General parity-odd CMB bispectrum estimation
aa r X i v : . [ a s t r o - ph . C O ] M a y Prepared for submission to JCAP
General parity-odd CMB bispectrumestimation
Maresuke Shiraishi, a,b
Michele Liguori a,b and James R. Fergusson c a Dipartimento di Fisica e Astronomia “G. Galilei”,Universit`a degli Studi di Padova, via Marzolo 8, I-35131, Padova, Italy b INFN, Sezione di Padova,via Marzolo 8, I-35131, Padova, Italy c Centre for Theoretical Cosmology,Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Abstract.
We develop a methodology for estimating parity-odd bispectra in the cosmicmicrowave background (CMB). This is achieved through the extension of the original sepa-rable modal methodology to parity-odd bispectrum domains ( ℓ + ℓ + ℓ = odd). Throughnumerical tests of the parity-odd modal decomposition with some theoretical bispectrumtemplates, we verify that the parity-odd modal methodology can successfully reproduce theCMB bispectrum, without numerical instabilities. We also present simulated non-Gaussianmaps produced by modal-decomposed parity-odd bispectra, and show the consistency withthe exact results. Our new methodology is applicable to all types of parity-odd temperatureand polarization bispectra. ontents Bispectrum estimation of the cosmic microwave background (CMB) is one of the most power-ful ways to explore the non-Gaussianity of primordial fluctuations. While standard single-fieldslow-roll inflation predicts a tiny amount non-Gaussianity (NG) of the primordial curvatureperturbations [1, 2], this is no longer true for a large number of extensions of the sim-plest inflationary paradigm (see e.g., refs. [3, 4] and references therein). Measurements ofprimordial NG thus provide a stringent test of the standard single-field slow roll scenario,and allow to put stringent constraints on alternative models. The most stringent constraintson primordial NG to date have been obtained through bispectrum measurements of
Planck temperature data [5]. Future analyses, including correlations with E-mode polarization (andthus additional CMB bispectra of the type h T T E i , h T EE i and h EEE i ), will bring in furtherimprovement on the current observational bounds [6, 7].All CMB NG searches so far have been focused on parity-even bispectra, in whichthe condition ℓ + ℓ + ℓ = even is enforced. This is because, as long as we consider thebispectrum of primordial curvature perturbations, parity cannot be broken, due to the spin-0nature of the scalar mode. On the other hand, several interesting models predict bispectragenerated by vector or tensor perturbations. In these cases the parity-even condition mighthave to be removed, since the vector or tensor modes can create parity-odd NG due to theirspin dependence. For example, Early Universe models with some parity-violating or parity-odd sources, such as the gravitational and electromagnetic Chern-Simons actions [8–12], orlarge-scale helical magnetic fields [13, 14], generate NG with sizable CMB bispectrum signalsin parity-odd configurations ( ℓ + ℓ + ℓ = odd) [15–18]. Vector or tensor modes also induceB-mode polarization. B-mode bispectra can thus be useful to prove tensor NG [18, 19]. Atthe same time, the parity-odd property of the B-mode field can generate ℓ + ℓ + ℓ = oddconfigurations in h T T B i , h T EB i , h EEB i and h BBB i bispectra, even when primordial NGhas even parity. B-mode bispectra are also generated via secondary CMB lensing effects [20].These theoretical predictions motivate us to investigate the CMB signals in ℓ + ℓ + ℓ =– 1 –dd triangles using observational data; hence, in this paper, we want to develop a generalframework for parity-odd bispectrum estimation.CMB bispectrum estimation is generally aimed at measuring the so called non-linearparameter f NL . This can be done optimally by mean of the following estimator [21]: E = 1 N Y n =1 X ℓ n m n (cid:18) ℓ ℓ ℓ m m m (cid:19) B ℓ ℓ ℓ " Y n =1 a O ℓ n m n C ℓ n ! − C ℓ m ,ℓ m C ℓ C ℓ a O ℓ m C ℓ , (1.1)where B ℓ ℓ ℓ is a theoretical template of the CMB angle-averaged bispectrum, a O ℓm are theobserved CMB multipoles, C ℓ is the CMB power spectrum, and C ℓ m ,ℓ m = (cid:10) a G ℓ m a G ℓ m (cid:11) is the covariance matrix, obtained from simulated Gaussian maps a G ℓm . Finally, N ≡ X ℓ ℓ ℓ B ℓ ℓ ℓ C ℓ C ℓ C ℓ , (1.2)is a normalization factor. The estimated f NL parameter basically measures the degree ofcorrelation between the theoretical template under study and the three-point function ex-tracted from the data. The input B ℓ ℓ ℓ and C ℓ , as well as the Monte Carlo simulationsused for covariance matrix calculations, include all realistic experimental features such asinstrumental beam, mask and noise. Note that the form of the estimator written above isderived under the “diagonal covariance approximation” i.e. we are replacing the general C − filtering of the multipoles, where C is the (in realistic experimental conditions non-diagonal) a ℓm covariance matrix, with a much simpler 1 /C ℓ filtering. This in principle implies someloss of optimality. In the context of Planck data analysis, it was however shown [5] that itis possible in practice to retain optimality using the simplified estimator above, provided theCMB map is pre-filtered by mean of a recursive inpainting technique. For this reason, wewill work in diagonal covariance approximation throughout the rest of this work (in any caseall of our derivation readily applies to the full-covariance expressions, by simply operatinga a ℓ m /C ℓ → ( C − a ) ℓm replacement). One important and well-known practical issue withthe estimator of eq. (1.1) is that its brute force numerical computation leads to O ( ℓ )operations. This requires huge CPU time and, for the large ℓ max achieved in current andforthcoming observations, it makes a direct approach of this kind totally unfeasible. A similarissue also appears when simulating NG maps with a given bispectrum, using the followingformula originally introduced in ref. [22]: a NG ℓ m = 16 Y n =2 X ℓ n m n a G ∗ ℓ n m n C ℓ n (cid:18) ℓ ℓ ℓ m m m (cid:19) B ℓ ℓ ℓ . (1.3)Such numerical issues can be solved if the theoretical bispectrum is given by a separable formin terms of ℓ , ℓ and ℓ . Using a general technique originally introduced in ref. [23], and oftendubbed the KSW method, the estimator can then be written in terms of a separate productof filtered maps in pixel space, thus massively reducing the computational cost to O ( ℓ )operations. For the parity-even case, many bispectra can be directly written in separableform. In particular, the so called local, equilateral and orthogonal bispectra, encompassinga vast number of NG scenarios, can be described in terms of separable templates. The KSWapproach is directly applicable in this case. On the other hand, the parity-odd bispectra– 2 –ere under study originate from complicated spin and angle dependences in the vector ortensor NG, or coming from lensing effects, and hence they are generally given by a com-plex non-separable form. A natural way to circumvent this issue is to adopt the separablemodal methodology, originally developed by [24–26] for parity-even templates, extending it toparity-odd bispectrum domains. In the modal approach, a general non-separable bispectrumshape is expanded in terms of a suitably constructed, complete basis of separable bispectrumtemplates in harmonic or Fourier space. Provided we use enough templates in the expansion(with convergence speed depending on the choice of basis and the shape of the bispectrumto expand) we can always reproduce the starting template with as high as needed degree ofaccuracy, and the new expanded shape will be separable by construction.In order to extend the methodology to parity-odd bispectra, we will have to introducea new weight function to account for spin dependence, and redefine a reduced bispectrumwhich is not restricted by ℓ + ℓ + ℓ = even. After getting analytical expressions for ourparity-odd estimator, we will numerically implement it for the three Early Universe modelsdescribed in [16–18]. This will allow us to confirm that the modal decomposition can besuccessfully applied to parity-odd bispectra. We also use the modal technique to produceNG maps including the bispectra under study.This paper is organized as follows. In the next section, we summarize the originalmodal decomposition for the parity-even case. In section 3, we extend it to parity-oddmodels. In section 4, we discuss the numerical implementation of the method, showingseveral applications, and we draw our conclusions in the final section. Before moving to the discussion on the parity-odd case, we here summarize the original modalmethodology, which was applied to the estimation of parity-even bispectra in WMAP and
