Stochastic Inflation with Interacting Noises

This paper extends the stochastic $\delta N$ formalism to interacting inflationary theories by incorporating one-loop quantum field theory corrections into the noise amplitude, demonstrating that the standard noise term is modified by the fractional correction to the power spectrum, a result applied to the three-phase SR-USR-SR scenario relevant for primordial black hole formation.

The Gist

The Big Picture: The Universe as a Noisy Drunk Walk

Imagine the early universe as a giant, rolling ball (the "inflaton field") trying to roll down a hill. This rolling motion is what we call Inflation. It's the period when the universe expanded incredibly fast, smoothing out wrinkles and setting the stage for galaxies to form.

Usually, physicists treat this ball as if it's rolling on a perfectly smooth, predictable track. But in reality, the track is bumpy. There are tiny, random jiggles caused by quantum mechanics. Think of these jiggles as noise.

In the standard model, these jiggles are like white noise on a radio: constant, predictable, and harmless. They are like a gentle, steady rain. Physicists have a tool called the Stochastic $\delta N$ formalism to calculate how these jiggles affect the universe. It's like a weather forecast that predicts how the rain will change the landscape.

The Problem:
This paper argues that in certain extreme scenarios (specifically when trying to create Primordial Black Holes, or PBHs), the "rain" isn't gentle or steady anymore. The rain becomes a storm. The jiggles start interacting with each other, getting louder and more chaotic. The standard "gentle rain" math breaks down.

The authors ask: What happens to our weather forecast if the rain suddenly becomes a hurricane?

---

The Core Idea: The "Noise" Gets a Makeover

In the standard theory, the size of the random jiggles (the noise amplitude) is fixed at a specific value: $H/2\pi$. It's like saying, "Every step the ball takes, it gets pushed sideways by exactly 1 inch."

However, the authors show that when the universe goes through a special phase called Ultra-Slow-Roll (USR) (where the ball rolls on a nearly flat part of the hill), the physics changes. The "noise" isn't just random anymore; it starts interacting.

The Analogy: The Crowd at a Concert

• Standard Theory (Free Noise): Imagine a crowd of people standing in a field, each tossing a coin. The coins land randomly. The total noise is just the sum of all those independent coin tosses. It's predictable.
• This Paper (Interacting Noise): Now, imagine the crowd starts shouting at each other. If one person tosses a coin, the noise makes the person next to them toss a coin harder. The noise amplifies itself. The "coin tosses" are no longer independent; they are a chaotic, interacting storm.

The paper calculates exactly how much louder this "storm" gets. They find that the size of the jiggles is modified by a factor related to how much the "storm" amplifies the energy of the universe.

The Three-Phase Rollercoaster (SR-USR-SR)

To test this, the authors look at a specific setup used to explain how Primordial Black Holes (tiny black holes formed right after the Big Bang) are created. They call it the SR-USR-SR model.

1. Phase 1 (SR - Slow Roll): The ball rolls down a steep hill. This creates the seeds for the galaxies we see today (the Cosmic Microwave Background). The noise here is normal.
2. Phase 2 (USR - Ultra-Slow Roll): The ball hits a flat plateau. It slows down, but the "wind" (quantum fluctuations) starts blowing harder and harder. This is where the noise gets "interacting" and chaotic. This is the "storm" phase.
3. Phase 3 (SR - Slow Roll): The ball rolls down the other side. The storm settles, but the damage (or creation) is done.

The authors used advanced quantum math (called the in-in formalism) to calculate exactly how much the noise amplitude increased during that flat plateau phase.

The Main Result: A New Formula for the "Step Size"

The paper's biggest discovery is a new formula for the size of the random steps the universe takes.

• Old Formula: Step Size = $H/2\pi$ (The standard, gentle rain).
• New Formula: Step Size = $\frac{H}{2\pi} \times \sqrt{1 + \text{Correction}}$

The "Correction" part depends on how long the "storm" (the USR phase) lasted and how sharp the transition was.

• If the storm was short and mild, the correction is tiny, and the old formula works fine.
• If the storm was long and intense, the correction is huge. The "Step Size" becomes much larger than expected.

