Enhanced tensor non-Gaussianiaties in presence of a source

We address the possibility of having an enhanced signal for tensor non-Gaussianities in presence of a source, as a signature of Primordial Gravitational Waves. We employ a nearly model-independent framework based on Effective Field Theory of inflation and compute tensor non-Gaussianities therefrom sourced by particle production during (p)reheating to arrive at an enhanced signal strength. We obtain the model-independent non-linearity parameters and compare the results with the latest Planck data to find out the possible constraints on the parameters. We also find that squeezed limit bispectra are more enhanced and do a primary analysis of the prospects of detection in upcoming CMB missions.


I. INTRODUCTION
Even after the profound advancement in the Cosmic Microwave Background (CMB) observations for nearly two decades, Primordial Gravitational Waves (PGW)the so-called tensor modes of perturbations -still remain as the holy grail of early universe Cosmology. The latest bound on the amplitude of two-point correlation function of tensor modes i.e tensor-to-scalar ratio is r < 0.064 from Planck 2018 data [1]. All it gives us is an impression that the signal strength of power spectrum for PGW, if exists, would be really tiny, making it a daunting task for next-generation CMB missions to detect it some day. Despite this, from theoretical point of view, PGW encodes crucial information about early universe Cosmology. PGW generated due to vacuum fluctuations during inflation is directly related to inflationary energy scale. In absence of any conclusive evidence of two-point function for PGW until now, the community got curious about the three-point function that reflects the non-Gaussian features of PGW, primarily because it has potential to serve as an addition probe of PGW. Over the last few years there has been some theoretical progress in this direction. In [2,3] the three-point function for tensor modes is calculated for general single field slow roll inflationary models. This analysis is further generalized in [4,5]. For a recent review the reader can refer to [6]. These analysis are for tensor modes generated by vacuum fluctuation. However, it has been pointed out in a previous article by the present authors [5] in a modelindependent framework based on EFT of inflation, and also by others following particular models, that the amplitude of bispectrum generated by vacuum fluctuations is generically small.
Apart from vacuum fluctuations, PGW can also be generated by some sources that may be present during the early epoch. However, even if they are produced, the two-point correlation function can not distinguish between different origins of PGW. So, one needs * abhiatrkmrc@gmail.com † supratik@isical.ac.in to go beyond two-point statistics and investigate for non-Gaussian features of PGW which has different momentum dependence for different sources and hence can distinguish among different sources and vacuum. Of late this revelation has served as a strong motivation to explore non-Gaussian statistics of PGW from possible sources. Subsequently, the possibilities of producing comparatively large signal using different sources have been investigated to some extent, for example, using axion as a source [7,8], or using extra spin particles during inflation [9]. The current observations are unable to detect any significant signal of tensor non-Gaussianities. Latest constraints on the amplitude of three-point function with 1σ error are f T N L = 600 ± 1500 from WMAP [10] and f T N L = 800 ± 1100 from Planck 2018 [11] for equilateral momentum configuration and on the amplitude for tensor-scalarscalar three point function are f T SS N L = 84±49 at 68%C.L. [12]; Nonetheless, the methodology for bispectrum estimation is established by adding B-mode polarization information [6]. Upcoming CMB mission LiteBIRD [13,14] targets to improve the results by three orders of magnitude. CMB-S4 [15] may improve the tensor-scalar-scalar cross correlation result by an order of magnitude. The dedicated gravitational waves detector LISA [16] can directly probe the bispectrum of gravitational waves. So it is important to do a theoretical analysis on generic aspects of tensor non-Gaussian statistics and interpret the constraints in the light of upcoming observations.
In this article we intend to take up our previous modelindependent analysis [5] based on EFT of inflation and extend it to possible sources. We want to explore if it is possible to enhance the bispectrum of PGW due to (p)reheating process and if there is any chance of detection in the upcoming surveys. To this end we will make use of the EFT of inflation [17] and EFT of (p)reheating [18]. As in the case of our previous analysis [5], the present analysis is more or less model-independent in the sense that it is developed from EFT framework with no particular model as such. In particular, we would be interested in proposing model-independent templates for non-linearity parameter f N L and investigate if one can have enhanced signal for the same that may fall within the reach of upcoming CMB missions.

