De Sitter Quantum Loops as the origin of Primordial Non-Gaussianities

It was pointed out recently that in some inflationary models quantum loops containing a scalar of mass $m$ that couples to the inflaton can be the dominant source of primordial non-Gaussianities. We explore this phenomenon in the simplest such model focusing on the behavior of the primordial curvature fluctuations for small $m/H$. Explicit calculations are done for the three and four point curvature fluctuation correlations. Constraints on the parameters of the model from the CMB limits on primordial non-Gaussianity are discussed. The bi-spectrum in the squeezed limit and the tri-spectrum in the compressed limit are examined. The form of the $n$-point correlations as any partial sum of wave vectors get small is determined.


I. INTRODUCTION
An inflationary era in the early universe is a popular possibility for the solution of the horizon and flatness problems [1]. It also provides an elegant mechanism to generate primordial density perturbations that, in the early universe, have wavelengths well outside the horizon. In the most simple inflationary cosmology where only a single inflaton field with a standard kinetic term plays a role, the density perturbations are almost Gaussian and any non-Gaussianities will be unobservable for the forseable future [2].
The inflationary era occurs in the early universe when the energy density is (temporarily) dominated by vacuum energy resulting in a scale factor that expands exponentially with time. This exponential expansion inflates the size of regions that were in causal contact to enormous size solving the horizon problem. It also increases exponentially the physical wavelength of perturbations with fixed comoving wavelength causing the density perturbations that arise from quantum fluctuations and are relevant for large scale structure and the cosmic microwave background radiation (CMB) to have wavelengths that are well outside the horizon when inflation ends. At the end of the inflationary era the universe reheats to a conventional radiation (or matter) dominated universe. arXiv:1806.05194v1 [hep-ph] 13 Jun 2018 Galaxies are biased objects and as such the power spectrum for fluctuations 1 in their number density can be enhanced, at low wave vectors, if there are non-Gaussian primordial curvature correlations, (generated in the inflationary era) that are enhanced as a single wave vector or partial sum of wave vectors go to zero [3][4][5][6]. For the three and four point primordial curvature correlations the dominant enhancements occur respectively in the squeezed and compressed limits.
These enhancements in the galaxy number density power spectrum (and bi-spectrum [7]) at very small wave vectors are sometimes called scale dependent bias and have been studied extensively in quasi single field inflation (QSFI [8]) which contains an additional scalar degree of freedom s with mass m. They cannot arise from non-linear gravitational evolution [9].
In QSFI the primordial non-Gaussianity results from tree diagrams in this theory. Recently a model was constructed where it is quantum loop diagrams in de-Sitter space that give rise to the non-Gaussian correlations that enhance the galaxy number density power spectrum at small wave vectors [10].
In this paper we follow up on this observation considering the simplest model where loop diagrams dominate the non-Gaussianities. This model contains the inflaton field and an additional massive scalar s with mass m that we take to be small compared with the Hubble constant during inflation. A Z 2 symmetry in the system forbids the tree-level contribution from s to the correlation functions of the curvature perturbation. We compute the curvature three point correlation (bi-spectrum) and four point correlation (tri-spectrum) for general wave vectors. When all ratio's of wave vectors are not unusually large or small the bispectrum and tri-spectrum respectively have the same form as local non-Gaussianity and τ N L non-Gaussianity 2 . In the squeezed and compressed limits the bi-spectrum and tri-spectrum have the familiar enhancements that give rise to an enhanced galaxy power spectrum at low wave vectors. We also discuss the form of the n-point curvature fluctuations as a partial sum of wave vectors goes to zero. We plot the enhancement of the two-point power spectrum of the galactic halo distributions generated by the quantum loops in this model using a simplified threshold model [13] for the galaxy halo number density.
The rest of the paper is organized as the following. In Sec. II we discuss the details of the model. In Sec. III we present the general three and four-point correlation functions of the curvature perturbation. Details of the calculations are relegated to appendices. In Sec. IV we calculate the constraints on the parameters of the model from CMB limits on the primordial bi-spectrum and tri-spectrum [11,12]. In contrast to most models the constraint from the limit on the tri-spectrum is stronger. We conclude in Sec. V.

