astro-ph.CO 2026-02-09 0

A non-perturbative framework for N-point functions of locally non-Gaussian fields

We present a non-perturbative approach to correlation functions and polyspectra of locally non-Gaussian fields and develop a simple semi-perturbative framework that does not rely on the local expansion. As an example, we apply it to locally non-Gauss…

Authors: Hardi Veermäe

A non-perturbative framework for N-point functions of locally non-Gaussian fields
Prep ared for submission to JCAP A non-p erturbative framew o rk fo r N -p oint functions of lo cally non-Gaussian fields Ha rdi V eerm¨ ae a a Keemilise ja Biolo ogilise F ¨ u ¨ usik a Instituut, R¨ av ala pst. 10, 10143 T allinn, Estonia E-mail: hardi.v eermae@cern.ch Abstract. W e presen t a non-p erturbativ e approach to correlation functions and p olysp ectra of lo cally non-Gaussian fields and dev elop a simple semi-p erturbativ e framew ork that does not rely on the local expansion. As an example, we apply it to lo cally non -Gaussian fields p os- sessing exponential tails and derive some exact analytic results in the strongly non-Gaussian limit. Con ten ts 1 In tro duction 1 2 N-p oin t functions of non-Gaussian fields 2 2.1 Lo cally non-Gaussian fields 3 2.2 A reduction of integration v ariables for lo cal non-Gaussianity 4 3 A semi-p erturbativ e framew ork 5 3.1 Kibble–Slepian decomp osition 6 3.2 A diagrammatic in terpretation of the decomp osition 8 3.3 2-p oin t functions and non-Gaussian p ow er sp ectra 11 3.4 Bisp ectra 13 3.5 4P: T risp ectra 14 4 Exp onen tially tailed lo cally non-Gaussian fields 15 5 Summary 20 1 In tro duction Cosmic structures ha v e their origins in random fields. The prev alent picture of the early Univ erse is that of a hot, expanding Univ erse with tiny curv ature perturbations on an oth- erwise homogeneous and isotropic F riedmann-Rob ertson-W alk er background. These fluctu- ations are the seeds of present-da y astronomical structures; they are visible in the cosmic micro w a v e background (CMB) [ 1 – 3 ], and on smaller scales, they can source cosmological GW backgrounds in the form of scalar-induced gravitational w a ves (SIGWs) [ 4 – 10 ] or col- lapse into primordial blac k holes (PBHs) [ 11 – 14 ], thereby participating in the genesis of dark matter [ 15 – 17 ]. All these phenomena dep end on the statistical c haracteristics of curv ature p erturbations. CMB observ ations sho w that primordial curv ature p erturbations were tiny and nearly Gaussian [ 3 ]. As long as the non-Gaussianity (NG) remains small, many problems, suc h as the formation of cosmic structures, PBHs, or scalar-induced GWs, are analytically tractable. W eak NG effects can b e treated p erturbatively , but a fully non-p erturbative treatment is curren tly limited to lattice studies [ 18 , 19 ], although some appro ximate non-p erturbative estimates exist, indicating that the p erturbativ e approaches may break do wn earlier than an ticipated for SIGWs [ 20 ]. Thus, ev en if exact estimates of observ ably in teresting quan tities are out of reach computationally , having exact statistical quan tities at hand can assist in testing the v alidit y of the p erturbative regime. N -p oin t functions and the corresp onding p olysp ectra are p otential candidates for this. This study aims to expand the non-perturbative to olb ox for dealing with non-Gaussian fields. W e will fo cus on cases where the NG is lo cal, that is, when the NG random field ζ ( x ) can b e constructed from a Gaussian one ζ G ( x ) and the relation dep ends only on the field at a given p oint in space, that is, ζ ( x ) = F ( ζ G ( x )), where F is some non-linear real function. In this setup, it is natural to study the abstract properties of the non-Gaussian field in co ordinate space and then translate it in to F ourier space, whic h is more commonly used – 1 – when describing the statistical characteristics of ζ ( x ), such as its sp ectra or bisp ectra. This approac h allo ws us to pro vide a simple but rigorous exact form ulation for N -point correlation functions. Starting from the exact formulation, w e will construct a series expansion in the p o w ers of the Gaussian correlation function without relying on the expansion of F ( ζ G ( x )). As a result, it is p ossible obtain a semi-p erturbative treatment applicable in cases where F ( ζ G ) is non-analytic or where a series expansion of F ( ζ G ) is not viable. The main motiv ation of this work is early univ erse cosmology , where NG ma y b e strong at small scales and affect the scalar-induced GWs and the formation of PBHs. The results are fairly general, as they hold for generic locally non-Gaussian fields. They can serv e as non- p erturbativ e appro ximations in cases where the field admits a lo cal description in a range of scales of in terest. The pap er is structured as follows: In section 2 , we pro vide definitions and present a simple exact formulation of N -p oin t functions. In section 3 , a p erturbative expansion to all orders is derived, and a metho d for effectiv ely resumming the series co efficients is provided. As an example, a class of theoretically well-motiv ated models of locally non-Gaussian fields with an exp onential tail is considered, and its effect on the p o w er sp ectrum is examined in section 4 . W e conclude in section 5 . 