The apparent fine-tuning of the cosmological, gravitational and fine structure constants

doi:10.1016/j.physa.2015.09.090

Physica A: Statistical Mechanics and its Applications

Volume 443, 1 February 2016, Pages 355-357

https://doi.org/10.1016/j.physa.2015.09.090 Get rights and content

Highlights

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Beck’s evaluation of $Λ$ and its implications are discussed.
•
Inter-relating the electron radius, Planck length and fine structure constant.
•
Relevance to a paper by C. Beck in Beck (2009) is discussed.
•
Relevance to a paper by D.N. Page in Page (2009) is discussed.
•
Relevance to the anthropic principle and multiverse scenario is discussed.

Abstract

A numerical coincidence relating the values of the cosmological, gravitational and electromagnetic fine structure constants is presented and discussed in relation to the apparent anthropic fine-tuning of these three fundamental constants of nature.

Introduction

Understanding the physics of the cosmological constant, $Λ$ , remains one of the outstanding challenges in science. If $Λ$ were significantly smaller than its measured value [1], [2], [3], it would be too small to detect with present technology, whereas its upper bound is constrained by arguments based on the anthropic principle and cosmology [4], [5], [6], [7], [8], [9], [10], [11]. Using a set of four statistical axioms, Beck [12] has argued that the value of $Λ$ is determined by gravitational and electromagnetic interactions rather than short-range forces. He thereby obtained a formula for $Λ$ in terms of the low energy value of the electromagnetic fine structure constant, $α$ , the gravitational constant, $G$ , Planck’s constant $ℏ$ , the electronic mass, $m_{e}$ , and charge, $e$ . Here we consider Beck’s result and a numerical coincidence that inter-relates the measured values of $Λ$ , $G$ and $α$ . Together they suggest that the apparent fine-tuning of $α$ would also ensure that $Λ$ and $G$ have the values that we measure for our universe.

Section snippets

Discussion

Beck’s formula for $Λ$ can be written in the following form (the subscript $B$ distinguishes it from the measured value of $Λ$ ): $2 π t_{e} Λ_{B}^{1 / 2} = {(\frac{L_{P}}{r_{e}})}^{2} .$

Here $L_{P} = \sqrt{h G / c^{3}} = 4.05 \times 10^{- 35} m$ is the Planck length, $r_{e} = e^{2} / 4 π ε_{0} m_{e} c^{2} = 2.82 \times 10^{- 15} m$ is the classical electron radius, $t_{e} = r_{e} / c$ is the corresponding time. Beck’s value of $Λ$ , which agrees with that obtained from astrophysical data to within experimental uncertainty, involves the ratio $N = {(r_{e} / L_{P})}^{2} \sim 10^{40}$ . This large number determines the relative strengths of the

Conclusions

There are well-reasoned arguments that the multiverse concept [29] is unscientific. Similar criticisms have been made of the anthropic principle and the use of numerical coincidences. However, Eqs. (1), (2) inter-relate, albeit empirically, a set of three precisely measured fundamental constants of nature to a surprising degree of accuracy. The history of science provides a reminder of the usefulness of empirical hypotheses: for example, a numerical law for the wavelengths of the spectral

Acknowledgments

Discussions with Ed Copeland, Richard Hill, Paul Saffin, Tony Padilla and Frazer Pearce are acknowledged.

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View full text

The apparent fine-tuning of the cosmological, gravitational and fine structure constants

Highlights

Abstract

Introduction

Section snippets

Discussion

Conclusions

Acknowledgments

Physica A

Phys. Lett. B

Measurements of Ω and Λ from 42 high-redshift supernovae

Astrophys. J.

Observational evidence from supernovae for an accelerating universe and a cosmological constant

Astron. J.

BVRI light curves for 22 Type Ia supernovae

Astron. J.

Anthropic principle and the structure of the physical world

Nature

The Anthropic Cosmological Principle

Anthropic bound on the cosmological constant

Phys. Rev. Lett.

The cosmological constant problem

Rev. Modern Phys.

The cosmological constant and cold dark matter

Nature

An anthropic argument for a cosmological constant

Mon. Not. R. Astron. Soc.

The cosmological constants

Nature

Dirac’s cosmology and Mach’s principle

Nature

On the change of physical constants

Phys. Rev.

Measurements of $Ω$ and $Λ$ from 42 high-redshift supernovae