II. THE MODEL
The model contains two scalar fields, the inflaton φ and another scalar field s of mass m. We assume the inflation and s fields have a φ → −φ symmetry and s → −s symmetry. We also assume a shift symmetry for φ that is only broken by the inflation potential. Then the lowest dimension term that contains interactions of φ with s is During inflation φ has a background value φ 0 (t) that depends on time. Its magnitude relative to H 2 is fixed by the CMB temperature fluctuations to be,φ 0 /H 2 3.5 × 10 4 .
This model has a fine tuning. The physical s mass m is a sum of two terms one the mass parameter µ from the s potential and the other from the interaction above. Explicitly, To get the enhancements we mentioned in the introduction we need m/H < 1 where H is the Hubble constant during inflation. But to get observable non-Gaussianities we need the second term in the equation above to be much greater than H 2 , hence the tuning. Finally we assume that there exists an inflaton potential that gives an exceptable region in r, n s space and is flat enough that m can be approximated as a constant for the calculation of the non-Gaussianities generated during inflation.
We proceed along the lines of the effective field theory of inflation working in the gauge where φ(x) = φ 0 (t) [14]. A Goldstone mode π, due to the breaking of the time translation symmetry, is introduced to describe the curvature perturbation as follows, where the evolution ofφ 0 is neglected and the relation between π and the curvature perturbation ζ in this model can be written as A three-point and a four-point interaction between π and s are induced with where τ is the conformal time defined as τ = −e −Ht /H and η µν = diag(1,-1,-1,-1). Here we include the √ −g factor into the Lagrangian.
The mode expansion of π and s are defined as Assuming the Bunch-Davies vacuum, the mode functions can be written as and and H (2) α is the second Hankel function with index α. Throughout this paper we focus on the region where m 2 /H 2 is small compared with unity. For |kτ | 1, and We are interested in the multi-point correlation functions of the curvature perturbation far outside the horizon (τ 0), so we need to evaluate ζ( x 1 , 0) · · · ζ( x n , 0) . This can be calculated using [15] where the I in the superscript stands for interaction picture.

III. GENERAL FORM OF THE THREE AND FOUR-POINT CORRELATION FUNCTIONS OF ζ
A. Three-point correlation function There are two diagrams generating one-loop contributions to the three-point function of π, which are shown in Fig. 1. We will show that with the saturation of the current limits of f NL and τ NL the contribution from Fig. 1(a) is much larger than from the Fig. 1(b). The calculation of the resulting bi-spectrum is performed in Appendix A and there it was found that for m/H 1 This is the general leading order expression for the bi-spectrum of ζ .
A factor (k i /k j ) ν 1 + νln(k i /k j ) + ..... So such factors can be set to unity when all the ratios of the k's are not very small or large. Then This is the form that the bi-spectrum has in local non-Gaussianity.
Next we consider the squeezed region first where k ≡ k 1 k 2 k 3 ≡ q, it is easy to see that in this limit the bi-spectrum goes to This differs from what one gets from local non-Gaussianity by a factor of (q/k) 2ν .
Provided non of the ratio's of the k's or partial sums of the k's is anomalously large this has the same form as τ N L (g N L = 0) non-Gaussianity, In the compressed region k 1 k 2 | k 1 + k 2 | ≡ q and k 3 k 4 q, it simplifies into The scaling of the four point curvature fluctuation in the compressed limit (∼ q −3+4ν ) can be read out from the scaling dimension of the operator s 2 since this determines the form of its two point correlation evaluated on the boundary τ = 0 of de-Sitter space. [16].