II. EFT, GRAVITON LAGRANGIAN AND (P)REHEATING
As mentioned, since our intention is to analyse the scenario in a more or less model independent framework, we make use of the EFT of inflation following our previous analysis [5], that was originally developed in [17,19]. In this approach, the inflaton field φ is a scalar under all diffeomorphisms but δφ breaks the time diffeomorphism. Using this symmetry of the system and unitary gauge where δφ = 0, the Lagrangian can be written as [17] The dots at the end of the Lagrangian represent higher order terms. As pointed out in [17], this is purely gravitational Lagrangian where δK ν µ is the extrinsic curvature; the scalar perturbation is not explicit but can be reintroduced using Stückleberg trick.
In Unitary gauge the perturbed metric can be written as, is scalar perturbation and γ ij (t, x) is tensor perturbation which is transverse and traceless satisfying, γ ii = 0 and ∂ j γ ij = 0. In terms of γ ij the Lagrangian (1) takes the form is the non-trivial sound speed of tensor perturbation due to the presence ofM 3 . Eq (2) is the most general third order Lagrangian for single field inflation. It has been shown that the term proportional toM 9 along with the Einstein term contribute to tensor bispectrum [5]. For our present investigation, our intention is to add, on top of this, the EFT of (p)reheating that was developed in [18]. Here, apart from the inflaton fluctuation, one more degree of freedom is considered. This approach also assumes that the background breaks the time diffeomorphism spontaneously and the construction of the Lagrangian is similar as [17]. For (p)reheat field χ it can be written as, Here with time reparametrization invariance, parameter α 4 has been set to zero [18]. Note that the (p)reheat particles also have non-trivial sound speed In our analysis we consider α 1 and α 2 to be time independent and hence the sound speed is also time independent.

III. TWO-POINT CORRELATION FUNCTION
With (p)reheating particles as source the equation of motion for γ ij is given by, Π ab ij is the transverse traceless projection tensor that sources the (p)reheat particles. Written explicitly, So the transverse traceless part of energy momentum tensor becomes The solution for Eq (5) can be obtained by Green's function method, where the expression for Green's function G k (τ, τ ) is given by, It is worthwhile to mention that in (9) the non trivial sound speed of tensor fluctuation plays a crucial role in determining the Green's function and hence the powerspectrum. This will be obvious from the following analysis. In what follows we employ the method of [20] to calculate the two-point correlation function for our setup of nontrivial contribution from the EFT parameters. Using this Green's function the power spectrum for tensor modes sourced by (p)reheat field turns out to be In order to evaluate the correlation functions we need to analyze the dynamics of χ particles. Varying (3) with respect to χ one arrives at the following equation of parametric oscillator where, χ c = aχ(α 1 + α 2 ) and the frequency of the oscillator is given by This clearly shows the nontrivial modifications to the frequency that arises due to the EFT of (p)reheating. Consequently, the solution for (11) becomes where α and β are the Bogolyubov coefficients.
To proceed further, we need to have explicit time dependence of ω(k, τ ); so we set α3 α1+α2 = g 2 (φ−φ 0 ) 2 , wherė φ = φ(t = 0). This choice can be identified as the leading order expansion of the parameter α 3 (t) as in [18]. Considering de-sitter background and with slow roll approximation we can assume that, φ(t) = φ 0 +φ 0 t. Further, non-adiabatic condition leads to a constraint g >> H 2 φ [20]. With these approximations the Bogolyubov coefficients turn out to be and ).
With these initial conditions, we will now work in the non-relativistic limit as the Bogolyubov coefficients contain exponential momentum suppression, for which ω(|k − p|) − ω(p) = 0 and ω 2 = g 2φ Consequently, the two-point correlation function looks The τ → 0 limit of the above Green's function is given by, Hence, upon performing the p and τ integration we get, The role of non-trivial sound speeds c γ and c χ are now crystal-clear from (17). They can be used to tune the signal strength of the two-point function. For example, it can be enhanced in the limit c γ → 0 or c χ → 0 or c γ , c χ → 0. So, it is expected that they will play crucial role in determining the signal strength of three-point correlation functions as well. However, we will concentrate on this in the next section. The total power spectrum for tensor modes reads It can be verified that the function gets maximum value at c γ kτ 0 = 2.46. In order to compare with the existing results in the literature, we take the same representative values for the parameter as in [20]: g = 1, H = 10 13 GeV/c 2 , M p = 2.48 × 10 18 GeV/c 2 anḋ φ = √ 2 HM p where, = 0.005. As a result, the tensor power spectrum becomes In the existing literature (e.g., [20]), the second term in the parenthesis was generically small. However, in the present analysis, it can be significantly large due to nontrivial sound speeds. For example, if the second term is order of one, it will be a relevant contribution due to the (p)reheating process. Fig 1 demonstrates the comparative values of the two sound speeds in order to achieve this.
Let us explain it with a particular example. If we take a representative value for the tensor-to-scalar ratio as r ≈ 0.06 that is close to the upper bound set by the latest Planck 2018 data [1], then for c γ = 1, c χ ≈ 0.02 the second term will be O(1). Consequently, we can get a significant contribution from (p)reheating. The reason for this is that for c χ < 1 the resonance band become broadened and there is an enhancement in particle production as discussed in [18]. On the other hand according to [21] small sound speed of tensor fluctuation is also responsible for large signal because non canonical inflationary case is responsible for a saw-tooth like profile of inflaton which moves the system to broad parametric resonance and significant particle production occurs. Note that in the above analysis we did not consider the non-adiabatic scenario as it is shown in [20] that this regime produces same result as the adiabatic regime.

