Testing Nambu-Goto approximation of cosmic string by lattice field simulations

Problem Statement
The precise calculation of gravitational waves (GWs) from cosmic string networks is critical for both theoretical understanding and experimental detection by facilities such as pulsar timing arrays (PTAs), LIGO-Virgo, and LISA. Currently, most predictions rely on the Nambu-Goto (NG) approximation, which models cosmic strings as infinitely thin objects. While this approximation is effective when the string width is negligible, its validity has never been systematically verified against first-principles field theory simulations. Specifically, it remains unclear whether the NG approximation holds for strongly coupled local strings where the string core scale may interact with dominant GW-emitting modes, and how particle emission competes with GW emission in energy loss.

Methodology
The authors perform large-scale, zero-temperature Abelian-Higgs lattice simulations to model cosmic string networks under varying gauge couplings ($e$). The study utilizes the Lagrangian of the Abelian-Higgs model, where the ratio of vector to scalar masses, $\beta = m_v^2/m_s^2 = 2e^2/\lambda$, determines the string properties.

• Simulation Setup: The simulations assume a radiation-dominated FLRW background using conformal time. The authors employ a leap-frog algorithm to solve the equations of motion for the scalar and gauge fields, ensuring the Gauss constraint is satisfied.
• Parameters: Three gauge coupling scenarios are tested: a near-global case ($e=0.0005$), a moderate local case ($e=0.05$), and a strongly coupled local case ($e=\sqrt{\lambda}/2 \approx 0.316$), where $m_v \sim m_s$.
• Comparison Framework: The GW spectra obtained directly from lattice simulations are compared against spectra reconstructed using the NG approximation. The NG model assumes GW emission is dominated by specific loop features (cusps, kinks, or kink-kink collisions).
• Metrics: The authors define a difference function, $f_{diff}$, to quantify the deviation between lattice and NG spectra specifically within the power-law region ($10^{-1.3} < k/m_s < 1$), avoiding infrared and ultraviolet artifacts. They also extract a universal transfer function, $F(x, y)$, from the lattice data to extrapolate spectra over cosmological times.
• Energy Analysis: The study calculates the ratio of GW energy density ($\rho_{gw}$) to particle energy density ($\rho_p$) to determine the dominant energy loss channel.

Key Contributions and Results

1. Validity of the NG Approximation:

   • Near-Global and Moderate Local Strings: For the near-global case ($e=0.0005$) and the moderate local case ($e=0.05$), the lattice results show excellent agreement with the NG approximation in the power-law region. The deviation ($f_{diff}$) is less than 0.1. The dominant emission mechanism appears to depend on the coupling: kink-kink collisions favor the near-global regime, while cusps are more significant for $e=0.05$.
   • Strongly Coupled Local Strings: For the case where $m_v/m_s \sim 1$ ($e=\sqrt{\lambda}/2$), the NG approximation breaks down significantly. The $f_{diff}$ values rise substantially (exceeding 0.14), indicating that the approximation fails when the dominant GW-emitting modes probe the microscopic string core. In this regime, finite-width effects and gauge-field degrees of freedom cannot be neglected.

2. Energy Loss Channels:

   • The simulations confirm that particle emission significantly dominates the energy loss of the string network. The ratio of GW energy to particle energy is approximately $10^{-3}$ to $10^{-2}$ for both near-global and local string scenarios.
   • In the near-global case, light field modes are efficiently excited, leading to a higher particle number density compared to the local case, where the larger gauge-boson mass suppresses low-momentum modes.

3. Spectral Characteristics and Extrapolation:

   • As the gauge coupling increases, the GW spectrum shifts toward higher momenta. For strongly coupled local strings, the spectral peak approaches the string core scale ($k/m_s \sim O(1)$).
   • The authors extract a universal transfer function from the lattice data, allowing for the extrapolation of GW spectra to broader frequency ranges beyond the immediate dynamical limits of the simulation.
   • For a symmetry breaking scale of $f_a = 2 \times 10^{15}$ GeV, the resulting GW spectra are potentially detectable by current PTA experiments and future space-based detectors (LISA, TianQin, Taiji).

Significance
The paper quantitatively establishes the regime of validity for the Nambu-Goto approximation in cosmic string GW predictions. It demonstrates that the approximation is reliable only when the dominant GW-emitting structures are well-separated from the string width scale. The breakdown of the NG model for strongly coupled local strings ($m_v/m_s \sim 1$) highlights the necessity of field-theoretic simulations for accurate predictions in these regimes. Furthermore, by confirming that particle radiation is the primary energy loss mechanism (dominating GW emission by two orders of magnitude), the study provides a crucial numerical benchmark for future observational and theoretical investigations into cosmic string networks and their associated dark matter candidates (such as axion-like particles and dark photons). The extracted transfer function offers a bridge for analytical extrapolations over cosmological times, refining the connection between particle model parameters and GW detectability.