2 N-p oin t functions of non-Gaussian fields A Gaussian field ζ G ( x ) with a v anishing mean is completely c haracterised b y its t w o-point function 1 ⟨ ζ G ( x 1 ) ζ G ( x 2 ) ⟩ = ξ ( x 1 , x 2 ) . (2.1) Defining the op erator ˆ ξ G b y ( ˆ ξ G ζ G )( x ) = R d 3 y ξ G ( x , y ) ζ G ( y ) and its inv erse ˆ ξ − 1 through ˆ ξ G ˆ ξ − 1 G = 1, then any exp ectation v alues of a functional F [ ζ G ] of ζ G can then b e expressed via the statistical path integral ⟨F [ ζ G ] ⟩ = 1 Z Z D ζ G F [ ζ G ] exp  − 1 2 ζ G ˆ ξ − 1 G ζ G  , (2.2) where the normalisation factor Z = (det 2 π ˆ ξ G ) 1 / 2 is given b y the functional determinant of ˆ ξ G . As usual, the path in tegral ( 2.2 ) is understo o d as the con tinuum limit of multidimensional Gaussian integrals in the discretised spacetime. In that con text, ˆ ξ G directly generalises the notion of a cov ariance matrix of a multiv ariate Gaussian distribution to the field ζ G . W e further note that suc h sto chastic path integrals can b e estimated numerically and are b etter b eha v ed than their quantum coun terparts due to their exp onential damping at large ζ G . W e will consider cases where the non-Gaussian field ζ can b e constructed from a Gaus- sian field ζ G via a general non-linear (and p ossibly non-lo cal) functional F ζ ( x ) = F [ ζ G ; x ] . (2.3) The n -point function of suc h fields can b e estimated using the path in tegral ( 2.2 ), that is, as * Y i ζ ( x i ) + = 1 Z Z D ζ G " Y i F [ ζ G ; x i ] # e − 1 2 ζ G ˆ ξ − 1 ζ G . (2.4) 1 W e will denote Gaussian fields and v ariables related to them by the subindex G . – 2 – W e stress that non-linearity is the decisive obstacle here. Any linear functional would yield another Gaussian v ariable whose PDF can b e deriv ed from its 2-p oint function. That is, if the observ ables can b e constructed from a set of linear com binations of the Gaussian field, ζ i ≡ Z d 3 x L i ( x ) ζ G ( x ) , (2.5) then their description reduces to a Gaussian one characterized b y the co v ariance matrix ⟨ ζ i ζ j ⟩ , which will completely determine the statistical features of the ζ . 2.1 Lo cally non-Gaussian fields The discussion ab ov e is v ery general. Therefore we will restrict our attention to (i) lo c al ly non-Gaussian fields, that can b e defined via ζ ( x ) = F [ ζ G ( x )] , (2.6) with F a non-linear function, that satisfy the prop ert y of (ii) b eing statistic al ly homo gene ous and isotr opic ξ G ( x 1 , x 2 ) = ξ G ( | x 1 − x 2 | ) . (2.7) Homogeneit y and isotropy make it conv enient to work in momen tum space, where the 2-p oin t function is describ ed by the dimensionless p o w er sp ectrum P G ( k ) 2 ⟨ ζ G ( k ) ζ G ( k ′ ) ⟩ = (2 π ) 3 δ ( k + k ′ ) 2 π 2 k 3 P G ( k ) . (2.9) The p ow er sp ectrum and the correlation function are related by ξ G ( x ) = Z ∞ 0 d k k sin( k x ) k x P G ( k ) , P G ( k ) = 2 k 2 π Z ∞ 0 d x x sin( k x ) ξ G ( x ) . (2.10) The 2-p oin t function ( 2.9 ) is anti-diagonal in momentum space. The eigenfunctions of ˆ ξ G are ζ G ( k ) ± ζ G ( − k ), and it can b e straightforw ardly in v erted so that, in momen tum space, w e hav e that ζ G ˆ ξ − 1 G ζ G = Z d 3 k (2 π ) 3 k 3 2 π 2 P G ( k ) ζ G ( k ) ζ G ( − k ) . (2.11) This, together with the fact that spatial homogeneity forces the linear ev olution of the field to b e describable b y multiplicativ e op erators, mak es it natural to describ e curv ature p erturba- tions in momentum space at the linear level. Lo cal non-linear mo difications ( 2.6 ) will couple all scales, and thus it is generally necessary to resort to p erturbation theory when estimating exp ectation v alues ( 2.4 ). Ev aluating the path integral ( 2.4 ) in momentum space offers a non-p erturbativ e solution, but, in general, it can b e computationally unfeasible. Ho w ev er, the computation can b e significantly simplified in p osition space. 