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Testing CCC+TL Cosmology with Galaxy Rotation Curves

This paper proposes that galaxy rotation curves and other cosmological observations can be explained without dark matter or dark energy by modeling the effects of evolving coupling constants (CCC) combined with a tired light redshift mechanism, where local variations in the coupling parameter $\alpha$ create the illusion of dark matter and dark energy.

Original authors: Rajendra P. Gupta

Published 2026-07-21

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CC BY 4.0

Original authors: Rajendra P. Gupta

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, expanding balloon. For decades, astronomers have been trying to figure out what's inside that balloon and how it's moving. They know the balloon is stretching, but when they look at the galaxies spinning inside it, something strange happens. The outer edges of these galaxies are spinning way too fast to be held together by the visible stars and gas alone. It's like a spinning pizza dough that should fly apart, yet it stays in a perfect circle. To explain this, most scientists have proposed a "cosmic glue" called dark matter—an invisible substance that adds extra gravity to keep the galaxies from flying apart. They also use dark energy to explain why the universe's expansion is speeding up. But here's the catch: nobody has ever actually found a particle of this "glue," even after hunting for it in deep underground labs and giant particle smashers.

This paper steps into that mystery with a different idea. Instead of adding invisible glue, the author asks: "What if the rules of the game are changing?" Specifically, what if the fundamental constants of nature—the numbers that define how strong gravity is or how fast light travels—are slowly shifting as the universe expands? This concept is called Covarying Coupling Constants (CCC). The paper also mixes in an old idea called Tired Light, which suggests that light loses a tiny bit of energy as it travels through space, making distant objects look redder without needing as much expansion. The big question is: Can these shifting rules explain the fast-spinning galaxies without needing any dark matter at all?

The author, Rajendra P. Gupta, proposes a fascinating solution. He suggests that the "illusion" of dark matter and dark energy is actually caused by these changing constants. He calls this new effect $\alpha$ -matter and $\alpha$ -energy. Think of it like a video game where the physics engine is subtly glitching. In the middle of a galaxy, where the stars are packed tightly together (high density), the rules stay normal, and everything looks standard. But as you move out to the edges of the galaxy, where the stars are sparse, the "glitch" kicks in. The changing constants create extra gravitational pull that mimics the effect of dark matter, keeping the outer stars spinning fast without needing any invisible particles.

In this paper, Gupta tests this idea using a massive database of galaxy rotation curves called SPARC, which contains data on 175 different galaxies. He doesn't try to prove the theory from scratch; instead, he works backward. He takes the observed speed of the stars in these galaxies and asks, "If our theory about changing constants is true, how much actual, visible (baryonic) matter must be there to make this happen?"

The results are surprisingly promising. By applying his model, Gupta found that he could match the observed rotation curves of several galaxies (like NGC 3198 and NGC 6503) using only the visible matter, provided he adjusted a single "turn-off" parameter. This parameter acts like a switch: inside a certain radius, the universe behaves normally; outside that radius, the "glitch" (the $\alpha$ -effect) takes over, providing the extra gravity needed to explain the fast spins. He found that the density at which this switch flips is remarkably consistent across different galaxies, hovering around $2.09 \times 10^{-24}$ g cm $^{-3}$ to $7.96 \times 10^{-24}$ g cm $^{-3}$ .

The paper suggests that this approach might even explain why galaxies at the very beginning of the universe (high redshift) look different. In the early universe, the ratio of this " $\alpha$ -matter" to normal matter was lower, meaning the "glitch" wasn't as strong. This would explain why those ancient galaxies seem to have less dark matter and spin more like normal, Keplerian systems, which matches recent observations.

However, the author is careful not to declare victory just yet. He admits that his model uses a simplified "spherical" shape for galaxies, while real galaxies are often flat disks or have complex shapes. He notes that the fit isn't perfect for every single galaxy (for instance, NGC 1090 didn't fit well at smaller radii). The paper presents this as a "proof of concept"—a demonstration that it is possible to explain these cosmic mysteries without dark matter, simply by letting the laws of physics evolve. It suggests that what we call dark matter might just be the universe's way of telling us that the constants of nature aren't as constant as we thought. While it doesn't rule out dark matter entirely, it offers a compelling alternative that fits the data without needing to find a particle that has remained invisible for decades.

Technical Summary: Testing CCC+TL Cosmology with Galaxy Rotation Curves

Problem Statement
Galaxy rotation curves present a primary astrophysical challenge to the standard $\Lambda$ CDM cosmology, typically interpreted as evidence for dark matter. While cosmological observations (e.g., Type Ia supernovae, CMB) are often fitted using the Friedmann equations assuming a homogeneous and isotropic universe, astrophysical systems like galaxies are gravitationally bound and isolated from the Hubble flow. The standard model requires dark matter to explain the flatness of rotation curves at large radii, yet dark matter particles remain undetected. Furthermore, recent observations of high-redshift galaxies suggest declining rotation curves, challenging the universality of flat curves. This paper investigates whether the "illusion" of dark matter and dark energy can be explained not by new particles or modified gravity (such as MOND), but by the covarying coupling constants (CCC) within an expanding universe, combined with the "tired light" (TL) redshift mechanism.

