⚛️ general relativity

Cosmological Correlator Discontinuities from Scattering Amplitudes

This paper establishes a method to compute cosmological correlators in de Sitter space by relating their energy discontinuities to unitarity cuts of flat-space scattering amplitudes via auxiliary propagators, thereby enabling the reconstruction of these observables through dispersion relations and sum rules.

Original authors: Chandramouli Chowdhury, Sadra Jazayeri, Arthur Lipstein, Joe Marshall, Jiajie Mei, Ivo Sachs

Published 2026-02-04

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CC BY 4.0

Original authors: Chandramouli Chowdhury, Sadra Jazayeri, Arthur Lipstein, Joe Marshall, Jiajie Mei, Ivo Sachs

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

Imagine the early universe as a giant, expanding balloon. Physicists want to understand what happened inside this balloon by studying "cosmological correlators." Think of these correlators as the echoes left behind by particles interacting in that early, expanding space. Calculating these echoes is usually incredibly difficult, like trying to predict the exact sound of a drumbeat inside a storm.

However, this paper introduces a clever shortcut. The authors discovered that these complex "echoes" from the expanding universe are actually built from much simpler "flat space" building blocks—specifically, the scattering amplitudes (collision results) of particles in a static, non-expanding universe.

Here is the breakdown of their discovery using simple analogies:

1. The "Cosmological Dressing" Trick

Imagine you have a standard Lego model built on a flat table (representing the flat universe). To turn this model into a representation of the expanding universe, you don't have to rebuild it from scratch. Instead, you just attach special "helper pieces" (called auxiliary propagators) to the connection points of your Lego model.

The paper confirms that if you take a flat-space diagram and attach these specific helper pieces, you instantly get the correct formula for the cosmological echo. It's like taking a 2D drawing and instantly turning it into a 3D hologram just by adding a few specific stickers.

2. The "Cutting" Game

The main breakthrough of this paper is figuring out how to find the discontinuities (sudden jumps or breaks) in these cosmological echoes. In physics, finding these jumps is often the key to solving the whole puzzle.

The authors found two ways to "cut" the problem to find these jumps, and both involve the flat-space Lego models:

Cutting the Internal Connections (The "y" variables):
Imagine the Lego model has internal beams connecting the pieces. If you want to know how the echo changes when you tweak the energy flowing through these internal beams, you simply take the flat-space model and cut the internal beams. In physics terms, this is called a "unitarity cut." You then attach your special "helper pieces" to this cut model. It's like taking a bridge, snapping the middle support, and seeing how the traffic (energy) flows differently.
Cutting the Helper Pieces (The "x" variables):
Now, imagine you want to know how the echo changes based on the energy of the cars entering and leaving the bridge (the external energy). To find this, you don't cut the bridge itself. Instead, you cut the special "helper pieces" you attached earlier. It's a bit like realizing that if you cut the support cables holding up your 3D hologram, you reveal the hidden information about the original 2D drawing.

3. The "Sum Rules" (The Magic Balance)

Because of these cutting rules, the authors discovered a new set of "sum rules." Think of this as a magical balance scale. If you take the cosmological echo and flip the signs of the energy variables (like turning positive numbers into negative ones) and add them all up, the result must be zero.

This is a powerful constraint. It's like a puzzle where, no matter how you arrange the pieces, the total weight must always equal zero. This rule helps physicists check if their calculations are correct and even helps them build the correct answer from scratch (a process called "bootstrapping").

4. Reconstructing the Whole from the Pieces

Finally, the paper shows how to use these "cuts" (the jumps and breaks) to rebuild the entire cosmological echo from scratch. They use a mathematical tool called dispersion relations.

Imagine you have a shattered vase. Usually, gluing it back together is a nightmare. But this paper says: "If you know exactly how the vase broke (the discontinuities), you can mathematically reconstruct the whole vase without needing the original blueprint." They take the "broken pieces" derived from the flat-space models and use them to rebuild the full cosmological correlator.

Summary

In short, this paper says:

Don't reinvent the wheel: You can turn flat-space particle collision diagrams into cosmological universe diagrams by adding special "helper" pieces.
Cut to learn: To find the tricky "jumps" in these universe diagrams, you just need to cut either the internal parts of the flat diagram or the helper pieces attached to it.
Rebuild from the break: Once you know where the diagram breaks (the discontinuities), you can mathematically reconstruct the entire cosmological signal.

This provides a new, much simpler toolkit for physicists to calculate the history of the early universe, turning a complex 4D problem into a manageable series of cuts and rebuilds based on simpler flat-space physics.

Technical Summary: Cosmological Correlator Discontinuities from Scattering Amplitudes

Problem Statement
Cosmological correlators (in-in correlators) are the fundamental observables of quantum field theory in de Sitter (dS) space, describing the statistics of fluctuations generated during cosmic inflation. Traditionally, these are computed using the Schwinger-Keldysh formalism or by squaring the cosmological wavefunction and integrating over boundary values. While recent work has revealed that these correlators possess a mathematical structure simpler than wavefunction coefficients—reminiscent of flat-space scattering amplitudes—computing them remains challenging. Existing methods, including those based on symmetries, factorization, and differential equations, have not fully exploited the connection to flat-space amplitudes to systematically compute discontinuities or reconstruct correlators from data. The authors aim to bridge this gap by establishing a direct link between the discontinuities of cosmological correlators and the unitarity cuts of flat-space Feynman diagrams.

Methodology
The paper leverages the "cosmological dressing rules," a framework that uplifts flat-space Feynman diagrams to de Sitter in-in correlators by attaching auxiliary propagators to interaction vertices. The authors analyze the discontinuities of these dressed diagrams with respect to two types of energy variables:

Internal Energies ( $y$ -variables): Sums of boundary momenta flowing through internal propagators.
External Energies ( $x$ -variables): Magnitudes of boundary momenta flowing through external propagators.

The core methodological insight is that discontinuities in the complex energy plane can be computed by manipulating the auxiliary propagators and the underlying flat-space diagrams:

$y$ -discontinuities: These are obtained by applying the dressing rules to the unitarity cuts of the corresponding flat-space Feynman diagrams. The discontinuity acts on the flat-space part of the integrand, replacing internal propagators with on-shell delta functions ( $\delta_+$ ).
$x$ -discontinuities: These arise from cutting the auxiliary propagators attached to the flat-space diagrams. Mathematically, this corresponds to replacing the auxiliary propagator terms $(p^2 - x^2)^{-1}$ with delta functions $\delta_+(p^2 - x^2)$ .

By sequentially cutting all auxiliary propagators, the authors demonstrate a procedure to recover the flat-space scattering amplitude (including energy conservation) from the cosmological correlator. Furthermore, they utilize the analyticity of the correlators in the complex energy plane to construct dispersion relations, allowing the reconstruction of the full correlator from its discontinuities.

Key Contributions and Results

Mapping Discontinuities to Cuts: The paper establishes that discontinuities of cosmological correlators can be directly "uplifted" from flat space.
- Discontinuities with respect to internal energies are computed by dressing the unitarity cuts of flat-space diagrams with auxiliary propagators.
- Discontinuities with respect to external energies are computed by cutting the auxiliary propagators themselves.
Sum Rules: The ability to recover the flat-space limit via sequential discontinuities implies highly non-trivial sum rules for cosmological correlators. Specifically, the sum of correlators over all sign flips of the external energy variables vanishes (e.g., $\sum_{\{s_i=\pm 1\}} (\prod s_i) C_n(s_1 x_1, \dots) = 0$ ). The authors note that these constraints are satisfied by correlators but not generally by wavefunction coefficients, highlighting the distinct mathematical structure of in-in correlators.
Explicit Examples: The authors illustrate these rules at tree-level and 1-loop for conformally coupled scalar theories ( $\phi^3$ $ϕ^{3}$ and $\phi^4$ $ϕ^{4}$ ):
- Tree-level exchange ( $\phi^3$ ): They show how the discontinuity with respect to $y^2$ yields a dressed unitarity cut, and how double discontinuities in $x$ variables recover the flat-space exchange amplitude.
- 1-loop bubble ( $\phi^4$ ): They compute the discontinuities with respect to both $x$ and $y$ variables, demonstrating how the $y$ -discontinuity is UV finite and how the sum rule is satisfied by the integrated expression.
- 1-loop triangle ( $\phi^4$ ): They analyze a 6-point correlator, showing how sequential discontinuities (involving both $x$ and $y$ cuts) relate to off-shell flat-space triangle diagrams and satisfy the derived sum rules.
Dispersion Relations: The authors provide a method to reconstruct cosmological correlators from their discontinuities using dispersion relations. They demonstrate this for the bubble graph, showing how the full correlator (including UV regularization) can be recovered from its discontinuities and boundary terms, offering a new algorithmic approach to computation.

Significance and Claims
The paper claims to provide a "powerful new machinery" for computing cosmological observables. By systematically reconstructing correlators from data uplifted from flat space (specifically, from unitarity cuts and discontinuities), the authors offer an alternative to traditional Schwinger-Keldysh calculations. The significance lies in:

Simplification: Revealing that the complex structure of dS correlators can be derived from the simpler structure of flat-space amplitudes via dressing rules.
New Constraints: Identifying sum rules that constrain the space of possible correlators, which could be used for "bootstrapping" these observables.
Algorithmic Reconstruction: Proposing a two-step algorithm: (1) uplift discontinuities from flat space, and (2) reconstruct the correlator via dispersion relations.

The authors remain modest regarding the scope, noting that while the results are demonstrated for conformally coupled scalars, they hold more generally. They acknowledge that a more systematic derivation using the "largest time equation" and extensions to massless theories with infrared divergences will be presented in a follow-up paper [47]. They also suggest that these findings hint at a deeper relationship between flat-space holography and de Sitter holography, a question they intend to explore in future work.