Modal Bispectrum Pipeline in Cosmology

• The modal bispectrum pipeline is a framework that expands the bispectrum in separable basis functions to compress complex three-point configurations.
• It projects both theory and observations into a lower-dimensional modal space, enabling faster covariance estimation and likelihood analysis.
• Its applications in CMB and large-scale structure studies yield near-optimal parameter constraints with significantly reduced data complexity.

A modal bispectrum pipeline is a bispectrum-estimation and inference framework in which the bispectrum is expanded in a finite set of separable basis functions and the data are compressed into modal coefficients rather than treated as the full set of triangle configurations. In cosmology this architecture has been developed for CMB bispectrum estimation and reconstruction, extended to the matter and galaxy bispectra, and adapted to pipelines that begin with numerical inflationary Lagrangians and end with observational constraints (Fergusson et al., 2010, Regan, 2017, Byun et al., 2022, Zhang et al., 24 Dec 2025).

1. Definition and historical scope

The bispectrum is the Fourier- or harmonic-space three-point function. In redshift-space large-scale structure it is defined by

$$\langle \delta_s(\mathbf{k}_1)\delta_s(\mathbf{k}_2)\delta_s(\mathbf{k}_3)\rangle = (2\pi)^3\delta_D(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B^s(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3),$$

while in the CMB the rotationally invariant content is carried by the reduced bispectrum $b_{\ell_1\ell_2\ell_3}$. A modal pipeline replaces direct estimation of $B$ or $b_{\ell_1\ell_2\ell_3}$ over all configurations by estimation of a lower-dimensional coefficient vector in a separable basis (Regan, 2017, Fergusson et al., 2010).

This strategy emerged in CMB non-Gaussianity work as a response to the intractability of general non-separable bispectra. The late-time mode-expansion estimator was already used to reconstruct the WMAP bispectrum with $l<500$ and $n=31$ orthonormal 3D eigenmodes, while simultaneously constraining local, equilateral, constant, flattened, warm-inflation, and feature models (Fergusson et al., 2010). In large-scale structure the same logic was transferred first to the matter bispectrum and then to anisotropic redshift-space galaxy bispectra, where the principal obstacle is not only the number of triangle bins but also the angular dependence induced by the line of sight (Regan, 2017, Byun et al., 2022).

The defining feature of the pipeline is therefore not a single estimator formula but a modular sequence: choose a separable basis, project theory onto that basis, estimate the corresponding coefficients from data through filtered fields, and perform covariance estimation and likelihood analysis in the compressed mode space.

2. Modal representation and basis construction

The fundamental modal ansatz is an expansion of the bispectrum in separable, symmetric basis functions. In one common large-scale-structure form,

$$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$

with

$$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$

The basis is defined on the allowed triangle domain, and its non-orthogonality is encoded in the mode-coupling matrix $\gamma_{nm}=(Q_n|Q_m)$, which is inverted to obtain orthonormal coefficients (Regan, 2017).

Closely related CMB formulations expand either the primordial shape function or the late-time reduced bispectrum. In CMB-BEST,

$$(k_1k_2k_3)^2 B_\zeta(k_1,k_2,k_3) = \sum_{p_1,p_2,p_3}\alpha_{p_1p_2p_3}\,q_{p_1}(k_1)q_{p_2}(k_2)q_{p_3}(k_3),$$

with symmetrized 3D basis functions $b_{\ell_1\ell_2\ell_3}$0 on the tetrapyd and coefficients obtained by solving $b_{\ell_1\ell_2\ell_3}$1 in a weighted inner product on the tetrapyd domain (Sohn et al., 2023). Earlier CMB modal work used polynomial and eigenmode constructions directly in $b_{\ell_1\ell_2\ell_3}$2-space, with an orthonormal basis $b_{\ell_1\ell_2\ell_3}$3 obtained from a non-orthogonal polynomial basis $b_{\ell_1\ell_2\ell_3}$4 by Gram–Schmidt or Cholesky rotation (Fergusson et al., 2010).

The basis is problem-dependent. Smooth CMB templates can be represented with monomials or Legendre polynomials; resonant and linearly oscillatory bispectra motivate sine–cosine or targeted oscillatory bases; anisotropic galaxy bispectra require basis functions that depend on both $b_{\ell_1\ell_2\ell_3}$5 and the line-of-sight cosine $b_{\ell_1\ell_2\ell_3}$6 (Sohn et al., 2023, Byun et al., 2022). For redshift-space galaxy bispectra, Byun and Krause constructed separable custom modes $b_{\ell_1\ell_2\ell_3}$7 from 24 one-dimensional building blocks $b_{\ell_1\ell_2\ell_3}$8, yielding 83 exact tree-level custom modes, or 247 modes after including Alcock–Paczyński derivatives (Byun et al., 2022).

A further generalization is the treatment of multiple correlated fields and parity sectors. General modal estimation for cross-bispectra introduces separate bases for $b_{\ell_1\ell_2\ell_3}$9, $B$0, $B$1, and $B$2, accommodates both even and odd $B$3, and permits either joint or independent amplitude estimation in the physical field basis rather than in pre-rotated orthogonal combinations (1904.02599).

3. Data-side estimation and reconstruction

On the data side, a modal pipeline turns separability into filtered maps or filtered fields. In redshift-space galaxy applications a practical implementation begins with the FKP-weighted field

$B$4

from which one constructs $B$5, FFTs to $B$6, and then defines line-of-sight multipole-weighted fields

$B$7

Modal fields are then

$B$8

and the compressed coefficients follow from real-space integrals of triple products, followed by multiplication by $B$9 to obtain $b_{\ell_1\ell_2\ell_3}$0 (Regan, 2017).

Because redshift-space distortions single out the line of sight, permutation symmetry is broken if one expands only $b_{\ell_1\ell_2\ell_3}$1 with respect to one side. The construction in (Regan, 2017) resolves this by expanding a symmetric combination of multipoles,

$b_{\ell_1\ell_2\ell_3}$2

so that the same symmetric basis $b_{\ell_1\ell_2\ell_3}$3 used in the isotropic case can be retained.

High-resolution CMB implementations use an analogous filtered-map architecture. CMB-BEST defines projected mode functions

$b_{\ell_1\ell_2\ell_3}$4

and filtered maps

$b_{\ell_1\ell_2\ell_3}$5

or their $b_{\ell_1\ell_2\ell_3}$6 generalization after decorrelating temperature and polarization. The data enter through cubic and linear modal coefficients,

$b_{\ell_1\ell_2\ell_3}$7

and the estimator is evaluated as a finite-dimensional contraction of $b_{\ell_1\ell_2\ell_3}$8, $b_{\ell_1\ell_2\ell_3}$9, and the Fisher matrix (Sohn et al., 2023).

In the numerical-inflation pipeline of “A Modal Approach to Constrain Inflation through Numerical Bispectra,” the data-side representation is the same orthonormal Planck modal basis used for observations, but the theory-side coefficients originate from Primodal rather than from an analytic template. Primodal computes $l<500$0 directly from the in-in formalism through separable 1D mode expansions of the time-dependent integrand; those coefficients are transformed to the Planck primordial basis through $l<500$1, projected to CMB space through $l<500$2, orthonormalised with $l<500$3, and then compared directly with the Planck data coefficients $l<500$4 (Zhang et al., 24 Dec 2025).

4. Compression, covariance, and estimator efficiency

The central practical value of a modal bispectrum pipeline is dimensionality reduction. In redshift-space galaxy surveys the full bispectrum multipoles inhabit $l<500$5 triangle configurations per multipole, making covariance estimation from mocks difficult because one needs $l<500$6 for stable covariance inversion. The modal approach compresses this to $l<500$7 numbers per multipole; from previous real-space work, as few as $l<500$8 modes can recover near full parameter constraints, while $l<500$9 modes provide excellent convergence for a wide family of shapes (Regan, 2017).

For anisotropic redshift-space galaxy bispectra, the compression can be quantified more sharply. Byun and Krause found that all three compression methods they studied recover the full information content of the bispectrum well, but the modal decomposition is the most efficient: only 14 modal expansion coefficients are necessary to obtain constraints within 10% of the full bispectrum result, and 42 coefficients are sufficient to reach 2%; with AP parameters fixed, 13 original custom modes already reach 2% for $n=31$0 (Byun et al., 2022).

In CMB work, compression may be performed partly in wavelet space rather than purely in mode space. The combined wavelet–modal pipeline used 15 wavelet scales, giving 680 independent cubic statistics, and estimated the inverse covariance in that reduced space from about $n=31$1 Gaussian simulations. This yielded an error bar for local non-Gaussianity close to optimal and improved over earlier modal-only WMAP analyses (Regan et al., 2013).

These examples illustrate a common structure. The pipeline replaces a large, highly correlated data vector by a smaller, usually much better conditioned representation. A reduced covariance matrix is cheaper to estimate, cheaper to invert, and easier to validate numerically. In large-scale-structure applications this is one reason the modal approach is repeatedly described as a natural route to joint power-spectrum–bispectrum analyses (Regan, 2017, Byun et al., 2022).

5. Scientific use: forecasting, inference, and theory comparison

Once theory and data have both been projected into the same mode space, the bispectrum enters inference through low-dimensional likelihoods or Fisher matrices. In redshift-space galaxy analyses the modal coefficients $n=31$2 become the compressed data vector, and theoretical coefficients $n=31$3 are obtained by projecting $n=31$4 or $n=31$5 onto the same basis. A Gaussian likelihood then compares $n=31$6 to the theory vector, with covariance estimated from mocks (Regan, 2017).

In CMB feature searches, the pipeline is often organized around Fisher matrices of modal amplitudes rather than around direct map-level simulations. “Joint resonant CMB power spectrum and bispectrum estimation” uses a modal expansion of oscillatory bispectra into linear oscillation modes, constructs mode Fisher matrices, and samples a multivariate Gaussian for the sine and cosine amplitude spectra across frequency. This allows efficient evaluation of peak PDFs, CDFs, and look-elsewhere effects for joint power-spectrum–bispectrum searches (Meerburg et al., 2015).

In numerical inflation studies, the modal pipeline supports direct parameter constraints without an intermediate analytic template family. The Primodal+Modal framework compares full numerical bispectrum predictions to Planck modal coefficients through the consistency-level indicator

$n=31$7

and rejects parameter points for which $n=31$8 lies outside the $n=31$9 interval. Applied to IR DBI inflation, this yields the template-free constraints $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$0 and $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$1 at 95% with Planck $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$2 data (Zhang et al., 24 Dec 2025).

A related use of modal methodology is theory benchmarking. MODAL-LSS was designed to reconstruct the full bispectrum of any 3D density field and to compare fast dark-matter codes against high-accuracy N-body simulations. In that context the modal coefficients support shape and amplitude correlators between simulation and theory, while the “three-shape” benchmark model,

$$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$3

provides a calibrated phenomenological description across scales and redshifts considered (Hung et al., 2019, Lazanu et al., 2015).

A plausible implication is that modal compression is especially relevant for next-generation redshift-space PNG analyses. Euclid-like mock validation with binned $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$4 and $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$5 found that the bispectrum quadrupole is key, that $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$6 alone reduces $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$7 by $$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$8–$$B(k_1,k_2,k_3) \approx w^{-1}(k_1,k_2,k_3)\sum_n B_n\,Q_n(k_1,k_2,k_3),$$9 relative to $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$0, and that joint analyses further tighten constraints by $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$1–$$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$2 (Collaboration et al., 20 May 2026). This suggests that an efficient modal representation of $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$3 is a natural extension when the full joint covariance must be controlled.

6. Limitations, variants, and extensions

Modal bispectrum pipelines are not a single approximation but a stack of approximations whose status depends on the application. In redshift-space galaxy work common assumptions include the global plane-parallel line of sight, Gaussian covariance, periodic boxes, and tree-level or mildly nonlinear modeling. Byun and Krause explicitly assume a periodic box, global plane-parallel LOS, and Gaussian covariance with simple shot noise; they note that survey geometry, wide-angle effects, non-Gaussian covariance, and Fingers-of-God would require extensions of the basis and covariance model (Byun et al., 2022).

Numerical-inflation implementations inherit a different set of limitations. The 2025 Primodal+Modal pipeline is tree-level only, single-field and adiabatic, restricted by finite modal truncation and finite $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$4-range, and introduces basis-transformation errors at the $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$5 level when converting between Primodal and Planck bases. Its DBI application was based on a grid scan rather than a full MCMC exploration, and reheating was absorbed into $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$6 rather than modeled explicitly (Zhang et al., 24 Dec 2025).

CMB pipelines also expose design choices in the basis itself. CMB-BEST emphasizes that accuracy is controlled by the primordial basis expansion and retains the line-of-sight projection exactly rather than expanding the late-time bispectrum into a second modal basis; at the cost of higher computational complexity, this is intended to retain KSW-level exactness for any template that is well represented in the chosen primordial basis (Sohn et al., 2023). Other variants combine modal decomposition with wavelet compression to make inverse-covariance weighting tractable under realistic masking and anisotropic noise (Regan et al., 2013).

The formalism extends naturally beyond a single auto-bispectrum. General modal estimation for cross-bispectra accommodates $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$7, $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$8, $$Q_n(k_1,k_2,k_3) = q_{n_1}(k_1)q_{n_2}(k_2)q_{n_3}(k_3)+\text{5 permutations of }k_i.$$9, and $\gamma_{nm}=(Q_n|Q_m)$0, including both even and odd parity, and supports both amplitude estimation and full bispectrum reconstruction in the physical field basis (1904.02599). This makes modal pipelines relevant not only for scalar auto-bispectra but also for temperature–polarization cross-bispectra, parity-odd signatures, and multi-field generalizations.

Across these variants, the common principle remains stable: modal decomposition converts an intrinsically high-dimensional three-point statistic into a structured coefficient space in which estimation, covariance modeling, null tests, and likelihood evaluation become computationally feasible without discarding the bispectrum as an observable.