Incorporating curved geometry in cosmological simulations

Imagine you are trying to bake a perfect, round cake (our Universe) inside a square baking pan (the computer simulation).

For decades, cosmologists have been baking these "Universe cakes" in square pans. Why? Because the computer code they use is built to handle square boxes with repeating patterns (like a tiled floor). If you try to bake a round cake in a square pan, the edges get weird, and the cake doesn't look right.

But here's the problem: Our Universe might not be flat. It might be slightly curved, like the surface of a sphere (closed) or a saddle (open). If the Universe is curved, baking it in a square, flat box introduces errors, especially when we try to look at things very far away, like light traveling from the beginning of time to our eyes today.

This paper by Julian Adamek and Renan Boschetti is like a clever new baking technique. They figured out how to bake a perfectly round, curved cake inside a square pan without the edges ruining the taste.

Here is how they did it, broken down into simple analogies:

1. The "Hole in the Dough" Trick

Imagine a giant, flat sheet of dough (the flat computer simulation). Now, imagine you cut a perfect circle out of the middle of that dough. You have a hole.

Instead of trying to force the curved cake to fit the square edges, they take a piece of curved dough (representing a curved Universe) and drop it right into that hole.

The Magic: They don't just jam it in; they use a special "glue" (mathematical rules called junction conditions) to make sure the curved dough fits perfectly into the flat dough. The transition is seamless.
The Result: Inside the hole, you have a perfect, curved Universe. Outside the hole, the square pan still works perfectly for the computer. The computer thinks it's still in a flat world, but the "hole" is secretly curved.

2. The "Einstein & Straus" Sandwich

This idea isn't entirely new; it's like a cosmic sandwich proposed way back in the 1940s by Einstein and Straus.

Layer 1 (The Center): A curved, expanding bubble of space where we live and where we observe the stars.
Layer 2 (The Middle): A vacuum layer (empty space) that acts as a buffer.
Layer 3 (The Outside): The flat, square computer world.

The authors figured out exactly how to calculate the "recipe" for the middle layer so that the physics works perfectly from the center all the way to the square edges.

3. Why Does This Matter? (The "Ruler" Problem)

Why go through all this trouble?

Imagine you are measuring the distance to a star.

In a Flat Universe: If you draw a circle around you, the distance around the edge is exactly $\pi$ times the distance across.
In a Curved Universe: That rule breaks! If the Universe is curved like a sphere, the distance around the circle is shorter than $\pi$ times the diameter. If it's curved like a saddle, it's longer.

If you use a "flat" simulation to measure light traveling billions of light-years, your ruler is wrong. You might think a galaxy is 10 billion light-years away, but because you ignored the curvature, it's actually 10.5 billion.

The authors' method fixes the ruler. It allows them to simulate light traveling through a curved space and tell us exactly what an observer would see, accounting for the fact that space itself is bent.

4. The "Time Travel" Glitch

There was one more tricky part: Time.
In a curved universe, time doesn't tick at the same speed everywhere compared to the flat computer world. It's like how a clock on a mountain runs slightly faster than one at sea level.

The authors had to write a complex "translation manual" (mathematical transformations) to make sure that when the simulation says "13 billion years have passed," the observer inside the curved bubble actually experiences the correct amount of time. They did this with extreme precision, calculating effects up to the "second order" (meaning they didn't just guess; they calculated the tiny, tiny ripples in time and space).

The Big Picture: What Did They Prove?

They ran three tests:

The "Empty" Test: They simulated a universe with no stars, just empty space, to see if the geometry worked. It did.
The "Realistic" Test: They added a cosmological constant (Dark Energy) to see if the expansion worked. It did.
The "Clumpy" Test: They added matter (stars and galaxies) to see if the universe would clump together naturally. It did.

The Conclusion:
Their new method proves that you can simulate a curved universe inside a standard computer code. They showed that if you ignore curvature, your measurements of the distant universe will be off by about 1% to 10% (depending on how curved it is).

Why should you care?
We are currently building massive telescopes (like the Vera C. Rubin Observatory) that will map the entire sky. To understand what we see, we need perfect simulations. If the Universe is curved, and we use flat simulations, we might misinterpret the data and get the wrong answers about the nature of Dark Energy, the mass of neutrinos, or even the Big Bang itself.

This paper gives us the "curved baking pan" we need to make sure our cosmic recipes turn out right.

Here is a detailed technical summary of the paper "Incorporating curved geometry in cosmological simulations" by Julian Adamek and Renan Boschetti.

1. Problem Statement

Cosmological simulations are essential for forward-modelling large-scale structure observations. However, standard $N$ -body simulations typically assume a spatially flat (Euclidean) geometry with periodic boundary conditions. This creates a fundamental limitation when testing theories or observational constraints that allow for non-zero spatial curvature ( $\Omega_K \neq 0$ ).

The paper identifies two distinct effects of curvature that current simulations struggle to model simultaneously:

Expansion History: Curvature affects the expansion rate and growth of structure. This is often handled via the "separate universe" approach, where a flat sub-volume evolves with a modified expansion history.
Non-Euclidean Geometry: Curvature alters distance-volume relations and the geometry of the past light cone over cosmological scales. The separate universe approach fails here because it cannot account for the non-Euclidean metric (e.g., the circumference of a circle not being $\pi \times$ diameter) required for accurate light-cone observables (like angular clustering or distance-redshift relations) over large fields of view.

Existing codes rely on periodic boundaries, which enforce flatness beyond the box size. Modifying boundary conditions to be non-periodic is technically difficult and breaks established code infrastructure. The authors aim to solve this by embedding a curved region within a flat simulation box while maintaining periodic boundaries.