Why does this matter?
If the steps are bigger, the ball is much more likely to roll off the track entirely. In cosmology, "rolling off the track" means creating a Primordial Black Hole.

The Consequences: Why We Should Care

The authors show that if you ignore this "interacting noise" and use the old, simple formula, you might be completely wrong about how many black holes were formed.

1. The Probability Shift: Because the "noise" is stronger, the probability of the universe creating a black hole changes non-linearly. It's not just a little bit more likely; it can be much more likely.
2. The "Diffusion" Effect: Imagine a drop of ink in water. If the water is still, the ink spreads slowly. If the water is turbulent (the interacting noise), the ink spreads wildly and unpredictably. The authors show that this "turbulence" changes the statistical distribution of where the ink (the universe's density) ends up.

Summary in a Nutshell

• The Setup: The universe expands, and quantum jiggles create the seeds for everything.
• The Twist: In the specific scenario where we try to make tiny black holes, these jiggles stop being independent and start "talking" to each other, creating a chaotic storm.
• The Fix: The authors updated the math to account for this storm. They found the "jiggles" are actually much stronger than we thought.
• The Impact: This changes our predictions for how many primordial black holes exist. If the storm was strong enough, we might have way more black holes than the standard model predicts.

The Takeaway:
Just as a weather forecast needs to account for a hurricane, not just a drizzle, our models of the early universe need to account for "interacting noise." The authors have provided the new "weather map" for these extreme cosmic storms, ensuring our predictions for the universe's structure are accurate.

Technical Summary

1. Problem Statement

The Stochastic $\delta N$ formalism is a powerful non-perturbative tool used to calculate cosmological correlators (such as the power spectrum and bispectrum) and model the probability distribution function (PDF) of field fluctuations during inflation. It is particularly useful for studying rare events like Primordial Black Hole (PBH) formation, which occur in the non-linear tails of the distribution.

However, the standard formulation of this formalism relies on a critical assumption: the underlying theory is a free theory (or effectively free in the slow-roll limit). Under this assumption, the stochastic noise term acting on the long-wavelength modes has a fixed amplitude of $H/2\pi$ (where $H$ is the Hubble parameter) and is purely Gaussian.

The Core Issue: In realistic inflationary scenarios designed to produce PBHs (e.g., Ultra-Slow-Roll or USR inflation), the scalar field potential involves significant interactions. These interactions lead to quantum loop corrections in the curvature power spectrum. Consequently, the stochastic noise is no longer a simple free Gaussian noise; it becomes interacting and non-Gaussian. The standard formalism fails to account for these modifications, potentially leading to incorrect predictions for the PDF and PBH abundance.

2. Methodology

The authors extend the Stochastic $\delta N$ formalism to accommodate interacting theories by integrating Quantum Field Theory (QFT) corrections directly into the noise amplitude.

• Generalized Langevin and Fokker-Planck Equations:
  The authors reformulate the Langevin equations for the long-wavelength field $\phi$ and its conjugate momentum $\Pi$. Instead of a fixed noise amplitude, they introduce time-dependent amplitude functions $\tilde{f}(N)$ and $\tilde{g}(N)$:
  $$\frac{d\phi}{dN} = \Pi + \tilde{f}(N)\xi(N)$$
  $$\frac{d\Pi}{dN} = -3\Pi - \frac{3V'(\phi)}{V(\phi)} + \tilde{g}(N)\xi(N)$$
  Here, $\xi(N)$ is a normalized Gaussian white noise, while $\tilde{f}(N)$ encodes the interaction effects.

• Adjoint Fokker-Planck Equation:
  They derive the corresponding adjoint Fokker-Planck equation to track the evolution of the probability distribution $P(\phi, \Pi, N)$. They utilize the method of characteristics and Fourier transforms to solve for the moments of the number of e-folds ($N$), which are the building blocks for the power spectrum.

