⚛️ phenomenology

Strongly mixed cosmological collider at unequal sound speeds

This paper derives exact solutions for the curvature power spectrum and bispectrum in a two-field inflationary model with arbitrary constant mixing and unequal sound speeds, revealing how the sound-speed ratio governs distinct enhancement regimes, oscillatory behaviors, and modifications to cosmological collider signals.

Original authors: Javier Huenupi, Gonzalo A. Palma, Spyros Sypsas

Published 2026-10-02

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CC BY 4.0

Original authors: Javier Huenupi, Gonzalo A. Palma, Spyros Sypsas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of our universe, a fraction of a second after the beginning, space itself underwent a period of explosive expansion known as cosmic inflation. During this brief era, the universe was not empty; it was filled with quantum fields, invisible substances that ripple and fluctuate like waves on a pond. While one field, the curvature field, drove the expansion and left the imprint of density that eventually formed galaxies, other fields, known as entropy fields, may have existed alongside it. These extra fields did not just sit still; they interacted with the curvature field, exchanging energy and influence. If these interactions were strong, they would have left a unique signature in the distribution of matter we see today. Scientists call this the "cosmological collider," a concept that treats the early universe as a giant particle accelerator, where the collisions of these fields reveal the existence of heavy particles that are far too massive to be created in any laboratory on Earth.

For years, researchers studying these signals have relied on a simplifying assumption: that all these fields travel at the same speed. In the physics of the early universe, this speed is not the speed of light, but a specific rate at which disturbances in the field propagate. By assuming these speeds were identical, scientists could solve the equations governing the fields' behavior, but this assumption ignored a crucial possibility. What if the fields traveled at different speeds? This difference would mean that the waves in one field would cross a critical boundary, known as a sound horizon, at a different time than the waves in the other field. This separation in time could fundamentally alter how the fields mix and how their final signals are recorded.

A team of physicists has now removed this simplifying assumption and solved the problem for the general case where the fields travel at different speeds. They developed a new mathematical framework to track the evolution of two interacting fields, one representing the curvature of space and the other an extra, hidden field. Their work reveals that the ratio of their speeds acts as a powerful control knob for the final signal. When the hidden field travels much slower than the curvature field, the signal can be dramatically amplified, growing by a factor that depends on the strength of their interaction and the mass of the hidden particle. In some specific cases, this amplification grows so large that it follows a precise power law, while in others, it grows logarithmically or oscillates in a bounded pattern. Conversely, when the hidden field travels much faster, the effect on the final signal diminishes rapidly, effectively decoupling the two fields despite their interaction.

The researchers did not stop at the basic power of the signal; they also calculated the three-point correlations, known as the bispectrum, which describe how three different points in the early universe are linked. This calculation is essential for identifying the specific shape of the signal that future telescopes might detect. They found that the difference in speeds changes both the amplitude and the phase of the oscillatory features in the signal. The phase is particularly important because it carries information about the timing of the interactions, distinguishing between different types of particle dynamics. Remarkably, the frequency of these oscillations remains fixed by the mass of the hidden particle, but the height and timing of the peaks shift depending on the speed hierarchy. In one scenario they analyzed, reducing the speed of the hidden field relative to the curvature field increased the signal's amplitude by a factor of approximately twenty-five, a change so significant that it could bring previously undetectable signals within the reach of upcoming observations.

This work provides a complete, exact description of how these fields behave without relying on approximations that treat the mixing as weak. The authors derived precise formulas that connect the final observed signal to the underlying properties of the fields, including their masses, how strongly they mix, and their relative speeds. These formulas are valid for both light and heavy hidden particles and cover all possible ratios of their speeds. The results show that the history of the universe's expansion is encoded in the relationship between these speeds. If the hidden field crosses its horizon first, it has a window of time to evolve and grow before the curvature field catches up, leading to a stronger final imprint. If the curvature field crosses first, the hidden field is suppressed, and the signal remains close to what it would be if the fields did not interact at all.

The implications of these findings extend to the search for new physics in the cosmos. Future surveys of the cosmic microwave background and the large-scale structure of the universe will be sensitive enough to detect these subtle variations. By measuring the amplitude and phase of the primordial signals, astronomers will be able to infer not just the mass of the particles that existed in the early universe, but also how fast they moved. This adds a new layer of detail to the cosmological collider program, allowing scientists to distinguish between different theoretical models of the early universe. The work confirms that the relative speed of propagation is an essential ingredient in the story of cosmic origins, one that must be accounted for to correctly interpret the faint whispers of the universe's birth.

Technical Summary: Strongly Mixed Cosmological Collider at Unequal Sound Speeds

Problem Statement
The cosmological collider program aims to probe the particle content of the early universe through primordial non-Gaussianity, specifically the bispectrum of curvature perturbations. While previous analytic treatments have largely assumed that additional massive fields (entropy modes) propagate at the same speed as the curvature perturbation, this assumption ignores the physical reality that sound speeds ( $c_\phi$ and $c_\sigma$ ) may differ. When sound speeds are unequal, the curvature and entropy fluctuations cross their respective sound horizons at different times. This separation alters the coupled evolution of the fields, modifying the transfer of power and the phase of oscillatory features in the bispectrum. Existing analytic solutions for strong linear mixing were limited to the equal-speed case, necessitating a generalization to arbitrary constant sound speeds to accurately predict collider signals in regimes where mixing is not perturbatively small.

