Stochastic Particle Acceleration during Pressure-Anisotropy-Driven Magnetogenesis in the Pre-Structure Universe

Abstract

We investigate whether stochastic acceleration associated with pressure-anisotropy-driven magnetogenesis can generate a dynamically significant population of cosmic rays (CRs) prior to nonlinear structure formation. As magnetic fields amplify in the early Universe, the associated increase in gyrofrequency enhances pitch-angle scattering, potentially shortening the stochastic acceleration time. We derive an analytic criterion for efficient cosmological acceleration by comparing the acceleration timescale with the Hubble time, which defines a critical magnetic field and a corresponding CR turn-on redshift $z_{\rm on}$. For representative parameters, we find $z_{\rm on}\sim1.7$. To quantify the resulting particle population, we solve a Fokker-Planck equation for the isotropic proton distribution in the redshift interval $z=10\rightarrow z_{\rm on}$. Throughout most of this epoch, adiabatic expansion dominates over stochastic energization and the distribution remains close to a cooling Maxwellian. However, as the system approaches the turn-on epoch, the stochastic acceleration time decreases, allowing a mild suprathermal tail to develop. Even under optimistic assumptions corresponding to the strong-scattering limit, the maximum attainable proton energy reaches at most $\mathcal{O}(10^2)\,\mathrm{GeV}$. These results indicate that efficient CR production in the intergalactic medium is intrinsically tied to the onset of structure-formation shocks, while earlier microinstability-driven stochastic processes can provide at most a modest pre-acceleration.

Figures (5)

• Figure 1: Redshift evolution of the magnetic field $B(z)$ (black) compared with the analytic critical field $B_{\rm crit}(z)$ (red). Their intersection defines the CR turn-on redshift $z_{\rm on}$. The curves are shown for fiducial parameters $B_0=10^{-18}$ G, $\delta\Delta/\Delta_0=0.1$, $\Delta_0\sim2\beta^{-1}=10^{-2}$, and $\epsilon=0.2$.
• Figure 2: Dependence of the CR turn-on redshift $z_{\rm on}$ on (a) the turbulence amplitude $\epsilon$ and (b) the pressure-anisotropy perturbation $\delta\Delta$. All other parameters are fixed to the fiducial values used in Fig. \ref{['fig:f1']}. Over the explored parameter range, $z_{\rm on}$ varies only weakly, remaining close to $z_{\rm on}\sim1.7$.
• Figure 3: Redshift evolution of the maximum attainable proton energy $E_{\max}(z)$ computed from Eq. (\ref{['eq:Emax_eps']}) for turbulence amplitudes $\epsilon=0.05$ (black), $0.1$ (red), and $0.2$ (blue). Near the turn-on epoch $z_{\rm on}\sim1.7$, $E_{\max}$ reaches $\sim10^2\,\mathrm{GeV}$.
• Figure 4: Proton spectra $p^{4}f(p,z)$ from the Fokker-Planck model for $z=10,5, 2$ and $1.7$. Panel (a): $\epsilon=0.05$; panel (b): $\epsilon=0.2$. The initial distribution at $z=10$ is Maxwellian with $k_B T_0=0.86~{\rm eV}$.
• Figure 5: Same as Fig. \ref{['fig:f4']}, but including a weak pre-existing suprathermal power-law tail ($s=4.2$) in the initial distribution.