2. Methodology

The authors propose a fully relativistic framework based on embedding a spherical patch of a curved Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime into a spatially flat exterior.

A. The Embedding Scheme (Einstein-Straus Construction)

Inner Region ( $r < r_1$ ): A curved FLRW patch (closed universe, $K > 0$ ) containing the observer and the region of interest.
Intermediate Region ( $r_1 < r < r_2$ ): A vacuum region described by the Schwarzschild metric. This acts as a "hole" in the flat exterior.
Outer Region ( $r > r_2$ ): A spatially flat FLRW spacetime.
Boundary Conditions: The entire system is placed inside a cubic simulation box with periodic boundaries. The flat exterior allows the use of standard periodic boundary conditions, while the inner curved patch models the non-Euclidean geometry.

B. Gauge Transformation and Initial Conditions

LTB Solution: The authors utilize the exact Lemaître-Tolman-Bondi (LTB) solution for pressureless dust to describe the transition between the curved patch and the vacuum.
Coordinate Transformation: Simulations (specifically the code gevolution) operate in the Poisson gauge adapted to the flat exterior. The LTB solution is naturally in the comoving synchronous gauge.
- The authors derive a second-order gauge transformation between these two gauges. This is crucial for high accuracy, especially when curvature is large, as it accounts for the coupling between curvature and perturbations.
- This transformation provides the Bardeen potentials ( $\phi, \psi$ ) and the coordinate shifts required to set up the simulation.
Relativistic Mass Defect: A key technical contribution is the handling of the "relativistic mass defect." In a curved geometry, the proper volume differs from the Euclidean volume. To satisfy the Hamiltonian constraint and periodic boundaries, the authors show that a mass correction must be applied to the particles. They distribute this mass defect uniformly across all particles in the simulation to ensure the displacement field remains compatible with periodic boundaries.

C. Perturbations

For matter perturbations, the authors use an approximate method where the mode functions in the curved patch are treated as Fourier modes of the flat exterior, scaled appropriately. While a rigorous treatment would require hyperspherical harmonics (discrete spectrum), this approximation is sufficient for the scales of interest in their current implementation.

3. Key Contributions

First Fully Relativistic Curved Simulation Framework: This is the first method to run cosmological $N$ -body simulations that fully incorporate non-Euclidean spatial geometry for the construction of light-cone observables, rather than just modifying the expansion history.
Second-Order Gauge Transformation: The derivation of the gauge transformation from LTB (synchronous) to Poisson gauge to second order ensures that the initial conditions and the evolution of potentials are accurate even for large curvature values.
Resolution of Boundary Conditions: The embedding technique allows curved cosmologies to be simulated using standard periodic boundary codes (like gevolution) without modifying the core solver, requiring changes only to initial conditions and post-processing.
Exact Schwarzschild Solution in Poisson Gauge: As a side product, the authors derive an exact analytical solution for the Schwarzschild metric in the Poisson gauge (adapted to the FLRW background), expressed via Weierstrass elliptic functions.

4. Numerical Results

The authors validated their framework using the relativistic particle-mesh code gevolution through three experiments:

Einstein-de Sitter Test ( $\tilde{\Omega}_K = -0.25$ ):
- Simulated a highly curved universe without matter perturbations.
- Result: The distance-redshift relation ( $d_A(z)$ ) for observers at the center and near the edge of the patch matched the exact curved FLRW prediction to within 0.1% up to $z \approx 1.7$ .
- Comparison: The "separate universe" approximation (flat geometry with curved expansion) showed errors of $\sim 1\%$ at $z=1$ , confirming that geometric effects are significant.
Curved $\Lambda$ CDM Test ( $\tilde{\Omega}_K = -0.1$ ):
- Included a cosmological constant.
- Result: Agreement with the exact curved model was better than 0.1% for observers at different locations. The Hubble diagram remained isotropic, satisfying the cosmological principle within the patch.
- Time Dilation: The simulation correctly accounted for the time dilation between the observer's proper time and the simulation's coordinate time.
Clustering Statistics with Perturbations:
- Added matter perturbations and measured the angular power spectrum ( $C_\ell$ ) in redshift bins.
- Result: The observed clustering statistics were consistent with linear theory predictions (from CLASS) and were statistically indistinguishable for observers at the center versus the edge of the patch. This confirms that the simulation correctly reproduces the cosmological principle (isotropy and homogeneity) despite the embedding.

5. Significance and Implications

Forward Modelling Accuracy: The study demonstrates that ignoring spatial geometry in forward modelling (using the separate universe approximation) introduces systematic errors of $\sim 1-10\%$ in distance-redshift relations and angular clustering for realistic curvature values. This is critical for future surveys (e.g., Euclid, Rubin, DESI) aiming for sub-percent precision.
Neutrino Mass and Inflation: Accurate curvature constraints are vital for determining neutrino masses and testing inflationary paradigms. This framework allows for rigorous testing of these parameters without assuming flatness a priori.
Code Compatibility: The methodology is designed to work with existing Newtonian and relativistic codes with minimal changes (initial conditions and post-processing), making it accessible for the broader cosmological community.
Future Work: The authors note that while they currently focus on closed universes, the method can be extended to open universes (requiring a compensating mass shell). They also plan to improve the initial condition generation to use true hyperspherical harmonics for perturbations on the curvature scale.

In summary, this paper provides a rigorous, relativistic solution to the problem of simulating curved cosmologies, bridging the gap between theoretical curvature constraints and the practical limitations of periodic $N$ -body simulations.