Loop-Induced Stochastic Bias at Small Wavevectors
Michael McAneny, Alexander K. Ridgway, Mikhail P. Solon, Mark B. Wise
CCALT-TH-2017-071
Loop-Induced Stochastic Bias at Small Wavevectors
Michael McAneny, Alexander K. Ridgway, Mikhail P. Solon and Mark B. Wise
Walter Burke Institute for Theoretical Physics,California Institute of Technology, Pasadena, CA 91125
Primordial non-Gaussianities enhanced at small wavevectors can induce a powerspectrum of the galaxy overdensity that differs greatly from that of the matter over-density at large length scales. In previous work, it was shown that “squeezed”three-point and “collapsed” four-point functions of the curvature perturbation ζ can generate these non-Gaussianities and give rise to so-called scale-dependent andstochastic bias in the galaxy overdensity power spectrum. We explore a third wayto generate non-Gaussianities enhanced at small wavevectors: the infrared behaviorof quantum loop contributions to the four-point correlations of ζ . We show thatthese loop effects lead to stochastic bias, which can be observable in the context ofquasi-single field inflation. I. INTRODUCTION
The inflationary paradigm [1] proposes an era in the very early universe during whichthe energy density is dominated by vacuum energy and the universe undergoes exponentialexpansion. Such a period elegantly explains why the universe is close to flat and the nearisotropy of the cosmic microwave background (CMB). It also provides a simple quantummechanical mechanism for generating energy density perturbations which have an almostscale-invariant Harrison-Zel’dovich power spectrum.The simplest inflation models consist of a single scalar field φ , called the inflaton, whosetime-dependent vacuum expectation value drives the expansion of the universe. The quan-tum fluctuations in the Goldstone mode π associated with the breaking of time translationinvariance by the inflaton [2] source the energy density fluctuations. In the simplest of thesesingle field models, the density perturbations are very nearly Gaussian [3]. One way to gen-erate measurable non-Gaussianities is to introduce a second field s that interacts with theinflaton field during the inflationary era. A simple realization of such a model is quasi-singlefield inflation (QSFI) [4].These non-Gaussianities affect the correlation functions of biased tracers of the underlyingmatter distribution such as galaxies. It was first pointed out in [5] and [6] that the powerspectrum of the galaxy overdensity can become greatly enhanced relative to the Harrison-Zel’dovich spectrum on large scales if the primordial mass density perturbations are non-Gaussian. These enhancements are known as scale-dependent bias and stochastic bias andwere systematically explored in the context of QSFI in [7] and [8]. We refer to these effects as “enhancements” even though for certain model parameters they can interferedestructively with the usual Gaussian primordial density fluctuations. a r X i v : . [ a s t r o - ph . C O ] D ec FIG. 1. One-loop contribution to the collapsed trispectrum of the primordial curvature perturba-tion. Dashed lines represent π , and solid lines represent s . The enhancements studied in [5] and [6] result from tree-level contributions to the three-and four-point functions of π that are in their “squeezed” and “collapsed” limits. In thispaper, we consider quantum contributions to the correlation functions of π which can alsogive rise to these long-distance effects. We find that the infrared region of loop integralscan induce sizable stochastic bias on large scales without introducing any scale-dependentbias. In section II we illustrate this loop effect using a higher dimension operator thatwould appear in a generic effective theory of multi-field inflation. In section III we showthat the loop effect can be observable in the context of QSFI and estimate the distancescale at which the loop contribution to the galaxy power spectrum could exceed the usualHarrison-Zel’dovich one. II. LOOP-INDUCED STOCHASTIC BIAS
