Primordial Physics in the Nonlinear Universe: mapping cosmological collider models to weak-lensing observables

This paper presents a suite of cosmological simulations incorporating primordial non-Gaussianities from collider physics to characterize their nonlinear signatures in late-time observables and forecasts competitive constraints on these signals using future Vera C. Rubin Observatory weak-lensing data.

The Gist

The Big Picture: A Cosmic Time Machine

Imagine the universe as a giant, expanding balloon. About 13.8 billion years ago, this balloon was tiny and underwent a period of incredibly fast growth called inflation. During this split-second, the universe wasn't just a smooth, featureless soup; it had tiny ripples and bumps.

Most of the time, physicists assume these ripples were perfectly random, like static on an old TV screen. This is called a "Gaussian" distribution. But, what if the universe had a secret? What if, during that inflationary era, heavy, exotic particles (too heavy to ever build in a lab on Earth) interacted with the energy driving the expansion?

If those particles existed, they would have left a very specific, non-random fingerprint on the universe. This fingerprint is called Primordial Non-Gaussianity (PNG). It's like finding a specific pattern in the static that tells you exactly what kind of radio station was broadcasting before the static started.

The Problem: The "Cosmological Collider" is Broken

Physicists call the study of these heavy particles "Cosmological Collider Physics." Think of the early universe as the ultimate particle accelerator, smashing things together at energies we can never reach on Earth.

The problem is that we usually study these fingerprints using simple math (linear theory) that only works when the universe is smooth and calm. But the universe we see today is messy. Gravity has pulled matter together into clumps, galaxies, and clusters. This is the "nonlinear regime."

Imagine trying to understand the ingredients of a cake by looking at a smooth batter (easy math). But the cake has been baked, and the ingredients have swirled, melted, and reacted with each other (messy reality). The old math breaks down when you try to look at the finished cake. Until now, we couldn't simulate these exotic "collider" fingerprints in the messy, nonlinear universe to see how they would actually look today.

The Solution: A New Recipe for the Universe

The authors of this paper built a new "kitchen" (a computer simulation) that can bake cakes with these exotic ingredients.

1. The Challenge: To simulate the universe, you need to start with the right initial ingredients (the "Initial Conditions"). For standard random noise, this is easy. But for these exotic "collider" patterns, the math is incredibly complex. It's like trying to mix a specific flavor into a giant vat of soup by stirring every single drop individually. It would take a computer longer than the age of the universe to do it.
2. The Trick: The authors developed a clever shortcut. Instead of stirring every drop, they figured out a way to break the complex flavor pattern into simple, separable layers (like peeling an onion). This allowed them to generate the initial conditions for 30 different types of exotic particle interactions in a fraction of the time.
3. The Simulation: They ran over 30 different simulations, each starting with a different "collider" fingerprint, and let gravity do its work for billions of years to see what the universe looks like now.

The Detective Work: Looking for Clues in the Dark

Now that they have these simulations, they asked: "How can we see these fingerprints today?"

They focused on Weak Lensing. Imagine looking at a distant galaxy through a funhouse mirror made of invisible glass (dark matter). The gravity of the matter in between bends the light, distorting the shape of the galaxy. By measuring these distortions across the whole sky (like the upcoming LSST telescope will do), we can map the "lens" (the matter distribution).

The authors looked at two main things in their simulated universe:

• The Clumps (Halos): How many massive galaxy clusters formed?
• The Shapes: How are these clusters arranged?

The Big Discovery:
They found that the "collider" fingerprints don't just change the smooth background; they drastically change the number of massive galaxy clusters that form.

• Analogy: Imagine you are baking cookies. If you add a specific secret spice (the PNG), you don't just change the taste; you might end up with 20% more giant, chewy cookies and fewer tiny crumbs.
• The simulations showed that these exotic interactions can change the number of massive galaxy clusters by 10% to 25%. This is a huge signal!

The Verdict: We Can See It with New Eyes

The paper concludes that by using the next generation of telescopes (like the Vera C. Rubin Observatory), we can measure these distortions in the light of billions of galaxies.

