Searching for Unparticles with the Cosmic Microwave Background

Abstract

Multi-field models of inflation typically assume that interactions between particles can be treated perturbatively. Strongly-coupled models provide an intriguing alternative and may offer novel inflationary phenomenology. We study the "unparticle" scenario, where the inflaton is weakly mixed with a strongly-coupled sector, specified by a (gapless) conformal field theory. For certain choices of conformal scaling dimension, $Δ$, the exchange of unparticles leads to distinctive non-Gaussian features in the primordial curvature distribution, including bispectra with enhanced squeezed limits and oscillations close to the equilateral regime. Efficiently analyzing these models using Cosmic Microwave Background (CMB) data is a challenge since the shapes are non-factorizable in momenta and often highly degenerate with single-field self-interactions. Here, we overcome these limitations using a library of tools, including neural-network factorization schemes and optimal CMB estimators. Our pipeline condenses 161 non-separable templates into just 7 factorizable forms, with negligible loss of signal-to-noise. We apply the model to the \textit{Planck} data, asking two key questions: (1) can we detect unparticles? (2) can we distinguish them from single-field self-interactions? Across $1\leq Δ\leq 9$, we find a maximal signal-to-noise of $1.2σ$, implying no evidence for new physics. We also place the first CMB constraints on the modified consistency-condition-satisfying orthogonal bispectrum with $f^{\rm orth^*}_{\rm NL} = -12\pm12$. While many unparticle models are degenerate with single-field shapes, values of $Δ$ close to half-integers have very different shapes, offering an intriguing future discovery channel. The methods developed herein can be directly applied to other classes of templates, motivating the exploration of models beyond the standard weakly-coupled paradigm.

Figures (6)

• Figure 1: Types of inflationary bispectra. The EFT of inflation provides a basis for describing self-interactions of curvature perturbations. Often, one assumes that the microscopic physics underlying this self-interaction is weakly coupled, sourced by a mediator particle of generic mass and spin. In the massless limit, where the mediator cannot be integrated out, this results in local non-Gaussianity, whilst the heavy-mass case leads to equilateral signatures. Alternatively, the microscopic dynamics can also arise from the exchange of an operator in a strongly coupled sector. In this case (which has rarely been considered in the literature) we can parameterize the correlation function of the strongly coupled theory by its mass gap and the scaling dimension of the operator. In this work, we focus on a gapless model known as the unparticle scenario, which is determined only by its scaling dimension, $\Delta$.
• Figure 2: Schematic outline of this work. Given a set of primordial bispectrum templates, computed analytically using bootstrap techniques, we use linear optimization to obtain a small set of basis templates that describe a wide span of models. Using the techniques of Philcox:2025bbo, these are expanded in factorizable form using machine learning techniques, allowing them to be integrated in KSW-type CMB estimators. Finally, we use the PolySpec package to place constraints on the underlying template amplitudes, $f_{\rm NL}^{\Delta}$. Though we focus on unparticle templates in this work, this outline could be applied to any suite of models.
• Figure 3: Overlap of unparticle shapes with standard templates. We plot three-dimensional cosine between the unparticle shape, $S_\Delta$, and various standard bispectrum shapes as a function of the conformal scaling dimension, $\Delta$. These cosines are obtained via a suitable inner-product in primordial-space, applying a mild $k$-weighting to mimic the Planck sensitivity. The 'orthogonal*' template refers to the shape of Senatore:2009gt (used in most large-scale structure studies), which has the same squeezed-limit scaling as the equilateral shape. Notably, models with $\Delta<2$ exhibit an enhanced squeezed limit, though unitarity requires a lower bound of $\Delta\geq 1$.
• Figure 4: Information loss from basis projection. We plot the theoretical errorbars on the unparticle shape from an idealized forecast ($\sigma^{\rm true}$) and after projecting onto the orthogonalized basis ($\sigma^{\rm SVD}$). The blue lines show the results for a single-template analysis of $f_{\rm NL}^{\Delta}$, whilst the red marginalize over the two self-interaction shapes (via $f_{\rm NL}^{\rm eq}$ and $f_{\rm NL}^{\rm orth^*}$). The dotted lines show the errorbars after projecting the models onto the neural-factorized basis -- the imperfect reconstruction results in a slight loss of information for some choices of $\Delta$. In all cases, we compute the errorbar using a three-dimensional Fisher forecast, which allows us to directly analyze both factorized and unfactorized templates.
• Figure 5: CMB constraints on $f_{\rm NL}^{\Delta}$. Left panel: Normalized constraints on $f_{\rm NL}^{\Delta}$ obtained from the Planck PR4 temperature and E-mode polarization dataset. We show results for both the raw and self-interaction-marginalized datasets, which are obtained using the orthogonalized and factorized templates discussed in the main text. Dotted lines show the results obtained from the mean of $100$ FFP10/npipe simulations. Note that the constraints with $\Delta\leq 2$ and $\Delta>2$ use different basis sets for maximal efficiency. As shown in Fig. \ref{['fig: corrmat']}, the individual constraints are highly degenerate, and the small amplitudes are driven by the non-detection of $f_{\rm NL}^{\rm eq}$. We find no evidence for unparticles in Planck data. Right panel: $1\sigma$ errors on the unparticle template amplitude. The points show results computed using 100 FFP10/npipe simulations, whilst the lines show the theoretical errors obtained from the (two-dimensional) Fisher matrix. The close agreement between points and lines indicate that our analysis is close-to-optimal. The constraints are strongest for certain values of $\Delta$ (around half-integers) where the equilateral cosine in Fig. \ref{['fig: cosines']} is small; these points also represent the most promising discovery space, given the limited correlations with self-interaction shapes.