Consider a theory of inflation that consists of two fields, the inflaton φ and a massivescalar s . Working in the gauge where φ ( x ) = φ ( t ), the Lagrangian describing the Goldstonemode π due to the breaking of time translational invariance and s can be written as L = 12 g µν ∂ µ π∂ ν π + 12 g µν ∂ µ s∂ ν s − m s + 1Λ g µν ∂ µ π∂ ν πs + . . . , (2.1)where the action is S = (cid:82) d x √− g L . The dimension six operator in (2.1) induces the one-loop contribution to the four-point function of π depicted in Fig. 1. The complete theoryincludes additional interactions denoted by the ellipsis above [9, 10] , which will give riseto other one-loop contributions that are comparable to or may even dominate this diagram.The goal of this section is to illustrate the infrared behavior of loop contributions to thecorrelation functions of π , which have interesting implications for the correlation functionsof galaxies. For simplicity, we only consider the interaction given in (2.1) and leave a morecomplete study to future work.We focus on the “collapsed” limit of the diagram, which occurs when the external wavevec-tors come in pairs that are nearly equal and opposite, as shown in Fig. 1 with q (cid:28) k i . Thiscontribution to the four-point function has previously been computed in [11], where therole of conformal symmetry was emphasized. In this section, we review this calculation anddescribe its effect on the power spectrum of galaxy overdensities. For example, the interaction 2 ˙ φ ∂ τ πs / Λ will also appear. To begin, we express the quantum fields π and s in terms of creation and annihilationoperators π ( x , τ ) = (cid:90) d k (2 π ) a ( k ) π k ( η ) e i k · x + h . c . , s ( x , τ ) = (cid:90) d k (2 π ) b ( k ) s k ( η ) e i k · x + h . c . , (2.2)where k = | k | , and η = kτ for conformal time τ <
0. The mode functions satisfy theequations of motion of the free theory with appropriate boundary conditions and are π k ( η ) = Hk / π ( η ) , π ( η ) = 1 √ iη ) e − iη , (2.3) s k ( η ) = Hk / s ( η ) , s ( η ) = − ie i (2 − ν ) π √ π − η ) / H (1) − ν ( − η ) , (2.4)where ν = 3 / − (cid:112) / − m /H and H (1) z is the Hankel function of the first kind. We assumethat the mass m of the field s is much less than the Hubble constant H during inflation, orequivalently ν (cid:28) We are interested in this region of parameter space because it leads tothe largest infrared enhanced contributions to the four-point function.Let us now compute the contribution in Fig. 1 to the collapsed trispectrum of the pri-mordial curvature perturbation ζ = − ( H/ ˙ φ ) π . The primordial curvature trispectrum T ζ isdefined by (cid:104) ζ k ζ k ζ k ζ k (cid:105) c = T ζ ( k , k , k , k )(2 π ) δ ( k + k + k + k ) (2.5)where the subscript c denotes the connected part of the four-point function. In Fig. 1 k = − k + q and k = − k − q . The collapsed configuration T coll ζ occurs when q (cid:28) k i .Using the in-in formalism [12] and introducing the variables η = k τ and η (cid:48) = k τ (cid:48) wefind T coll ζ = 32 (cid:18) H Λ (cid:19) (cid:18) H ˙ φ (cid:19) k k (cid:90) d p (2 π ) | p + q | p (cid:90) −∞ dηη (cid:90) k k η −∞ dη (cid:48) η (cid:48) e (cid:15) ( η + η (cid:48) ) Im [ F ( η )] × Im (cid:20) F ( η (cid:48) ) s (cid:18) | p + q | k η (cid:19) s ∗ (cid:18) | p + q | k η (cid:48) (cid:19) s (cid:18) pk η (cid:19) s ∗ (cid:18) pk η (cid:48) (cid:19)(cid:21) + (cid:0) k ↔ k (cid:1) (2.6)where F ( η ) = π (0) (cid:0) [ ∂ η π ∗ ( η )] − [ π ∗ ( η )] (cid:1) . (2.7)In Eq. (2.6), (cid:15) is an infinitesimal positive quantity that regulates the time integrations inthe distant past and we have expanded in q (cid:28) k i .The dominant contribution of the loop integral in (2.6) comes from p ∼ q . Moreover,the time integrals are dominated at late times η , η (cid:48) ∼ −
1. We can thus use the small η expansion of the s mode function s ( η ) η → (cid:39) b ( − η ) ν , | b | = 2 − ν Γ(3 / − ν ) /π ν → (cid:39) / T coll ζ (cid:39) (cid:18) H Λ (cid:19) (cid:18) H ˙ φ (cid:19) k k ) ν I ν ( q ) J (2.9) In (2.1), the mass m includes contributions from terms such as ( ˙ φ / Λ ) s . Tuning is required for m (cid:28) H . where I ν ( q ) = (cid:90) d p (2 π ) | p + q | − ν p − ν ν → (cid:39) π ν q − ν , (2.10) J = (cid:90) −∞ dηη e (cid:15)η ( − η ) ν Im [ F ( η )] = 2 − − ν Γ(2 + 2 ν )1 − ν ν → (cid:39) . (2.11)In (2.10) we have kept only the term singular in ν . Note that our result is finite because wefocused on the relevant region p ∼ q (cid:28) k i and neglected the region of large loop momentawhich is not as important in the limit q →
0. The UV divergence due to the region of largeloop momentum would be rendered finite by a counterterm.Our final result for the four-point function of the curvature perturbation for m (cid:28) H and q (cid:28) k i is T coll ζ (cid:39) π ν (cid:18) H Λ (cid:19) (cid:18) H ˙ φ (cid:19) k k q (cid:18) q k k (cid:19) ν . (2.12)The factors of wavevector magnitudes in (2.12) essentially follow from the form of s ( η )expanded for small η in the limit m (cid:28) H , and from dimensional analysis. For m (cid:28) H thefour-point function is enhanced by 1 /ν (cid:39) H /m . This arises because for small m/H thethe mode function s ( η ) falls off slowly as the mode k redshifts outside the de-Sitter horizon.Note also that there is no IR divergence in the loop integration since the s field is massive.Three- and four-point curvature fluctuations generated by loop effects have been consideredin Refs. [13–16] using the δN formalism. It would be interesting to see if this method canreproduce (2.12).We now qualitatively discuss the effects of (2.12) on the galaxy power spectrum. Tobegin, the matter overdensity δ R averaged over a spherical volume of radius R is related tothe primordial curvature fluctuation via δ R ( k ) = 2 k m H T ( k ) W R ( k ) ζ k (2.13)where W R ( k ) is the window function, T ( k ) is the transfer function, Ω m is the ratio of thematter density to the critical density today, and H is the Hubble constant evaluated today.We consider an expansion for the galaxy overdensity δ h in terms of δ R of the followingform δ h ( x ) = b δ R ( x ) + b ( δ R ( x ) − σ R ) + b ( δ R ( x ) − δ R ( x ) σ R ) + . . . , (2.14)where σ R = (cid:104) δ R ( x ) δ R ( x ) (cid:105) and the constants b , b , and b are bias coefficients (for a morecomplete treatment, see [17]). The bias coefficients can be determined from data or computedusing a specific model of galaxy halo formation that expresses the galaxy overdensity in termsof δ R . The two-point function of the galaxy overdensity is then: (cid:104) δ h ( x ) δ h ( y ) (cid:105) = b (cid:104) δ R ( x ) δ R ( y ) (cid:105) + b b (cid:0) (cid:10) ( δ R ( x ) − σ R ) δ R ( y ) (cid:11) + (cid:10) δ R ( x )( δ R ( y ) − σ R ) (cid:11) (cid:1) + b (cid:10) ( δ R ( x ) − σ R )( δ R ( y ) − σ R ) (cid:11) + . . . (2.15)A similar expression could be derived for the galaxy-matter cross-correlation (cid:104) δ h ( x ) δ R ( y ) (cid:105) .Ignoring other contributions to the non-Gaussianities of ζ besides the one given in (2.12),the term proportional to b in (2.15) yields a contribution to the galaxy power spectrum ofthe form P hh ( q ) ∼ /q − ν , but not to the galaxy-matter cross-correlation P hm ( q ). Hence thisloop contributes to stochastic bias, but not to scale-dependent bias. Note that in the absenceof primordial non-Gaussianity, P hh ( q ) ∼ q , so the trispectrum contribution is enhanced bya relative factor of q − ν and dominates as q → III. LOOP-INDUCED STOCHASTIC BIAS IN QUASI-SINGLE FIELDINFLATION
In this section, we show that loop-induced non-Gaussianities in QSFI [4] can give riseto stochastic bias that is potentially observable given the stringent constraints from CMBdata on non-Gaussianities. The model we consider consists of an inflaton φ and a massivescalar s with the symmetries φ → φ + c , φ → − φ , and s → − s . These symmetries arebroken by the potential of φ as well as by the lowest dimension operator that couples φ and s , g µν ∂ µ φ∂ ν φs/ Λ. The Lagrangian written in terms of the Goldstone mode π is L = 12 g µν ∂ µ π∂ ν π (cid:18) s (cid:19) + 12 g µν ∂ µ s∂ ν s − µHτ s∂ τ π − m s − V (4) s (3.1)where the kinetic mixing term is parameterized by the coupling µ = 2 ˙ φ / Λ and we haveignored higher order terms in the potential for s . Similar to the previous section, we focushere on the region where m (cid:28) H and µ (cid:28) H , which gives the most significant longwavelength enhancement to the galaxy power spectrum.Due to the kinetic mixing, π and s share a set of creation and annihilation operators: π ( x , τ ) = (cid:90) d k (2 π ) (cid:16) a (1) ( k ) π (1) k ( η ) e i k · x + a (2) ( k ) π (2) k ( η ) e i k · x + h . c . (cid:17) (3.2) s ( x , τ ) = (cid:90) d k (2 π ) (cid:16) a (1) ( k ) s (1) k ( η ) e i k · x + a (2) ( k ) s (2) k ( η ) e i k · x + h . c . (cid:17) . (3.3)The mode functions π ( i ) k = ( H/k / ) π ( i ) and s ( i ) k = ( H/k / ) s ( i ) are difficult to solve forexactly. However, analytic progress can be made by considering series solutions. It caneasily be checked that the most general series solutions to the mode equations derived from(3.1) are π ( i ) ( η ) = ∞ (cid:88) n =0 (cid:104) a ( i )0 , n ( − η ) n + a ( i ) − , n ( − η ) n + α − + a ( i )+ , n ( − η ) n + α + + a ( i )3 , n ( − η ) n +3 (cid:105) (3.4) s ( i ) ( η ) = ∞ (cid:88) n =0 (cid:104) b ( i )0 , n ( − η ) n + b ( i ) − , n ( − η ) n + α − + b ( i )+ , n ( − η ) n + α + + b ( i )3 , n ( − η ) n +3 (cid:105) (3.5)where α ± = 3 / ± (cid:112) / − µ /H − m /H and b ( i )0 , = 0. For ease of notation we denote a ( i ) r, and b ( i ) r, as a ( i ) r and b ( i ) r . In Ref. [8], it was shown that the non-Gaussianities can be well FIG. 2. One-loop contribution to the collapsed trispectrum of the primordial curvature perturba-tion in QSFI. Dashed lines represent π , and solid lines represent s . approximated by a finite set of combinations of the power series coefficients when µ, m (cid:28) H .The combinations of power series coefficients needed to compute the loop in Fig. 2 areRe (cid:104) a ( i )0 b ∗ ( i ) − (cid:105) (cid:39) − µH µ + m ) , Im (cid:104) a ( i )0 b ∗ ( i )3 (cid:105) = µH µ + m ) , (cid:12)(cid:12) b ( i ) − (cid:12)(cid:12) (cid:39) , (3.6)Im (cid:104) a ( i )0 b ∗ ( i ) − (cid:105) = Im (cid:104) a ( i )0 b ∗ ( i )0 , (cid:105) = Im (cid:104) a ( i )0 b ∗ ( i ) − , (cid:105) = Im (cid:104) a ( i )0 b ∗ ( i )+ (cid:105) = 0 , (3.7)which were determined in [8]. The repeated superscripts ( i ) are summed over i = 1 ,
