Scale Dependent Local Non-Gaussianity from Loops
PPreprint typeset in JHEP style - HYPER VERSION
UCI-TR-2009-15, UH511-1141-09, MIFP-09-37, arXiv:0909.2040
Scale Dependent Local Non-Gaussianity from
Loops
Jason Kumar , Louis Leblond , , Arvind Rajaraman Department of Physics and Astronomy, University of Hawaii,Honolulu, HI 96822, USA George P. & Cynthia W. Mitchell Institute for Fundamental Physics,Texas A&M University, College Station, TX 77843, USA Perimeter Institute, 31 Caroline St, Waterloo, On, N2L 2Y5, Canada Department of Physics and Astronomy, University of California,Irvine, CA 92697, USA
Abstract:
We analyze multi-field inflationary systems which yield strongly scaledependent non-Gaussianity with a shape that is very close to the local shape. Asin usual multi-field models, the non-Gaussianity arises from the non-linear transferof scalar field fluctuations to curvature perturbations. Here we consider modelsin which higher order terms (loops) dominate over the lowest order source of non-linearity. The magnitude of non-Gaussianity depends on an infrared cutoff which isdetermined by our observational probes measuring non-Gaussianity. In our models,the running is positive and large ( n NG ∼ .
2) on CMB scales. The magnitude of thebispectrum is maximally of order O (100), and grows on small scales. This can leadto interesting signals for large scale structure. Keywords:
Effective Field Theory, Cosmology, Inflation. a r X i v : . [ a s t r o - ph . C O ] S e p ontents
1. Introduction 12. Scale Dependence from Loops 33. Multi-Field Model 4
4. The Power Spectrum 95. Higher Point Functions 10
6. Conclusions 15A. A Specific Model 17B. Numerical Evaluation of the Integral 19
1. Introduction
With the advent of precise cosmological data, it is now possible to constrain models ofinflation by the measured magnitude and scale-dependence of correlated temperatureperturbations in the cosmic microwave background (CMB) and from tracking densityperturbations in dark matter from measuring the Large Scale Structure (LSS) ofour universe. In these observations, it is found that the primordial perturbationscoming from inflation are Gaussian to a remarkable accuracy, in agreement with thepredictions of most single field models of inflation.Non-Gaussianity (NG) can be quantified by the magnitude of the bispectrumdenoted f NL (this is usually quoted at the equilateral point in momentum spacewhere all three momenta are equal). For most slow-roll models, f NL is smaller than1 [1, 2]. By comparison, the most recent constraints from WMAP5 [3] data are − < f NL <
80 for the local shape and − < f equiNL <
435 for the equilateralshape [4]. The Planck satellite is expected to improve the bounds to ∆ f NL < . The running is positive(or blue which means that the NG grows as k increases) and can be achieved whilekeeping the power spectrum nearly scale invariant. It arises from loops (or higherorder terms in the local ansatz) and the shape of the bispectrum is very well ap-proximated by the local shape multiplied by a logarithm. We provide a consistentsetup where the 1-loop effect dominates the bispectrum while giving a subdominantcontribution to the power spectrum, and where higher loop contributions can be ne-glected. Since the running is positive, we can engineer a set-up where the curvatureperturbation on CMB scales are extremely Gaussian while having a detectable NGon LSS scales.Running NG has already been considered in the context of DBI inflation [13, 14].This model can have a strong NG signal due to a small and varying sound speed forthe inflaton fluctuations [15]. The amplitude of the 3-pt can strongly run with scaleif the sound speed varies but the running of the sound speed is exactly cancelled bythe quickly varying Hubble constant along the trajectory. This is the key point ofthis type of model where the potential is steep but the inflaton moves slowly becauseof a speed limit. This causes the power spectrum to be scale invariant while thebispectrum can run wildly [16, 17].The prospect of detecting large NG with large scale structure data has spurredmuch activity recently. LoVerde et al [18] have examined the possibility of usingcluster counts and the galaxy bispectrum to constrain running f NL . It was alsorealized in [19, 20], that NG of the local shape can induce a scale dependence of thegalaxy/halo bias (see also [21, 22, 23, 24, 25, 26]). This effect can be easily found inthe data and it results in a competitive bound on NG with local shape − < f NL <
70 [21]. At the time of this writing, there exists no significant experimental bound onthe running of NG with scale. Recently, Sefusatti et al [27] argued that Planck couldbound n NG , the running of non-Gaussianity, with a precision ∆ n NG ∼ . .
