⚛️ general relativity

Rainbow McVittie Horizons in an Expanding Universe

This paper derives the thermodynamic and Friedmann dynamics of a spatially flat McVittie spacetime within Gravity's Rainbow, demonstrating observational viability by constraining a low-energy rainbow deformation parameter using cosmic-chronometer and Pantheon+ data to yield consistent cosmological parameters.

Original authors: Amani Ashour, Ahmed Farag Ali

Published 2026-08-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Amani Ashour, Ahmed Farag Ali

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 universe as a giant, stretching balloon. For decades, physicists have used a set of rules called General Relativity to describe how this balloon inflates and how gravity works on its surface. These rules are incredibly successful, but they hit a wall when we try to look at the very smallest scales (like inside a black hole) or the very first moments of the Big Bang. It's like trying to use a map of a city to navigate a single atom; the map is too big, and the details get blurry. To fix this, scientists have proposed "Gravity's Rainbow." Think of this not as a colorful arc after a storm, but as a pair of magical, energy-dependent glasses. If you wear these glasses, the shape of space and time looks different depending on how much energy your particle has. A high-energy particle sees a slightly different universe than a low-energy one. The big question is: Does this rainbow effect actually change how our universe expands today, or is it just a fancy mathematical trick that disappears when we look at the big picture?

This paper takes a deep dive into that question by mixing two very different cosmic ideas: the "McVittie spacetime" and "Gravity's Rainbow." The McVittie spacetime is a special cosmic model that describes a single, heavy object (like a giant star or black hole) sitting in the middle of an expanding universe. It's the ultimate test of how local gravity (the heavy object) plays nice with global expansion (the stretching balloon). The authors wanted to see what happens if you put this heavy object inside a "Rainbow" universe. They built a new mathematical model to see if the universe could expand differently because of these energy-dependent glasses, and then they checked if that new expansion matches what we actually see in the sky.

The researchers found that you can indeed build a consistent model where the universe expands with these rainbow effects, but it requires a very specific setup. They discovered that if you just slap the rainbow rules onto the equations without care, you accidentally create "radial momentum," which is like forcing the universe to have a wind blowing in or out that shouldn't be there. To fix this, they had to re-arrange their math, effectively changing the way they measure time and distance so that the "wind" cancels out. This allowed them to create a "non-accreting" model, meaning the central object doesn't suck in extra matter from the expanding universe, keeping the system clean and stable.

Once they had this clean model, they asked: "Does this rainbow universe look like the one we live in?" To find out, they compared their model against real-world data. They used two massive datasets: 32 measurements of how fast the universe is expanding at different times (called cosmic chronometers) and 1,580 observations of exploding stars called supernovae (from the Pantheon+ dataset). They also included a strict rule from the Planck satellite about how much matter is in the universe.

The results were surprisingly precise. The data gave them a value for a "rainbow parameter" (let's call it $\epsilon$ ) of $0.0106$ with an uncertainty range of roughly $-0.0231$ to $+0.0234$ . In plain English, this number tells us how much the rainbow effect changes the expansion rate. The most important finding is that the "zero" value—which represents our standard, non-rainbow universe (General Relativity)—sits right inside the most likely range of their results. In fact, the data slightly prefers the standard universe without rainbow effects. While the rainbow model isn't completely ruled out, the "best fit" suggests that if rainbow effects exist at the low energies we can measure today, they are tiny, almost invisible corrections. The paper concludes that while the math works beautifully, the universe we observe seems to be well-described by the old, standard rules, leaving the "rainbow" as a subtle possibility rather than a dramatic new discovery.

Technical Summary: Rainbow McVittie Horizons in an Expanding Universe

Problem Statement
The paper addresses the challenge of constructing a consistent cosmological model within the framework of Gravity's Rainbow that unifies local gravitational structures (specifically the McVittie spacetime) with global cosmic expansion. While Gravity's Rainbow introduces energy-dependent metrics via functions $f(E/E_P)$ and $g(E/E_P)$ , applying these deformations to the McVittie metric—a solution describing a central mass in an expanding universe—presents significant theoretical hurdles. A direct application of rainbow factors to the time and spatial coordinates of the McVittie metric generates unphysical radial momentum densities, violating the non-accreting perfect-fluid condition essential for the standard McVittie solution. Furthermore, the paper seeks to determine if low-energy rainbow deformations can be observationally constrained using current cosmological data without breaking the consistency of the thermodynamic and geometric framework.

Methodology
The authors employ a rigorous geometric and thermodynamic approach, followed by a phenomenological data analysis:

Geometric Construction:
- The authors first demonstrate that a direct ansatz inserting rainbow factors $f(t)$ and $g(t)$ into the standard McVittie metric creates a non-vanishing radial momentum constraint source, $D_j(K^j_r - \delta^j_r K) \neq 0$ , which violates the non-accreting perfect-fluid sector.
- To resolve this, they introduce a consistent time-dependent rainbow parametrization. They define new variables: an effective scale factor $b(t) = a(t)/g(t)$ and a new time coordinate $d\tau = dt/f(t)$ .
- In these variables, the metric reduces exactly to the standard McVittie form with a constant central mass $m$ . This construction isolates the kinematical reparametrization, ensuring the matter content remains a non-accreting perfect fluid with homogeneous density $\rho(t)$ and inhomogeneous pressure $p(t,r)$ .
Thermodynamic Reconstruction:
- The authors derive the Friedmann dynamics and apparent-horizon thermodynamics for this rainbow-parametrized McVittie spacetime.
- They calculate the Misner–Sharp energy, surface gravity ( $\kappa_A$ ), and horizon temperature ( $T_A$ ).
- Using the unified first law ( $dE = A\Psi + W dV$ ) and the Clausius relation ( $\delta Q = T_s dS$ ), they demonstrate that projecting these laws onto the trapping horizon reconstructs the modified Friedmann equations. This establishes a closed thermodynamic system where the horizon dynamics are consistent with the Einstein equations.
Phenomenological Closure and Data Analysis:
- To test the theory observationally, the authors introduce a low-energy deformation ansatz. They define an effective Hubble rate $H_{RG}(z)$ that deviates from the standard $\Lambda$ CDM model by a linear term in redshift: $H_{RG}(z) \approx H_{\Lambda CDM}(z)(1 + \epsilon z)$ .
- This parameter $\epsilon$ is linked to the microscopic rainbow functions $f$ and $g$ and the probe energy $E_0$ via the relation $\epsilon = n(\alpha_n + n\beta_n)(E_0/E_P)^n$ .
- The model is constrained using:
  - 32 cosmic-chronometer measurements ( $H(z)$ data).
  - 1580 Pantheon+ supernova light curves (using the full released covariance matrix).
  - A Gaussian prior on the matter density $\Omega_m$ from Planck 2018.

Key Contributions

Consistent Non-Accreting Sector: The paper provides the first derivation of a non-accreting perfect-fluid McVittie solution within Gravity's Rainbow. It proves that a direct application of rainbow factors fails, but a specific reparametrization of time and scale factor restores the standard McVittie geometry in the "rainbow frame," preserving the constant central mass and homogeneous density.
Thermodynamic Consistency: It establishes that the rainbow-deformed McVittie spacetime satisfies the unified first law and Clausius relation on the apparent horizon, reproducing the Friedmann dynamics without introducing ad-hoc thermodynamic corrections.
Observational Constraints: The study presents the first joint constraints on the low-energy rainbow deformation parameter $\epsilon$ using a combination of cosmic chronometers and the full-covariance Pantheon+ dataset.

Results

Parameter Constraints: The joint analysis yields the following best-fit parameters:
- Hubble constant: $H_0 = 68.75 \pm 2.63 \text{ km s}^{-1} \text{ Mpc}^{-1}$ .
- Matter density: $\Omega_m = 0.3157 \pm 0.0070$ .
- Rainbow deformation parameter: $\epsilon = 0.0106^{+0.0234}_{-0.0231}$ (68% profile).
Statistical Significance: The inclusion of the rainbow parameter $\epsilon$ improves the minimum statistic by only $\Delta\chi^2 \approx 0.204$ relative to the nested General Relativity (GR) limit ( $\epsilon = 0$ ).
Model Selection: Information criteria (AIC and BIC) favor the nested GR limit ( $\epsilon = 0$ ). The fitted value of $\epsilon$ is statistically consistent with zero, placing the General Relativistic limit well within the preferred parameter region (within one standard deviation).
Degeneracy: The analysis highlights a degeneracy between the matter density $\Omega_m$ and the rainbow parameter $\epsilon$ , as well as an identifiability limit where current data can only constrain the specific combination $\alpha_n + n\beta_n$ of the microscopic rainbow functions, not the functions individually.

Significance and Claims
The paper claims to establish the observational viability of small, late-time rainbow corrections while simultaneously demonstrating that current data does not require them. By constructing a mathematically consistent McVittie sector in Gravity's Rainbow, the authors provide a rigorous benchmark for testing energy-dependent gravity.

The authors emphasize that their results are modest: the study does not claim to detect rainbow gravity but rather sets a bound on the effective redshift response of the universe. The work clarifies that a single prescribed energy history in this framework is geometrically equivalent to a reparametrized standard geometry; thus, physical rainbow effects require additional structure (e.g., comparing distinct energy sectors or energy-dependent couplings) to be distinguishable from coordinate transformations. The paper concludes that while the formalism is consistent, the current observational constraints place the General Relativistic limit firmly within the preferred region, limiting the magnitude of any potential late-time rainbow deformation to the percent level or less.

Technical Summary: Rainbow McVittie Horizons in an Expanding Universe

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