IV. CMB CONSTRAINTS
In the previous section it was noted that for ν 1, the bi-spectrum of this model is the same as the local non-Gaussianity model (for typical wave vectors). Therefore, we can use the observed limit of the f local NL to estimate the constraint on this model the CMB observation of the non-Gaussianity. In the limit where the bi-spectrum reduces to local non-Gaussianity, 2.14 × 10 −9 . (4. 2) The 2σ constraint [12] on f local NL is about |f local NL | < 10. Therefore, we have Similarly for the tri-spectrum τ NL can be estimated as The constraint on τ NL from the Planck observation of the CMB spectrum [11] is  which is much larger than unity unless ν is exceptionally small 3 . Similar conclusions hold for diagram 2c and for other correlation functions.
The constraints from f local NL and τ NL are shown in Fig. 3, where one can see that the constraint from τ NL is always stronger. This result is in contrast to many other models, where the constraint from f NL is stronger. The main reason is the following. In the models where there are tree-level contributions to non-Gaussianity, f NL is usually proportional to a single power of the small coupling of the new interaction and τ NL proportional to the square of it. Whereas in this model f NL is proportional to the third power of the coupling and τ NL proportional to the fourth power of it. Therefore, relatively in this model τ NL is more important.
Since there is some small time dependence inφ 0 , Eq. (2.2) implies that the s mass depends on time The slow role parameter η must be small enough that this time dependence can be neglected in our computations.

V. CONCLUDING REMARKS
We have examined the primordial curvature perturbations in an inflationary cosmology where the inflaton φ couples to an additional scalar field s with mass m H through the non-renormalizable interaction g µν ∂ µ φ∂ ν φ/4Λ 2 . In this model primordial non-Gaussianities arise from quantum loop diagrams (with a virtual s in the loop) in de-Sitter space. The primordial curvature fluctuation bi-spectrum and tri-spectrum in this model were computed.
Typically they respectively have the form of local non-Gaussianity and τ N L non-Gaussianity, although in squeezed and compressed limits there are deviations from that form.
It is not difficult given the work of this paper to deduce the form of the n > 4 curvature perturbations.. In the situation where all the wave vectors and their partial sums of wave vectors are of order k, and in the limit where a single partial sum of wave vectors | k 1 +. . .+ k j | = q k it becomes, In this model due to the IR behavior of the compressed tri-spectrum (3.6), the long distance behavior of the power spectrum for fluctuations in the galaxy number density is enhanced by a factor of q 4−4ν compared to the Harrison-Zel'dovich spectrum (which goes, apart from a small tilt, as q). Following the same procedure as described in Refs. [7,10] we estimate this power spectrum and its ratio to the leading Harrison-Zel'dovich contribution is shown in Fig. 4 using the curvature bi-spectrum and tri-spectrum calculated in this paper.
One can see that if the current constraint from τ NL is saturated the power spectrum of the galactic halo distribution can differ significantly from the Harrison-Zel'dovich spectrum at

MBW is supported by the DOE Grant DE-SC0011632 and the Walter Burke
Institute for Theoretical Physics.
The Hankel function H (2) α (kτ ) in general can be expanded into two series that The important property is that the coefficients in each series share the same phase. Then after some straightforward calculation the leading contribution of the bi-spectrum of ζ can be written as The UV cut-off of each τ i integral is determined by the point where the integrant start to oscillate that (A14) Now we can do the τ i integrals explicitly and we have where and The p integral is also controlled by the infrared contributions. One can see that the integrant of the p integral has three branch points, and the integral is supported by the domain around these branch points. For example, for the branch point at p = 0, the integral is mainly supported at region around p < ∼ ν × min(k 1 , k 3 ). Therefore, one can see that the three regions are well separated. Therefore, the leading order contribution of the integral can be separated to three parts that where k max is short for max(k 1 , k 2 , k 3 ).
With the permutation (and noting that diagram a dominates) after straightforward calculation one can show that the leading contribution to the bi-spectrum is Eq. (A19) gives the general leading order expression for the bi-spectrum of ζ in the region that m/H 1.