3) fora local (equilateral) shape of non-Gaussianity.In our models, we find NG with a (nearly) local shape with a scale dependencesuch that the NG signal grows on small scales. The magnitude of the bispectrumgrows with k with a model independent running of n NG ∼ . . n s . Wefind that f NL ∼
100 can be achieved in principle. We also calculate the trispectrum τ NL , which also runs. Before getting into the details, we summarize the basic ideaand results. There has been much recent work in calculating the bispectrum and trispectrum in multi-fieldinflation, for some recent references see [6, 7, 8, 9, 10, 11, 12]. – 2 – . Scale Dependence from Loops
Local shape NG can be obtained in multi-field models of inflation, where each field isGaussian but a non-linear relation between the inflaton perturbations and curvatureperturbations induces NG. The original definition of the local ansatz for the curvatureperturbation was done in real space [28] ζ ( (cid:126)x, t ) = ζ Gauss + 35 f NL ( ζ Gauss − (cid:10) ζ Gauss (cid:11) ) , (2.1)where ζ Gauss is the Gaussian piece of the curvature perturbation. f NL in this formulais by definition scale invariant. In momentum space, the above ansatz leads to thefollowing bispectrum (cid:10) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:11) = 35 f NL (cid:10) ζ (cid:126)k ζ (cid:126)k ( ζ (cid:63) ζ ) (cid:126)k (cid:11) = (2 π ) δ ( (cid:88) (cid:126)k i ) 310 f NL ( P ζ ) (cid:80) k i (cid:81) k i , (2.2)where ( ζ (cid:63) ζ ) (cid:126)k denotes a convolution, P ζ is the power spectrum (which is assumed tobe scale invariant, for simplicity) and P k i Q k i defines the local shape. Many multi-fieldmodels (such as curvatons [29, 30]) have local scale invariant NG of this type. TheNG can also be scale dependent even if the shape is nearly local; for example, thisis expected to happen when the NG is generated throughout the whole trajectory asopposed to simply at some fixed later time, such as in curvaton models. A particularmodel with this feature was considered by Byrnes et al [31, 32], where the scale-dependence arises from the dependence of f NL on the (time-dependent) slow-rolland Hubble parameters. In their case, the NG decreases on small scales.We instead look for scale dependence coming from loops and higher order terms.Indeed, it was realized early on [33] that an additional contribution to the bispectrumin the ansatz Eq. (2.1) comes from (cid:10) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:11) = (cid:18) f NL (cid:19) (cid:10) ( ζ (cid:63) ζ ) (cid:126)k ( ζ (cid:63) ζ ) (cid:126)k ( ζ (cid:63) ζ ) (cid:126)k (cid:11) . (2.3)This higher order contribution to the bispectrum has a structure similar from a loopcontribution as it involves an integral over internal momenta. The integral convergesin the UV but contains IR divergences if the power spectrum is nearly scale invariant.One can ‘regulate’ this divergence by introducing an IR cutoff in momenta 1 /L . These loops have been called c-loops [34]. They must not be confused with q-loops, or loopscoming from the expansion of the quantum evolution operator prior to horizon crossing [35]. Therehas been much discussion recently on the physical significance of the IR divergences in loop calcula-tion in inflation. For c-loops, this IR cutoff is physical and depends on the observational probe andon how we measure the zero mode of curvature perturbations. We will justify this point of view inmore detail in Sec. (3.1). – 3 –oing so, the shape of this term is close to local up to a log [33, 36] (cid:10) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:11) ∝ ln(Min[k i ]L) (cid:80) k (cid:81) k . (2.4)If this term dominates the bispectrum, we will have a scale dependence with a runningof order n NG ∼ kL . As we will show later, the cutoff L is well approximated by thesize of the universe today such that ln kL ∼ n NG ∼ .
3. Multi-Field Model
A simple way to move beyond single field slow-roll and generate NG is to havemultiple fields. This type of model can quickly become very complicated and inorder to simply illustrate the main physical effect of interest (namely large scaledependent NG from loops), we will consider a very simplified set-up. More generalmodels and in-depth analysis of the model we present is left for future work. Considera model of hybrid inflation with two real light scalar fields ( φ and χ ) and a waterfallfield T which ends inflation when it becomes tachyonic and condenses. In this paper,we will consider a rather general action, a more detailed and worked example is givenin Appendix A. The action is (we follow the notation of [40]): S = 12 (cid:90) √ g [ M p R − ( ∂φ ) − ( ∂T ) − ( ∂χ ) − V ] ,V = V inf ( φ ) + V hid ( χ ) + V mess ( φ, χ, T )] . (3.1)– 4 –he only coupling between φ and χ are through the tachyon which acts as a mediatoror messenger. The form of V mess is taken to be V mess ∝ T f ( φ, χ ) + O ( T n ) ; n > . (3.2)The function f interpolates from large and positive values (in Hubble units) duringinflation to negative values after the system crosses a critical line in field space.Therefore during inflation, T has a large positive mass, its vev is driven to zero andits potential vanishes. Because of its large mass, this field will not fluctuate and itcan be integrated out of the theory. In this model, inflation ends suddenly when themass of the tachyon vanishes, which occurs on a line in field space parameterized by f ( φ e , χ e ) = 0 , (3.3)where the index “ e ” denotes the value of the fields at the end of inflation. During theinflationary phase, φ and χ have no direct coupling. To simplify further, we assumethat V hid ( χ ) (cid:28) V inf ( φ ) and we refer to φ as the inflaton from now on. The Hubblescale is then approximately given by H ≈ V inf M p (3.4)and χ is a “hidden” field during inflation which fluctuates but without much impacton the total energy density of the Universe. Nevertheless, its quantum fluctuationsare still important as they will be felt as ripples on the surface of