Separating the Inseparable: Constraining Arbitrary Primordial Bispectra with Cosmic Microwave Background Data

Abstract

To efficiently probe primordial non-Gaussianity using Cosmic Microwave Background (CMB) data, we require theoretical predictions that are factorizable, \textit{i.e.}\ those whose kinematic dependence can be separated. This property does not hold for many models, hindering their application to data. In this work, we introduce a general framework for constructing separable approximations to primordial bispectra, enabling direct CMB constraints on arbitrary models including those computed using numerical tools. In contrast to other approaches such as modal decompositions, we learn the basis functions directly from the data, allowing high-fidelity representations with just a handful of terms. This is practically implemented using machine-learning techniques, utilizing neural network basis functions and a loss function designed to mimic the CMB cosine similarity. We validate our pipeline using a variety of input bispectra, demonstrating that the approximations are $>99.5\%$ correlated with the truth with just three terms. By incorporating the neural basis into the \textsc{PolySpec} code, we derive KSW-type CMB estimators, which reproduce local- and equilateral-type non-Gaussianity to within $0.1σ$. As a proof-of-concept, we constrain two inflationary bispectra from the `cosmological collider' scenario; these feature an additional strongly-mixed particle sector and cannot be computed analytically. By combining the numerical predictions from \textsc{CosmoFlow} with our factorizable approach (with just three terms), we place novel constraints on the collider models using \textit{Planck} PR4 data, finding no detection of non-Gaussianity. Our method facilitates detailed studies of the inflationary paradigm, connecting modern theoretical tools with high-resolution observational data.

Figures (6)

• Figure 1: Convergence of the primordial-space bispectrum approximations as a function of the number of terms in the factorizable representation, $N$. We show results using both linear- (left) and log-transformed (right) neural networks for two types of permutation structure. For all templates considered (defined in §\ref{['sec: app']}), we obtain high-accuracy approximations using $N\geq 4$, all with cosine similarities above $99\%$. Due to the differing template shapes, some decompositions are easier to learn than others, with, for example, the local shape being recovered at extremely high accuracy. The optimal choice of network architecture and symmetry assumptions also depends on the model in question, though all choices eventually lead to convergence. For the local and equilateral shapes, we find excellent results at $N=1$ (cyclic) and $N=3$ (full), recovering the standard analytic decompositions. For the collider shapes, which do not have an explicit analytic form, we can obtain $>99.9\%$ accuracy approximations at $N=3$, allowing for their efficient estimation from CMB data in §\ref{['sec: cmb-constraints']}.
• Figure 2: Comparison between the true (left) and approximated (middle) bispectrum shapes for the Collider-I model discussed in §\ref{['subsec: models']}. The right panel shows the fractional difference, multiplying by $50$ for visibility. The factorizable approximation is calculated using $N=3$, assuming cyclic symmetries. The top panels show the primordial-space templates as a function of the dimensionless variables $k_2/k_1$ and $k_3/k_1$ (with $k_1 = 0.1\,\mathrm{Mpc}^{-1}$), with the top left, center bottom and top right edges corresponding to squeezed, flattened and equilateral configurations. In the bottom panel, we show the same results, but project out the equilateral template using \ref{['eq: deproj']}. We find excellent agreement across the whole shape-space, with the factorizable representation achieving a cosine of 99.96%.
• Figure 3: As Fig. \ref{['fig: primordial-coll']} but for the primordial Collider-II template. We again find excellent agreement, achieving a cosine of 99.95%.
• Figure 4: CMB correlation matrix for the neural, ISW-lensing, equilateral, and local $f_{\rm NL}$ amplitudes (where 'neural' represents the factorized basis introduced in this work). The left (right) panel shows results using the factorized cyclic representation trained on the local (equilateral) shape with $N=1$ ($N=3$). Each value indicates the theoretical cosine between two templates in a Planck analysis, including the observational mask, beam, and noise (computed using PolySpec). In both cases, the factorizable representations are almost perfectly correlated with the target shapes, and show similar correlations with the other shapes of interest. This is a non-trivial test of the pipeline since we compare two-dimensional CMB predictions, whilst the neural templates were trained only on three-dimensional correlators.
• Figure 5: Comparison of Planck PR4 non-Gaussianity constraints obtained using standard templates (parametrized by $f_{\rm NL}^{\rm loc}, f_{\rm NL}^{\rm equil}$) to those obtained using the pipeline of this work (parametrized by $f_{\rm NL}^{\rm neural}$). In the left (right) panel, we use the machine-learning model to obtain a separable approximation to the local (equilateral) shape; if our pipeline is accurate, $f_{\rm NL}^{\rm neural}$ should exhibit strong degeneracies with $f_{\rm NL}^{\rm loc}$ ($f_{\rm NL}^{\rm equil}$). Here, each shape is analyzed independently, i.e. the constraints are unmarginalized. We use $N=1$ for the local model and $N=3$ for the equilateral model, testing both full (blue) and cyclic (red) basis function expansions. Results are obtained using the full temperature-plus-polarization dataset, and errors are computed from 400 FFP10/npipe simulations. In all cases, we find excellent agreement between the standard constraints and those from our pipeline, validating our approach.