IV. THREE-POINT CORRELATION FUNCTION
Having convinced ourselves about the role of the nontrivial sounds speeds on the signal strength, let us now move forward to calculate the three-point function for (p)reheating-sourced gravitational waves. The expres-sion for three-point function is given by where s i are helicity indices and e si ij are polarisation tensors. To fix the representation of polarisation tensors we take a particular k i basis and consider that this basis is lying on (x, y) plane. In doing so we will not lose any generality because of the momentum conserving δ function. In what follows we will choose the representation adapted in [22] : k 1 = k 1 (1, 0, 0), k 2 = k 2 (cos θ 1 , sin θ 1 , 0), . With this choice the polarisation tensors can be written as, Consequently, the total three-point function gives us, where the subscripts "vac" and "so" stand for "vacuum" and "source" (here, (p)reheating) respectively and these abbreviations would be used in the rest of the article. As already mentioned, the vacuum solution has been explored at length in a previous article by the present authors [5] and is given as, We will calculate the contribution from source term here. In evaluating the three-point function, we will use the same approximation of adiabatic regime as in the case of two-point function. By employing this approximation, the source part of the three-point function takes the form where the terms A k and B k have very tedious expressions. For completeness, we summarise them below: and As mentioned, the resulting three-point function (24) is the sumtotal of (25) and (26).
Let us now critically investigate for the results thus obtained. To do so, we will have the following observations. First, from the expression of A k and B k we can see that they can be written as, 3 and f i (k) and g i (k) encodes all the momentum dependence and relevant prefactors. It is evident from the above expression that for a small c χ we can neglect the first term proportional to c χ . Secondly, the term (c γ k i τ 0 cos(c γ k i τ 0 ) − sin(c γ k i τ 0 )) can be expanded for small c γ and can be written as, (c γ k i τ 0 ) 3 . Further, in order to get an idea about the momentum dependence of the bispectra we are working in a limit where we can keep up to c 3 γ term and can neglect c 2 χ term. The resultant contributions have been pictorially depicted in Fig 2. The figure shows the momentum dependence of the bispectra as a function of k1 k2 and k3 k2 . The essential conclusion that can be readily obtained from the above figure is that for k1 k2 → 0 and 0.7 < k3 k2 ≤ 1 we get large amplitude for the bispectra. Also we get positive contribution for squeezed and equilateral limit and much larger amplitude for the bispectra which cannot be achieved in case of vacuum. This was the primary goal of the present article. We shall elaborate more on this in the following section.