• QFT In-In Formalism:
  To determine the modified noise amplitude $\tilde{f}(N)$, the authors employ the in-in (Schwinger-Keldysh) formalism. They calculate the two-point correlation function of the noise $\langle \xi_\phi(N) \xi_\phi(N') \rangle$ by relating it to the two-point function of the curvature perturbation $\mathcal{R}$ (or the Goldstone boson $\pi$).
  Crucially, they include one-loop corrections arising from the interaction Hamiltonian (derived via the Effective Field Theory of Inflation). This allows them to map the fractional loop correction in the power spectrum ($\Delta P_\mathcal{R} / P_\mathcal{R}^{(0)}$) directly to the noise amplitude.

• Specific Model Application (SR-USR-SR):
  The formalism is applied to the SR-USR-SR (Slow-Roll – Ultra-Slow-Roll – Slow-Roll) model, a standard setup for PBH formation. They analyze the transition from the USR phase to the final SR phase, characterized by a "sharpness parameter" $h$.

3. Key Contributions and Results

A. Modified Noise Amplitude

The primary theoretical result is the derivation of the corrected noise amplitude. The authors show that the amplitude of the stochastic noise is modified from the standard $H/2\pi$ to:
$$\tilde{f}(N) = \frac{H}{2\pi} \sqrt{1 + \frac{\Delta P_\mathcal{R}}{P_\mathcal{R}^{(0)}}}$$
where:

• $P_\mathcal{R}^{(0)}$ is the tree-level (free theory) power spectrum.
• $\Delta P_\mathcal{R}$ represents the one-loop quantum correction to the power spectrum.

This result implies that the "quantum jumps" in the stochastic evolution are not fixed but depend on the interaction strength and the loop corrections of the underlying theory.

B. Loop Corrections in SR-USR-SR

For the specific SR-USR-SR setup, the authors calculate the fractional one-loop correction induced by small-scale modes becoming super-horizon during the USR phase. The correction is given by:
$$\frac{\Delta P_\mathcal{R}}{P_\mathcal{R}^{(0)}} \approx \frac{6}{h(h^2 + 24h + 180)} \Delta N e^{6\Delta N} P_\mathcal{R}^{(0)}$$
Key findings from this expression:

• Exponential Sensitivity: The correction scales exponentially with the duration of the USR phase ($\Delta N$).
• Sharpness Dependence: It depends non-linearly on the transition sharpness parameter $h$. For sharp transitions ($|h| \gg 1$), the correction grows linearly with $h$.
• Sign: In this specific setup, the correction is negative ($\Delta P_\mathcal{R} < 0$), meaning the effective noise amplitude is smaller than $H/2\pi$.

C. Impact on PDF and PBH Abundance

The authors demonstrate how this modified noise amplitude alters the Fokker-Planck equation and the resulting Probability Distribution Function (PDF).

• Non-Perturbative Effects: Even small loop corrections can lead to non-perturbative changes in the tail of the PDF.
• First Passage Time: In the diffusion-dominated regime (relevant for PBH formation), the mean and variance of the first passage time (the time to reach a critical field value) are modified by the factor $(1 + \Delta P_\mathcal{R}/P_\mathcal{R}^{(0)})^{-1}$.
• Implication: Ignoring these corrections can lead to significant over- or under-estimation of the PBH abundance, as the tail of the distribution is exponentially sensitive to the noise amplitude.

4. Significance

• Theoretical Consistency: The paper bridges the gap between perturbative QFT calculations (loop corrections) and non-perturbative stochastic methods. It validates that the stochastic $\delta N$ formalism can reproduce standard QFT results if the noise amplitude is correctly renormalized.
• Refinement of PBH Predictions: Since PBH formation relies on the extreme tail of the curvature perturbation distribution, the standard assumption of a fixed $H/2\pi$ noise is insufficient for models with strong interactions (like USR). This work provides the necessary framework to include quantum corrections self-consistently.
• Control of Perturbativity: The authors highlight that large loop corrections (where $|\Delta P_\mathcal{R}/P_\mathcal{R}^{(0)}| \sim 1$) signal a breakdown of perturbation theory. Their formalism helps identify the parameter space (e.g., limits on $\Delta N$ and $h$) where the stochastic approach remains valid and where higher-order loops must be considered.

In summary, this work generalizes the stochastic inflation formalism to interacting theories, proving that quantum loop corrections manifest as a renormalization of the noise amplitude, which has profound implications for the statistical prediction of primordial structures like black holes.