[Paper Review] Fine-structure constant variability: surprises for laboratory atomic spectroscopy and cosmological evolution of quasar spectra

Jacob D. Bekenstein|arXiv (Cornell University)|Jan 29, 2003

Relativity and Gravitational Theory1 references3 citations

TL;DR

This paper challenges the conventional assumption that atomic spectra evolve with cosmological time solely through changes in the fine-structure constant α, showing that in a dynamical α framework, the Dirac Hamiltonian's form alters, leading to a distinct dependence of spectral line shifts on quantum numbers. This implies that combining data from different fine-structure multiplets in quasar absorption spectra can bias estimates of α variability, and the standard analysis method may significantly underestimate or misestimate the true cosmological variation.

ABSTRACT

Calculation of the Dirac hydrogen atom spectrum in the framework of dynamical fine structure constant (alpha) variability discloses a small departure in the laboratory from Sommerfeld's formula for the fine structure shifts, possibly measurable today. And for a distant object in the universe, the wavelength shift of a spectral line specifically ascribable to cosmological alpha variation is found to depend differently on the quantum numbers than in the conventional view. This last result clashes with the conventional wisdom that an atom's spectrum can change with cosmological time only through evolution of the alpha parameter in the energy eigenvalue formula, and thus impacts on the Webb group's analysis of fine structure intervals in quasar absorption lines (which has been claimed to disclose cosmological alpha evolution). In particular, analyzing together a mix of quasar absorption lines from different fine structure multiplets can bias estimates of cosmological alpha variability.

Motivation & Objective

• To investigate how dynamical variation of the fine-structure constant α affects atomic energy levels and spectral line shifts in the laboratory and cosmological settings.
• To challenge the conventional assumption in quasar spectroscopy that α variation alters atomic spectra only by scaling the laboratory α value in energy formulas.
• To assess the impact of modified Dirac Hamiltonian dynamics on the interpretation of fine-structure multiplets in quasar absorption lines.
• To determine whether current multi-multiplet analyses of quasar spectra are biased by assuming a conventional α-dependent spectral shift formula.
• To provide a corrected spectral shift formula that accounts for the cosmological 'fossil' value of α and its dynamical evolution.

Proposed method

• Derives the modified Dirac Hamiltonian in a scalar-tensor theory where α is a dynamical field governed by a scalar field ψ, leading to non-standard energy level splittings.
• Computes the fine-structure shifts in hydrogen using perturbation theory, showing a small deviation from Sommerfeld’s formula due to the presence of α* — the value of α when vacuum permittivity was unity.
• Derives a new formula (Eq. 33) for the wavelength shift of a quasar absorption line relative to its laboratory counterpart, incorporating the dynamical α dependence via α*.
• Compares the new formula (33) with the conventional formula (34) used in the Webb group’s analysis, highlighting differences in the coefficient of the n⁻⁴ term.
• Analyzes the discrepancy between the two formulas across various transitions, identifying cases where the conventional formula underestimates or even inverts the sign of the shift.
• Demonstrates that while single-multiplet comparisons are insensitive to the difference between dynamical and phase-transition α variability, multi-multiplet analyses are highly sensitive to the correct shift formula.

Experimental results

Research questions

• RQ1 How does a dynamical fine-structure constant α, governed by a scalar field ψ, alter the energy level structure of the hydrogen atom compared to the standard Sommerfeld formula?
• RQ2 What is the correct expression for the cosmological redshifted wavelength shift of a spectral line when α varies dynamically, and how does it differ from the conventional formula?
• RQ3 Why does combining data from different fine-structure multiplets in quasar spectra introduce bias in the estimation of α variability if the correct dynamical α model is not used?
• RQ4 In what ways do the predictions of the new spectral shift formula (33) differ quantitatively from the conventional formula (34), particularly for transitions with different n and j quantum numbers?
• RQ5 Can the discrepancy between the two formulas be used to discriminate between dynamical α variability and phase-transition-based α variability in cosmological data?

Key findings

• The laboratory spectrum of hydrogen in a dynamical α framework deviates slightly from Sommerfeld’s formula due to the presence of the cosmological 'fossil' α* value, potentially measurable with high-precision spectroscopy.
• The wavelength shift of a quasar absorption line due to cosmological α variation depends on quantum numbers in a way that differs from the conventional formula, particularly in the coefficient of the n⁻⁴ term.
• For the 2p₃/₂ → 4s₁/₂ transition, the conventional formula underestimates the shift by a factor of 6, and for 2p₃/₂ → 3p₁/₂, it overestimates by a factor of 31.5 and incorrectly predicts the sign.
• The new formula (33) shows that the shift depends on α* and α_l in a non-trivial way, making it essential to use this form when analyzing multi-multiplet quasar data.
• While single-multiplet comparisons cannot distinguish between dynamical and phase-transition α variability, multi-multiplet analyses are sensitive to the correct shift formula and can thus discriminate between the two paradigms.
• A reliable estimate of cosmological α variability from quasar spectra requires using the corrected formula (33), as the conventional approach (34) can yield significantly biased results.