Cosmology often feels like a treasure hunt along a foggy shoreline, where equations sketch the tides of time and the shape of space. A Brown University team turns the hunt inward, asking not for a single grand law but for the way a whole family of diagrams stitches together the universe’s wavefunction, diagram by diagram. They study a simple scalar field in a cosmological setting and ask what happens if you look at the building blocks themselves—the individual graphs that contribute to the wavefunction. The answer is surprising: zeros—points where contributions vanish—are not random glitches. They are structured, repeatable features that obey geometric rules and echo patterns already seen in flat-space scattering. In short, the zeros whisper a hidden order into the cosmos.
These hidden zeros appear across particle physics and string theory, but the Brown team—led by Shounak De with collaborators Shruti Paranjape, Andrzej Pokraka, Marcus Spradlin, and Anastasia Volovich—shows they also organize cosmological observables. The work, carried out at Brown University and the Kavli Institute for Theoretical Physics, uncovers three distinct families of zeros for the wavefunction coefficients of a massless scalar with a cubic interaction. The authors connect these zeros to a rich geometric picture built from graph associahedra, unveiling a cosmological cousin to the ABHY geometry that physicists use to understand scattering amplitudes. The upshot is not only a deeper mathematical story but a bridge between how the early universe encodes its correlations and how particles scatter in flat spacetime.
Three kinds of zeros in the cosmological wavefunction
The first kind, which they call wavefunction zeros, lives in chain-like graphs. If you label the graph’s sites and energies in a certain way, a precise set of linear relations among the energy factors forces the stripped wavefunction to vanish. It’s not a single point but a locus in the space of external energies. Near such a zero, the wavefunction splits neatly into a product of smaller chains. It’s as if the cosmic chain coagulates just enough to fall apart into two shorter chains, revealing a simple, universal factorization even in a complicated diagram.
Second are factorization zeros. In the flat-space story of amplitudes, zeros appear on the residues of poles, and the physics of unitarity shows how amplitudes factorize there. Here, stripped wavefunctions exhibit a similar near-zero factorization when an interior site’s energy factor is set to zero. The result is a clean decomposition of the wavefunction into subgraphs whose strip-by-strip behavior mirrors the familiar pole-factorization of scattering amplitudes—only now in a cosmological, time-nontranslation-invariant setting. It’s a near-zero cousin of the usual pole story, but with the same kinesthetic logic: the whole is more than the sum of its parts, and the parts reveal themselves when you tease apart a substructure.
Third are parametric zeros. These arise not from a particular energy relation but from the geometry itself: a cosmological limit in which the graph associahedron—an object that encodes how tubings of a graph can sit inside the graph—flattens, reducing its dimension. In practical terms, the canonical form associated with the stripped wavefunction vanishes because the polytope loses a direction. The result is a new family of zeros tied to how the geometry of the diagram degenerates in the cosmological limit. Parametric zeros sit at the intersection of combinatorics, geometry, and physics, showing how a shape in a high-dimensional space can dictate the behavior of a quantum field’s cosmological imprint.
The geometry that hosts the zeros
Geometry, not chance, organizes these patterns. The stripped wavefunction ψ̃G associated to a single graph G is the canonical form of a polytope called a graph associahedron AG. The face structure of AG mirrors the combinatorics of tubings inside the graph. In the cosmological limit, this polytope becomes a non-simple object—what the authors call the cosmological graph associahedron, ˜AG—whose degenerations encode the parametric zeros. In other words, the zeros are the footprints of a geometric collapse in the very space that encodes how a graph sits inside a network of tubings.
The construction is technical but elegant: one fixes 2-tubes as the fundamental building blocks and uses a truncation procedure on a simplex to realize AG. The energy variables of the graph play the role of hyperplane-defining coordinates, and the limit δτ → 0 pushes the simple, δ-deformed polytope toward ˜AG. In this cosmological limit, the stripped wavefunction becomes the canonical function of ˜AG. The geometry, therefore, doesn’t merely accompany the physics; it dictates where and how the zeros appear.
Parametric zeros are a vivid example. For several elementary graphs—the 4-chain, the 4-star, the 3-gon—the paper shows that as the cosmological geometry collapses along a chosen direction, the polytope shrinks to a lower-dimensional Minkowski summand or even to a line segment. Those degenerations line up with the predicted parametric zeros. The polytope’s flattening is not a metaphor; it is a precise geometric operation that leaves a measurable imprint on the wavefunction coefficients. This is how the authors connect the language of polytopes to the language of quantum fields in the early universe.