• The Result: The authors predict that these new telescope measurements will be almost as good at detecting these ancient particle interactions as the best measurements we have from the Cosmic Microwave Background (the "baby picture" of the universe).
• Why it matters: This gives us a second, independent way to test the laws of physics at energies we can't reach on Earth. It's like having two different detectives solving the same crime; if they both find the same clue, we know for sure the criminal was there.

Summary in a Nutshell

• The Mystery: Did heavy, unknown particles exist in the first split-second of the universe?
• The Old Way: We tried to find them in the smooth "baby picture" of the universe, but we were hitting a wall.
• The New Way: The authors built a super-advanced computer simulation that can handle the messy, adult universe.
• The Clue: They found that these particles would leave a massive mark on how many giant galaxy clusters exist today.
• The Future: Upcoming telescopes will be able to spot this mark, potentially revealing the existence of particles that are impossible to create in any lab on Earth.

In short, they built a time machine to simulate the universe's messy teenage years, proving that we can still read the diary of the universe's infancy by looking at the scars it left on the galaxy clusters of today.

Technical Summary

1. Problem Statement

The Limitation of Linear Regime Analysis:
Primordial non-Gaussianities (PNGs) encode information about high-energy particle physics during inflation (up to $10^{15}$ GeV), specifically through "cosmological collider" signatures where the inflaton interacts with heavy particles. Historically, constraints on these signatures (parameterized by $f_{NL}$) have relied on the Cosmic Microwave Background (CMB) and perturbative analyses of the late-time universe.

• The Gap: Perturbative methods are restricted to the quasi-linear regime ($\delta \ll 1$). They cannot access the strongly nonlinear regime of structure formation (where $\delta \gtrsim 1$), which contains significant cosmological information and is probed by upcoming surveys like the Vera C. Rubin Observatory's LSST.
• The Simulation Challenge: Generating initial conditions (ICs) for N-body simulations with arbitrary bispectra is computationally expensive ($O(N_{grid}^6)$). While algorithms exist for separable bispectra (e.g., local, equilateral, orthogonal), cosmological collider models produce non-separable bispectra (often involving oscillatory terms from particle exchange), making them impossible to simulate using standard methods.

2. Methodology

The authors developed a novel pipeline to simulate the full nonlinear evolution of arbitrary cosmological collider bispectra and forecast constraints using weak lensing.

A. Generating Initial Conditions (The Core Innovation)

To overcome the non-separability of collider bispectra, the authors extended the basis decomposition method of Sohn et al. (2023/2024):

1. Basis Decomposition: They approximate the target bispectrum shape function $S(k_1, k_2, k_3)$ as a sum of separable terms:
   $$S(k_1, k_2, k_3) \approx \sum \alpha_{ijk} q_i(k_1)q_j(k_2)q_k(k_3)$$
2. Custom Basis Functions: They constructed a specific set of basis functions $q_i(k)$ combining:
   • Four monomial terms ($k^{-1}$ to $k^2$) to exactly reproduce standard templates (local, equilateral, orthogonal).
   • Modified Legendre polynomials $P_{i-1}(\tilde{k})$ for higher-order modes.
   • Crucial Modification: They explicitly subtracted constant terms from the Legendre polynomials to prevent infrared (IR) divergences in the one-loop power spectrum, ensuring the primordial power spectrum remains unperturbed by the simulation method.
3. Kernel Construction: The decomposed coefficients are used to construct a separable kernel $K_{12}$, allowing the generation of non-Gaussian potentials via three Fast Fourier Transforms (FFTs), reducing complexity to $O(N_{grid}^3 \log N_{grid})$.
4. Validation: They verified that the approximated templates match the analytic targets with high correlation ($r \gtrsim 0.99$) and that the induced one-loop power spectrum corrections are subdominant to the tree-level signal.

B. Simulation Suite

• Code: Used the Ulagam simulation suite with the PkdGrav3 N-body code.
• Setup: 100 snapshots from $z=127$ to $z=0$ in a $(1 \text{ Gpc}/h)^3$ volume with $512^3$ particles.
• Models: Simulated >30 templates covering three classes of collider physics:
  1. Scalar Exchange: Quasi-Single Field (QSF), Scalar I, Scalar II.
  2. Spin Exchange: Heavy Spin Collider (HSC) with spins $s=1$ to $4$.
  3. Nontrivial Sound Speed: Equilateral Collider (EC), Low-Speed Collider (LSC), Multi-Speed Collider (MSC).