Planck data [5, 24–26].The parity-even angle-averaged bispectrum, B ( e ) ℓ ℓ ℓ , respects the following selectionrules: ℓ + ℓ + ℓ = even , | ℓ − ℓ | ≤ ℓ ≤ ℓ + ℓ . (2.1)Due to rotational invariance, all the physical information for parity-even bispectra is encodedin the so-called reduced bispectrum, b ( e ) ℓ ℓ ℓ , defined by B ( e ) ℓ ℓ ℓ ≡ h ℓ ℓ ℓ b ( e ) ℓ ℓ ℓ . (2.2)In this expression, the model-independent weight function h ℓ ℓ ℓ , defined in terms of Wigner3j-symbols as h ℓ ℓ ℓ ≡ r (2 ℓ + 1)(2 ℓ + 1)(2 ℓ + 1)4 π (cid:18) ℓ ℓ ℓ (cid:19) , (2.3)enforces the parity-even condition and the triangle inequalities (2.1). As we were mention-ing in the previous section, the reduced bispectrum arising from many inflationary modelscannot be directly written in separable form, although this is an essential requirement for f NL estimation. The modal methodology is based on expanding the reduced bispectrum as– 3 – sum over separable basis templates “bispectrum modes” Q ijk ( ℓ , ℓ , ℓ ): v ℓ v ℓ v ℓ p C ℓ C ℓ C ℓ b ( e ) ℓ ℓ ℓ = X ijk α Qijk Q ijk ( ℓ , ℓ , ℓ ) , (2.4) Q ijk ( ℓ , ℓ , ℓ ) ≡ q { i ( ℓ ) q j ( ℓ ) q k } ( ℓ )= 16 q i ( ℓ ) q j ( ℓ ) q k ( ℓ ) + 5 perms in i, j, k , (2.5)where C ℓ denotes the CMB angular power spectrum, and a weight function v ℓ = (2 ℓ + 1) / is introduced to remove an overall ℓ − / scaling in the bispectrum estimator functions; wealso have used the generic notation { a, b, c } to indicate permutations over the indices a , b , c : A { a A b A c } = A a A b A c + 5 perms in a, b, c . The q p ( ℓ ) quantities are built, starting fromgeneric functions such as e.g. monomial of different degree, or cosine and sine plane waves,through an orthonormalization procedure, so that: h q p ( ℓ ) , q r ( ℓ ) i = δ pr , (2.6)where δ pr is a Kronecker delta, and the scalar product h· , ·i between two (parity-even) func-tions of ℓ , ℓ , ℓ is defined in analogy to the correlator between two bispectrum shapes. (cid:10) F, F ′ (cid:11) e ≡ X ℓ ℓ ℓ (cid:18) h ℓ ℓ ℓ v ℓ v ℓ v ℓ (cid:19) F ( ℓ , ℓ , ℓ ) F ′ ( ℓ , ℓ , ℓ ) . (2.7)Note that the Q ijk ( ℓ , ℓ , ℓ ) templates, constructed with this procedure, form a completebut not orthonormal basis. We can obtain a basis of orthonormal templates R n ( ℓ , ℓ , ℓ )through a rotation in bispectrum space: R n ( ℓ , ℓ , ℓ ) = X p λ np Q p ( ℓ , ℓ , ℓ ) . (2.8)In the last expression, for convenience we have labeled the triples ijk , defining a given Q or R template, by mean of a single index n . The rotation matrix λ is a lower triangular matrix,obtained as: γ − = λ ⊤ λ , (2.9)where γ is the matrix of scalar products of the templates Q n : γ pr ≡ h Q p , Q r i e . (2.10)Then, the modal coefficients α Qn of eq. (2.5) are computed as: α Qn = X p ( λ ⊤ ) np α Rp , (2.11)where α Rn are the coefficients of the bispectrum expansion using the orthonormal basis R n as v ℓ v ℓ v ℓ p C ℓ C ℓ C ℓ b ( e ) ℓ ℓ ℓ = X n α Rn R n ( ℓ , ℓ , ℓ ) . (2.12)Due to orthonormality we get: α Rn = * v ℓ v ℓ v ℓ b ( e ) ℓ ℓ ℓ p C ℓ C ℓ C ℓ , R n ( ℓ , ℓ , ℓ ) + e . (2.13)– 4 –sing the expressions above, and the spin-0 Gaunt integral formula, Z d ˆ n Y ℓ m (ˆ n ) Y ℓ m (ˆ n ) Y ℓ m (ˆ n ) = h ℓ ℓ ℓ (cid:18) ℓ ℓ ℓ m m m (cid:19) , (2.14)it was originally shown in ref. [24] that the estimator (1.1) can be written as: E = 1 N X n α Qn β Qn . (2.15)In the past expression the normalization can be conveniently written in terms of the α R expansion coefficients as N = P n ( α Rn ) (it can be easily verified using orthonormality of R n ). Moreover we have defined a quantity β Qn related to the observational data as: β Qn ↔ ijk ≡ Z d ˆ n h M O { i (ˆ n ) M O j (ˆ n ) M O k } (ˆ n ) − D M G { i (ˆ n ) M G j (ˆ n ) E M O k } (ˆ n ) i , (2.16)with M i (ˆ n ) ≡ X ℓm q i ( ℓ ) a ℓm v ℓ √ C ℓ Y ℓm (ˆ n ) , (2.17)being a map generated from given a ℓm and from the modal function q i ( ℓ ). Likewise, theformula for generating NG maps, (1.3), can also be simplified as a NG ℓm = √ C ℓ v ℓ X n ↔ ijk α Qn (cid:20) q { i ( ℓ ) Z d ˆ n Y ℓm (ˆ n ) M G j (ˆ n ) M G k } (ˆ n ) (cid:21) ∗ . (2.18)Note