2 Whether we are working in momentum or p osition space is to b e understo od from the argument. The momen tum and p osition space representations of the field are related through the F ourier transformation, ζ G ( k ) = Z d 3 xζ G ( x ) e − i kx , ζ G ( x ) = Z d 3 k (2 π ) 3 ζ G ( k ) e i kx . (2.8) These relations, as well as Eq. ( 2.10 ) are also v alid for non-Gaussian fields. – 3 – Determining the lo calit y of non-Gaussian fields By Eq. ( 2.6 ) ab ov e, a lo cally non-Gaussian field can b e defined via an auxiliary Gaussian field and the lo cal mapping F . Such a definition of lo cality merely stipulates the existence of ζ G and F . How ev er, it is not at all clear how to generally demonstrate that a non-Gaussian field is lo cal if ζ G and F are not given b eforehand. Let us tak e a brief detour and consider this question in more detail b efore resuming the discussion of N -p oint functions. In general, when ζ is lo c al ly NG in the sense of Eq. ( 2.6 ), there must exist an in v erse function F − 1 that maps the non-Gaussian field in to a Gaussian one as ζ G ( x ) = F − 1 [ ζ ( x )], implying that such an approach only w orks when F is in v ertible. Moreov er, the follo wing holds: If F is monotonous, it c an b e r e c onstructe d fr om the 1-p oint distribution of ζ ( x ) . T o see this, consider that when ζ ( x ) = F [ ζ G ( x )] and F is monotonously gro wing, then the cumulativ e distribution functions C ( ζ ) ≡ P ( ζ ( x ) ≤ ζ ) and C G ( ζ G ) = P G ( ζ G ( x ) ≤ ζ G ) = [1 + erf ( ζ G / √ 2 ξ 0 )] / 2 must satisfy , C ( F ( ζ G )) = C G ( ζ G ) ⇔ F ( ζ G )) = C − 1 ( C G ( ζ G )) , (2.12) that is, F is determined b y the one-p oint distribution of the non-Gaussian field. Ab ov e, ξ 0 is the v ariance of the Gaussian field. Since C ( ζ ), C G ( ζ G ) are monotonously gro wing, so is F . The case where F is monotonously decreasing can b e treated analogously by using C − 1 G (1 − C ( ζ )). The situation is muc h more complicated when F is not inv ertible, and thus there is no ob vious wa y to go from ζ ( x ) to ζ G ( x ). Although the ab ov e construction ( 2.12 ) can still b e used to obtain a Gaussian one-p oint distribution b y defining the monotonously increasing function ¯ F ( ζ G )  = C − 1 ( C G ( ζ G )). Ho w ev er, since F ( ζ G ) is not monotonous, F ( ζ G )  = ¯ F ( ζ G ) and ¯ F − 1 ( ζ ) will not yield the initial auxiliary Gaussian field. It is thus also not guaranteed that ¯ F − 1 ( ζ ) w ould b e a Gaussian random field at all; that is, even if the 1-p oint distribution is Gaussian, the n -p oint function might not b e. 2.2 A reduction of in tegration v ariables for lo cal non-Gaussianity The computation of n -p oint functions of lo cally non-Gaussian fields is a balancing act betw een c ho osing the momentum representation, which significantly simplifies the linear theory , and the p osition represen tation in which the NG is quantified b y the remark ably simple relation ζ ( x ) = F [ ζ G ( x )] ( 2.6 ). Belo w, we will first compute the p osition space N -p oin t functions of ζ ( x ) and then conv ert them to momentum space by ev aluating their F ourier transform. W e need to estimate the exp ectation v alues for a Gaussian field ζ G ( x ) at n p oints x i . T o shorten notation for later conv enience, w e will denote the Gaussian fields at x i as ζ i ≡ ζ G ( x i ) and the correlation function as 3 ξ ij ≡ ⟨ ζ i ζ j ⟩ ≡ ξ G ( | x i − x j | ) , ξ 0 ≡ ⟨ ζ 2 i ⟩ ≡ ξ G (0) . (2.13) The ζ i admit the form ( 2.5 ) with L i ( x ) = δ ( x i − x ). The distribution of ζ i will b e entirely determined by the finite co v ariance matrix ξ ij and the path integral ( 2.4 ) is reduced to a m ulti-dimensional Gaussian av erage. The n -p oint function is given by 3 W e drop the subindex G denoting Gaussian v ariables only in the quan tities ζ i , ξ ij and ξ 0 . – 4 – * Y i ζ ( x i ) + = 1 p det(2 π ξ ij ) Z exp  − 1 2 ζ i ( ξ − 1 ) ij ζ j  Y i F ( ζ i )d ζ i ≡ G n ( ξ ij ) , (2.14) where ( ξ − 1 ) ij is the inv erse co v ariance matrix. This simple but exact expression provides the foundation on which the rest of the analysis will b e built. The function G n maps Gaussian n -point functions to non-Gaussian ones. It is a function of the cov ariance matrix or 1+ ( n − 1)( n − 2) / 2 v ariables ξ 0 and ξ ij with 1 ≤ i < j . Imp ortan tly , it do es not dep end explicitly on co ordinates; this dep endence arises implicitly through ξ ij . Therefore, G n is indep endent of the shap e of the p ower sp e ctrum and depends only on the in tegrated pow er of fluctuations ξ 0 = R d ln k P G . This is a consider- able simplification, as it allo ws us to factorise the problem: the mapping G n can b e computed without fixing the p ow er sp ectrum. Once G n is known for a mo del F of NG, an y Gaussian p o w er sp ectrum can b e conv erted in to a non-Gaussian n -p oint function * n Y i =1 ζ ( k i ) + ′ = Z G n ( ξ G ( | x