reheating. Indeed,at different point in space, the (slightly) different value of χ e will mean differentcritical value φ e for the inflaton resulting in more or less inflation in these differentregions. This correlates directly in curvature perturbations (See Fig. (1)) Since thequantum perturbations of χ mainly affect the surface of reheating, this system iswell amenable to analysis through the δN (or separable universe) formalism [46].The idea is that the curvature perturbation on large scales is simply given by theperturbation in the number of efolds for each trajectories ζ ( (cid:126)x, t ) = δN ( (cid:126)x, t ) , (3.5)where the curvature perturbation ζ is given by fluctuations of the scale factor a ( (cid:126)x, t ) = a ( t ) e ζ ( (cid:126)x,t ) and the difference in number of efolds is from a initial flat hypersurfaceto a uniform energy density final hypersurface. This formula does not take into ac-count possible interactions between the various fields inside the horizon (on smallscale) and it is only valid after horizon crossing where the evolution of the curvatureperturbation is classical . The δN formalism will not account correctly for multi-field effects for modes inside the horizon.In our case, because the fields are uncoupled during inflation, we can solve for δφ and δχ are horizonexit independently and follow the subsequent evolution of ζ with the δN formalism. – 5 – igure 1: This figure depicts the trajectory in field space. The blue (dashed) line denotethe surface of reheating defined by f ( φ e , χ e ) = 0 and it is assumed to be thin. The classicaltrajectory is in the φ direction (red/dotted line) but both δφ and δχ will induce curvatureperturbations. The surface where inflation ends Eq. (3.3) is not a uniform energy density hy-persurface and a correction term must be included as discussed in [47, 44]. Thecorrection term is very small in the hybrid scenario where the potential is very flatand it will be dropped in what follows. The number of efolds is given by dN = − Hdt .For the case where the classical trajectory is determined by a single field φ , one has N = − (cid:90) φ e ( χ ) φ ∗ H ˙ φ dφ (cid:48) , (3.6)where the critical value of φ depends on the value of the field χ at the end ofinflation (we dropped the subscript e and χ = χ e unless otherwise specified ) and ∗ refers to horizon crossing for a given mode. By varying φ ∗ → φ ∗ + δφ and then φ e ( χ + δχ ) = φ e + γδχ + γ ,χ δχ / · · · with γ ( χ ) = ∂φ e ∂χ (3.7)where we denote the zero mode of χ by χ , that is χ ( (cid:126)x, t ) = χ ( t ) + δχ ( (cid:126)x, t ) (fornotational simplicity, the bar is omitted in any derivative subscript). We get atsecond order (using H ˙ φ = − N (cid:48) ) δN = N (cid:48) δφ (cid:12)(cid:12) ∗ − N (cid:48) γδχ (cid:12)(cid:12) e + 12 N (cid:48)(cid:48) δφ (cid:12)(cid:12) ∗ − N (cid:48) γ ,χ δχ (cid:12)(cid:12) e − N (cid:48)(cid:48) γ δχ (cid:12)(cid:12) e , (3.8)where (cid:48) denotes derivatives with respect to φ . This can be reproduced using theformula of Vernizzi and Wands [47], for the case (cid:15) χ (cid:28) (cid:15) φ albeit they implicitly The field χ is evolving stochastically and the value of the field at the end of inflation is the sumof all fluctuations created for each mode as they exit the horizon. – 6 –ssume that all fields obey their equation of motion which is not true here for thefield χ . It is simple to show that N (cid:48) = ∂N/∂φ = 1 / √ (cid:15) φ M p where the slow-rollparameters are (cid:15) φ = 12 M p (cid:18) V inf ,φ V (cid:19) , (cid:15) χ = 12 M p (cid:18) V hid ,χ V (cid:19) . (3.9)The terms with N (cid:48)(cid:48) involve derivatives of slow-roll parameters and will therefore besuppressed. To simplify the formula and the analysis we will consider the case wherethe slow-roll parameter at horizon crossing and at the end are equal, (cid:15) φe = (cid:15) φ ∗ . Thisis not true in many models and we will discuss at the end how that would affect ourresults. We thus drop all subscript referring to the time of evaluation. The mean ofEq. (3.8) is non-zero and as it is we will generate a one-pt function. To ensure thatthe mean is zero we can subtract a constant piece (keeping only the leading terms) ζ = N (cid:48) δφ − N (cid:48) γδχ − N (cid:48) γ ,χ δχ + 12 N (cid:48) γ ,χ (cid:10) δχ (cid:11) , (3.10)which is of the form Eq. (2.1). ζ = ζ + ζ − (cid:104) ζ (cid:105) . (3.11)Note that this series terminates if1. the function γ is such that γ ,χχ and higher derivatives are small.2. N (cid:48)(cid:48) and higher derivative contributions are small.In this type of model, the function γ could be anything and in the case where γ ,χ δχ > γ the quadratic piece in δχ will dominate over the linear piece (in δχ ) whichensures that the loop contribution to the bispectrum will dominate (cid:10) ζ (cid:11) ∝ γ ,χ (cid:10) ( δχ ) (cid:11) , (3.12)as we advocated earlier. In order for the power spectrum to be nearly scale invariantwe will still need the δφ piece to be the dominant contribution to the power spectrum.There is no contradiction since the linear perturbation in φ does not contribute to thebispectrum (or gives a very small slow-roll suppressed contribution). Furthermore, inthe case where the higher derivatives of γ are suppressed, the higher loop contributioncan be neglected, ensuring a consistent truncation.Another important point is that for the loop to dominate, the zero mode of χ atthe end of inflation ( χ e which is the mean averaged over the size of the universe atthe end of inflation) must be smaller then the 1- σ deviation value of the perturbationaround the mean. Taking the quantum perturbation to be of order δχ ∼ H , we musthave χ e < δχ . This is better seen in a specific model such as the one presented in– 7 –ppendix A. There, we use a model where φ e = f ( χ ) such that γ ∝ χ and γ ,χ ∼ cstand the series truncate. In such models it is clear that the quadratic term dominateover the linear piece when γ ,χ δχγ ∼ δχχ > . (3.13)It is then clear that the field χ has to behave stochastically