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Impact of inhomogeneous curvature on growth rate measurements from magnitude fluctuations

Using full General Relativity numerical simulations, this paper demonstrates that inhomogeneous curvature induces systematic offsets in growth rate measurements that are currently sub-dominant to statistical errors at low redshifts ( $z \lesssim 0.2$ ), validating the adequacy of standard FLRW models for current peculiar velocity experiments while highlighting the need for refined theoretical models in future high-redshift datasets.

Original authors: Andrew Nguyen, Chris Blake, Hayley J. Macpherson

Published 2026-06-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Nguyen, Chris Blake, Hayley J. Macpherson

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

The Big Picture: Is the Universe a Smooth Cake or a Bumpy Muffin?

Imagine the Universe as a giant loaf of bread. For a long time, cosmologists have assumed this bread is perfectly smooth and uniform, like a well-mixed cake batter. This idea is called the FLRW model. It assumes that if you zoom out far enough, the universe looks the same everywhere, expanding evenly like a rising cake.

However, we know the universe isn't actually a smooth cake. It's more like a bumpy muffin. It has raisins (galaxies), chocolate chips (dark matter), and empty air pockets (voids). These lumps create "bumps" in the fabric of space itself, known as inhomogeneous curvature.

The big question this paper asks is: Does the fact that the universe is "bumpy" mess up our measurements? Specifically, does it make us miscalculate how fast the universe's structure is growing?

The Experiment: Measuring the "Bumpiness"

To answer this, the authors didn't just look at the real sky; they built a virtual universe inside a supercomputer.

The Simulation (The Virtual Muffin): They used a powerful method called "Numerical Relativity" to simulate a universe that follows the strict laws of Einstein's gravity, including all the bumps and curves caused by matter. This is the "realistic" version.
The Comparison (The Smooth Cake): They also ran a simulation based on the old, smooth "FLRW" model, which ignores the bumps.
The Test (The Flashlight): They placed 20 virtual observers in these universes and shone "flashlights" (light rays) back in time to see how bright distant objects appeared.

The Key Concept: Why Things Look Brighter or Dimmer

In this study, the scientists looked at magnitude fluctuations. Think of this like looking at a lighthouse from a boat.

The Standard View (Peculiar Velocities): Usually, if a lighthouse seems brighter or dimmer than expected, we think it's because the lighthouse or the boat is moving toward or away from us. This is called a "peculiar velocity." It's like the Doppler effect with sound (a siren changing pitch as it passes).
The New View (Lensing and Curvature): But in a bumpy universe, the light doesn't just travel in a straight line. The "bumps" in space (gravity) can act like a magnifying glass or a funhouse mirror, bending the light and changing how bright the lighthouse looks. This is gravitational lensing.

The paper asks: If we only look at the movement (velocity) and ignore the magnifying glass effect (lensing/curvature), do we get the wrong answer about how fast the universe is growing?

The Findings: It Depends on How Far You Look

The researchers compared the "Realistic Bumpy Universe" against the "Smooth Universe" at different distances (redshifts).

1. Close to Home (Low Redshift, $z \lesssim 0.2$ ):

The Analogy: Imagine looking at a lighthouse just a few miles away. The water is calm, and the air is clear.
The Result: The "bumpy" effects are tiny. The light hasn't traveled far enough to get distorted by many bumps.
Conclusion: For nearby galaxies, the old "smooth" model works perfectly fine. If you ignore the bumps, your calculation of how fast the universe is growing is still accurate enough. The error is smaller than the natural noise in our measurements.

2. Far Away (High Redshift, $z \gtrsim 0.3$ ):

The Analogy: Now imagine looking at a lighthouse 50 miles away through a stormy sea with giant waves. The light has to travel through a lot of turbulence.
The Result: As the light travels further, it passes through more "bumps." The gravitational lensing (the magnifying glass effect) starts to dominate. The "bumpy" universe looks significantly different from the "smooth" one.
Conclusion: If we try to measure the growth rate of the universe using distant objects but only use the old "smooth" model, we will get a massive error.
- At $z = 0.3$ , the error is about 8%.
- At $z = 0.5$ , the error explodes to over 100%.
- Essentially, if we look far enough away and ignore the "bumps," we might think the universe is growing twice as fast (or slow) as it actually is.

The Bottom Line

For current surveys: We are mostly looking at the "nearby" universe. The "bumps" don't matter much yet. The standard smooth models are good enough for today's data.
For future surveys: As we build better telescopes to see deeper into the universe (further away), the "bumps" will become a major problem. To get accurate results in the future, scientists will need to update their math to include the effects of gravitational lensing and the uneven curvature of space, rather than assuming the universe is a perfectly smooth cake.

In short: The universe is a bumpy muffin, not a smooth cake. For close-up views, the bumps don't matter. But if you try to see the whole muffin from far away, you have to account for the bumps, or you'll get the recipe wrong.

