Circular reasoning: Solving the Hubble tension with a non-$π$ value of $π$ (2403.20219v1)

Abstract: Recently, cosmology has seen a surge in alternative models that purport to solve the discrepancy between the values of the Hubble constant $H_0$ as measured by cosmological microwave background anisotropies and local supernovae, respectively. In particular, many of the most successful approaches have involved varying fundamental constants, such as an alternative value of the fine structure constant and time-varying values of the electron mass, the latter of which showed particular promise as the strongest candidate in several earlier studies. Inspired by these approaches, in this paper, we investigate a cosmological model where the value of the geometric constant $\pi$ is taken to be a free model parameter. Using the latest CMB data from Planck as well as baryon-acoustic oscillation data, we constrain the parameters of the model and find a strong correlation between $\pi$ and $H_0$, with the final constraint $H_0 = 71.3 \pm 1.1 \ \mathrm{ km/s/Mpc}$, equivalent to a mere $1.5\sigma$ discrepancy with the value measured by the SH0ES collaboration. Furthermore, our results show that $\pi = 3.206 \pm 0.038$ at $95 \%$ C.L., which is in good agreement with several external measurements discussed in the paper. Hence, we conclude that the $\pi \Lambda$CDM model presented in this paper, which has only a single extra parameter, currently stands as the perhaps strongest solution to the Hubble tension.

• The paper presents a novel approach by treating the geometric constant π as a free parameter in the ΛCDM model to address the Hubble tension.
• It combines Planck, BAO, and SH0ES data to yield H0 = 71.3 ± 1.1 km/s/Mpc and a variable π value of 3.206 ± 0.038 at 95% confidence.
• The study suggests that reexamining fundamental geometric constants may open new avenues for resolving discrepancies in cosmological measurements.

An Examination of the Proposal to Resolve the Hubble Tension through a Non-Standard Value of $\pi$

Introduction

The paper "Circular reasoning: Solving the Hubble tension with a non-$\pi$ value of $\pi$" asserts a novel approach to reconcile the well-known Hubble tension by considering the geometric constant $\pi$ as a free parameter in cosmological models. Hubble tension denotes the discrepancy between the value of the Hubble constant $H_0$ obtained from measurements of the Cosmic Microwave Background (CMB) and values obtained through local observations, such as supernovae.

Theoretical Foundation

In the pursuit to resolve the Hubble tension, the authors expand upon previous methodologies that modify fundamental constants, such as the fine structure constant and electron mass. This paper takes an unconventional course, proposing that the value of $\pi$, a geometric constant, can vary and should be treated as a parameter within the $\Lambda$CDM model. This modified model, referred to as $\pi\Lambda$CDM, derives theoretical support from the concept of extra dimensions pointed out in theories like Kaluza-Klein and string theories. In these frameworks, what we perceive as fundamental constants could feasibly differ or "run" due to the characteristics of higher-dimensional spaces.

Methodology and Results

Utilizing data from the Planck CMB measurements and Baryon Acoustic Oscillations (BAO), the authors perform parameter inference to ascertain the optimal value of $\pi$ that minimizes the tensions in $H_0$. When only Planck and BAO data are used, a moderate resolution of the Hubble tension is achieved; however, the inclusion of the SH0ES prior on $H_0$ further reduces the tension to an insignificant level, specifically a $1.5\sigma$ discrepancy, indicating satisfactory convergence between different $H_0$ values.

The derived value of the Hubble constant, when incorporating all datasets including SH0ES, is $H_0 = 71.3 \pm 1.1$ km/s/Mpc. Simultaneously, the study postulates a value for $\pi$ of $3.206 \pm 0.038$ at a 95% confidence level, deviating from its traditional Euclidean value.

Implications and Future Work

This model, with its single additional degree of freedom, demonstrates a potential in offering a less contentious solution to the Hubble tension than models introducing numerous additional parameters. In particular, the correlation found between $\pi$ and $H_0$ suggests that modifications of geometric structure constants could be a viable direction for resolving cosmological discrepancies.

However, the notion of a variable $\pi$ raises numerous theoretical questions. It calls for an investigation into potential impacts on established physical laws, such as those describing wave mechanics where $\pi$ plays a fundamental role. Moreover, the rejection of $\pi$'s constancy might imply broad, profound amendments to our understanding of spatial structure in physics, possibly requiring reconsideration of various physical phenomena that rely on geometric invariants.

The authors acknowledge the primitive nature of their approach, committing to future work that extends to configurations involving time-dependent $\pi$ values or exploring other constants. This further research could potentially clear the "dark ages" of $\pi$, grappling with constraints from cosmic history to reinforce their framework.

Conclusion

In conclusion, while the paper offers an intriguing hypothesis and delivers numerical resolution to a significant problem in cosmology—the Hubble tension—it naturally attracts skepticism by challenging what is conventionally a fixed numerical constant. Future explorations in this vein might bring about new insights, not just into the Hubble tension, but into the very fabric of cosmological theories and the constants that underpin them. As with any model proposing fundamental shifts, robust theoretical justifications and comprehensive empirical validations are paramount to transition from speculative ideas to accepted scientific advancement.