Modal Bispectrum Estimator

• The Modal Bispectrum Estimator is a framework that defines the bispectrum as the three-point correlation function with efficient, separable mode expansions.
• It decomposes non-separable, oscillatory bispectrum templates into basis functions to enable rapid evaluation and robust constraints on primordial non-Gaussian signals.
• The method incorporates orthogonalization and statistical corrections, including look-elsewhere adjustments, to probe heavy particle signatures during inflation.

The Modal Bispectrum Estimator is a computational and statistical framework employed in cosmological data analysis, particularly for searching and quantifying specific forms of primordial non-Gaussianity in the cosmic microwave background (CMB) and large-scale structure (LSS) through the bispectrum—the three-point correlation function in Fourier space. Modal methods enable highly efficient, separable representation and estimation of general, potentially non-separable bispectrum templates, making them essential for probing subtle signals generated by, for example, massive particles during inflation as posited by the cosmological collider program (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024).

1. Theoretical Foundations: Non-Gaussian Signatures and Bispectrum Templates

Primordial non-Gaussianities encode beyond-minimal physics of inflation, with the bispectrum $B(k_1,k_2,k_3)$ capturing leading departures from Gaussian statistics. Generic scenarios with extra fields during inflation predict distinctive bispectrum shapes. For massive exchange fields of mass $m$ and spin $s$, the curvature bispectrum admits "collider" terms with oscillatory and angular dependence, formulated as functions of triangle momenta and parametrized by the mass parameter $\mu=\sqrt{m^2/H^2-9/4}$ and, for nontrivial interactions, relative sound speeds $c_s$ (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024). The bispectrum thus encodes a wealth of high-energy physics.

Characteristic bispectrum templates include:

• Oscillatory squeezed-limit shapes from heavy (typically $m > 3H/2$) fields: $$S(k_s,k_s,k_l) \propto (k_l/k_s)^{1/2} \cos[\mu\ln(k_l/k_s)+\delta]$$.
• Spin-dependent signatures: $S^\text{spin-$s$}(k_s,k_s,k_l)\sim P_s(\hat{k}_l\cdot\hat{k}_s)(k_l/k_s)^{1/2}\cos[\mu\ln(k_l/k_s)+\delta]$, with $P_s$ a Legendre polynomial.

These theoretically-motivated templates are often non-separable and highly oscillatory, making brute-force likelihood evaluation in multipole space impractical ($\mathcal{O}(\ell_\text{max}^5)$). Modal estimation addresses this via separable mode expansions.

2. Modal Expansion: Separable Basis Decomposition

The Modal Bispectrum Estimator proceeds by expanding arbitrary bispectrum shapes in a finite, symmetric, separable basis:

$$S(k_1,k_2,k_3) = \sum_{n=0}^{n_\text{max}} \alpha_n Q_n(k_1,k_2,k_3),$$

where each symmetrized basis function $$Q_{prs}(k_1,k_2,k_3) = [q_p(k_1)q_r(k_2)q_s(k_3) + 5\,\text{perms}]/6$$. The $q_p(k)$ are typically smooth Fourier-Legendre polynomials on the physical tetrahedral domain (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025). This expansion enables the efficient construction and application of otherwise intractable templates, including those with complex oscillatory structure or nontrivial angular dependence.

In the context of CMB analyses, the primordial bispectrum is projected to observational multipole space,

$$b_{\ell_1\ell_2\ell_3} = (2/\pi)^3 \int dr\, r^2 \prod_{i=1}^3 \left[ \int dk_i\, k_i^2\, j_{\ell_i}(k_i r)\, T_{\ell_i}(k_i) \right] S(k_1,k_2,k_3),$$

where $j_{\ell}(kr)$ are spherical Bessel functions and $T_\ell(k)$ the linear transfer functions.

3. Modal Estimation Pipeline: CMB and LSS Implementation

The operational pipeline for the Modal Bispectrum Estimator consists of the following stages (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024):

• Basis coefficient computation: For each shape $S(k_1,k_2,k_3)$ of interest (collider templates, equilateral, orthogonal, etc.), compute the expansion coefficients $\{\alpha_n\}$.
• Filtered map construction: Define filtered maps $M_p(\Omega)$ in the observed sky by convolving the CMB (or LSS) data with the radial and spherical basis modes.
• Cubic and linear statistics: Compute for each modal combination a cubic term and a linear term (for mean-field subtraction):

$$\beta_n = \int d\Omega [M_p M_r M_s - \langle M_p^G M_r^G \rangle M_s],$$

where $M_p^G$ is evaluated on Gaussian Monte Carlo simulations, implementing noise and mask corrections.

• Estimator and variance: The unbiased estimator for $f_\text{NL}$ (the amplitude of non-Gaussianity) and its variance are

$$\hat f_\text{NL} = \frac{\sum_n \alpha_n \beta_n}{\sum_n \alpha_n^2}, \quad \operatorname{Var}(\hat f_\text{NL}) = \frac{1}{\sum_n \alpha_n^2}.$$

This approach enables rapid evaluation (typically requiring 100–200 modes for convergence) of $f_\text{NL}$ for a wide class of templates directly in CMB or LSS data. Fast computation is crucial for statistically intensive parameter scans (e.g., over mass and sound speed) and for correcting the look-elsewhere effect in significance estimation.

