Spectrum of radiation from global strings and the relic axion density

Technical Summary: Spectrum of Radiation from Global Strings and the Relic Axion Density

Problem Statement
The production of axions via topological defects (global cosmic strings) is a primary candidate for explaining the observed Cold Dark Matter (CDM) density. However, the predicted axion mass ($m_a$) and decay constant ($f_a$) remain highly uncertain due to conflicting theoretical scenarios regarding the spectrum of radiation emitted by these strings.

• Scenario A: Based on the Kalb-Ramond (KR) action treating strings as Nambu-like objects coupled to an antisymmetric tensor field, this scenario predicts a "soft" spectrum where radiation peaks near the fundamental mode of string perturbations ($q > 1$).
• Scenario B: Supported by various numerical field theory simulations, this scenario suggests a "hard" spectrum with a logarithmic divergence ($q \approx 1$), implying significant emission at high frequencies up to the string core width.

The discrepancy between these scenarios leads to vastly different predictions for the relic axion density. Furthermore, existing numerical simulations often lack the dynamic range ($\log(\Delta/\delta) \approx 70$) required to fully resolve the string core versus the perturbation scale, and they may inadvertently measure the string's self-field rather than the emitted axion radiation.

Methodology
The authors employ a two-pronged approach combining analytical modeling and numerical simulation:

1. Analytical Framework:

   • The authors derive expressions for the relic axion density ($\Omega_a$) considering two production channels: axions emitted directly by the long string network and axions emitted by the decay of string loops.
   • They introduce a spectral function $g(z)$ to model the radiation spectrum, parameterized by a power-law index ($q$ or $p$) and a cutoff ($n_*$ or $m_*$).
   • They define a spectral factor $G_2(q, n_*)$ which encapsulates the dependence of the relic density on the radiation spectrum. This factor is calculated for both power-law and exponential fall-off models.
   • The model incorporates the "chopping efficiency" ($c$) and the fraction of energy ($F_\ell$) emitted into loops versus directly into axions, acknowledging the uncertainty in these parameters between Nambu-string and field-theory simulations.

2. Numerical Simulations of Perturbed Straight Strings:

   • The authors perform high-resolution numerical simulations of the $U(1)$ Goldstone model using a grid with up to $801^3$ points to resolve the string core and radiation.
   • Field Decomposition: A critical methodological innovation is the explicit separation of the total field phase ($\alpha$) into the string self-field ($\alpha_{str}$) and the radiated axion field ($\Delta\alpha$). The authors demonstrate that standard simulation outputs often conflate these two, dominated by the self-field near the string core.
   • Subtraction Techniques: They compare two methods for isolating the axion radiation:
     1. Direct subtraction of a static string ansatz ($\alpha \approx \theta + \Delta\alpha$).
     2. Calculation of the momentum flux projection ($P_\alpha$) of the energy-momentum tensor.
   • Masking: To ensure the measured spectrum represents free axions and not the string's self-field, the authors apply a cylindrical mask (excising a region $r < r_{crit} \approx 10\Delta x$) around the string before performing spectral analysis.

Key Results

1. Spectral Characterization:

   • The simulations reveal that without masking, the measured spectrum is dominated by the string's self-field, which exhibits a power-law fall-off of $k^{-2.25}$.
   • After applying the self-field subtraction and masking, the spectrum of the emitted axion radiation ($\Delta\alpha$) is found to be compatible with an exponential fall-off, consistent with the predictions of the Nambu-like Kalb-Ramond action.
   • The results explicitly contradict the "hard" spectrum ($q \approx 1$) often reported in unmasked network simulations. The authors argue that previous "hard" spectra were artifacts of measuring the self-field rather than the radiated axions.

2. Dependence on Spectral Parameters:

   • The relic density is shown to be highly sensitive to the spectral index $q$ and the cutoff $n_*$, particularly in the range $1 < q < 2$.
   • For $q \le 1$, the density is dominated by high-frequency modes (core width), whereas for $q > 1$, it is dominated by low-frequency modes (perturbation wavelength).

3. Revised Axion Mass Predictions:

   • The authors quantify the uncertainties in the relic density calculation. They find that the predicted axion mass range is broader than that derived from the Initial Misalignment Mechanism (IMM).
   • Scenario 1 (Soft Spectrum): If the primary decay mode is via a soft spectrum from loops produced near the backreaction scale, the authors estimate $m_a \approx 160 \, \mu\text{eV}$ (frequency $f \approx 38 \, \text{GHz}$).
   • Scenario 2 (Hard Spectrum from Simulations): If axions are emitted directly by the network using spectra from typical unmasked field theory simulations, $m_a \approx 4 \, \mu\text{eV}$ ($f \approx 1 \, \text{GHz}$).
   • Scenario 3 (Oscillating String with Corrected Spectrum): Using the authors' corrected spectra for an oscillating string (exponential/soft), the estimate shifts to $m_a \approx 125 \, \mu\text{eV}$ ($f \approx 30 \, \text{GHz}$).

Significance and Claims
The paper claims that the uncertainty in the relic axion density is largely driven by the inability of current numerical simulations to distinguish between the string's self-field and the emitted axion radiation.

• Correction of Methodology: The authors assert that a "significant correction" (excising a sizeable region around the string) is required to analyze axions correctly. They argue that standard field theory simulations often fail to do this, leading to the erroneous conclusion of a hard spectrum.
• Resolution of Discrepancy: By demonstrating that the radiation spectrum is exponential (soft) rather than hard, the paper suggests that the tension between the Kalb-Ramond action predictions and field theory simulations may be resolved by better data analysis techniques rather than a fundamental failure of the Nambu-like approximation.
• Impact on Detection: The findings imply that if the soft spectrum is correct, the axion mass is likely higher ($\sim 100\text{--}160 \, \mu\text{eV}$) than the values derived from uncorrected simulations, shifting the target frequency for axion detection experiments to the $\sim 30\text{--}40 \, \text{GHz}$ range.

The authors conclude with modesty, acknowledging that significant uncertainties remain regarding the fraction of energy emitted into loops ($F_\ell$) and the exact parameters of the radiation spectrum, but emphasize that the spectral shape is the dominant factor in determining the viable axion mass window.