Search for Quintessence-Like Pseudoscalar Dark Energy Effects on $^{56}\text{Fe}$ Nuclear Transition Energies in Supernova 1991T

Technical Summary: Quintessence-Like Pseudoscalar Dark Energy Effects on $^{56}$Fe Nuclear Transition Energies in Supernova 1991T

Problem Statement
The nature of dark energy remains a primary unsolved problem in physics. While the cosmological constant ($\Lambda$) within the $\Lambda$CDM model fits observations well, it faces theoretical challenges (e.g., the cosmological constant problem, coincidence problem) and observational anomalies (e.g., Hubble tension, cosmic birefringence). Dynamical dark energy models, specifically those involving a pseudoscalar field (quintessence), have been proposed to address these issues. If such a field exists and couples to the hadronic sector, it could induce variations in fundamental constants, specifically the pion mass ($m_\pi$), which in turn affects nuclear charge radii and energy levels. While local terrestrial experiments (e.g., Oklo natural reactor, atomic clocks) constrain the temporal evolution of these fields, they cannot probe spatial variations. This paper aims to constrain the evolution of a dynamical pseudoscalar quintessence-like field, $Q(t)$, by comparing astrophysical gamma-ray energies from a distant source with terrestrial values, thereby probing both temporal and spatial variations.

Methodology
The authors utilize the peculiar Type Ia supernova SN 1991T, located 44.0 million light-years away, as an astrophysical probe. The study focuses on the gamma-ray de-excitation spectra of $^{56}$Fe nuclei, produced by the decay chain $^{56}\text{Ni} \to {}^{56}\text{Co} \to {}^{56}\text{Fe}$, observed by the COMPTEL instrument aboard the Compton Gamma Ray Observatory.

1. Theoretical Framework:

   • The dark energy field $Q(t)$ is modeled as a light pseudoscalar (axion-like particle) with an inverse power-law tracking potential (Ratra-Peebles potential).
   • The field couples to the QCD vacuum angle, inducing a variation in the pion mass: $\delta m_\pi / m_\pi \propto Q(t)$.
   • Variations in $m_\pi$ alter the nuclear charge radius ($r_0$), which shifts nuclear energy levels. The authors derive the relationship between the fractional energy shift of nuclear states ($\delta E/E$) and the time derivative of the field ($\dot{Q}_0$).
   • Two nuclear models are employed to estimate systematic uncertainties: a Liquid Drop Model (treating excited states as surface phonons) and a Rigid Rotor Model (treating the nucleus as a deformed rotor). The Liquid Drop model is selected for final calculations as the results are relatively insensitive to the model choice (ratio $\approx 0.88$).

2. Data Analysis:

   • The authors re-analyzed published COMPTEL data of SN 1991T, focusing on the $^{56}$Fe gamma-ray lines at 847 keV and 1238 keV.
   • Data was fitted using Gaussian line shapes convolved with the detector's energy resolution function.
   • Corrections were applied for the Doppler redshift of the host galaxy (NGC 4527, $z \approx 0.005791$).
   • Systematic errors, including potential background shifts from neutron capture on aluminum ($^{27}\text{Mg}$) and calibration artifacts, were evaluated. A conservative systematic shift of 6 keV was added in quadrature to statistical errors.

3. Constraint Derivation:

   • The measured fractional energy shift ($\delta E/E$) was used to calculate the fractional rate of change of the pion mass ($\dot{m}_\pi/m_\pi$).
   • Using the derived relationship between pion mass variation and the field evolution, limits were placed on $\dot{Q}_0$.
   • These limits were translated into constraints on the kinetic energy density fraction ($\Omega_{KE}$) and the equation of state parameter ($w_Q$) for the tracking model.

Key Results

• Energy Shift: The analysis yielded an average fractional energy shift for the first and second excited states of $^{56}$Fe of $\delta E/E = -0.006 \pm 0.008$ (including statistical and systematic errors). This result is consistent with zero, indicating that the astrophysical gamma-ray energies are consistent with terrestrial laboratory values within the current precision.
• Pion Mass Variation: Assuming the shift is caused by the dynamical field, the observed deviation implies a limit on the fractional rate of change of the pion mass at the 68% Gaussian limit:
  $$3 \times 10^{-11} \ge \frac{\dot{m}_\pi}{m_\pi} \ge -15 \times 10^{-11} \, \text{yr}^{-1}$$
  This limit is five orders of magnitude larger than atomic clock limits and seven orders of magnitude larger than Oklo reactor limits, reflecting the different nature of the probe (non-local vs. local).
• Field Evolution ($\dot{Q}_0$): For tracking models, the upper limit on the rate of change of the quintessence-like field is determined to be:
  $$-1 \times 10^7 \le \dot{Q}_0 \le 7 \times 10^7 \, \text{GeV/yr}$$
  (The upper bound is derived assuming specific parameters: decay constant coefficient $\alpha = 0.1$ and potential power $p=1$).
• Cosmological Parameters: The derived upper limit on $\dot{Q}_0$ results in a fractional kinetic energy density $\Omega_{KE} \le 0.029$ and an equation of state $-1.0 \le w_Q \le -0.92$. These values are consistent with the cosmological constant limit ($\dot{Q}_0 = 0, \Omega_{KE} = 0, w = -1.0$).

Significance and Claims
The paper claims to have developed a novel framework for relating deviations in astrophysical gamma-ray spectra to a pseudoscalar quintessence field. By utilizing SN 1991T, the study provides a unique constraint that incorporates spatial variation of the dark energy field, a capability lacking in local terrestrial experiments like the Oklo reactor or atomic clocks which only probe temporal evolution.

The authors emphasize that while the results are consistent with the standard cosmological constant model (implying no significant deviation was detected), the methodology successfully establishes a new class of constraints on pseudoscalar dark energy models. The derived limits on $\dot{Q}_0$ and the associated equation of state parameters help narrow the viable parameter space for tracking models, specifically constraining the decay constant coefficient and the tracking power potential. The work demonstrates that extragalactic gamma-ray spectroscopy can serve as a viable probe for testing the spatial homogeneity of dark energy and the stability of fundamental constants over cosmic distances.