and is in no way followinga classical equation of motion. The fact that χ has essentially no effect on theinflationary dynamics prior to the reheating tells us that the stochastic behavior isunimportant during inflation. The value of χ e , being stochastic, could have any valueand it is therefore a free parameter. Before going into more details of the calculation,we need to discuss the choice of IR cutoff in the loop calculation. There has been much discussion in the literature about the choice of cutoff thatshould be used in loop calculations. For the calculation of quantum loops in thein-in formalism (prior to horizon exit), the correlations function of scalars appearto be sensitive to this choice of cutoff, and there is no clear understanding of howthis cutoff should be set. But for the c-loops which we consider in this paper, thesituation is considerably simpler and there is a natural choice of cutoff [48]. We willdefine the observed zero modes of the fields φ, χ as φ = 1 L (cid:90) L/ − L/ d xφ , χ = 1 L (cid:90) L/ − L/ d x χ , (3.14)where L is the largest scale over which we have measured the fields. The perturba-tions of the fields are then defined as δφ = φ − φ , δχ = χ − χ .When computing correlators of δN , we are actually interested in the correlationsfunctions of the perturbations e.g. (cid:104) δφ (cid:126)k δφ (cid:126)k (cid:105) . From the definition of the perturba-tions, we see that the effect of subtracting the zero mode is to remove all Fouriermodes with momentum k > L − . Hence δφ (cid:126)k = φ (cid:126)k for k > L − , and zero otherwise.Similarly, we find (cid:104) δφ (cid:126)k δφ (cid:126)k (cid:105) = (cid:40) (2 π ) δ ( (cid:126)k + (cid:126)k )2 π P ∗ k k > L − k < L − . (3.15)The effect is to include a cutoff L − on any momentum integral. Due to the cutoff,the correlation functions will have an explicit dependence on L . This can be tracedback directly to the fact that we are calculating correlation functions of perturbationslike δφ = φ − φ , which have a direct dependence on L through φ . In this formalism,it is clear that all the dependence on L comes from the variation in the zero modeas a function of L as was discussed in more details in [48] (see also [49]).– 8 –o summarize, there is a natural cutoff L determined by the biggest scale onwhich we are able to measure the background zero mode of curvature. This is max-imally the size of the universe today L ∼ /H . This coincides with the lowest k perturbations that are possible to observe now. Since there are about 5 efolds be-tween when the lowest observable wavenumber leaves the horizon and when CMBscales leave the horizon, we have k CMB L ∼ e . LSS are about two orders of magni-tude greater than CMB scales, giving k LSS L ∼ e .
4. The Power Spectrum
We will first consider the two-point function (cid:104) ζ k ζ k (cid:105) . For the scalar fields, we have (cid:10) δχ (cid:126)k (cid:11) = (cid:10) δφ (cid:126)k (cid:11) = (2 π ) δ ( (cid:88) i (cid:126)k i ) P ( k ) ,P ( k ) = 2 π P k , (4.1)and we consider a model where these expectation values are approximately constantand where P is scale invariant (independent of k ). We will also assume that anyintrinsic 3-pt functions are negligible, (cid:104) ( δφ k ) n (cid:105) ≈ (cid:104) ( δχ k ) n (cid:105) ≈ n odd. InFourier space the curvature perturbation is given by (from Eq. (3.10)) ζ (cid:126)k = N (cid:48) δφ (cid:126)k − N (cid:48) γδχ (cid:126)k − N (cid:48) γ ,χ (cid:90) d (cid:126)k (cid:48) (2 π ) δχ (cid:126)k − (cid:126)k (cid:48) δχ (cid:126)k (cid:48) + 12 N (cid:48) γ ,χ (cid:10) δχ (cid:126)k (cid:11) , (4.2)The “tree-level” contribution to the power spectrum arises from linear terms in theexpansion of δN , and it is easily seen to give (cid:10) ζ (cid:126)k (cid:11) tree = N (cid:48) ( (cid:10) δφ (cid:126)k (cid:11) + γ (cid:10) δχ k (cid:11) ) , = N (cid:48) (1 + γ )(2 π ) δ ( (cid:88) i (cid:126)k i ) P ( k ) . (4.3)However, there is also a “one-loop” contribution which arises from the non-linearterms in the δN expansion which leads to (cid:10) ζ (cid:126)k ζ (cid:126)k (cid:11) loop = N (cid:48) γ ,χ (cid:90) d (cid:126)k (cid:48) (2 π ) d (cid:126)k (cid:48)(cid:48) (2 π ) (cid:10) δχ (cid:126)k − (cid:126)k (cid:48) δχ (cid:126)k (cid:48) δχ (cid:126)k − (cid:126)k (cid:48)(cid:48) δχ (cid:126)k (cid:48)(cid:48) (cid:11) , = N (cid:48) γ ,χ π ) δ ( (cid:88) i (cid:126)k i ) (cid:90) d k (cid:48) (2 π ) (2)(2 π P ) | (cid:126)k − (cid:126)k (cid:48) | k (cid:48) , (4.4)where the factor of (2) is from the combinatorics. For a scale invariant power spectra P , the integral is approximately (cid:90) k /L d (cid:126)k (cid:48) (2 π ) | (cid:126)k − (cid:126)k (cid:48) | k (cid:48) , (4.5)– 9 –here we use k as the upper limit because for k (cid:48) > k the denominator goes as k (cid:48) n with n >
3, and the integrand drops rapidly. The integrand has two simple poleswhich give logarithmic divergences. We regulate these by putting an IR cutoff onthe integral. Hence for this example, we get (cid:90) k /L d (cid:126)k (cid:48) (2 π ) | (cid:126)k − (cid:126)k (cid:48) | k (cid:48) ∼ kL )2 π . (4.6)This contribution will depend on the IR limit of the momentum integration. Thislimit is given by the size of the observable universe today, L ∼ H − as we discussedin Sec. (3.1). Modes of longer wavelength are already summed in the backgroundvalue of the field. We thus find (cid:10) ζ (cid:126)k (cid:11) loop = N (cid:48) γ ,χ (2 π ) δ ( (cid:88) i (cid:126)k i ) 2 π P ln( kL ) k . (4.7)Combining these terms yields (cid:10) ζ k (cid:11) = (2 π ) δ ( (cid:88) i (cid:126)k i ) 2 π P ζ k , (4.8)= N (cid:48) (2 π ) δ ( (cid:88) i (cid:126)k i ) P (cid:2) γ + γ ,χ P ln( kL ) (cid:3) . (4.9)We have defined the power spectrum for curvature with the superscript ζ . Thespectral index n s − d ln P ζ d ln k is n s − γ ,χ P γ + γ ,χ P ln kL . (4.10)Note that the log contribution is positive (blue) and if this is the only contribution,we cannot match to the currently observed value of n s ∼ .