2. Theabove expressions are valid for µ/H , m/H (cid:28) η = k τ and η (cid:48) = k τ (cid:48) , we find T coll ζ = 2 V (4)2 (cid:18) H ˙ φ (cid:19) k k (cid:90) d p (2 π ) | p + q | p (cid:90) −∞ dηη (cid:90) k k η −∞ dη (cid:48) η (cid:48) Im (cid:2) ( π ( i ) (0) s ∗ ( i ) ( η )) (cid:3) × Im (cid:20) [ π ( j ) (0) s ∗ ( j ) ( η (cid:48) )] s ( k ) (cid:18) | p + q | k η (cid:19) s ∗ ( k ) (cid:18) | p + q | k η (cid:48) (cid:19) s ( l ) (cid:18) pk η (cid:19) s ∗ ( l ) (cid:18) pk η (cid:48) (cid:19)(cid:21) + (cid:0) k ↔ k (cid:1) . (3.8)Similar to before, the dominant contribution to the loop integral occurs for loop momenta p ∼ q (cid:28) k i and the time integrals are dominated by late times. We can immediately expandthe s mode functions to find T coll ζ (cid:39) V (4)2 (cid:18) H ˙ φ (cid:19) k k ) α − I α − ( q ) K ( µ, m ) , (3.9)where I ν ( q ) is given in (2.10) and K ( µ, m ) = (cid:90) −∞ dη ( − η ) − α − Im (cid:2) ( π ( i ) (0) s ∗ ( i ) ( η )) (cid:3) . (3.10)It was shown in [8] that the most important contribution to (3.10) is obtained by cutting offthe lower bound of the integral at η which is around horizon crossing. Inserting the powerseries expansions of the mode functions in (3.4) and (3.5), we find K ( µ, m ) (cid:39) (cid:104) a ( i )0 b ∗ ( i )3 (cid:105) Re (cid:104) a ( j )0 b ∗ ( j ) − (cid:105) (cid:90) η dη ( − η ) − α − (cid:39) −
23 (3 µ/ H ( µ + m ) , (3.11)where we have neglected contributions from higher powers of η which are suppressed in thelimit α − (cid:28)
1. Note that this piece most singular in α − is insensitive to the choice of η .Our final result for the four-point function of the curvature perturbation for m , µ (cid:28) H and q (cid:28) k i is then T coll ζ (cid:39) π V (4)2 (cid:18) H ˙ φ (cid:19) k k q (cid:18) q k k (cid:19) α − (3 µ/ H ( µ + m ) . (3.12)In (3.12), the factors of wavevector magnitudes and α − − from the integral I α − are thesame as those in (2.12) from the integral I ν . These features are characteristic of quantummechanical effects from the exchange of a massive particle [11, 18].We now consider the long wavelength enhancement to the galaxy power spectrum re-sulting from this collapsed primordial trispectrum. In our numerical evaluation, we makethe simplifying assumption that galaxies form at points in space at which the smoothedmatter overdensity is greater than a threshold density at the time of collapse δ c ( a coll ), i.e. n h ( x ) ∝ Θ H ( δ R ( x , a coll ) − δ c ( a coll )) = Θ H ( δ R ( x ) − δ c ), where δ c ≡ δ c ( a coll ) /D ( a coll ). Wefurther assume that δ c ( a coll ) = 1 .
686 [19], all halos collapse instantaneously at redshift z = 1 .
5, and their number density does not evolve in time after collapse. This correspondsto a value of δ c = 4 . δ h ( x ) = ( n h ( x ) − (cid:104) n h (cid:105) ) / (cid:104) n h (cid:105) .With this threshold collapse model, the bias coefficients are given by (see e.g. [20]) b = e − δ c σ R √ πσ R (cid:104) n h (cid:105) , b = δ c σ R e − δ c σ R √ πσ R (cid:104) n h (cid:105) , b = (cid:18) δ c σ R − (cid:19) e − δ c σ R √ πσ R (cid:104) n h (cid:105) (3.13)where (cid:104) n h (cid:105) = erfc (cid:0) δ c / ( √ σ R ) (cid:1) /
