Isochrones in primordial magnetic field evolution

Technical Summary: Isochrones in Primordial Magnetic Field Evolution

Problem Statement
The evolution of primordial magnetic fields (PMFs) during the radiation-dominated era of the Universe is governed by turbulent decay and an inverse cascade, which increases the field's characteristic length scale while decreasing its strength. A central diagnostic tool in this field is the relationship between the Alfvén speed ($v_A$) and the magnetic integral scale ($\xi_M$). Theoretical models suggest that at sufficiently late times, PMF evolution follows "isochrones" defined by the relation $v_A = \xi_M/t^*$, where $t^*$ is the time elapsed since generation.

However, a discrepancy exists between this theoretical expectation and numerical simulations. Different magnetogenesis mechanisms produce varying initial length scales and field strengths, leading to initial conditions that do not lie on the expected asymptotic isochrones at the nominal simulation start time ($t=0$). Furthermore, the actual decay time of magnetohydrodynamic (MHD) turbulence is known to exceed the nominal Alfvén time by a factor dependent on the Lundquist number ($Lu$). The paper addresses the challenge of reconciling diverse initial conditions with universal isochrones and determining the "proper time" required to align early-time simulation data with late-time theoretical predictions.

Methodology
The authors employ two-dimensional numerical simulations of decaying MHD turbulence using the Pencil Code. The study focuses on the 2D case, where the conserved quantity is anastrophy ($I_A = \langle A_z^2 \rangle$), rather than magnetic helicity (3D helical) or the Hosking integral (3D non-helical).

• Initial Conditions: Simulations vary the initial position of the magnetic energy spectrum peak ($k_B$) and the initial field strength ($B_0$). Three primary initial configurations (Points A, B, and C) are tested to explore different combinations of initial $v_A$ and $\xi_M$.
• Velocity Fields: Most runs start with zero initial velocity ($u_0=0$). A subset of runs introduces a finite initial velocity field ($u_0 \neq 0$) with a relative amplitude $u_0/B_0 \approx 0.3$ to test its effect on the evolutionary track.
• Spectral Shapes: The study examines the impact of the initial spectral slope, comparing standard inertial range slopes ($\alpha_{ine} = -2$) against cases with very short or absent inertial ranges ($\alpha_{ine} = -20$).
• Resolution and Parameters: Simulations are performed with resolutions ranging from $1024^2$ to $8192^2$ mesh points, covering Lundquist numbers from $Lu \approx 70$ to $Lu \approx 1100$. The magnetic Prandtl number is fixed at $Pr_M = 1$.

Key Contributions and Results

1. Definition of Proper Time and Isochrones:
   The paper demonstrates that while initial conditions can be arbitrary, they can be mapped onto universal isochrones by resetting the nominal time axis. The authors define a "proper time" ($t_{proper}$) for each simulation. This is achieved by fitting the instantaneous Alfvén time ($t_A = \xi_M/v_A$) to the nominal time $t$ using a linear relation $t_{fit}^A = mt + t_0$.

   • The slope $m$ yields the decay factor $C_M = 1/m$, which quantifies how much longer the decay time is compared to the Alfvén time. The study finds $C_M$ ranges from roughly 10 to 24, increasing with $Lu$ and saturating near 50 for very high $Lu$.
   • The intercept $t_0$ (scaled by $C_M$) provides an offset time ($t_{offset}$) that, when added to the nominal time, aligns the simulation's trajectory with the theoretical isochrone.

2. Universal Tracks:
   By applying these time corrections, the authors show that evolutionary tracks from simulations with vastly different initial $v_A$ and $\xi_M$ values collapse onto parallel isochrones. This confirms that the magnetic field parameters lie on universal tracks even at early times, provided the time axis is properly shifted.

3. Impact of Initial Velocity:
   The presence of an initial velocity field of realistic strength ($u_0/B_0 \approx 0.3$) is found to produce a "straighter" evolutionary track in the $v_A$ vs. $\xi_M$ diagram. In contrast, runs starting with zero velocity exhibit an initial adjustment phase where the velocity field builds up, causing a slight upward bend in the track before settling into the asymptotic decay behavior.

4. Anastrophy Conservation:
   The study quantifies the decay of anastrophy due to finite resistivity. The decay exponent $s$ (where $I_A \propto t^{-s}$) is found to decrease with increasing Lundquist number, following approximately $s \approx 3.1 Lu^{-2/3}$. For high $Lu$, anastrophy is nearly conserved over cosmological timescales.

5. Spectral Evolution:
   The simulations confirm that the energy spectrum shifts to lower wavenumbers (inverse cascade) in a shape-invariant manner. The study also notes that even with a mismatched initial spectral slope (e.g., lacking an inertial range), the system quickly develops the expected inertial range, and the $(\xi_M, v_A)$ point eventually aligns with the track of a standard run, albeit with a slightly different conserved anastrophy value.

Significance and Claims
The paper claims to resolve the apparent conflict between diverse initial magnetogenesis scenarios and the concept of universal isochrones. The authors argue that the restriction on permissible initial conditions is not physical but rather a consequence of an arbitrary time definition. By determining the proper time offset and the decay factor $C_M$, any initial field configuration can be mapped to the universal evolutionary track.

The work emphasizes that comparing different magnetogenesis mechanisms requires resetting the nominal time such that the initial value of $v_A t$ agrees with the initial scale $\xi_M$. The authors conclude that while artifacts related to finite resolution and the initial adjustment phase (in zero-velocity runs) exist, the underlying physics of decaying MHD turbulence leads to predictable, universal isochrones that are independent of the specific generation mechanism, provided the time axis is correctly calibrated. The study does not claim to definitively identify the origin of extragalactic magnetic fields but provides the necessary theoretical framework to constrain such origins against observational data.