96 [3]. For now, we simplyimpose that the log contribution contribute no more than a percent correction to n s γ ,χ P < ∼ − , (4.11)which in turn implies that the non-linear contribution to the 2-point function mustbe subleading if log( kL ) ∼
5. Higher Point Functions
We now compute the 3-point function (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) . Again, we find that this correlationfunction can easily be computed by expanding δN in terms of δφ and δχ . Since δφ – 10 –nd δχ are Gaussian fields, the only non-trivial contributions will come from non-linearities in the δN expansion. As in the case of the 2-point function, there is anatural separation into “tree-level” and “loop” contributions [50]. The contributionwhich is of lowest order in γ ,χ is (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) tree = − γ N (cid:48) γ ,χ γ (3) (cid:90) d (cid:126)k (cid:48) (2 π ) (cid:104) δχ (cid:126)k δχ (cid:126)k δχ (cid:126)k − (cid:126)k (cid:48) δχ k (cid:48) (cid:105) , = − γ N (cid:48) (2 π ) δ ( (cid:88) i (cid:126)k i ) γ ,χ γ (2 π P ) (cid:80) i k i (cid:81) i k i . (5.1)The next term in the γ ,χ expansion is (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) loop = − γ N (cid:48) γ ,χ γ (cid:90) d (cid:126)k (cid:48) d (cid:126)k (cid:48)(cid:48) d (cid:126)k (cid:48)(cid:48)(cid:48) (2 π ) (cid:104) ( δχ (cid:126)k − (cid:126)k (cid:48) δχ (cid:126)k (cid:48) )( δχ (cid:126)k − (cid:126)k (cid:48)(cid:48) δχ (cid:126)k (cid:48)(cid:48) )( δχ (cid:126)k − (cid:126)k (cid:48)(cid:48)(cid:48) δχ (cid:126)k (cid:48)(cid:48)(cid:48) ) (cid:105) , = − γ N (cid:48) (2 π ) δ ( (cid:88) i (cid:126)k i ) 18 γ ,χ γ (cid:90) d (cid:126)k (cid:48) (2 π ) (cid:32) (2 π P ) k (cid:48) | (cid:126)k + (cid:126)k (cid:48) | | (cid:126)k − (cid:126)k (cid:48) | + 7 perms (cid:33) , = − γ N (cid:48) (2 π ) δ ( (cid:88) i (cid:126)k i ) 18 γ ,χ γ (2 π P ) B ( (cid:126)k , (cid:126)k , (cid:126)k ) . (5.2)Now the loop integral involves two different momenta B ( (cid:126)k , (cid:126)k , (cid:126)k ) = (cid:90) d (cid:126)k (cid:48) (2 π ) (cid:32) k (cid:48) | (cid:126)k + (cid:126)k (cid:48) | | (cid:126)k − (cid:126)k (cid:48) | + 7 perms (cid:33) . (5.3)Diagrammatically this is equivalent to a triangular loop of scalars (see Fig. (2)). Wenote that near the poles at (cid:126)k (cid:48) = 0 , (cid:126)k , − (cid:126)k , we get logarithmic divergences which arecut off by the IR scale L . This logarithmic dependence breaks scale invariance. Soour shape B is a function of three variables which we choose to simply be the normof all three vectors k , k , k . An estimate of the shape can be obtained by simplyevaluating the integral around each poles, cutting off the momentum integration inthe infrared at scale 1 /L . So for example, the integrand (cid:90) d (cid:126)k (cid:48) (2 π ) k (cid:48) | (cid:126)k + (cid:126)k (cid:48) | | (cid:126)k − (cid:126)k (cid:48) | (5.4)has a pole around (cid:126)k (cid:48) = 0, and the integrand falls off rapidly when k (cid:48) becomes of thesame order as k or k . Hence we can approximate the integral around that pole as (cid:90) d (cid:126)k (cid:48) (2 π ) k (cid:48) | (cid:126)k + (cid:126)k (cid:48) | | (cid:126)k − (cid:126)k (cid:48) | = ln(Min(k , k )L)2 π k k + · · · . (5.5)The same thing can be done for the other poles and for the various permutations.There are also points in parameter space where the integrand has a pole of order– 11 – igure 2: The 1-loop diagram. In our case, each vertex is accompanied by a factor of N (cid:48) γ ,χ while each internal propagator is given by π P p . More detailed Feynman rules foruse with the δN expansion (which we are not carefully describing here) can be found in[51].