how the use of the complete but not orthogonal basis Q n allows to develop afast estimator by employing separability and reducing the total scaling from ∼ ℓ in thebrute force approach, to ∼ ℓ operations in the angular integral (2.16). On the otherhand, the use of the orthonormal basis R n provides a straightforward way to compute theestimator’s normalization, as a well as a more transparent way to present several results,since the measured coefficients of the R n expansion are uncorrelated. In this section we will discuss the extension of the modal methodology to parity-odd bispectra,which constitutes the original contribution of this work. A general parity-odd angle-averagedbispectrum, B ( o ) ℓ ℓ ℓ , satisfies ℓ + ℓ + ℓ = odd , | ℓ − ℓ | ≤ ℓ ≤ ℓ + ℓ . (3.1)In the standard modal methodology, outlined in the previous section, these signals cannotbe picked up because the averaged bispectrum (2.2) is automatically restricted only to ℓ + ℓ + ℓ = even configurations, due to the parity-even selection rule of the weight function h ℓ ℓ ℓ . In order to build a parity-odd modal pipeline we first need to redefine the reducedbispectrum, and to introduce an h ℓ ℓ ℓ function which includes spin-dependence and covers ℓ + ℓ + ℓ = odd configurations. This will naturally lead to a redefinition of the scalarproduct in parity-odd domains. – 5 – .1 Separable bispectrum Let us introduce a new h ℓ ℓ ℓ function by including three spin values x , y and z , satisfying x + y + z = 0: h { xyz } ℓ ℓ ℓ ≡ h x y zℓ ℓ ℓ + 5 perms in x, y, z , (3.2) h x y zℓ ℓ ℓ ≡ r (2 ℓ + 1)(2 ℓ + 1)(2 ℓ + 1)4 π (cid:18) ℓ ℓ ℓ x y z (cid:19) . (3.3)Note that h { xyz } ℓ ℓ ℓ coincides with the parity-even geometrical function h ℓ ℓ ℓ for x = y = z = 0,but it does not vanish in ℓ + ℓ + ℓ = odd for other specific x , y and z . Using this function,we shall define a parity-odd reduced bispectrum, b ( o ) ℓ ℓ ℓ , as B ( o ) ℓ ℓ ℓ ≡ h { xyz } ℓ ℓ ℓ b ( o ) ℓ ℓ ℓ , (3.4)where b ( o ) ℓ ℓ ℓ is symmetric under the permutation of ℓ , ℓ and ℓ due to h { xyz } ℓ ℓ ℓ = ( − ℓ + ℓ + ℓ h { xyz } ℓ ℓ ℓ and B ( o ) ℓ ℓ ℓ = ( − ℓ + ℓ + ℓ B ( o ) ℓ ℓ ℓ , and takes only purely imaginary values. A general bispec-trum estimator takes into account all triangles for which ℓ , ℓ , ℓ ≥
2. In the parity-odd casewe also have to satisfy the selection rules, eq. (3.1). These conditions specify the allowedrange of x , y and z as ( x, y, z ) = ( ± , ± , ∓
2) and its permutations.The modal decomposition of the parity-odd bispectrum reads v ℓ v ℓ v ℓ i p C ℓ C ℓ C ℓ b ( o ) ℓ ℓ ℓ = X n ↔ ijk α Qn Q n ( ℓ , ℓ , ℓ ) , (3.5)where the left-hand side becomes real thanks to a factor 1 /i . As Q n , we can safely adoptthe modal basis used in the original parity-even modal methodology, since b ( o ) ℓ ℓ ℓ is still asmooth function in the ℓ -space tetrahedron as shown in figure 2. Hence, α Qn , Q n ∈ R holdsalso in this parity-odd case. The modal coefficient is given by α Qn = X p γ − np * v ℓ v ℓ v ℓ b ( o ) ℓ ℓ ℓ i p C ℓ C ℓ C ℓ , Q p ( ℓ , ℓ , ℓ ) + o , (3.6)where h· , ·i o is the inner product in the parity-odd space, which will be defined in the nextsubsection, and γ np ≡ h Q n , Q p i o . In principle, this γ np does not coincide with the parity-evencounterpart, even if we start from the same Q n templates, due to the difference of the innerproduct.Likewise, the decomposition with the orthonormal basis, R n = P np =0 λ np Q p , can beexpressed as v ℓ v ℓ v ℓ i p C ℓ C ℓ C ℓ b ( o ) ℓ ℓ ℓ = X n ↔ ijk α Rn R n ( ℓ , ℓ , ℓ ) , (3.7)where the rotation matrix λ np given by ( γ − ) np = P r ( λ ⊤ ) nr λ rp again differs from the parity-even counterpart. Therefore the R n orthonormal basis will also be different.