i − x j | )) | x n =0 n − 1 Y i =1 e − i x i k i d 3 x i , (2.15) where the ’ indicates that we impose P n i =1 k i = 0 and omit the δ -function that arises due to homogeneit y , that is, the full n -p oint function reads * n Y i =1 ζ ( k i ) + = (2 π ) 3 δ n X i =1 k i ! * n Y i =1 ζ ( k i ) + ′ . (2.16) T ogether with the F ourier transformation, this amoun ts to 4 n − 6 integrals when n > 2: n when ev aluating G n and 3 n when ev aluating the F ourier transformation, which is reduced b y 6 when accoun ting for homogeneity and isotropy . F or 2-p oint functions, the n um b er of in tegrals is 3. On the other hand, the num b er of terms grows exp onentially with the p erturbative order, and with it, the num b er of momen tum integrals also increases. F or non-Gaussian p o w er sp ectra, that is, 2-point functions, the n umerical ev aluation of ( 2.14 ) and ( 2.15 ) can comp ete with p erturbation theory even at the lo w est order. F or bi- and trisp ectra, n umerical estimates based on conv en tional p erturbation theory are exp ected to outp erform the proposed non-p erturbativ e approach at the first few orders. 3 A semi-p erturbative framew ork Aside from numerical estimates, p osition space n -p oint functions are also easier to study p erturbativ ely . The p erturbativ e approac h is set up b y expanding ( 2.6 ) in p o w ers of the Gaussian field ζ ( x ) = ∞ X n =1 F NL ,n ( ζ n G ( x ) − ⟨ ζ n G ( x ) ⟩ ) , (3.1) where the subtracted exp ectation v alues ensure that ⟨ ζ ( x ) ⟩ = 0. In p erturbative estimates, w e will follo w conv ention and assume that F NL , 1 = F ′ [0] = 1, so that ζ = ζ G at the 0’th order. – 5 – The next three indices are often denoted as F NL , 2 = (3 / 5) f NL ≡ F NL , F NL , 3 = (3 / 5) 2 g NL ≡ G NL , F NL , 4 = (3 / 5) 3 h NL ≡ H NL . W e will occasionally use the notation in capital letters to refer to lo w er-order p erturbative corrections. Belo w, w e will deriv e the series expansion of G n in ξ ij and sho w ho w eac h order is related to the series expansion ( 3.1 ) of F . W e will show that F NL ,n can b e resummed at ev ery order of ξ ij and that a p erturbative approac h do es not rely on F b eing analytic. The series expansion in ξ ij is not restricted to p osition space and can be straigh tforw ardly conv erted to momen tum space by replacing m ultiplication with conv olutions; e.g., the term ξ n ij is replaced b y the n -th con v olution p ow er of ( 2.9 ). 3.1 Kibble–Slepian decomp osition An y approach based on the series expansion ( 3.1 ) of F [ ζ G ] will yield a series expansion ξ ij . Th us, it is natural to start with an expansion of ξ ij ( i  = j ) of the multidimensional PDF in ( 2.14 ). In the limit ξ ij → ξ 0 δ ij , i.e., at the 0-th order, the v ariables ζ i b ecome independent, and the exp ectation v alue factorises into v anishing one-p oint functions 4 and thus will also v anish itself. The series expansion in correlation functions is obtained as a direct application of the Kibble–Slepian form ula [ 21 , 22 ]. First, we must introduce some notation to formulate it. Consider a n × n symmetric matrix ψ ij with unit diagonal elements, ψ ij : ψ ii = 1 , ψ ij = ψ j i (3.4) and define the set N of n × n symmetric multiplicit y matrices ν ij with a v anishing diagonal ( ν ii = 0) and with the off-diagonal entries being non-negative in tegers, suc h that N = { ν ij ∈ N n × n : ν ij = ν j i , ν ii = 0 } . (3.5) F or a giv en N ∈ N , we denote s i = P j ν ij . The Kibble–Slepian formula then states that 1 det( ψ ij ) exp  1 2 z i z i − 1 2 z i ( ψ − 1 ) ij z j  = X ν ij ∈N ˜ H s 1 ( z 1 ) . . . ˜ H s n ( z n ) Y i 0, the shift F [ ζ 1 ] → F [ ζ 1 ] − ⟨ F [ ζ 1 ] ⟩ can b e omitted when ev aluating the co efficients C n b ecause ⟨ ¯ H s ( ζ 1 ) ⟩ = 0 when s ≥ 1. T o make con tact with the F NL ,n expansion ( 3.1 ), w e apply the identit y  ζ n 1 ¯ H m ( ζ 1 )  = ( ξ n − m 2 0 n ! ( n − m )!! n ≥ m, n ≡ m mo d 2 0 , (3.12) where !! denotes the double factorial. This gives C s = X m ≥ 0 ( s + 2 m )! s !