r ) = dH ∗ dt ∗ F ( σ r ) = 0 , (3.15)which implies that F ( σ r ) is scale independent. Taking into account that σ r is a function of σ ∗ , wehave N ,σ ( t ∗ ) = ∂σ r ∂σ ∗ ∂N ( σ r ) ∂σ r , (3.16) N ,σσ ( t ∗ ) = ∂ σ r ∂σ ∗ ∂N ( σ r ) ∂σ r + (cid:18) ∂σ r ∂σ ∗ (cid:19) ∂ N ( σ r ) ∂σ r , (3.17) N ,σσσ ( t ∗ ) = ∂ σ r ∂σ ∗ ∂N ( σ r ) ∂σ r + 3 ∂σ r ∂σ ∗ ∂ σ r ∂σ ∗ ∂ N ( σ r ) ∂σ r + (cid:18) ∂σ r ∂σ ∗ (cid:19) ∂ N ( σ r ) ∂σ r . (3.18)Since ∂N ( σ r ) ∂σ r , ∂ N ( σ r ) ∂σ r and ∂ N ( σ r ) ∂σ r are scale independent, one obtains d ln N ,σ ( t ∗ ) d ln k = c s η σσ , (3.19) d ln N ,σσ ( t ∗ ) d ln k = 2 c s η σσ + c s η N ,σ ( t ∗ ) N ,σσ ( t ∗ ) , (3.20) d ln N ,σσσ ( t ∗ ) d ln k = 3 c s η σσ + 3 c s η N ,σσ ( t ∗ ) N ,σσσ ( t ∗ ) + c s ξ N ,σ ( t ∗ ) N ,σσσ ( t ∗ ) , (3.21)where the slow-roll equation of motion for σ is considered and η σσ ≡ V ′′ eff ( σ ∗ )3 H ∗ , η ≡ V ′′′ eff ( σ ∗ )3 H ∗ , ξ = V (4)eff ( σ ∗ )3 H ∗ . (3.22)From the above results, the spectral indices of ∆ R , f NL and g NL are respectively given by n s ≡ d ln ∆ R d ln k = 1 − ǫ H − κ + 2 c s η σσ , (3.23) More precisely, the perturbation mode with k exits horizon when c s k = a ∗ H ∗ , and then d ln k = (1 − ǫ H − κ ) H ∗ dt ∗ ≃ H ∗ dt ∗ . – 6 – f NL ≡ d ln | f NL | d ln k = c s η N ,σ ( t ∗ ) N ,σσ ( t ∗ ) , (3.24)and n g NL ≡ d ln | g NL | d ln k = 3 c s η N ,σσ ( t ∗ ) N ,σσσ ( t ∗ ) + c s ξ N ,σ ( t ∗ ) N ,σσσ ( t ∗ ) . (3.25)We see that the spectral indices of both f NL and g NL are suppressed by a factor of c s for c s ≪ f NL and g NL are defined by α f NL ≡ dn f NL d ln k = ( κ + 2 ǫ H − c s η σσ − c s η ) n f NL − n f NL , (3.26) α g NL ≡ dn g NL d ln k = ( κ + 2 ǫ H − c s η σσ ) n g NL − n g NL + 1 g NL (cid:20) f NL ( c s η σσ − c s η + n f NL ) n f NL − ξ ξ r T (cid:21) , (3.27)where η = V ′ eff V (4)eff H V ′′′ , ξ = V ′ eff V (5)eff H V (4)eff . (3.28)The spectral indices of f NL and g NL are nice parameters to characterize the scale dependences ofthese two non-Gaussianity parameters only when n f NL and n g NL are much less than unity. In this subsection we will consider a simple example in which the DBI isocurvature field has an effectivepolynomial potential V eff ( σ ) = 12 m σ + λσ n , (3.29)and then η σσ = m H ∗ h n ( n − s i , (3.30) η = η σσ σ ∗ n ( n − n − s/
21 + n ( n − s/ . (3.31)Combing the normalization of curvature perturbation, we find n f NL f NL = sign( N ,σ ) 56∆ R ( c s η σσ ) H ∗ / πc s σ ∗ n ( n − n − s/
21 + n ( n − s/ , (3.32)Here ∆ R is normalized to be 4 . × − by WMAP in [47]. The slow-roll condition for σ is c s η σσ ≪ H ∗ / πc s is the amplitude of quantum fluctuation of σ and it should be less than σ ∗ . For c s η σσ ∼ − , H ∗ / πc s σ ∗ ∼ − and s & − , we have | n f NL f NL | ∼ O (10) which is detectable by PLANCK. In thiscase, if c s . − , η σσ & σ is not less than the Hubbleparameter during inflation.Taking into account the typical value of σ in Sec. 2, the above equation becomes n f NL f NL = sign( N ,σ )( c s η σσ ) / d ( s ) , (3.33)where d ( s ) = 5 √ R s ns/
21 + n ( n − s/ n ( n − n − s/
21 + n ( n − s/ n = 4 and s ≫ d ( s ) ≃ . × , the scale dependence of f NL canbe detected by PLANCK if c s η σσ & . .001 0.01 0.1 1 10 100500100050001 ´ ´ s d H s L Figure 1:
The function of d ( s ). The solid and dashed curves correspond to n = 4 and n = 8 respectively.
4. Discussions
In this paper we calculate the spectral indices of f NL and g NL and their runnings generated by a DBIisocurvature field on the super-horizon scales. We find that the indices are suppressed by the speed ofsound c s . This suppression effect is not surprised and one can understand it from the effect of Lorentzboost. Here 1 /c s is nothing but the Lorentz boost factor. The time interval in the frame where braneis moving is dilated by a factor 1 /c s compared to that in the frame where brane is at rest. Because thecoordinate σ is transverse to the motion direction of brane, the value of σ does not change. Thereforethe velocity of σ is suppressed by a factor c s . That is why there is a factor c s on the right handside of Eq. (2.12). On the other hand, the scale dependences of the non-Gaussianity parameters comefrom the small variation of isocurvature field σ from t k to t k + dk , where t k and t k + dk correspond to thetime when perturbation modes k and k + dk exit horizon during inflation respectively. Therefore thevariation of isocurvature field σ is suppressed by c s , and hence the spectral indices of f NL and g NL are suppressed by c s as well.Even though the scale dependences of non-Gaussianity parameters generated by the DBI isocur-vature field are probably detectable, its effective of mass is required to be comparable or larger thanthe Hubble parameter during inflation if c s ≪ f NL cannot help us to distinguish them. But once the scale dependences of the non-Gaussianity parameters are detected, we can get more information about the isocurvature field, suchas how it interacts with itself. Acknowledgments
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