From cosmology to scattering amplitudes
One of the paper’s striking moves is a bridge between cosmology and flat-space scattering amplitudes, via a precise map between the cosmological wavefunction coefficients of chain graphs and the color-ordered amplitudes of Tr(ϕ3) theory. The authors show that a chain’s stripped wavefunction exactly matches an (n+1)-point amplitude in Tr(ϕ3) theory when you identify a particular set of kinematic variables: the planar Xi,j variables map to energy factors of the chain’s tubes, and the non-planar variables map to combinations of those energies plus the total energy. In other words, a single diagram’s cosmological wavefunction already contains the seeds of the full amplitude’s structure.
Concretely, the five-term structure that appears in the stripped 4-chain wavefunction mirrors the five terms of the 5-point Tr(ϕ3) amplitude. The parametric zeros of the chain translate into zeros of the amplitude in specific regions of the kinematic mesh, just as a rectangle in the kinematic lattice signals a zero in the amplitude. This is more than a curiosity: it shows that the zeros discovered in the cosmological setting are not isolated artifacts but parts of a shared geometric story with flat-space physics. The ABHY associahedron—which encodes amplitudes in a purely geometric way—also appears as a cousin in the cosmological narrative, with its cosmological counterpart, the cosmohedron, arising when you sum over graphs. In this language, zeros are not exceptions; they are integral features of a unified geometric framework that spans cosmology and particle scattering.
The paper’s Section 5 makes this connection concrete. It reviews the zeros of Tr(ϕ3) amplitudes, organized on a kinematic mesh, and then shows how the stripped chain wavefunction maps into the (n+1)-point amplitude. The translation is not symmetric in all directions—cyclic symmetry is broken in the cosmological ABHY construction—but the underlying mechanism is the same: linear conditions on kinematic or energy variables carve out zero loci that are both geometric and physically meaningful. The upshot is a two-way street: zeros discovered in a single graph’s wavefunction illuminate zeros of a full amplitude, and vice versa, hinting at a shared, deeper structure tying cosmos to collider physics.
Why this changes how we think about the universe
Zeros in physics often feel like edge cases, yet they carry essential information about the structure of a theory. The fact that the zeros of a single diagram’s cosmological wavefunction dovetail with the zeros of a full amplitude—across two very different physical settings—suggests that geometry can govern both time-asymmetric cosmology and time-symmetric scattering. It is as if the universe speaks through a common language of shapes, where a polytope’s facets and a diagram’s tubings determine when a contribution vanishes and when it splits into simpler pieces.
Moreover, the geometry-driven view of zeros opens practical doors. If zeros are fixed by the combinatorics of tubings and the way polytopes collapse, they could serve as a building block for reconstructing wavefunction coefficients diagram by diagram, or for cross-checking complex calculations by exploiting near-zero factorizations. The cosmological ABHY story provides a scaffold that could organize not just the zeros but the entire wavefunction in a way that mirrors how amplitudes are organized in flat space. And because the cosmological wavefunction reduces to familiar flat-space objects under the right transform, these insights have the potential to unify how we think about correlators across epochs of the universe.
What comes next for cosmology and math
The paper points to several exciting directions. Extending the analysis beyond the cubic Tr(ϕ3) setting—toward more general polynomial interactions and richer cosmologies—could reveal whether these zero patterns survive in more realistic models of the early universe. The interplay between graph tubings, cosmohedra, and the adjoint polynomial hints at a deeper combinatorial backbone for cosmological correlators that could be exploited computationally or conceptually. The authors even gesture toward a stringy, cluster-like formulation of wavefunction coefficients that could parallel the ABHY program for flat-space amplitudes, opening the door to a cosmological version of the “positive geometry” revolution in scattering theory.
Physically, the zeros are tied to soft and near-soft limits of the wavefunction: the chain wavefunction zeros point to multi-soft kinematic configurations, inviting a more thorough exploration of soft limits in cosmology. On the mathematical side, the distinction between simple polytopes and their non-simple cosmological limits offers a playground to study how flattening a polytope interacts with the analytic structure of its canonical form. The Brown–Kavli team’s work is not just an isolated observation; it is an invitation to a broader program that could reveal how geometry encodes the universe’s deepest patterns, from the math that underlies the wavefunction to the amplitudes that describe particle scattering.
In the end, the zeros are not silentvoids; they are signs pointing to a harmony between cosmology and quantum field theory. They suggest that, underneath the messiness of diagrams and spacetimes, there is a kind of geometric poetry—one that we are only beginning to learn how to read. The researchers at Brown University and the Kavli Institute for Theoretical Physics have given us a detailed map of part of that poetry, and it is a map that will likely be consulted for years as we try to understand what the universe is saying through its most fundamental patterns.
Credits: The study was conducted by researchers at Brown University and the Kavli Institute for Theoretical Physics, with Shounak De as lead author and colleagues Shruti Paranjape, Andrzej Pokraka, Marcus Spradlin, and Anastasia Volovich contributing to the work.