– 6 – .2 Inner product The scaling behavior of h { xyz } ℓ ℓ ℓ is similar to h ℓ ℓ ℓ , hence we can define the inner productbetween two real functions, defined on a parity-odd domain, in total analogy to the parity-even case: (cid:10) F, F ′ (cid:11) o ≡ X ℓ + ℓ + ℓ =odd h { xyz } ℓ ℓ ℓ v ℓ v ℓ v ℓ ! F ( ℓ , ℓ , ℓ ) F ′ ( ℓ , ℓ , ℓ ) . (3.8)Here note that we must restrict the summation range as ℓ + ℓ + ℓ = odd by hand since h { xyz } ℓ ℓ ℓ does not vanish also in ℓ + ℓ + ℓ = even. In order to obtain the γ and λ matrices, which are necessary to generate the modalexpansion, we have to compute a large number of scalar products between basis templates(more precisely, due to symmetry of γ , we need to compute n max ( n max +1) / n max is the number of templates used in the numerical expansions, which is typically of order10 − , depending on the bispectrum we want to expand) . This poses a practical problem,since the number of operations in a brute-force approach scales like ∼ ℓ for each elementof γ . Fortunately, when the functions F ( ℓ , ℓ , ℓ ) and F ′ ( ℓ , ℓ , ℓ ) can be written in separableforms as F ( ℓ , ℓ , ℓ ) = f { i ( ℓ ) f j ( ℓ ) f k } ( ℓ ) and F ′ ( ℓ , ℓ , ℓ ) = f ′{ i ′ ( ℓ ) f ′ j ′ ( ℓ ) f ′ k ′ } ( ℓ ) (whichis by construction always the case for the basis templates), we can follow an approach similarto the one introduced in ref. [22], in order to speed up the numerical computation of eq. (3.8).We start by rewriting the scalar product in terms of angular integrals: (cid:10) F, F ′ (cid:11) o = 8 π Z − dµ h {− x Y ( o ) { i { i ′ { x − y Y ( e ) jj ′ y − z } Y ( e ) k } k ′ } z } + {− x Y ( e ) { i { i ′ { x − y Y ( o ) jj ′ y − z } Y ( e ) k } k ′ } z } + {− x Y ( e ) { i { i ′ { x − y Y ( e ) jj ′ y − z } Y ( o ) k } k ′ } z } + {− x Y ( o ) { i { i ′ { x − y Y ( o ) jj ′ y − z } Y ( o ) k } k ′ } z } i , (3.9)where we have introduced a map depending on spin and parity as x Y ( o/e ) ii ′ y ( µ ) ≡ X ℓ =odd / even r ℓ + 14 π f i ( ℓ ) f ′ i ′ ( ℓ ) v ℓ x λ ℓy ( µ ) , (3.10)with x Y ℓy (ˆ n ) ≡ x λ ℓy ( µ ) e iyφ . To obtain this form, we have used the spin-dependent Gauntintegral, Z d ˆ n − s Y ℓ m (ˆ n ) − s Y ℓ m (ˆ n ) − s Y ℓ m (ˆ n ) = h s s s ℓ ℓ ℓ (cid:18) ℓ ℓ ℓ m m m (cid:19) , (3.11)and the fact that the φ dependence in each integrand cancels out due to the spin conservation,namely, x + y + z = 0. Since x λ ℓy ( µ ) behaves like the Legendre polynomial, the µ integrals canbe numerically estimated with a high degree of accuracy through Gauss-Legendre quadrature.The total number of angular integrals to compute is 6 × O ( ℓ ) to O (10 × ℓ max ), once the set of all x Y ( o/e ) ii ′ y ’s is pre-computed. As mentioned at beginning, the reduced form (3.9) can then be used to speed up Note that the same scalar product includes both parity-odd and parity-even configurations, so in principleit can be used also to expand parity-even bispectra. – 7 –he calculation of γ np , which is explicitly written down as γ np = (cid:10) Q n ↔ ijk , Q p ↔ i ′ j ′ k ′ (cid:11) o = 8 π Z − dµ h {− x ζ ( o ) { i { i ′ { x − y ζ ( e ) jj ′ y − z } ζ ( e ) k } k ′ } z } + {− x ζ ( e ) { i { i ′ { x − y ζ ( o ) jj ′ y − z } ζ ( e ) k } k ′ } z } + {− x ζ ( e ) { i { i ′ { x − y ζ ( e ) jj ′ y − z } ζ ( o ) k } k ′ } z } + {− x ζ ( o ) { i { i ′ { x − y ζ ( o ) jj ′ y − z } ζ ( o ) k } k ′ } z } i , (3.12)with x ζ ( o/e ) ii ′ y ( µ ) ≡ X ℓ =odd / even r ℓ + 14 π q i ( ℓ ) q i ′ ( ℓ ) v ℓ x λ ℓy ( µ ) . (3.13) Using the new parity-odd modal decomposition, and the spin-dependent Gaunt integral(3.11), the estimator (1.1) can be written as: E = iN X n α Qn β Qn , (3.14)where the normalization becomes negative because b ( o ) ℓ ℓ ℓ takes only purely imaginary values,so that N = − P np α Qn γ np α Qp , and β Qn ↔ ijk ≡ Z d ˆ n h {− x M O( o ) { i − y M O( e ) j − z } M O( e ) k } − D {− x M G( o ) { i − y M G( e ) j E − z } M O( e ) k } + {− x M O( e ) { i − y M O( o ) j − z } M O( e ) k } − D {− x M G( e ) { i − y M G( o ) j E − z } M O( e ) k } + {− x M O( e ) { i − y M O( e ) j − z } M O( o ) k } − D {− x M G( e ) { i − y M G( e ) j E − z } M O( o ) k } + {− x M O( o ) { i − y M O( o ) j − z } M O( o ) k } − D {− x M G( o ) { i − y M G( o ) j E − z } M O( o ) k } i . (3.15)In the parity-odd case, a map generated from a ℓm involves the spin and parity dependence: x M ( o/e ) i (ˆ n ) ≡ X ℓ =odd / even X m q i ( ℓ ) a ℓm v ℓ √ C ℓ x Y ℓm (ˆ n ) . (3.16)Equation (3.14) requires a comparable number of numerical operations with respect to theoriginal parity-even modal methodology, namely O ( ℓ ). The estimator can be written asusual in terms of the orthonormal basis as E = iN X n α Rn β Rn , (3.17)with N = − P n ( α Rn ) .In the same manner, the form of the simulated NG map (1.3) is also reduced to a NG ℓm = i √ C ℓ v ℓ X n ↔ ijk α Qn q { i ( ℓ ) Z d ˆ n {− x Y ∗ ℓm × h − y M G( o ) j − z } M G( o ) k } + − y M G( e ) j − z } M G( e ) k } i ∗ ( ℓ = odd) h − y M G( o ) j − z } M G( e ) k } + − y M G( e ) j − z } M G( o ) k } i ∗ ( ℓ = even) . (3.18)– 8 – F t h l max Weylpseudoscalarhelical PMF
Figure 1 . Signal-to-noise ratios of the parity-odd temperature bispectrum in the Weyl (red solidline), pseudoscalar (green dashed line) and helical PMF (blue dotted line) models, respectively. Thevalues are normalized at ℓ max = 200. In this section, we shall present numerical results from the modal decompositions of parity-odd temperature bispectra arising from some interesting Early Universe models. In thisanalysis, without loss of generality, we choose the spin set in the geometrical function as( x, y, z ) = (1 , , − Parity-odd signatures in the CMB bispectra [16–18] can arise when we consider Early Uni-verse models including Chern-Simons like actions in Weyl gravity ( f ( φ ) f W W ) [8, 9] or in-flationary models with rolling pseudoscalars ( φ e F F ) [10, 12], or when we have primordialparity-odd objects, like helical primordial magnetic fields (PMFs) (see e.g., refs. [13, 14]).These signatures arise not from scalar, due to their spin-0 nature, but from tensor perturba-tions. Tensor-mode temperature signals are transferred to CMB temperature fluctuations viathe Integrated Sachs-Wolfe (ISW) effect on large scales, while polarization signals appear onsmaller scales due to Thomson scattering effects [27]. Accordingly, the temperature bispectrainduced from the above NG sources can be large for ℓ . igure 2 . Three-dimensional representation of Weyl (left top panel), pseudoscalar (right top panel)and helical PMF (bottom panel) reduced bispectra in the tetrahedral domain (3.1). In order to removean ℓ − scaling, we rescale the shapes using a constant Sachs-Wolfe template [24]. doscalar and helical PMF models [16–18]) by mean of a brute force approach. The Fishermatrices for these models can be written, using the scalar product (3.8), as: F th = 16 * v ℓ v ℓ v ℓ b ( o ) ℓ ℓ ℓ i p C ℓ C ℓ C ℓ , v ℓ v ℓ v ℓ b ( o ) ℓ ℓ ℓ i p C ℓ C ℓ C ℓ + o . (4.1)As shown in this figure, all the signal-to-noise ratios are saturated for ℓ max & B ℓℓℓ = B ℓℓℓ ′ = B ℓℓ ′ ℓ = 0.In the following subsections, we will obtain modal decompositions for the three modelsabove, and compare their performances to the brute-force approach. For safety and furtheraccuracy, we will expand the bispectrum signals up to ℓ = 200, although the signal-to-noiseratios almost converge for ℓ max ∼ F th ( ℓ max = 200) = 1, asdescribed in figure 1. For ( x, y, z ) = (1 , , − γ np can explicitly written as γ np = 8 π Z − dµ − ζ ( o ) { i { i ′ (cid:16) − ζ ( o ) jj ′ ζ ( o ) k } k ′ }− + 2 − ζ ( o ) jj ′ − ζ ( o ) k } k ′ } (cid:17) + 16 π Z − dµ − ζ ( o ) { i { i ′ (cid:16) − ζ ( e ) jj ′ ζ ( e ) k } k ′ }− + − ζ ( e ) jj ′ − ζ ( e ) k } k ′ } (cid:17) + 16 π Z − dµ (cid:16) − ζ ( o ) { i { i ′ − − ζ ( e ) jj ′ ζ ( e ) k } k ′ } + ζ ( o ) { i { i ′ − ζ ( e ) jj ′ − ζ ( e ) k } k ′ }− (cid:17) + 8 π Z − dµ ζ ( o ) { i { i ′ − − ζ ( e ) jj ′ − ζ ( e ) k } k ′ } . (4.2)Starting from this γ , we can calculate the expansion coefficients α Qn by use of eq. (3.6) andthis γ np , and rotate to the orthonormal basis to get α Rn . We implemented a stable numericalalgorithm that perform these operations using about 200 modes in 5 CPU hours.Figure 3 depicts the spectrum of mode coefficients as a function of mode number each n .We plot as usual the α R coefficients, related to the orthonormal expansion. The interpreta-tion of the results in the rotated space is in fact more straightforward due to the fact that, asa consequence of orthonormality of the basis templates, the expansion coefficients are uncor-related. Here, we have decomposed using an hybrid basis involving polynomial eigenfunctionsand a special mode function enhanced at the squeezed limit (this is the same basis used forthe analysis of Planck data [5] using the parity-even pipeline; here we use however a smallernumber of modes, since we work at lower ℓ max , which allows for more rapid convergence). Inthe chosen mode ordering, the squeezed mode is located at n = 1. In the helical PMF case,the n = 1 mode is relatively large compared with the other modes due to the squeezed-limitamplification, while, as expected, it is not prominent in the equilateral-type bispectra in theWeyl and pseudoscalar cases. We find that α Rn ’s in these equilateral bispectra converge morerapidly compared with the squeezed-type one.The convergence of the expansion is assessed via computation of the correlation betweenthe full starting theoretical bispectrum and the modal-decomposed one. Using the orthonor-mal basis expansion, it is easy to verify that the correlation between the two is expressed as F th R / √ F th F R , where each Fisher matrix is given by eq. (4.1) and F th R = 16 * v ℓ v ℓ v ℓ b ( o ) ℓ ℓ ℓ i p C ℓ C ℓ C ℓ , n max X n =0 α Rn R n ( ℓ , ℓ , ℓ ) + o , (4.3) F R = 16 n max X n =0 ( α Rn ) . (4.4) These bispectrum amplitudes in the Weyl, pseudoscalar and helical PMF model are obtained when Λ r − =1 . × GeV, X = 9 . × and B / B / = 2 . r − is a combination of an energyscale of the dual cubic action and a tensor-to-scalar ratio in the Weyl gravity, X is a coupling parameter ofa pseudoscalar, and B / B / is a combination of the helical and non-helical magnetic field strengths per 1Mpc generated at the GUT epoch [16–18]. These parameter values will be detected at 68% CL through theparity-odd bispectrum estimation with noiseless large-scale ( ℓ . Planck . – 11 – α R n n Weylpseudoscalarhelical PMF Figure 3 . Modal coefficients in the orthonormal basis, α Rn , in the Weyl (red solid line), pseudoscalar(green dashed line) and helical PMF (blue dotted line) models, respectively. Here we decomposewith the polynomial eigenfunctions and an additional function ( n = 1) sensitive to the squeezed-limitsignals. Figure 4 shows numerical results of the correlations obtained from the modal decompositionsas a function of n max , where n max denotes the number of modes at which we truncate our nu-merical expansions. For this analysis, we have adopted both the hybrid polynomial-squeezedbasis described above and an additional plane-wave decomposition, also equipped with the n = 1 squeezed mode. Like the polynomial basis, also the plane-wave expansion was al-ready adopted in the context of WMAP and Planck data analysis for parity-even bispectra.From the figure we can observe that, in all three models, the polynomial decompositionsachieve more than 90% correlations already for n max &
70, while the plane-wave decomposedbispectra are less correlated with the theoretical templetes than the polynomial ones. Theconvergence speed seems to depend strongly on the bispectrum shape. The polynomial ba-sis can reconstruct the pseudoscalar bispectrum more rapidly, achieving 98% correlation for n max ≃
40. In another equilateral-type case, namely the Weyl model, the convergence is a bitweaker and the correlation reaches around 0.98 for n max ≃ n max & c o rr e l a ti on n max Weyl: polynomialplane wavepseudoscalar: polynomialplane wavehelical PMF: polynomialplane wave
Figure 4 . Correlations between exact bispectra and modal-decomposed ones up to n max , in the Weyl(red lines), pseudoscalar (green lines) and helical PMF (blue lines) models, respectively. Here, wecompare the differences in convergence between the polynomial + squeezed mode (solid lines) and theplane-wave + squeezed mode (dotted lines) decompositions. We now discuss the issue of simulation of NG maps including bispectra from each of ourthree theoretical models. The NG part of the map multipoles can be written in separableform for ( x, y, z ) = (1 , , − a NG ℓm ≡ ( a NG ooℓm + a NG eeℓm ( ℓ = odd) a NG oeℓm + a NG eoℓm ( ℓ = even) , (4.5)– 13 – l ( l + ) C l / ( π ) × T [ µ K ] lWeyl: n max = ∞ max = ∞ max = ∞ Gl Figure 5 . Power spectra of a G ℓm ’s in the Weyl (red lines), pseudoscalar (green lines) and helicalPMF (blue lines) models, respectively. Here, for comparison of convergence, we plot the exact results,namely n max = ∞ , estimated in the direct non-separable computations (solid lines), and the resultsobtained in the n max = 200 modal decompositions (dotted lines). C G ℓ denotes the Gaussian part of C ℓ . where a NG abℓm = i √ C ℓ v ℓ X n ↔ ijk α Qn Z d ˆ n × h q i ( ℓ ) − Y ℓm (cid:16) − M G( a ) j M G( b ) k + M G( a ) j − M G( b ) k (cid:17) + q i ( ℓ ) Y ℓm − M G( a ) j − M G( b ) k + q j ( ℓ ) − Y ℓm (cid:16) − M G( a ) k M G( b ) i + M G( a ) k − M