Technical Summary: Impact of Inhomogeneous Curvature on Growth Rate Measurements from Magnitude Fluctuations

Problem Statement
Modern cosmology relies on the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which assumes a homogeneous and isotropic universe. However, the actual large-scale structure of the Universe is non-uniform, leading to inhomogeneous curvature and scale factor variations. While previous studies have investigated "backreaction" effects on cosmic expansion, this paper addresses a specific, unexplored aspect: the impact of inhomogeneous curvature on the measurement of the growth rate of cosmic structure ( $f$ ) using magnitude fluctuations (peculiar velocities).

In standard analyses, magnitude fluctuations are interpreted as deviations in luminosity distance caused solely by the Doppler effect of peculiar velocities (PVs) relative to a homogeneous background. In a full General Relativity (GR) treatment, however, light propagation through an inhomogeneous spacetime introduces additional effects, including gravitational lensing and variations in curvature along the line of sight. The authors investigate whether neglecting these non-velocity effects in an FLRW framework introduces a systematic bias in inferred growth rates, particularly as future surveys extend to higher redshifts.

Methodology
The study employs full numerical relativity (NR) simulations to model the non-linear evolution of structure without enforcing a global FLRW background.

Simulations: The authors used the Einstein Toolkit to solve the Einstein field equations on a computational grid. Initial conditions were generated using a perturbed FLRW metric in the Newtonian gauge at $z_{ini}=20$ , with parameters matching the $\Lambda$ CDM model (though the simulations themselves do not include a cosmological constant, $\Lambda$ , due to current software limitations). The fiducial simulation used a resolution of $N=256$ in a box of $L=3072 \, h^{-1}\text{Mpc}$ .
Ray Tracing: Using the mescaline code, the authors performed general-relativistic ray tracing from 20 synthetic observers at $z=0$ back through the simulation volume. This generated light-cone slices containing redshifts and angular diameter distances without assuming a specific metric form.
Magnitude Fluctuation Extraction: Magnitude fluctuation maps ( $\delta m$ $δ m$ ) were constructed at fixed observed redshifts ( $z \in [0.1, 0.6]$ $z \in [0.1, 0.6]$ ). The authors compared two datasets:
1. Full NR ( $\delta m_{NR}$ ): Captures the total effect of inhomogeneous spacetime, including PVs, lensing, and curvature.
2. Velocity-Only ( $\delta m_{vel}$ ): Calculated using only the source and observer velocities relative to the simulation coordinates, effectively isolating the Doppler contribution and excluding lensing/curvature effects.
Analysis: The angular power spectra ( $C_\ell$ ) of these magnitude fluctuations were computed using spherical harmonic decomposition. The authors used a Fisher matrix formalism to forecast the systematic offset ( $\Delta f$ ) in the growth rate that would result from fitting the full NR data with a velocity-only (FLRW) template, comparing this offset against the forecasted statistical error ( $\sigma_f$ ) for a future all-sky survey.

Key Results

Low Redshift ( $z \lesssim 0.2$ ): The angular power spectra of the full NR simulation and the velocity-only measurement are in close agreement (average difference of 4.7% for $\ell > 1$ ). The systematic offset in the growth rate is sub-dominant to the statistical error (< 1% offset vs. 3–4% statistical error). This confirms that for current low-redshift peculiar velocity experiments, FLRW modeling is adequate.
Intermediate to High Redshift ( $z \gtrsim 0.3$ ): As redshift increases, the full NR power spectrum exhibits a significant excess over the velocity-only measurement. This excess is well-matched by theoretical predictions for weak gravitational lensing.
- At $z=0.3$ , the systematic offset reaches $\sim 8\%$ .
- At $z=0.4$ , the offset rises to $\sim 39\%$ .
- By $z=0.5$ , the offset exceeds $100\%$ , while the statistical error grows more gradually.
Convergence: The results were tested across three resolutions ( $N=128, 200, 256$ ). The angular power spectra and the inferred systematic offset converged at $N=256$ , confirming that the detected differences are physical effects of inhomogeneous curvature and lensing rather than numerical artifacts.

Significance and Claims
The paper concludes that the assumption of a homogeneous background introduces a systematic bias in growth rate measurements when applied to magnitude fluctuations at redshifts beyond $z \approx 0.2$ .

Current Surveys: For existing datasets restricted to $z \lesssim 0.1\text{--}0.2$ , the bias is negligible compared to statistical uncertainties, validating current FLRW-based analyses.
Future Surveys: As surveys push to intermediate redshifts ( $z \approx 0.3$ ) with improved precision, the contributions from gravitational lensing and inhomogeneous curvature become dominant over statistical errors. The authors claim that ignoring these effects in future high-precision growth rate measurements would lead to significantly biased constraints (potentially overestimating $f$ by order unity).
Theoretical Requirement: The study suggests that future theoretical models for interpreting magnitude fluctuations at higher redshifts must incorporate contributions from gravitational lensing and inhomogeneous curvature to achieve percent-level accuracy.

The authors note limitations, including the absence of collapsed structures (dark matter halos) in their simulations and the restriction to 2D redshift slices, but maintain that their findings robustly quantify the transition point where inhomogeneous effects become critical for cosmological inference.