Cosmological Collider Signal

• Cosmological Collider Signal is the distinctive non-Gaussian signature from virtual exchanges of heavy fields during inflation, encoding particle mass and spin information.
• It utilizes oscillatory bispectrum patterns with angular dependence and sound speed modifications to differentiate heavy field effects from single-field inflation models.
• Advanced data analysis techniques, including template orthogonalization and high-performance estimators, enable probing of energy scales around 10¹³ GeV with cosmic spectroscopy.

A cosmological collider signal is the distinctive non-Gaussian correlation imprinted in primordial cosmological observables—most notably the bispectrum of curvature perturbations—by the virtual exchange of massive fields with mass near or above the Hubble scale during inflation. The statistical signatures of these heavy particles combine scale-dependent oscillations, angular dependence, and possible couplings or sound speed effects, enabling a direct "cosmic spectroscopy" of inflationary particle content at 10¹³ GeV energies inaccessible to terrestrial colliders. Detection or stringent constraints of these signals provides a means to probe or falsify both minimal single-field inflation and extensions involving additional bosonic or spinning sectors.

1. Theoretical Mechanism and Signal Structure

The cosmological collider paradigm leverages the equivalence between inflationary de Sitter expansion and an ultra-high-energy collider, where quantum fluctuations and the rapid expansion allow for the non-perturbative pair creation of any field with mass $m \gtrsim 3H/2$ (with $H$ the inflationary Hubble rate). Such heavy quanta, though diluted quickly, can be exchanged in tree-level or loop diagrams for correlation functions of the inflaton fluctuation $\zeta$. In the in-in (Schwinger–Keldysh) formalism, these processes generate bispectra of the schematic form

$$\langle\zeta_{k_1}\zeta_{k_2}\zeta_{k_3}\rangle' = (2\pi)^4 \delta^3(\sum k_i) \frac{f_{\mathrm{NL}}\,\Delta_{\zeta}^2\,S(k_1,k_2,k_3)}{(k_1k_2k_3)^2}$$

where the dimensionless shape function $S$ contains the collider signal. For a single field of mass $m$ and spin $s$, in the squeezed limit $k_3 \ll k_1 \simeq k_2$, the bispectrum receives an oscillatory contribution: $$S_\text{col} \simeq P_s(\hat k_2 \cdot \hat k_3) \left( \frac{k_3}{k_1 + k_2} \right)^{1/2} \sum_a f_a(...) \cos\left[\mu \ln\left(\frac{k_3}{2c_s(k_1 + k_2)}\right) + \delta_a \right] + \text{perms}$$ where $\mu = \sqrt{m^2/H^2-9/4}$ sets the log-frequency of the oscillation, $P_s$ is the Legendre polynomial encoding the mediating spin (isotropic for $s=0$, dipolar for $s=1$, quadrupolar for $s=2$), $c_s$ is a sound-speed ratio, and $f_a$, $\delta_a$ are calculable form-factor and phase parameters (Suman et al., 21 Nov 2025, Anbajagane et al., 2 Sep 2025).

This oscillatory "clock signal" is non-analytic and nonlocal in momentum space, uniquely marking the non-EFT propagation of a Hubble-scale mass and allowing direct inference of $(m, s)$ from observational bispectra.

2. Template Construction and Orthogonalization

Collider bispectrum signals overlap strongly with standard single-field templates (local, equilateral, orthogonal), necessitating precise template construction to isolate signatures unique to heavy particle exchange. Suman et al. (Suman et al., 21 Nov 2025) define four families of collider templates:

• Scalar I: $s=0$, with interactions $\propto \zeta'^2\sigma$ and $\zeta'\sigma$
• Scalar II: $s=0$, with $(\partial\zeta)^2\sigma$ and $\zeta'\sigma$
• Spin-1: $s=1$, interactions $\propto \zeta' \partial\zeta\,\sigma_i$ and $\partial\zeta\,\sigma_i$
• Spin-2: $s=2$, with $\zeta' \partial^{ij}\zeta\,\sigma_{ij}$ and $\partial^{ij}\zeta\,\sigma_{ij}$

Each is parameterized by $(\mu, c_s)$ and normalized so $S(k, k, k) = 1$.

To project out contamination from equilateral and orthogonal single-field models, templates are orthogonalized via the inner product

$$\langle S^{(1)}, S^{(2)} \rangle = \iiint w(k_1,k_2,k_3) S^{(1)}(k_i) S^{(2)}(k_i) dV_k$$

with $w = 1/(k_1 + k_2 + k_3)$. The orthogonalized collider signal is then

$$|\tilde S_\text{col}\rangle = |S_\text{col}\rangle + \alpha |S^\text{equil}\rangle + \beta |S^\text{ortho}\rangle$$

with $(\alpha,\beta)$ determined by enforcing inner product orthogonality. This defines a shape-by-shape measurement of the "pure" collider effect, disentangled from the tightly constrained single-field sector (Suman et al., 21 Nov 2025).

3. Data Analysis Pipelines and Statistical Inference

Collider templates, after orthogonalization, are analyzed against observational datasets using high-performance pipelines such as the Modal estimator, which expands primordial bispectra into separable eigenmodes and projects onto observed CMB multipoles. This reduces the computational complexity from $O(\ell_\text{max}^5)$ to $O(n_\text{modes}\cdot \ell_\text{max}^2)$ and enables a systematic scan over $1 < \mu < 6$ and $10^{-2} < c_s < 10^2$ parameter space.

In Suman et al. (Suman et al., 21 Nov 2025), the Planck 2018 SMICA T+E map ($\ell_\text{max} \simeq 2000$) is employed with an optimal Komatsu–Spergel–Wandelt–type estimator, yielding signal-to-noise (SNR) measurements for each collider template and rigorous adjustment for the look-elsewhere effect due to multiple-parameter scanning.

Orthogonalized $f_{\rm NL}$ amplitudes provide a clean statistical measure for the presence of new physics, in contrast to standard non-Gaussian searches that may conflate signatures from inequivalent theoretical origins.

4. Empirical Constraints and Phenomenological Results

The main constraints in (Suman et al., 21 Nov 2025) are summarized as follows:

Template	Best-fit $f_{\rm NL}$	$(\mu, c_s)$	Raw SNR	Adjusted SNR (look-elsewhere)
Scalar II	$467 \pm 140$	$(1.85, 0.012)$	3.34σ	2.35σ ≈ 2.4σ
Spin-2	$-68 \pm 23$	$(1.08, 0.68)$	2.95σ	1.93σ
Spin-1	$-133 \pm 52$	$(3.8, 2.8)$	2.7σ	1.86σ

Orthogonalized templates show slightly reduced significance but maintain the qualitative hierarchy [(Suman et al., 21 Nov 2025), Table II].

These results represent the most significant cosmological collider hints to date, with the spin-0 Scalar II channel approaching the threshold for "evidence" but not discovery. The best-fit mass parameters correspond to $m/H \sim \sqrt{\mu^2 + 9/4}$, allowing direct inference of particle mass.

Future CMB experiments (Simons Observatory, CMB-S4) and galaxy/LSS surveys (Euclid, LSST, SPHEREx) are forecast to reduce uncertainties on collider $f_{\rm NL}$ by factors of several, potentially converting current hints into robust detections or else excluding significant portions of theory parameter space (Suman et al., 21 Nov 2025, Anbajagane et al., 2 Sep 2025).

5. Distinctive Features and Diagnostic Power

The collider signal uniquely features:

• Logarithmic oscillations in momentum ratios, e.g. $\cos[\mu\ln(k_3/k_1) + \delta]$
• Angular dependence set by spin: $P_s(\hat k_i \cdot \hat k_j)$ selects the spin $s$ of exchanged particle
• Boltzmann suppression for large masses: amplitude scales like $e^{-\pi\mu}$, making detection for $m \gg H$ challenging except in scenarios with chemical potential or resonance enhancements
• Sound-speed and phase effects: deviations from $c_s = 1$ shift oscillation phase and frequency, and explicit phase information can distinguish coupling structures and particle types (Qin et al., 2022).

Resonant or non-Bunch–Davies initial states can further lift the Boltzmann suppression, extending heavy state probe reach to $m/H \gg 1$ for substantial Bogoliubov excitation (Yin, 2023, Wang et al., 25 May 2025).

6. Implications for Early Universe Physics

The detection of a collider signal at high significance would constitute an unambiguous probe of new massive fields during inflation. The ability to extract $m/H$ and $s$ from the bispectrum constitutes "cosmic spectroscopy" at $\sim10^{13}$ GeV energies (Suman et al., 21 Nov 2025). Strong constraints or null results directly rule out classes of multi-field and non-minimal inflationary models. The orthogonalized template methodology enables unified analysis of CMB and LSS bispectrum data, maximizing the cross-survey sensitivity and interpretive clarity.

Current limits indicate that non-Gaussianity arising from single-field inflation is still consistent with Planck data, but robust collider searches are now sensitive to $f_{\rm NL} \sim$ few $\times 100$ for collider signals, with much improved reach anticipated soon (Suman et al., 21 Nov 2025, Sohn et al., 10 Apr 2024).

7. Future Directions

• Statistical Precision: Upcoming CMB-S4, Simons Observatory, and LSST/Euclid/21cm surveys will probe $f_{\rm NL}^{\rm coll}$ down to $\sim 10$ or lower across a range of masses and spins (Suman et al., 21 Nov 2025, Anbajagane et al., 2 Sep 2025).
• Theoretical Extensions: Template sets are expanding to include low-speed collider, equilateral collider, and multi-sound-speed models, as well as analytic control of loop-level and phase-resolved signals (Qin et al., 2022, Jazayeri et al., 2023).
• Systematics and Model Separation: Robust separation from Standard Model non-Gaussian "background" is achieved by combining spin/angular decomposition, phase measurement, and orthogonalization methods (Chen et al., 2016, Suman et al., 21 Nov 2025).
• Joint CMB–LSS Analyses: Cross-correlation of CMB and LSS non-Gaussianity is expected to be crucial for both sensitivity and foreground/degeneracy breaking (Suman et al., 21 Nov 2025, Sohn et al., 10 Apr 2024).

The cosmological collider program now operates at the scientific frontier of probing ultra-high energy beyond-Standard-Model sectors through precision cosmological statistics, linking particle physics, inflationary model building, and cosmological large-scale data analysis.