(2 m )!! ξ m 0 F NL ,s +2 m . (3.13) All the dep endence on F NL ,n has b een absorb ed into the C s co efficien ts. This also holds for the dep endence on ξ 0 , whic h determines the Gaussian fluctuations in a p oint. F rom the p ersp ective of p erturbation theory , the Kibble–Slepian decomp osition has r esumme d all 5 The p olynomials ¯ H m ( ζ 1 ) are orthogonal in the sense that  ¯ H n ( ζ 1 ) ¯ H m ( ζ 1 )  = n ! δ mn . – 7 – v ertices due to fluctuations at a single point. W e will demonstrate this explicitly in the next section 3.2 using diagrammatic techniques. W e included the factorial suppression when defining C n in Eq. ( 3.35 ), so that the leading term of C n is F NL ,n . Explicitly , the first four terms are C 1 = 1 +3 ξ 0 G NL +15 ξ 2 0 F NL , 5 +105 ξ 3 0 F NL , 7 + . . . , C 2 = F NL +6 ξ 0 H NL +45 ξ 2 0 F NL , 6 +420 ξ 3 0 F NL , 8 + . . . , C 3 = G NL +10 ξ 0 F NL , 5 +105 ξ 2 0 F NL , 7 +1260 ξ 3 0 F NL , 9 + . . . , C 4 = H NL +15 ξ 0 F NL , 6 +210 ξ 2 0 F NL , 8 +3150 ξ 3 0 F NL , 10 + . . . . (3.14) Note that the combinatorial term in Eq. ( 3.35 ) gro ws rapidly due to the double factorial in the denominator, so the conv ergence of the series is w orse than the conv ergence of F in the F NL , n series. Estimating the n -p oin t function by expanding it in F NL , n , as done in most of the literature, is thus likely to pro duce an asymptotic series that will fail at higher orders. An example of such a scenario will b e given in Sec. 4 . Since F NL ,n app ears in the n -p oin t functions only inside C s , it is possible to w ork directly with C s . Imp ortantly , C s uniquely determines F NL ,n at any giv en order via the in verse relation F NL ,n = X m ≥ 0 ( n + 2 m )! n !(2 m )!! ( − ξ 0 ) m C n +2 m . (3.15) Ov erall, as long as the p erturbative approach is well b ehav ed, there is no loss of information when swapping from C s to F NL ,m or back. W orking at the n -th p erturbativ e order, the first C s co efficien ts map one-to-one to the first n coefficients F NL ,m . Also, F NL ,m do es not dep end on C s , when s < m and vice versa. In particular, truncating either C s at the order s cut , implies that F NL ,s are truncated at s cut , that is, C s = 0 , s > s cut ⇔ F NL ,s = 0 , s > s cut . (3.16) W e will therefore b e w orking mostly with the co efficients C s , which can b e considered more fundamen tal to the n -p oint functions than F NL ,s . 3.2 A diagrammatic in terpretation of the decomp osition Diagrammatic tec hniques ha ve prov en to b e an extremely pow erful tool in perturbative field theory . It is therefore tempting to interpret the expansion ( 3.9 ) in terms of a diagrammatic formalism. Suc h techniques are not new in the con text of lo cally non-Gaussian random fields and hav e b een used for estimating corrections to SIGW backgrounds [ 23 – 26 ]. Eac h term in the decomp osition ( 3.9 ) can b e represented by a diagram consisting of v ertices lab elled b y i , lines b etw een v ertices lab elled by ij , and the multiplicit y of the lines giv en b y ν ij . In this wa y , eac h diagram can b e uniquely related to a matrix ν ij o v er whic h the sum is taken. The basic elements of the p erturbative expansion typically consist of vertices F NL ,n and lines that are represented by correlation functions ξ ij . Ho w ev er, ( 3.9 ) already reorganises the expansion and can b e obtained from the following F eynman rules that consist of i) lines with multiplicit y ν ij , (3.17) – 8 – ii) resummed v ertices 6 (3.18) of order s i ≡ P n j =1 ν ij that is given by the total m ultiplicit y of the legs entering the vertex, and the condition that iii) lines can only connect resummed vertices with different indices. Let us try to understand the combinatorial factors and ho w these rules arise from the p ow er series expansion ( 2.15 ). First, for the lines, the basic elemen t is the line with m ultiplicit y 1, whic h corresp onds to ξ ij . Lines with higher m ultiplicit y arise when 2 points are connected b y more than one correlation function, and the combinatorial factor in Eq. ( 3.17 ) is a symmetry factor that av oids ov er-counting. Second, a bare v ertex with n legs arises from the term F NL ,n ζ G ( x ) n in the expansion ( 3.1 ). Such vertices are represen ted b y the black dot in ( 3.18 ). As usual, correlation functions of Gaussian fields can b e computed b y Wick contracting all p ossible pairs of fields. Eac h con traction contributes a 2-point function. Thus, 2 m of the fields in each term F NL ,n ζ G ( x ) n can b e con tracted with itself, and thus they would contribute to vertices that connect to s = n − 2 m other vertices. It follows that each vertex of order s will receiv e contributions from terms F NL ,s +2 m ζ s +2 m i , where m ∈ N . The num b er of p ossibilities for connecting the ζ s +2 m i to s other v ertices b y con tracting them with some ζ j (with j  = i ) is ( s + 2 m )! / (2 m )! . On the other hand, the num ber of p ossible self contractions of the remaining 2 m fields in ζ s +2 m i is (2 m − 1)!!. The total m ultiplicit y factor ( s + 2 m )! (2 m )! × (2 m − 1)!! = ( s + 2 m )! (2 m )!! , (3.19) agrees with the one in Eq. ( 3.18 ). The 2 m self con tractions will additionally produce a factor of ξ m ii ≡ ξ m 0 . Com bining all p ossible contributions will therefore yield the sum in ( 3.18 ), whic h is identical to the expansion ( 3.13 ) of C s found using analytic tec hniques. W e remark that the expansion of C s in Eq. ( 3.13 ) agrees with the series expansions for resummed v ertices obtained using an alternative deriv ation based on diagrammatic tec hniques for SIGW [ 26 ]. Although the diagrammatic approach using bare quan tities agrees with the analytic one, w e m ust stress that the in tegral formula for C s in ( 3.9 ) is more general and can b e applied ev en when F ( ζ G ) do es not admit a series expansion or at v alues of ζ G at which the series div erges. As a result, the diagrammatic formulation outlined in Eq. ( 3.17 ), ( 3.18 ) form ulated using the exact non-p erturbative coefficients C s can b e applied if the expansion F NL ,n fails. 6 Strictly sp eaking, “resummed” is a misnomer here, since the deriv ation of Eq. ( 3.9 ) do es not rely on a resummation pro cedure. – 9 – The F eynman rules can be straigh tforw ardly translated to k -space. Below we summarise the rules for the resummed p erturbative formalism that yields N -p oint functions * Y i ζ ( x i ) + and * Y i ζ ( k i ) + ′ . (3.20) They read: i. Lines carry m ultiplicity and distance ( ν ij , x ij ) in x -space or m ultiplicity and an internal momen tum ( ν ij , q ij ) in k -space, with the momen tum directed from i → j . Each line corresp onds to 7 : ξ ν ij G ( x ij ) or P ∗ ν ij G ( q ij ) ≡ 2 π 2 q 3 ij P ∗ ν ij G ( q ij ) , (3.23) where P ∗ n G denotes the n -th con v olution pow er of the Gaussian p ow er sp ectrum P G and P ∗ ν ij G ( q ij ) is the dimensionless p ow er sp ectrum corresp onding to ξ ν ij G ( x ij ). ii. V ertices carry multiplicit y and p osition ( s i , x i ) in x -space or m ultiplicit y and an external momen tum ( s i , k i )) in k -space. In b oth cases, the F eynman rule is the same : s i ! C s i . (3.24) iii. The multiplicities satisfy: a) Multiplicities are not directional ν ij = ν j i ∈ N . b) The m ultiplicit y of lines connecting the same v ertex v anishes ν ii = 0, so such lines are not allo w ed. c) Multiplicities are ”conserved” at each v ertex s i = n X j =1 ν ij . (3.25) 7 There is some abuse of terminology when we define the conv olution for dimensionless p ow er sp ectra b ecause they are not given purely as the F ourier transform of the correlation function. By accoun ting for the definition ( 2.10 ) the pow er spectrum corresponding to a correlation function ξ ( x ) = ξ a ( x ) ξ b ( x ) giv en by the pro duct of correlation functions ξ a ( x ) and ξ b ( x ) is P ( k ) = ( P a ∗ P b )( k ) ≡ k 3 4 π Z d 3 q q 3 | k − q | 3 P a ( q ) P b ( | k − q | ) , (3.21) where P a , P b are p ow er sp ectra corresp onding to ξ a ( x ) and ξ b ( x ), resp ectively . This defines the con volution of dimensionless p ow er sp ectra used in Eq. ( 3.23 ). By making use of isotropy , the con volution in tegral can be recast as ( P a ∗ P b )( k ) = k 2 2 Z D k d q 1 d q 2 q 2 1 q 2 2 P b ( q 1 ) P a ( q 2 ) , = 4 Z ∞ 1 d t Z 1 − 1 d s 1 ( t 2 − s 2 ) 2 P b  k t + s 2  P a  k t − s 2  , (3.22) where the domain of in tegration D k in the first expression is giv en by q 1 , q 2 > 0 and | q 1 − q 2 | ≤ k ≤ q 1 + q 2 .The in tegration v ariables s and t are analogous to those that often app ear when computing scalar-induced GW signals (see e.g. [ 27 , 28 ]). – 10 – iv.( k ) In k -space, momen ta are conserv ed at e ac h vertex k i = n X j =1 q ij . (3.26) Eac h unconstrained internal momen tum (lo op) must b e in tegrated ov er R d q 3 ij / (2 π ) 3 . The momen ta (not explicitly shown ab ov e) can be tak en to satisfy q ij = − q j i to simplify b o okkeeping. iv.( x ) The analogue of the last rule is, that, in x -space, the lines carry the distance b etw een the vertices, x ij = | x i − x j | . 3.3 2-p oin t functions and non-Gaussian p ow er sp ectra Let us consider the non-linear 2-p oint function and the corresp onding p o w er sp ectrum. In this case, determining the non-linear p ow er sp ectrum has b een reduced to computing the function G 2 : [ − ξ 0 , ξ 0 ] → [ −G 2 ( ξ 0 ) , G 2 ( ξ 0 )] , (3.27) that, by Eq. ( 2.14 ), can b e expressed by the double integral 8 G 2 ( ξ 12 ) ≡ ⟨ F ( ζ 1 ) F ( ζ 2 ) ⟩ = Z d ζ 1 d ζ 2 F ( ζ 1 ) F ( ζ 2 ) 2 π p ξ 2 0 − ξ 2 12 e − 1 2 ( ζ 2 1 + ζ 2 2 ) ξ 0 − 2 ζ 1 ζ 2 ξ 12 ξ 2 0 − ξ 2 12 (3.28) Explicitly , expressed in terms of the Gaussian correlation function ξ G ( x ), the non-linear correlation function is ξ ( x ) ≡ ⟨ ζ ( x ) ζ (0) ⟩ = G 2 ( ξ G ( x )) (3.29) and the corresp onding non-linear p ow er sp ectrum is P ( k ) = 2 k 2 π Z ∞ 0 d x x sin( k x ) G 2 ( ξ G ( x )) . (3.30) Some of the basic prop erties of G 2 can b e laid out in general, without reference to a sp ecific realisation of ζ = F ( ζ G ). When the Gaussian field