4. Template Orthogonalization and Statistical Robustness

A central challenge is that collider templates can have large, nonphysical overlap (correlation) with standard equilateral and orthogonal EFT bispectrum shapes. To obtain robust limits or detections for genuinely new physics, collider templates are orthogonalized:

• Define inner products over the momentum tetrahedron as

$$\langle S^{(1)}, S^{(2)} \rangle = \int d^3 k_1 d^3 k_2 d^3 k_3\, \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3)\, \frac{S^{(1)} S^{(2)}}{k_1 + k_2 + k_3}$$

• For a collider template $S_\text{col.}$, construct an orthogonalized version $$\tilde S_\text{col.} = S_\text{col.} + x S^\text{equil} + y S^\text{ortho}$$, solving for $x, y$ such that it is uncorrelated with both equilateral and orthogonal shapes:

$$\langle \tilde S_\text{col.}, S^\text{equil} \rangle = 0, \quad \langle \tilde S_\text{col.}, S^\text{ortho} \rangle = 0$$

(Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025).

This orthogonalization is numerically implemented for each parameter combination $(\mu, c_s)$ in the scan. The resulting templates probe oscillatory signals uniquely associated with massive-particle exchange during inflation, without contamination from EFT backgrounds.

5. Results, Statistical Significance, and Look-Elsewhere Correction

Parameter scans in the Modal pipeline yield $f_\text{NL}$ amplitudes and associated raw significance levels ($|\hat f_\text{NL}|/\sigma_{f_{NL}}$) for each template and parameter set. Given the multidimensional nature of the scan (e.g., mass parameter $\mu$, sound speed ratio $c_s$), the global significance must be corrected for the look-elsewhere effect.

This is achieved by Monte Carlo simulation:

• Generate null (Gaussian) realizations; compute $\hat f_\text{NL}$ for each template and parameter combination.
• Build the distribution of $\max |\hat f_\text{NL}|/\sigma_{f_{NL}}$ under the null; estimate the $p$-value corresponding to the largest observed value.
• Convert this to an adjusted significance $\tilde{\sigma}$.

For instance, the most significant result in Planck data, after look-elsewhere correction, is $\tilde{\sigma}=2.35$ for a scalar-II (cubic $(\partial_i\phi)^2\sigma$) collider shape at $(\mu \approx 1.85, c_s\approx0.012)$ (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025). None of the scanned templates reach $3\sigma$ significance post-correction, but the Modal estimator robustly excludes $|f_\text{NL}| \gtrsim \text{few} \times 10^2$ for these shapes (Sohn et al., 2024).

6. Cross-Validation, Extensions, and Future Prospects

Cross-validation against independent pipelines (e.g., CMB-BEST) demonstrates consistency for oscillatory and angular-dependent collider templates (Suman et al., 26 Dec 2025, Sohn et al., 2024). Modal and CMB-BEST, though different in their mode decompositions and late-time projection methods, find $\hat f_\text{NL}$ in agreement within $1\sigma$ across parameter space.

Future prospects include:

• Application to next-generation CMB surveys (Simons Observatory, CMB-S4) and LSS datasets (Euclid, LSST), with improvements in sensitivity and error bar reduction (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025).
• Extension of the modal method to higher-order correlators (trispectrum), enabling spin-resolved searches and further EFT contamination suppression.
• Realization of combined-likelihood pipelines spanning CMB and LSS, exploiting shared modal bases for optimal extraction of collider signals across redshift and scale (Suman et al., 26 Dec 2025).
• Theoretical refinements of template construction, including full Hankel-function integrals and systematic inclusion of higher-spin and fermionic templates.

7. Summary Table: Modal Bispectrum Estimator Key Elements

Component	Description	References
Mode expansion	$S(k_1,k_2,k_3) = \sum_n \alpha_n Q_n(k_1,k_2,k_3)$, fully separable over $k$-tetrapyd	(Suman et al., 26 Dec 2025)
Orthogonalization	Remove degeneracies with equilateral/orthogonal shapes via Gram–Schmidt procedure in template space	(Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025)
Estimator formula	$$\hat f_\text{NL} = \sum_n \alpha_n\beta_n / \sum_n \alpha_n^2$$, with $\beta_n$ map statistics (cubic, linear terms)	(Suman et al., 26 Dec 2025)
Statistical test	Signal-to-noise ratio $|\hat f_\text{NL}|/\sigma_{f_{NL}}$; look-elsewhere–corrected via MC over parameter space	(Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025)
Numerical pipeline	Fast modal expansions (100–200 modes), per-mode projection to late time, robust mask/noise/anisotropy corrections, CMB/LSS ready	(Suman et al., 26 Dec 2025, Sohn et al., 2024)

The Modal Bispectrum Estimator is thus the state-of-the-art pipeline for separable, unbiased, and statistically robust measurement of primordial bispectra, especially those predicted by the cosmological collider framework, and will play a central role in next-generation searches for heavy particle physics during inflation (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024).