4. These poles occur in the squeezed limit where (cid:126)k = − (cid:126)k and hence (cid:126)k → (cid:126) k to a resolution ∼ /L and thebispectrum, while large, is finite and of order L / (3 k i ) in this limit. Hence thestronger poles that we have neglected are only important in the squeezed limit andthey give contributions of the same order as the log terms in that limit. The fullapproximative shape is B ( k , k , k ) ≈ π (cid:18) ln(Min(k , k )L) + 1 / k k + 2 perm . (cid:19) . (5.6)The 1 / k ∼ e / L . For larger k theshape is very well approximated by B ( k , k , k ) ≈ π ln(Min(k i )L) (cid:80) i k (cid:81) i k . (5.7)We show numerically in Appendix B that this is a good approximation. In Figure(3), we plotted the shape given by Eq. (5.6) in term of the usual variable x = k /k and x = k /k . When the bispectrum is scale invariant, k is fixed to 1 (arbitrarily)but here we plotted the shape for different value of k . As the figure clearly shows,the graph is very close to local and the magnitude grows as k increases. At theequilateral point k = k = k ≡ k , the loop contribution to the bispectrum simplifies– 12 – igure 3: Plot of the approximate shape B ( k , k x , k x ) x x k (with B ( k , k , k ) givenby Eq. (5.6)) in terms of x = k k and x = k k for k = 0 . k = 1 . k (1 − x ) < k x < k x dueto momentum conservation and to avoid overcounting identical triangle configurations (see[52]). The shape is clearly very close to local with the strongest signal in the squeezed limitwhen k = k x →
0. The overall magnitude of NG increases with the wavenumber k oras we consider smaller wavelengths. to (cid:10) ζ (cid:11) = − γ N (cid:48) (2 π ) δ ( (cid:88) i (cid:126)k i ) γ ,χ γ ln( kL )(2 π ) P k . (5.8)If we compare the standard parameterization for local non-Gaussianities (Eqns. (2.1)and (2.2)) at the equilateral point to Eq. (5.1) and Eq. (5.8) and using the approxi-mation P ζ ≈ N (cid:48) P , we have f NL ≈ − γ γ ,χ N (cid:48) (cid:18) γ ,χ γ ln( kL ) P (cid:19) , (5.9)where the first term is the tree-level contribution, and the second term is the one-loopcontribution.In the case of the two-point function, experimental bounds on the spectral indexrequired the loop-contribution to be subleading. But there is no such requirementfor the bispectrum. The loop contribution will dominate if γ ,χ γ P ln( kL ) > . (5.10)In this limit we have | f NL | ≈
56 ( γ ,χ P ) N (cid:48) P ln( kL ) < ∼
100 ln( kL ) , (5.11)– 13 –here we have utilized the bound γ ,χ P < − and the normalization P / ζ ∼ N (cid:48) P / ∼ − from COBE data. We thus find, in this scenario, that one caneasily generate local non-Gaussianity which is not ruled out by WMAP5 and canpotentially be probed at Planck. Note that the magnitude of the non-Gaussianityincreases logarithmically with momentum, suggesting that non-Gaussianity can havean important impact on the formation of structure at smaller scales. If we define therunning of f NL at the equilateral point n NG = d ln f NL d ln k (cid:12)(cid:12)(cid:12)(cid:12) k i = k (5.12)one gets in the loop dominated limit n NG (cid:39) kL ) . (5.13)In the limit where non-linearities dominate, the running of f NL is thus independentof N (cid:48) , γ and γ ,χ . As in the case of the 3-point function, the only non-vanishing contributions will arisefrom the non-linear dependence of δN on δχ , so we can ignore δφ fluctuations. Tosimplify notation, we define ζ (cid:126)k = Aδχ (cid:126)k + B (cid:90) d (cid:126)k (cid:48) (2 π ) δχ (cid:126)k − (cid:126)k (cid:48) δχ (cid:126)k (cid:48) − B (cid:10) δχ (cid:126)k (cid:11) , (5.14)where A = − N (cid:48) γ and B = − N (cid:48) γ ,χ . The last term ensures that we only keep theconnected part of every diagrams. The tree level contribution (the term of lowestorder in B ) is (cid:10) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:11) = A B (cid:90) d (cid:126)k (cid:48) (2 π ) d (cid:126)k (cid:48)(cid:48) (2 π ) (cid:10) δχ (cid:126)k δχ (cid:126)k δχ (cid:126)k − (cid:126)k (cid:48) δχ (cid:126)k (cid:48) δχ (cid:126)k − (cid:126)k (cid:48)(cid:48) δχ (cid:126)k (cid:48)(cid:48) (cid:11) + 5 perm , = (2 π ) A B δ (cid:16)(cid:88) (cid:126)k i (cid:17) [ P ( k + k ) P ( k ) P ( k ) + 11 perm] , = 4 A B (2 π ) δ (cid:16)(cid:88) (cid:126)k i (cid:17) T ( k i ) N (cid:48) , (5.15)where we have used that P ζ ∼ N (cid:48) P and the shape is given by T ( k i ) = (cid:18) (2 π P ζ ) ( k k k ) + 11 perm (cid:19) (5.16)with the notation k ij = | (cid:126)k i + (cid:126)k j | . The magnitude of the trispectrum is usually givenby two numbers ( τ NL and g NL ) corresponding to two distinct shapes: (cid:10) ζ (cid:11) = (2 π ) δ (cid:16)(cid:88) (cid:126)k i (cid:17) (cid:20) τ NL T ( k i ) + 5425 g NL ( P ζ ( k ) P ζ ( k ) P ζ ( k ) + 3 perm) (cid:21) . (5.17)– 14 –he lowest order contribution thus corresponds to g NL = 0 and τ NL = 4 A B /N (cid:48) .The 1-loop contribution comes from the following term (cid:10) ζ (cid:11) − loop = B (cid:90) d (cid:126)k (cid:48) · · · d (cid:126)k iv (2 π ) (cid:10) δχ (cid:126)k − (cid:126)k (cid:48) δχ (cid:126)k (cid:48) · · · δχ (cid:126)k − (cid:126)k iv δχ (cid:126)k iv (cid:11) , (5.18)= (2 π ) (16) B δ (cid:16)(cid:88) (cid:126)k i (cid:17) (cid:34)(cid:90) d k (cid:48) (2 π ) (2 π P ) k (cid:48) | (cid:126)k − (cid:126)k (cid:48) | | (cid:126)k + (cid:126)k − (cid:126)k (cid:48) | | (cid:126)k + (cid:126)k (cid:48) | + 5 perm (cid:35) . The integral over momentum is difficult in general, so we will only estimate its valueat the equilateral point | k i | = k (cid:10) ζ (cid:11) − loop = 16(2 π ) B δ (cid:16)(cid:88) (cid:126)k i (cid:17) (2 π P ) ln( kL )2 π T ( k i ) N (cid:48) (5.19)and thus τ NL = 4 B N (cid:48) (cid:0) A + 4 B P ln( kL ) (cid:1) , = γ γ ,χ N (cid:48) (cid:18) γ ,χ γ P ln( kL ) (cid:19) ,g NL = 0 . (5.20)We see that the trispectrum is dominated by the non-linear contributions in largelythe same regime as the bispectrum. Given the bound from n s −