is uncorrelated, the expectation v alue factorises ⟨ F ( ζ 1 ) F ( ζ 2 ) ⟩ = ⟨ F ( ζ 1 ) ⟩⟨ F ( ζ 2 ) ⟩ = 0 and v anishes b ecause ⟨ F ( ζ 1 ) ⟩ = 0 by construction, so that G 2 (0) = 0 (3.31) and uncorrelated p oints are therefore alwa ys mapp ed to uncorrelated ones. In the maxi- mally (anti)correlated case when ξ 12 = ( − ) ξ 0 , we must hav e ζ 1 = ( − ) ζ 2 and the double in tegral ( 3.28 ) is reduced to a single integral G 2 ( ξ 0 ) =  F ( ζ 1 ) 2  , G 2 ( − ξ 0 ) = ⟨ F ( ζ 1 ) F ( − ζ 1 ) ⟩ . (3.32) The Cauc h y-Sc hw artz inequality gives |G 2 ( ξ 12 ) | ≤ G 2 ( ξ 0 ) for all ξ 12 ∈ [ − ξ 0 , ξ 0 ] whic h corre- sp onds to the range rep orted in Eq. ( 3.27 ). As maximally correlated p oints are mapp ed to maximally correlated ones, then inequalit y is saturated when ξ 12 = ξ 0 . This do es, ho w ever, not hold in the maximally anti-correlated Gaussian case, as anti-correlated p oints in the Gaussian field do not hav e to translate into an ti-correlated points in the non-Gaussian field. A simple example is giv en b y ζ ( x ) ∝ ζ 2 G ( x ) − ⟨ ζ 2 G ( x ) ⟩ , that giv es G 2 ( ξ 12 ) ∝ ξ 2 12 so all p oints of – 11 – Figure 1 . Diagrams con tributing to the correlation function and the pow er sp ectrum. The line and the v ertices carry the same m ultiplicit y n ≡ ν 12 = s 1 = s 2 ∈ N and the same momen tum. the non-Gaussian field ζ ( x ) are p ositively correlated ( ξ ( x ) > 0) indep enden tly of the shap e of the Gaussian correlation function. In general, the in tegral ( 3.28 ) does not admit an analytic form, but it can b e ev aluated n umerically (see Sec. 4 ). Ho w ev er, for a near-Gaussian field ζ , it can be appro ximated p erturbativ ely via the expansion ( 3.9 ). The multiplicit y matrices for the biv ariate distribution are N =  0 n n 0  : n ∈ N +  , (3.33) so that each m ultiplicit y matrix/diagram is completely quan tified by a single integer, n ≡ ν 12 = s 1 = s 2 . The Kibble-Slepian formula reduces to Mehler’s formula p 2 ( ζ 1 , ζ 2 ; ξ 12 , ξ 0 ) p 2 ( ζ 1 ) p 2 ( ζ 2 ) = ∞ X n =1 ξ n 12 n ! ¯ H n ( ζ 1 ) ¯ H n ( ζ 2 ) , (3.34) and the non-Gaussian 2-point function reduces to a p ow er series of Gaussian 2-p oint functions, G 2 ( ξ 12 ) = ∞ X n =1 n ! C 2 n ξ n 12 . (3.35) The corresp onding diagrams are given in Fig. 1 . In momen tum space, the p ow ers of ξ n 12 are replaced by conv olution p o w ers of the Gaus- sian p ow er sp ectrum. Either b y taking the F ourier of ( 3.35 ) or by following the k -space F eynman rules for the series diagrams in Fig. 1 , the expansion ( 3.35 ) can b e recast as P ( k ) = ∞ X n =1 n ! C 2 n P ∗ n G ( k ) , (3.36) where P ∗ n G denotes the n -th conv olution p ow er of the Gaussian p ow er sp ectrum P G . W e remark that, by Eq. ( 3.21 ), the conv olutions tend to p ossess a k 3 ”causalit y” tail in the IR, e.g., P ∗ 2 G ( k ) ∼ k 3 R P 2 G ( p ) d p/p 4 . If P G ( k ) ∝ k n , with n < − 3, this tail dominates the non-Gaussian p o w er sp ectrum if C n are sufficien tly large. Indeed, from ( 3.30 ), we find in the k → 0 limit that P ( k ) k ≪ k IR ∼ G 2 ( ξ 0 )  k k IR  3 , k IR ≡  2 π Z ∞ 0 d x x 2 G 2 ( ξ G ( x )) G 2 ( ξ 0 )  − 1 / 3 , (3.37) where the IR scale k IR is determined b y the av erage v olume within which the field remains correlated (note that G 2 ( ξ G ( x )) / G 2 ( ξ 0 ) = ξ ( x ) /ξ (0) gives the correlation of the non-Gaussian field.) The k 3 scaling is exp ected to hold when k ≪ k IR . Sp ectra with a faster than k 3 IR- gro wth correspond to cases in whic h k IR div erges. This requires a specific shap e of G 2 ( ξ G ( x )), whic h is unlikely to o ccur if G 2 and ξ G ( x ) are unrelated. As a result, the k 3 IR-tails are exp ected to b e a fairly general feature of lo cally non-Gaussian fields. 8 W e keep the dep endence on ξ 0 in G 2 implicit. – 12 – Figure 2 . Diagrams con tributing to the 3-p oint function up to p ermutations of the v ertices. Momenta (or positions) at the v ertices and lines are not sho wn. 3.4 Bisp ectra The bisp ectrum corresp onds to the connected 3-p oint correlation function, and thus, b y ( 2.15 ), it can b e constructed from the Gaussian 2-p oint function as B ( k 1 , k 2 ) ≡ ⟨ ζ ( k 1 ) ζ ( k 2 ) ζ ( k 3 ) ⟩ ′ = Z d x 3 1 d x 3 2 e − i ( x 1 k 1 + x 2 k 2 ) G 3 ( ξ G ( x 1 ) , ξ G ( x 2 ) , ξ G ( | x 1 − x 2 | )) , (3.38) where k 1 + k 2 + k 3 = 0. Since the one-p oint function v anishes, there are no disconnected con- tributions, making the bisp ectrum the leading pure prob e of NG. Statistical isotropy allo ws us to express the bisp ectrum as a function of the tw o mo duli k 1 , k 2 and an additional angle ∠ ( k 1 , k 2 ) or moduli k 3 ≡ | k 1 − k 2 | so that the bisp ectrum can b e expressed either as a func- tion