V. ESTIMATION OF fNL
We are now in a position to formulate the templates for the nonlinearity parameter f N L . In what follows we shall make use of the same definition of the nonlinearity parameter as adopted in [5], namely, 6 5 . Also, the tensor modes generated due to vacuum fluctuation would in any case be small, the templates for which have already been proposed in the previous article [5]. Hence, in this article we would be interested only about the three-point function due to source term γ s1 (k 1 )γ s2 (k 2 )γ s3 (k 3 ) so in formulating the templates. As has been pointed out, we are interested about any significant enhancement of signal. Hence, we would consider the scenario where the three-point function due to source term would have dominant contribution to γ s1 (k 1 )γ s2 (k 2 )γ s3 (k 3 ) total in Eq (24) and would investigate if this is achievable with the parameters under consideration.
Like the vacuum solution, in the case of equilateral limit k 1 = k 2 = k 3 we have two independent non-linearity parameters. They are given by While calculating the f N L in squeezed limit one has to remember that (26) is not symmetric in k 1 , k 2 and k 3 but symmetric in k 2 and k 3 . As a result, one has to take two different limits of the expression: k 1 → 0 and k 2 → 0 and take their average to obtain the average value of f sq N L . Consequently, for the squeezed limit, we get the following independent non-linearity parameters 15688.5gφ ln To estimate the size of f N L we use the representative values of parameters chosen while estimating the power spectrum amplitude in Section III. This guarantees that we are working within the latest observational bound of tensor-to-scalar ratio. As we have stated earlier, the two point function is peaked at c γ k i τ 0 = 2.46 for c γ = 1 and c χ = 0.02. On top of that we have additional constraints on squeezed limit and equilateral limit bispectra [6,23,24] which will further constrain c χ . To estimate the squeezed limit f N L where one momentum is smaller than the other two momenta, we consider that c γ k i τ 0 = 2.46 is due to the larger momenta, the reason for the choice is if k i → 0 then c γ k i τ 0 will also be very small.We also consider that k large k small ≈ 10. The constraint on squeezed limit from Planck is 290 ± 180 [24]. Using the above approximations and the upper limit of observational value of f +++,sq Of course, these estimations are not too accurate as we have considered the coupling constant to be O(1) which may not be strictly valid. These estimations are done to demonstrate that using EFT in inflation and (p)reheating, large signal for tenor non-Gaussianities can be produced due to the presence of non trivial sound speed of χ particles.
The bottomline of the above analysis is that we can have an enhanced tensor non-Gaussian signal for (p)reheating with non-trivial sound speed c χ . In principle, the signal strength for squeezed limit bispectra can be as high as to reach pretty close to the upper bound of latest data release by Planck. Nevertheless, the sound speed for inflation degree of freedom c γ ≤ 1 as explained in section III. Smaller the value of c γ , higher is the signal strength. Thus, particle production from non-canonical inflation with c γ < 1 can enhance the tensor non-Gaussian signal further. A rather conservative statement would be that, even if the nonGaussian signal is not that high, this can fall well within the reach of next generation CMB missions. However, an actual comparison with the sensitivity of upcoming CMB missions can only confirm this.

VI. CONCLUSION
In this article we have presented a way to enhance the signal for tensor three-point function sourced by (p)reheating. Our analysis is nearly model-independent since we have used EFT of inflation and (p)reheating. Using EFT we have been able to deal with a non standard case for (p)reheating for which the sound speed of (p)reheat particle χ is different from unity. We have demonstrated that this non-trivial sound speed can actually enhance the signal of tensor non-Gaussianities which was not achievable in the vacuum as well as in the standard (p)reheating analysis. We have further been able to propose model-independent templates for the nonlinearity parameter f N L and found that, like the sourcefree case, here also squeezed limit bispectrum is stronger than equilateral limit. As a result, possibility of detection in future mission of the squeezed limit is higher along with the momentum range described in Section IV. We have done a primary analysis towards this direction by comparing the results with the latest constraints on the parameters with Planck data and found out the prospects in upcoming CMB surveys. However, an actual comparison with the sensitivity of upcoming CMB missions is beyond the scope of present article. We hope to address