1, the maximumvalue for τ NL in this loop dominated regime is τ NL ∼ γ ,χ P N (cid:48) P ln( kL ) < ln( kL ) . (5.21)Interestingly, the bound from WMAP5 on this parameter is | τ NL | < while Planckis expected to improve this bound up to | τ NL | <
6. Conclusions
We have studied a simple class of models in which non-Gaussianity is dominantlyproduced by higher-order non-linearities in the transfer of fluctuations from the fun-damental scalars to the curvature. These higher-order non-linear order contributionsare often referred to in the literature as “c-loops”, and can dominate the lowest or-der “tree-level” contribution in the limit where γ ,χ γ P ln( kL ) >
1, where γ and γ ,χ parameterize the non-linear transfer of fluctuations. In particular, f NL ∼
100 can beachieved in these models.We have also found in these models that the magnitude of non-Gaussianity isscale dependent, with n NG ∼ . n NG ∼ . τ NL ) that also runs.A number of open issues remain. In our model, we have assumed that the slow-roll parameter (cid:15) is constant throughout inflation. This was necessary in order tohave an observable effect from the end of inflation, but it requires tuning and itleads to a very flat power spectrum. It would be interesting to either relax thisassumption in our scenarios or to look at a completely different set-up where the NGis not generated at the end of inflation. We expect that we can relax this assumptionsince we could have a case where f NL is very small on CMB scales but grows to bedetectable on LSS scales. We note though that D-term inflation with a Coleman-Weinberg potential (as illustrated in Appendix A) has a natural regime with therequired flat potential, (cid:15) e ∼ (cid:15) f . From an effective field theory point of view (andfrom string theory models such as [40, 53]), the real tuning is in keeping all otherallowed terms (such as a mass term for φ ) subdominant to the Coleman-Weinbergpotential.We have also assumed that the fundamental scalars ( φ and χ ) are Gaussian, andthat all non-Gaussianity is induced by the non-linear transfer of δχ fluctuations tothe curvature. Non-trivial NG can also arise from non standard kinetic terms, ora steep potential for χ (which unlike the inflaton does not have to satisfy slow-rollconditions). Loop corrections then have a richer structure although the basic idearemains the same. Of particular interest are models like DBI inflation where thespectral index is nearly one and entropy modes being converted to curvature at theend of inflation can also be observable [54]. This scenario has been analyzed recentlyin [55] based on methods developed in [56, 57] (see also [58][59]) and a mixture ofequilateral and local NG has been found. It would be interesting to consider theregime where the loop dominate in this kind of models. Acknowledgments
We are particularly thankful to Bhaskar Dutta for early collaboration on this project.We are grateful to Niayesh Afshordi, Sarah Shandera, Martin Sloth, Xerxes Tata andAndrew Tolley for useful discussions. L.L. would like to thank the organizers of theworkshop on Effective Field Theory of Inflation at the Perimeter Institute and ofthe Phenomenology workshop at Cooks Branch Conservancy where part of this workwas presented. L.L. would also like thank the KITP and the Aspen Institute fortheir hospitality. LL is supported in part by NSF Grant No. PHY–0505757. AR issupported in part by NSF Grant No. PHY–0653656. This research was supported inpart by Perimeter Institute for Theoretical Physics. Research at Perimeter Instituteis supported by the Government of Canada though Industry Canada and by theprovince of Ontario through the Ministry of Research & Innovation.– 16 – . A Specific Model
The discussion in the text is very general and the ultimate goal of having dominantloop contribution in the bispectrum inducing a large running may be achievable in avariety of ways. Here we we illustrate the necessary ingredients with a specific model(based on [39]). Take an inflationary potential V inf = g ξ (cid:20) g π V CW ( x ) (cid:21) , (A.1) V CW ( x ) = ( x + 1) ln( x + 1) − x ln x + ( x − ln( x − − , where x = λ φ g ξ , ξ has mass dimension 2 and λ and g are dimensionless couplings.The reader will recognize this as the Coleman-Weinberg potential. There is a regimein parameter space where the inflaton does not move very much with φ ∗ ∼ φ e = g ξλ (A.2)and the slow-roll parameter is also nearly constant (cid:15) φ = (cid:18) g ln 2 π φ (cid:19) M p ∼ λ g (ln 2) M p π ξ . (A.3)The φ and χ power spectrum are simply given by P = H (2 π ) and they will remainapproximately constant until the end of inflation if η φ and η χ are much smaller than1. Now consider a simple potential for χV hid = ν χ / . (A.4)This potential drives χ to 0 but the field will fluctuate and acquire some stochasticvalue χ which in general will be non-zero (although small). The tachyon potential isof the form Eq. (3.2) with the surface of reheating defined by0 = f ( φ e , χ e ) = − g ξ + λ φ e + βχ e . (A.5)We choose a model such that χ e = χ e + δχ (cid:28) φ e . Since the function f is quadraticin both fields, the transfer function is simply γ = ∂φ e ∂χ (cid:12)(cid:12)(cid:12)(cid:12) e = − βλ χ e φ e , (A.6)and γ ,χ ∼ γχ while γ ,χχ ∼