of tw o momen ta B ( k 1 , k 2 ) or three mo duli B ( k 1 , k 2 , k 3 ). The mapping G 3 ( ξ 12 , ξ 13 , ξ 23 ) is non-p erturbatively giv en b y the 3-dimensional integral ( 2.14 ). P erturbativ ely , the 3-p oin t function can be split into terms that dep end on pro ducts of 2 or 3 different correlation functions, as depicted diagrammatically in Fig. 2 , mo dulo vertex p erm utations. Thus, the 3-p oin t function can b e expressed as G 3 ( ξ 12 , ξ 13 , ξ 23 ) = G (2) 3 ( ξ 12 , ξ 13 ) + G (2) 3 ( ξ 12 , ξ 23 ) + G (2) 3 ( ξ 23 , ξ 13 ) + G (3) 3 ( ξ 12 , ξ 13 , ξ 23 ) , (3.39) where G (2) 3 ( ξ 12 , ξ 13 ) = X s 2 ,s 3 ≥ 1 ( s 2 + s 3 )! C s 2 C s 3 C s 2 + s 3 ξ s 2 12 ξ s 3 13 , G (3) 3 ( ξ 12 , ξ 13 , ξ 23 ) = X ν 12 ,ν 13 ,ν 23 ≥ 1 s 1 ! s 2 ! s 3 ! ν 12 ! ν 13 ! ν 23 ! C s 1 C s 2 C s 3 ξ ν 23 23 ξ ν 13 13 ξ ν 23 12 (3.40) are obtained from diagrams that contain 2 and 3 legs, resp ectiv ely . In the 2-legged diagram, m ultiplicit y conserv ation Eq. ( 3.25 ) giv es ν 13 = s 3 , ν 12 = s 2 , so that s 1 = s 2 + s 3 . F or the 3-legged diagrams, w e hav e that s 1 = ν 12 + ν 13 , s 2 = ν 12 + ν 23 , s 3 = ν 13 + ν 23 . The bispectrum inherits the decomp osition in ( 3.39 ) in to terms dep ending explicitly on 2 or 3 momenta B ( k 1 , k 2 , k 3 ) = B (2) ( k 1 , k 2 ) + B (2) ( k 1 , k 3 ) + B (2) ( k 2 , k 3 ) + B (3) ( k 1 , k 2 , k 3 ) . (3.41) The B (2) do es not contain an y lo ops and thus it reduces to a sum of pro ducts of the conv o- lution p ow ers of the p o w er sp ectrum B (2) ( k 2 , k 3 ) = (2 π 2 ) 2 k 3 2 k 3 3 X s 2 ,s 3 ≥ 1 ( s 2 + s 3 )! C s 2 C s 3 C s 2 + s 3 P ∗ s 2 G ( k 2 ) P ∗ s 3 G ( k 3 ) (3.42) – 13 – Figure 3 . Diagrams con tributing to the 4-point function up to p erm utations of the v ertices. b ecause one can alw a ys eliminate the third co ordinate, e.g., ξ n 13 ξ m 23 → ξ n G ( x 1 ) ξ m G ( x 2 ) and then use that the F ourier transform of ξ n G ( x ) is (2 π ) 2 P ∗ n G ( k ) /k 3 , as in the case of the 2-p oint functions. This is not the case for the diagrams with 3 legs for which w e obtain B (3) ( k 1 , k 2 , k 3 ) = X ν 12 ,ν 13 ,ν 23 ≥ 1 s 1 ! s 2 ! s 3 ! ν 12 ! ν 13 ! ν 23 ! C s 1 C s 2 C s 3 × Z d 3 q (2 π ) 3 P ∗ ν 12 G ( q ) P ∗ ν 13 G ( q − k 1 ) P ∗ ν 23 G ( q + k 2 ) . (3.43) The B (3) term contributes only at the next-to-leading order. The sum in ( 3.43 ) runs o v er strictly p ositive ν ij ≥ 1, such that s i ≥ 1, and th us the lo west order term is proportional to C 3 2 (or F 3 NL , 2 ), while the leading term in ( 3.42 ) appears at the C 2 1 C 2 (or F NL , 2 ) order. Otherwise, Eq. ( 3.42 ) can b e absorb ed into Eq. ( 3.43 ) by including terms with ν ij = 0. 3.5 4P: T risp ectra The trisp ectrum is the connected component of the 4-p oin t function. In terms of multiplicit y matrices, the three terms of the disconnected comp onent arise from the subset N dis =            0 n 0 0 n 0 0 0 0 0 0 m 0 0 m 0     ,     0 0 n 0 0 0 0 m n 0 0 0 0 m 0 0     ,     0 0 0 n 0 0 m 0 0 m 0 0 n 0 0 0     : n, m ∈ N +        ⊂ N , (3.44) while the rest N con = N \ N dis giv e rise to the connected component. In this w a y , w e obtain the usual decomp osition into connected and disconnected comp onents, ⟨ ζ ( x 1 ) ζ ( x 2 ) ζ ( x 3 ) ζ ( x 4 ) ⟩ = G 2 ( ξ 12 ) G 2 ( ξ 34 ) + G 2 ( ξ 13 ) G 2 ( ξ 24 ) + G 2 ( ξ 14 ) G 2 ( ξ 23 ) + G 4 , con ( ξ 12 , ξ 13 , ξ 14 , ξ 23 , ξ 24 , ξ 34 ) , (3.45) – 14 – via p erturbation theory . The connected comp onent G 4 , con is a function of 7 v ariables ( ξ ij where i > j and ξ 0 ) that maps the Gaussian 2-p oin t functions in to the connected 4-p oin t function of the non-Gaussian field ζ . The F eynman diagrams con tributing to the 4P functions are listed in Fig. 3 , up to p erm utations. Analogously to the case of the 3-p oint function, the first 5 graphs can be obtained from the last by setting some ν ij to 0. Explicitly , the 4-p oin t function is giv en by the series of 6 v ariables, G 4 , con ( ξ 12 ,ξ 13 , ξ 14 , ξ 23 , ξ 24 , ξ 34 ) = X ν ij ∈N con s 1 ! s 2 ! s 3 ! s 4 ! C s 1 C s 2 C s 3 C s 4 ν 12 ! ν 13 ! ν 14 ! ν 23 ! ν 24 ! ν 34 ! ξ ν 12 12 ξ ν 13 13 ξ ν 14 14 ξ ν 23 23 ξ ν 24 24 ξ ν 34 34 = (2!) 2 C 2 1 C 2 2 X σ ξ σ 1 σ 2 ξ σ 1 σ 3 ξ σ 2 σ 4 + 3! C 3 1 C 3 X σ ξ σ 1 σ 2 ξ σ 1 σ 3 ξ σ 1 σ 4 + (2!) 4 C 4 2 X σ ξ σ 1 σ 2 ξ σ 2 σ 3 ξ σ 3 σ 4 ξ σ 1 σ 4 + (3!) 2 2 C 2 1 C 2 3 X σ ξ 2 σ 1 σ 2 ξ σ 1 σ 3 ξ σ 2 σ 4 + (2!) 2 3! C 1 C 2 2 C 3  1 2 X σ ξ σ 1 σ 2 ξ 2 σ 1 σ 3 ξ σ 3 σ 4 + X σ ξ σ 1 σ 2 ξ σ 1 σ 3 ξ σ 1 σ 4 ξ σ 2 σ 3  + 4! C 2 1 C 2 C 4 X σ ξ 2 σ 1 σ 2 ξ σ 1 σ 3 ξ σ 1 σ 4 + . . . . (3.46) where s i = P 4 j =1 ν ij . Let us define the multiplicit y 9 of a term as ν tot ≡ X i

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment