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Does time or space shape black hole accretion in Rainbow Gravity? Observational constraints from Pantheon+SH0ES, CC+BAO, and joint data

This paper utilizes Pantheon+SH0ES, CC+BAO, and joint observational data within a Rainbow Gravity framework coupled with generalized holographic dark energy models to demonstrate that temporal versus spatial metric modifications yield distinct cosmological constraints and black hole accretion histories, with the temporal modification (RF1) providing more stable results while the spatial modification (RF2) predicts a higher Hubble constant and slightly lower mass growth.

Original authors: Puja Mukherjee, Ujjal Debnath

Published 2026-08-25

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Puja Mukherjee, Ujjal Debnath

Original paper licensed under CC BY 4.0 (https://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 universe is expanding, and not just expanding, but accelerating. For decades, cosmologists have tried to explain this cosmic speed-up, usually by invoking a mysterious force called dark energy that pushes galaxies apart. But there is another possibility: perhaps the rules of gravity themselves are different than we think, especially when we look at the very small scales where quantum physics and the vastness of space-time meet. One such idea is "rainbow gravity," a theory suggesting that space and time do not look the same to every particle. Instead, the geometry of the universe changes depending on the energy of the object moving through it, much like how a prism splits white light into a spectrum of colors. This means that high-energy particles might experience a different version of reality than low-energy ones. Understanding how this strange idea plays out is crucial, because if gravity truly depends on energy, it could change how black holes grow and how the universe evolves.

In a recent study, researchers set out to test this concept by looking at how black holes swallow matter in a rainbow gravity universe. They focused on two specific ways this energy-dependent gravity could work. In the first scenario, the energy dependence changes the flow of time for the particles involved, while the second scenario changes the shape of space itself. To see which of these possibilities fits our reality, the team combined these theories with three different models of dark energy, each based on a unique way of counting the disorder, or entropy, of the universe. They then used a massive collection of real-world astronomical data—observations of exploding stars, the ages of ancient galaxies, and the large-scale structure of the cosmos—to see which mathematical models matched the observations best.

The researchers found that the universe seems to prefer the version of rainbow gravity where time is modified, rather than space. When they analyzed the data, the model that altered the temporal component of space-time provided a much tighter and more consistent fit to the observations. In this scenario, a key number describing the nature of dark energy settled very close to the value predicted by standard physics, suggesting that the universe is behaving in a way that is familiar, even within this exotic framework. In contrast, the model that modified space instead of time produced much looser results and suggested a universe that is expanding faster than current measurements typically indicate. This difference is significant because it implies that if rainbow gravity is real, its effects are likely felt most strongly in the passage of time, not in the stretching of space.

Beyond the expansion of the universe, the study also looked at how these theories affect black holes. Black holes are cosmic vacuum cleaners, growing as they pull in surrounding gas and dark energy. The team calculated how much mass these black holes would gain over billions of years under the different rainbow gravity scenarios. They discovered that in the time-modified scenario, black holes grow slightly more efficiently, gaining about 0.15 percent of their mass from the distant past until today. In the space-modified scenario, this growth is slightly slower, at about 0.10 percent. While these percentages seem tiny, they represent a measurable difference in the history of the universe. The study showed that the time-modified model is more stable and less sensitive to which specific data set is used, making it a more reliable description of how black holes evolve.

Ultimately, this research does not prove that rainbow gravity is the final answer, but it does provide a way to test it against the real universe. By comparing how different versions of the theory predict black holes should grow, the authors have opened a new door for observation. They suggest that future telescopes, capable of measuring the shadows of black holes or the gravitational waves they emit, could eventually confirm whether time or space is the true architect of these cosmic changes. For now, the evidence points toward a universe where the flow of time is the variable that bends under the weight of quantum energy, quietly shaping the growth of the darkest objects in the cosmos.

Technical Summary: Does Time or Space Shape Black Hole Accretion in Rainbow Gravity?

Problem Statement
This research investigates how specific formulations of Rainbow Gravity—a framework extending Doubly Special Relativity (DSR) to curved spacetimes where the metric depends on particle energy—influence the mass accretion of black holes. The study addresses a fundamental question: do quantum gravity corrections manifesting through temporal modifications of the metric differ phenomenologically from those arising through spatial modifications? To answer this, the authors couple two distinct Rainbow Function sets (RF1 and RF2) with three generalized Holographic Dark Energy (HDE) models (Tsallis, Rényi, and Sharma-Mittal) that incorporate non-extensive entropy corrections to the standard Bekenstein-Hawking area law. The goal is to determine how these theoretical choices affect observable cosmological parameters and the evolutionary history of black hole mass growth.

Methodology
The authors employ a multi-stage theoretical and statistical approach:

Theoretical Framework:
- Rainbow Gravity: Two specific function sets are analyzed. RF1 (based on the Ling model) modifies the temporal component of the metric ( $f \neq 1, g=1$ ), while RF2 (based on the Gangopadhyay & Dutta model) modifies the spatial component ( $f=1, g \neq 1$ ).
- Dark Energy Models: The study utilizes Tsallis, Rényi, and Sharma-Mittal HDE models. These models introduce non-extensive entropy parameters ( $\delta$ for Tsallis/Rényi, and $R$ for Sharma-Mittal) to generalize the standard holographic principle.
- Black Hole Solution: The analysis focuses on a 4-dimensional Schwarzschild rainbow black hole. The mass accretion rate ( $\dot{M}$ ) is derived using the energy-momentum tensor of the accreting fluid, linking the mass evolution to the equation of state ( $\rho + p$ ) and the modified Friedmann equations.
Observational Constraints:
- A comprehensive Markov Chain Monte Carlo (MCMC) analysis is performed to constrain the free parameters ( $H_0, \Omega_{m0}, \delta, \epsilon, R$ ).
- The analysis utilizes three complementary datasets: Pantheon+SH0ES (1701 Type Ia supernovae), Cosmic Chronometers (CC) (31 $H(z)$ measurements), and Baryon Acoustic Oscillations (BAO) (26 measurements).
- Both individual datasets and joint analyses (Pantheon+SH0ES + CC + BAO) are used to break parameter degeneracies.
- Model selection is evaluated using Akaike (AIC), Bayesian (BIC), and Deviance (DIC) information criteria.
Accretion Analysis:
- Using the best-fit parameters derived from the MCMC analysis, the authors compute the mass evolution of the black hole from high redshift ( $z=2$ ) to the present day ( $z=0$ ) for both RF1 and RF2 frameworks.

Key Contributions and Results

Parameter Constraints:
- RF1 (Temporal Modification): The Tsallis exponent $\delta$ is tightly constrained around the Bekenstein-Hawking limit ( $\delta = 2.0 \pm 0.03$ ) across all datasets. The Hubble constant ( $H_0$ ) from the joint analysis is approximately $70$ km/s/Mpc. The Sharma-Mittal parameter $R$ is consistent with zero, suggesting no need for additional entropy corrections beyond the Rényi limit.
- RF2 (Spatial Modification): Constraints on $\delta$ are significantly weaker (uncertainties $\sim 0.7$ ). The joint analysis yields a higher Hubble constant ( $H_0 \sim 74$ km/s/Mpc). The parameter $R$ is fixed to zero due to degeneracy with the rainbow functions.
Mass Accretion Dynamics:
- Both frameworks predict monotonic, quintessence-like mass growth ( $\rho + p > 0$ ), indicating that the black holes gain mass over cosmic time.
- Quantitative Difference: The RF1 framework results in a total mass increase of approximately 0.15% from $z=2$ to the present. In contrast, the RF2 framework yields a slightly lower increase of about 0.10%.
- Dataset Sensitivity: RF1 demonstrates robustness, with accretion histories showing strong agreement across different datasets (Pantheon+, CC+BAO, and Joint). RF2 exhibits greater sensitivity to the choice of dataset, particularly showing divergence when using CC+BAO constraints alone.
Model Comparison:
- Within the RF1 framework, the Tsallis HDE model tracks the standard $\Lambda$ CDM accretion history most closely.
- In the RF2 framework, the Tsallis model emerges as the most stable candidate, though all models show larger deviations from the $\Lambda$ CDM baseline compared to the RF1 case.

Significance and Claims
The paper claims that the choice of rainbow function—specifically whether the modification occurs in the temporal or spatial component of the metric—directly dictates the observable signatures of black hole accretion. The authors assert that:

Temporal vs. Spatial Distinction: RF1 (temporal modification) leads to more stable, dataset-independent predictions and slightly more efficient mass accretion. RF2 (spatial modification) results in suppressed mass growth and higher sensitivity to observational data choices.
Testing Quantum Gravity: The study establishes a direct link between the specific form of quantum gravity corrections (rainbow functions) and astrophysical observables (black hole mass growth), opening a new avenue for testing quantum gravity phenomenology through cosmological observations.
Entropy Constraints: The tight constraint of the Tsallis exponent $\delta$ to the standard Bekenstein-Hawking limit in the RF1 framework suggests that current observational data do not require significant deviations from standard area-law entropy in this specific context.

The authors conclude that while RF2 offers a phenomenologically richer landscape, RF1 provides a more reliable and stable description of black hole mass accretion under the current observational constraints. They note that future work could extend these findings to rotating or charged black holes and incorporate data from upcoming missions like DESI, Euclid, and the Roman Space Telescope to further tighten constraints.

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