Methodology
The study employs the CCC+TL cosmological model, a hybrid framework where fundamental constants (speed of light $c$ , gravitational constant $G$ , Planck constant $\hbar$ , and Boltzmann constant $k$ ) evolve with the scale factor $a(t)$ via a function $f(t) = \exp(\alpha(t-t_0))$ .

Theoretical Framework: The author derives modified Friedmann equations where the parameter $\alpha$ generates terms equivalent to "dark energy" ( $\alpha$ -energy) and "dark matter" ( $\alpha$ -matter). In this model, the critical density depends on both the Hubble constant and $\alpha$ .
Local Variation of Parameters: While $\alpha$ is treated as a constant on cosmological scales (determined by fitting Pantheon+ data), the paper posits that $\alpha$ (and the derived parameter $X = -\alpha/H_X$ ) varies locally due to the anisotropy of matter distribution. In high baryonic density regions (galaxy centers), $X \to 0$ (pure baryonic matter), while in low-density regions (outer radii), $X$ increases, introducing $\alpha$ -matter and $\alpha$ -energy contributions.
Data Analysis: The author analyzes galaxy rotation curves from the SPARC database (Spitzer Photometry and Accurate Rotation Curves).
- Procedure: Instead of adding dark matter halos to baryonic mass, the method works in reverse. It uses the observed rotation velocity $V_o(R)$ to calculate the total effective mass density.
- Turn-off Criterion: A "turn-off density" ( $\rho_t$ ) is defined as the threshold where non-Keplerian dynamics become perceptible. Below the corresponding turn-off radius ( $R_t$ ), the matter is assumed to be purely baryonic ( $X=0$ ). Beyond $R_t$ , the density profile is modified by the factor $(1-X(R))^2$ .
- Derivation: By assuming flat rotation curves at large radii, the author derives that the baryonic density must scale as $R^{-4}$ beyond $R_t$ , while the total density scales as $R^{-2}$ . The parameter $X(R)$ is solved numerically to fit the observed curves, allowing the reconstruction of the baryonic mass distribution ( $V_{bX}$ ) to compare with the SPARC database's baryonic estimates.

Key Contributions and Results

Reinterpretation of Dark Components: The paper demonstrates that the $\alpha$ -matter and $\alpha$ -energy terms emerging from the CCC equations can mathematically replicate the effects attributed to dark matter and dark energy in $\Lambda$ CDM. The $\alpha$ -matter term is shown to be proportional to the square root of the product of baryonic matter and $\alpha$ -energy, behaving dynamically.
Galaxy Rotation Curve Fits: The model was applied to a sample of seven galaxies (including NGC3198, D631-7, NGC6503, UGC09133, NGC3521, NGC1090, and NGC7814) from the SPARC database. Using a single free parameter, the turn-off density $\rho_t$ , the author successfully derived baryonic velocity curves ( $V_{bX}$ ) that closely match the composite baryonic curves provided in the SPARC database.
Density Scaling: The analysis confirms that for flat rotation curves, the total density scales as $R^{-2}$ , while the derived baryonic density scales as $R^{-4}$ beyond the turn-off radius, consistent with the Hernquist model.
Consistency with High-Redshift Observations: The model naturally explains why rotation curves at high redshifts ( $z \approx 2$ ) appear to decline (approaching Keplerian). As redshift increases, the ratio of $\alpha$ -matter to baryonic matter decreases, reducing the "dark matter" effect and leaving the rotation curve dominated by baryonic mass.
Galaxy Clusters: A preliminary application to galaxy cluster density profiles (using data from Mandelbaum et al.) showed a good visual fit with a turn-off density similar to that of individual galaxies, suggesting the model's applicability to larger scales.
Tully-Fisher Relation: The paper notes that the derived scaling ( $M_B \propto V^4$ ) is consistent with the Baryonic Tully-Fisher Relation (BTFR).

Significance and Claims
The paper claims to provide a proof-of-concept that the covarying coupling constant model, combined with tired light, is both cosmologically and astrophysically consistent.

Elimination of Dark Matter: The author argues that "dark matter" is not a physical particle but an emergent effect of varying coupling constants ( $\alpha$ -matter) and that "dark energy" is similarly an emergent $\alpha$ -energy term.
No New Physics Required: The study suggests that observed flat rotation curves do not require modified gravity theories (like MOND) or new particle physics. The CCC approach offers a unified explanation for cosmological and astrophysical data without invoking dark matter.
Relation to MOND: The paper observes that the turn-off acceleration ( $a_t$ ) derived from the CCC model is of the same order of magnitude as the MOND acceleration constant ( $a_0$ ). It hypothesizes that the MOND transition from Newtonian to non-Newtonian dynamics may be a manifestation of the CCC effect, where varying coupling constants alter the effective gravitational pull.
Modesty and Limitations: The author acknowledges that the current analysis relies on a simplified spherical approximation for galaxy morphology, whereas real galaxies are disk-like and oblate. The paper explicitly states that more realistic morphological models are required for better data fitting and that the current work serves as a foundational step to demonstrate the viability of the CCC+TL approach for astrophysical systems. The paper does not claim to have solved all issues regarding galaxy formation or cluster dynamics but presents these as future areas for investigation within this framework.

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