0. Note that γ and γ ,χ can both be either sign dependingon β . The curvature power spectrum is (from Eq. (4.8)) P ζ = H π ) (cid:15) φ M p (cid:18) γ + γ ,χ H ln kL (2 π ) (cid:19) . (A.7) By integrating the EoM of motion of φ , in the limit x →
1, one can check that φ ∗ ∼ φ e is agood approximation as long as ξM p λ (cid:29) √ N e π where N e is the number of efolds between horizoncrossing and the surface of reheating. – 17 –e are interested in the regime where1 > γ ,χ H ln kL (2 π ) > γ (A.8)and where the power spectrum P ζ , f NL and τ NL are well approximated by P ζ ∼ π ξ (2 ln 2) λ M p ,f NL = 5 ln 2 β H M p ln kL π λ g (2 π ) ξ ,τ NL = β M p H ln kL π (2 π ) λ g ξ . (A.9)Note that none of these observables depend on the precise value of ¯ χ at the end ofinflation although in order for the loop contribution it must be that the average valueof the zero mode of χ at the end of inflation is smaller than H . We show a point inparameter space (see Table (1)) where all the conditions mentioned in this sectionare respected. g λ β ξ/M p − × − . . × − Table 1:
A point in parameter space. λ was first chosen and ξ was solved for by matchingto COBE data. The parameter g is then constrained such that φ < M p by at least two orderof magnitude. β is a free parameters that determine the magnitude of NG. The stochasticvalue of χ at the end of inflation in this model is less than H although none of the observablesdepend on its precise value. One can check that for this choice of parameters: λ φ e (cid:29) λ (cid:48) χ , γ ,χ H ln kL (2 π ) = 0 .
03 and γ ∼ . (cid:15) φ ∗ ∼ (cid:15) φe ∼ − while (cid:15) χe ∼ − (for ν ∼ − giving η χ ∼ − ). ln kL P ζ f NL τ NL n s − n NG CMB scales ∼ × − . .
004 0 . ∼
10 2 × −
62 4 . × .
004 0 . Table 2:
Predicted value for various parameters. This point was deliberately chosen toillustrate the possibility of having a non-observable level of NG at CMB scale but with avery detectable signal for LSS.
As can be seen from Table (2), this simple model can lead to interesting observa-tional signatures whereas the CMB is very Gaussian, but significant NG appears forlarge scale structure. On the other hand, not everything is perfect since the spectralindex is nearly one in tension with the most current WMAP5 data. This is directconsequence of working with a model where (cid:15) φ ∗ ∼ (cid:15) φe . If this assumption is relaxed,– 18 – running will be induced but the non-Gaussian signal coming from the end of in-flation will also be reduced. This can potentially be compensated by varying otherparameters but this required more detailed analysis keeping track of the time of eval-uation for each quantity. We leave this for further work. Note also that the tensionbetween n s ∼ B. Numerical Evaluation of the Integral
The integral B ( (cid:126)k , (cid:126)k , (cid:126)k ) = (cid:90) d k (cid:48) (2 π ) (cid:32) k (cid:48) | (cid:126)k + (cid:126)k (cid:48) | | (cid:126)k − (cid:126)k (cid:48) | + 7 perms (cid:33) (B.1)can be evaluated numerically in Mathematica. There are three poles in the integrand,and the IR cutoff discussed in Sec. (3.1) is most easily implemented by setting theintegrand to zero whenever k (cid:48) is within 1 /L of a pole.On general grounds, one expects that the integral is can be written as B ( (cid:126)k , (cid:126)k , (cid:126)k ) = (cid:88) i ln( k i L ) F i ( | (cid:126)k | , | (cid:126)k | ) + F ( | (cid:126)k | , | (cid:126)k | ) + . . . , (B.2)where the neglected terms contain powers of 1 /k i L . For momenta relevant to CMB,these terms are very small. Note that both F and F are homogeneous Lorentzinvariant functions of the external momenta with degree -6 (i.e., are scale-invariant),and thus are completely determined by the norms of any two of the momenta.As argued in Sec. (5.1), the leading term is dominated by the poles of theintegrand, and we expect it to be of the form (cid:88) i ln( k i L ) F i ( (cid:126)k , (cid:126)k , (cid:126)k ) = 82 π (cid:80) i ln(Min(k α (cid:54) =i )L)k (cid:16)(cid:81) j k j (cid:17) . (B.3)Because F , are scale-invariant, we can numerically integrate the shape at variousscales and fit the result to our ansatz in order to determine the magnitude of F , .For example, we numerically integrated B at the equilateral limit k i = k forvarious values of k near the range kL ≈ B ( k, k, k ) × ( kL ) = c ln( kL ) + c . This fit yielded the expected c ∼ π , wherethe second term c was ∼
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