Methodology
The authors consider a two-field inflationary system with a canonically normalized curvature fluctuation $\phi$ and an isocurvature fluctuation $\sigma$ , coupled by a dimensionless quadratic mixing parameter $\lambda$ . The fields possess distinct, constant sound speeds $c_\phi$ and $c_\sigma$ . Working at leading order in slow-roll parameters with a constant Hubble rate $H$ , the authors employ a Laplace representation to solve the coupled linear equations of motion.

Laplace Representation: The coupled dynamics are mapped into Laplace space, where the system reduces from a set of coupled differential equations to a single second-order differential equation.
Heun Equation: For unequal sound speeds ( $r_c \equiv c_\sigma/c_\phi \neq 1$ ), this equation is identified as a general Heun equation with four regular singular points ( $u = \pm 1, \pm r_c$ ). This generalizes the associated Legendre equation (hypergeometric) found in the equal-speed limit.
Mode Construction: The authors construct canonically normalized mode functions by defining integration contours in the complex spectral plane that select the positive-frequency Bunch–Davies vacuum. These contours are chosen to ensure convergence and to correctly reproduce the incoming wave behavior (incoming curvature and entropy waves) at early times.
Exact Solutions: The normalized modes are expressed as contour integrals of Heun functions. These exact solutions are used to compute the late-time curvature power spectrum and the tree-level bispectrum for three representative cubic interactions ( $\sigma(\dot{\phi}-\lambda H\sigma)^2$ , $\sigma^2(\dot{\phi}-\lambda H\sigma)$ , and $\sigma^3$ ).

Key Contributions and Results

Exact Power Spectrum: The paper derives an exact, closed-form expression for the dimensionless curvature power spectrum $\Delta_\zeta(k)$ valid for arbitrary mixing $\lambda$ and sound-speed ratio $r_c$ . The result is expressed in terms of local Heun functions ($HeunG$) and Gamma functions (Eq. 1.8). This expression recovers the known equal-speed results as $r_c \to 1$ and the unmixed spectrum as $\lambda \to 0$ .
Sound-Speed Hierarchies: The authors analyze the spectrum in two limiting regimes:
- Small Ratio ( $r_c \ll 1$ ): The entropy field crosses its horizon first. Depending on the combination $\chi = \sqrt{\nu^2 + \lambda^2}$ $χ = ν^{2} + λ^{2}$ (where $\nu$ $ν$ relates to the entropy mass), the spectrum exhibits:
  - Power-law enhancement $\propto r_c^{-2\chi}$ if $\chi > 0$ .
  - Logarithmic growth $\propto \ln^2(1/r_c)$ if $\chi = 0$ .
  - Bounded oscillations in $\ln r_c$ if $\chi$ is imaginary.
- Large Ratio ( $r_c \gg 1$ ): The curvature field crosses first. The correction to the unmixed spectrum vanishes as $\lambda^2 \ln r_c / r_c^2$ , indicating a decoupling limit even for large mixing.
Bispectrum and Collider Signals: Using the exact modes, the authors compute the squeezed limit of the bispectrum ( $k_L \ll k_S$ $k_{L} ≪ k_{S}$ ).
- The squeezed limit retains the strict frequency set by the final entropy mass ( $\rho = \sqrt{\mu^2/H^2 - 9/4}$ ), but the amplitude and phase of the oscillatory signal depend non-trivially on the sound-speed ratio $r_c$ .
- The power-normalized amplitude can vary non-monotonically with $r_c$ . For example, in the $\sigma^3$ interaction with fixed mass and mixing, reducing $r_c$ from 2 to 0.5 can increase the oscillation amplitude by a factor of $\sim 25$ .
- The phase of the signal also shifts with $r_c$ , providing additional information to distinguish particle dynamics beyond just the oscillation frequency.
Numerical Verification: The analytic results are validated against independent numerical solutions of the coupled field equations in the time domain, showing agreement to better than $10^{-9}$ in relative magnitude for various parameter choices.

Significance
The paper establishes that the sound-speed hierarchy is an essential component of the relation between primordial signals and underlying particle dynamics in strongly mixed systems. By providing exact analytic solutions for unequal sound speeds, the work enables precise predictions for cosmological collider observables without relying on perturbative expansions in the mixing parameter.

The authors emphasize that the hierarchy dependence of the collider amplitude cannot be inferred from the power spectrum alone; the interaction region and the conversion of entropy to curvature must be treated simultaneously. The derived formulas allow for the construction of full-shape templates for future observational searches (e.g., in CMB and large-scale structure) that account for strong mixing and unequal propagation speeds. Furthermore, the normalized modes provide the necessary propagators for investigating quantum corrections and loop signals in these systems. The results confirm that while the oscillation frequency remains fixed by the mass, the amplitude and phase are sensitive probes of the sound-speed ordering and the strength of the mixing.

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