2. We use the BBKS approximation to the transfer function[21] and the top-hat window function W R ( k ) = 3(sin( kR ) − kR cos( kR )) / ( kR ) . Moreover,we take R = 1 . /h as the smoothing scale, and numerically we find σ R = 3 . (cid:104) δ R ( x ) δ R ( y ) (cid:105) gives the matter power spectrum P mm ( q ): P mm ( q ) = (cid:18) m H (cid:19) (cid:18) H ˙ φ (cid:19) C ( µ, m ) T ( q ) q , (3.14)where C ( µ, m ) = 1 / µ/ H / ( µ + m ) [8]. It then follows from (2.15) that theratio of the galaxy power spectrum to the matter power spectrum normalized by b is P hh ( q ) b P mm ( q ) = 1 + b b (cid:18) m H R (cid:19) (cid:18) H ˙ φ (cid:19) V (4)2 J π ( qR ) − α − T ( q ) (3 µ/ H ( µ + m ) C ( µ, m )(3.15) δ R ( x ) is the linearly evolved matter overdensity today. FIG. 3. These two tree-level diagrams involving the V (4) interaction can also contribute to scale-dependent and stochastic bias. However, these contributions are small compared to the loopcontribution in Fig. (2) due a suppression arising from the integration over additional hard externalwavevectors. where J = 12 π (cid:90) ∞ du T ( u/R ) W R ( u/R ) u . (3.16)The V (4) interaction in (3.1) also gives rise to the tree-level diagrams shown in Fig. 3 whichcontribute to the long wavelength enhancement of the galaxy power spectrum. However,these terms contain integrals with three transfer functions rather than two like in (3.16).This integral then gives ∼ J / rather than J . Numerically we find J ≈ . × − so thecontributions from these tree-level diagrams are suppressed, as can be seen in Fig. 4.One could also consider the contribution of the ( ∂π ) s/ Λ interaction in (3.1) to P hh ( q ).However, estimating f NL = 5 B ζ ( k, k, k ) / P ζ ( k ) from this interaction numerically, we findthat f NL < ∼ − for µ/H , m/H < ∼ .
4. This small f NL has a negligible contribution to P hh ( q ) compared to the loop contribution we have considered.We can constrain V (4) using the bounds on τ NL and g NL from Planck 2013 and 2015 [22,23]. The bound due to τ NL is estimated using (3.12), with factors of ( q/k ) α − set to 1 in orderto match the τ NL shape. The bound due to g NL is estimated using the tree-level four-pointdiagram with a single V (4) vertex, with factors of ( k i /k j ) α − set to 1 to match the g NL shape.We take τ σNL = 2 . × and g σNL = − . × as the maximum allowed values of τ NL and g NL at a 2 σ confidence level. We find that for most of the ( µ, m ) parameter space τ σNL gives the stronger constraints on V (4) . For µ/H = m/H = 0 .
274 (so that α − = 0 . τ σNL constraint yields V (4) ≤ . P hh ( q ) /b P mm ( q ). The enhanced behavior begins at around q ∼ (200 Mpc /h ) − and q ∼ (300 Mpc /h ) − for the values of V (4) that saturate the τ σNL (black curve) and τ σNL / τ σNL bound, and is significantlysmaller than the loop contribution shown in black.Finally we briefly comment on how our results depend on the parameters R and δ c . Theloop contribution to P hh ( q ) /b P mm ( q ) is insensitive to the choice of smoothing radius R .The tree-level contributions in Fig. 3 increase as R increases, yet even for R = 2 . /h ,we find that the loop contribution remains an order of magnitude larger than the tree-levelcontributions. Furthermore, since b /b ∼ δ c , the second term in (3.15) goes like δ c /q − α − .This implies that the characteristic scale q at which the long-wavelength enhancementsbecome significant depends on δ c like q ∼ δ / c . FIG. 4. The ratio P hh ( q ) /b P mm ( q ) is plotted for τ σNL = 2800 (Planck 2013) in black, and τ σNL / τ σNL bound. Note that the enhanced behavior beginsaround (200 Mpc /h ) − for the black curve, and around (300 Mpc /h ) − for the red curve. Moreover,note that the tree contributions in blue are very small compared to the loop contribution in black.We plot for µ/H = m/H = 0 . α − = 0 .
05. Moreover we take R = 1 . /h and δ c = 4 . IV. CONCLUDING REMARKS
We have shown, using a particular QSFI model, that one-loop diagrams involving anintermediate light scalar can give rise to significant stochastic bias at long wavelengths. Inthis model, the one-loop contribution to the four-point function of primordial curvatureperturbations induces a non-Gaussian contribution to the galaxy power spectrum P hh ( q )that is five times larger than the Gaussian one at q ∼ h/ (500 Mpc) for values of τ NL and g NL at only half their current 2 σ bounds. These non-Gaussianities could be observed inupcoming large-scale surveys [24–26].It would be interesting to study the effects of these loop contributions to the bias withinthe framework of the effective field theory of inflation. At a minimum, this would requirethe computation of the one-loop diagram presented in section II and the ones due to theinteraction L I ∼ ˙ πs . ACKNOWLEDGEMENTS
This work was supported by the DOE Grant DE-SC0011632 and by the Walter BurkeInstitute for Theoretical Physics. [1] A. A. Starobinsky, JETP Lett. , 682 (1979); A. Guth, Phys. Rev. D , 347 (1981);A. D. Linde, Phys. Lett. B , 389 (1982); , 431 (1982); A. Albrecht and P. Steinhardt, Phys. Rev. Lett. , 1220 (1982).[2] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JHEP , 014(2008) [arXiv:0709.0293 [hep-th]].[3] J. M. Maldacena, JHEP , 013 (2003) [astro-ph/0210603].[4] X. Chen and Y. Wang, JCAP , 027 (2010) [arXiv:0911.3380 [hep-th]].[5] T. J. Allen, B. Grinstein and M. B. Wise, Phys. Lett. B. , 66 (1987).[6] N. Dalal, O. Dor´e, D. Huterer and A. Shirokov, Phys. Rev. D , 123514 (2008)[arXiv:0710.4560 [astro-ph]].[7] D. Baumann, S. Ferraro, D. Green and K. M. Smith, JCAP , 001 (2013) [arXiv:1209.2173[astro-ph.CO]].[8] H. An, M. McAneny, A. K. Ridgway and M. B. Wise, arXiv:1711.02667 [hep-ph].[9] L. Senatore and M. Zaldarriaga, JHEP , 024 (2012) doi:10.1007/JHEP04(2012)024[arXiv:1009.2093 [hep-th]].[10] N. Khosravi, JCAP , 018 (2012) doi:10.1088/1475-7516/2012/05/018 [arXiv:1203.2266[hep-th]].[11] N. Arkani-Hamed and J. Maldacena, arXiv:1503.08043 [hep-th].[12] S. Weinberg, Phys. Rev. D , 043514 (2005) doi:10.1103/PhysRevD.72.043514 [hep-th/0506236].[13] H. R. S. Cogollo, Y. Rodriguez and C. A. Valenzuela-Toledo, JCAP , 029 (2008)doi:10.1088/1475-7516/2008/08/029 [arXiv:0806.1546 [astro-ph]].[14] Y. Rodriguez and C. A. Valenzuela-Toledo, Phys. Rev. D , 023531 (2010)doi:10.1103/PhysRevD.81.023531 [arXiv:0811.4092 [astro-ph]].[15] J. Kumar, L. Leblond and A. Rajaraman, JCAP , 024 (2010) doi:10.1088/1475-7516/2010/04/024 [arXiv:0909.2040 [astro-ph.CO]].[16] J. Bramante and J. Kumar, JCAP , 036 (2011) doi:10.1088/1475-7516/2011/09/036[arXiv:1107.5362 [astro-ph.CO]].[17] V. Desjacques, D. Jeong and F. Schmidt, arXiv:1611.09787 [astro-ph.CO].[18] M. Mirbabayi and M. Simonovi´c, JCAP , no. 03, 056 (2016) doi:10.1088/1475-7516/2016/03/056 [arXiv:1507.04755 [hep-th]].[19] J. E. Gunn and J. R. Gott, III, Astrophys. J. , 1 (1972). doi:10.1086/151605[20] S. Ferraro, K. M. Smith, D. Green and D. Baumann, Mon. Not. Roy. Astron. Soc. , 934(2013) doi:10.1093/mnras/stt1272 [arXiv:1209.2175 [astro-ph.CO]].[21] J. M. Bardeen, J. R. Bond, N. Kaiser and A. S. Szalay, Astrophys. J. , 15 (1986).[22] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. , A24 (2014)doi:10.1051/0004-6361/201321554 [arXiv:1303.5084 [astro-ph.CO]].[23] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. , A17 (2016)doi:10.1051/0004-6361/201525836 [arXiv:1502.01592 [astro-ph.CO]].[24] O. Dor´e et al. , arXiv:1412.4872 [astro-ph.CO].[25] P. A. Abell et al. [LSST Science and LSST Project Collaborations], arXiv:0912.0201 [astro-ph.IM].[26] R. Laureijs et al.et al.