C. Forecasting Framework

• Observables: Used the second and third moments of the weak lensing convergence field ($\kappa$) from the LSST Year-10 (Y10) dataset.
• Fisher Analysis: Computed the Fisher information matrix to forecast constraints on $f_{NL}$, assuming a single-parameter analysis (marginalizing over other cosmological parameters was not performed due to simulation suite size, representing a "best-case" scenario).

3. Key Contributions

1. First Nonlinear Collider Simulations: This is the first simulation suite to propagate the full domain of cosmological collider bispectra (not just the squeezed limit) into the deeply nonlinear regime of structure formation.
2. Generalized IC Generator: Developed a robust, publicly available algorithm (Aarambam) that can generate initial conditions for any bispectrum template by decomposing non-separable kernels into stable, separable basis functions.
3. Weak Lensing as a PNG Probe: Demonstrated that weak lensing (specifically higher-order moments of the convergence field) is a competitive probe for PNGs, offering constraints comparable to CMB data but with access to 3D volume information.
4. Comprehensive Model Library: Provided a systematic study of how various collider signatures (scalar mass, spin, sound speed) manifest in late-time observables.

4. Key Results

A. Impact on Nonlinear Structure

• Halo Abundance (Strongest Signal): The most significant impact of PNGs is on the halo mass function. For $f_{NL} = 100$, the change in halo abundance is 10–25%, an order of magnitude larger than changes in the power spectrum or bispectrum. This is driven by the alteration of the tails of the initial density distribution.
• Halo Bias: Different templates induce distinct scale-dependent biases. For example, local-type $f_{NL}$ shows the classic scale-dependent bias divergence on large scales, while collider models (like QSF and HSC) show more complex, template-specific scale dependencies.
• Matter Power Spectrum & Bispectrum:
  • The power spectrum shows a $\sim 1-2\%$ change for $f_{NL}=100$.
  • The matter bispectrum shows a 3–5% change, with the largest fractional changes occurring in equilateral and folded configurations (suppressed in the squeezed limit due to the dominance of gravitational nonlinearities there).

B. Forecasted Constraints (LSST Y10)

• Competitive Precision: Weak lensing constraints on $f_{NL}$ are predicted to be within 30–50% of current Planck 2018 CMB constraints for most models.
• Complementarity: Lensing constraints are complementary to CMB data. While CMB is limited by cosmic variance on large scales, lensing accesses smaller scales ($k \approx 1 h/\text{Mpc}$) and 3D volume, breaking degeneracies.
• Template Degeneracies:
  • Scalar-exchange models (QSF, SI, SII) are highly correlated with each other and with standard local/equilateral templates.
  • Models involving spin (HSC) and sound speed variations (LSC, MSC) are largely orthogonal to scalar models and to each other, offering unique handles to distinguish between different inflationary particle physics scenarios.

C. Data Products

• The simulation suite and the initial condition generator are publicly released, enabling the community to study nonlinear PNG effects without needing to develop complex IC generators from scratch.

5. Significance

This work bridges a critical gap between high-energy particle physics and late-time cosmology. By enabling the simulation of non-separable, oscillatory bispectra in the nonlinear regime, the authors unlock the potential to use the full power of next-generation surveys (like LSST) to probe the inflationary epoch.

• Paradigm Shift: It moves PNG analysis beyond the linear regime and simple templates, allowing for the testing of complex "cosmological collider" scenarios that were previously inaccessible to simulation-based inference.
• Multi-Probe Synergy: It establishes that weak lensing is a viable, high-precision probe for primordial physics, paving the way for multi-probe analyses (combining CMB, galaxy clustering, and weak lensing) to tightly constrain the nature of the inflaton and heavy particles in the early universe.
• Future Outlook: The authors note that incorporating more advanced statistics (e.g., persistent homology, peak/void statistics) could further improve constraints, suggesting that the full potential of weak lensing for PNGs has yet to be realized.