G( b ) i (cid:17) + q j ( ℓ ) Y ℓm − M G( a ) k − M G( b ) i + q k ( ℓ ) − Y ℓm (cid:16) − M G( a ) i M G( b ) j + M G( a ) i − M G( b ) j (cid:17) + q k ( ℓ ) Y ℓm − M G( a ) i − M G( b ) j i ∗ , (4.6)with a, b ∈ o, e .Figure 5 describes the power spectrum of a NG ℓm in each theoretical model. Here wecan compare the results from the separable modal decompositions for n max = 200 given by– 14 –q. (4.5) with the exact results from direct non-separable computations of eq. (1.3), whichshould be equivalent to the n max = ∞ modal decomposition. As expected, we find that, withpre-computed modal coefficients α , the separable modal approach drastically reduces theCPU time from 120 to 0.2 CPU hours. From figure 5, it is also clear that the modal resultsare in good agreement with the exact non-separable computation up to ℓ ≃
100 in all threemodels. This comes from the fact that, for n max = 200, the modal bispectrum reconstructsthe exact shape with more than 98% correlation in each model. In contrast, for ℓ & x M G( o/e ) i , and of the angular integrals of their products. We foundthat this issue can be circumvented by expanding the bispectra and generating maps with anangular resolution much larger than the required ℓ max for the analyis, and then smoothingthe maps by picking only multipoles up to ℓ max . Concerning the shapes that are specificallyunder study, all three spectra decay for ℓ >
100 due to the end of the tensor-mode ISWenhancement, so this approach is not unfeasible.Figure 6 shows the NG part of a simulated map for each of the three models we aretesting; the three maps have been obtained starting from the same Gaussian seed, and we haveconsidered multipoles up to ℓ = 100, in order to avoid the numerical instabilities mentionedabove. Despite the fact that there are several theoretical primordial scenarios predicting the existenceof parity-odd bispectra, no observational constraint on this type of NG has been placed sofar. Generally, parity-odd bispectra are written in non-separable form, and this has madedata analysis unpractical, due to large CPU-time requirements. This paper has developeda new framework for parity-odd CMB bispectrum estimation by extending the separablemodal decomposition methodology, already developed and used for parity-even analyses, toparity-odd domains. The analytical extension to the case of interest has been obtained bydefining a new reduced bispectrum and a new inner product weight function, in such away as to account for spin dependence, and to change selection rules in order to include ℓ + ℓ + ℓ = odd configurations. In this way, we can achieve separability of parity-oddNG estimators and get fast NG maps algorithm, in strict analogy to the parity-even modalexpansion procedure.Our parity-odd modal decomposition has been numerically implemented and tested byexpanding temperature bispectra predicted by several parity-odd Early Universe models. Wehave checked that the numerical algorithm is stable and achieves convergence using a rea-sonable number of templates in a few CPU-hours. The exact convergence efficiency dependson the bispectrum shape and the type of modal eigenfunctions. Using decomposed separablebispectra, we have also produced NG simulations and checked the consistency with the exactresults from a slow brute-force approach. As expected, we get massive computational gainswhen working with separable modal bispectra.The algorithm for bispectrum estimation developed in this paper is applicable to alltypes of parity-odd bispectra (i.e., bispectra enforcing the condition ℓ + ℓ + ℓ = odd). Ournumerical approach so far has included only temperature bispectra. Future interesting appli-cations will include actual estimation of parity-odd NG from CMB data, and the extensionof our method to polarized bispectra, which are generally predicted in parity-odd scenarios,with taking care of bias due to the experimental systematics or the imperfect sky coverage.– 15 – cknowledgments We are very grateful to Paul Shellard for many useful discussions. MS is supported in partby a Grant-in-Aid for JSPS Research under Grant No. 25-573. This work is supported inpart by the ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2.
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