tokens: perpendicular to the mean magnetic field. The spectral hardness does not arise because the particle energy spectrum itself is hard, i.e., with , but rather because the distribution is anisotropic. Within the nonthermal tail of accelerated particles, lower energy electrons tend to be more aligned with the local magnetic field than higher energy electrons, so their synchrotron emission is comparatively weaker. This energy-dependent anisotropy has the effect of hardening the synchrotron spectrum. 3 3 We expect similar conclusions to hold for electrons accelerated by reconnection with a strong guide field, even in a nonturbulent environment. Indeed, in our simulations of magnetically dominated turbulence, the acceleration of particles attaining the strongest anisotropy is primarily due to reconnection (Comisso Sironi 2019 ). In laminar reconnection, the guide field would play the role of the mean field of our turbulent setup. Our results have important implications for the interpretation of the hard radio spectra of pulsar wind nebulae (PWNe), where the observed radio spectral index is typically (Gaensler Slane 2006 ; Reynolds et al. 2017 ). In the notable case of the Crab Nebula, the radio spectral index is (Hester 2008 ; Bühler Blandford 2014 ). Turbulence, which develops as an effect of MHD instabilities in PWNe (Porth et al. 2014 ), has been suggested to accelerate the radio-emitting electrons (Lyutikov et al. 2019 ; Xu et al. 2019 ). However, these studies simply invoked a hard particle spectrum () over some energy range, without addressing from first principles the origin of the accelerated particles. We have shown that producing hard radio spectra does not necessarily require an acceleration mechanism that transfers most of the available energy to a small fraction of particles (which is the case when ). Instead, hard synchrotron spectra may be produced by a relatively soft population of nonthermal particles with an energy-dependent anisotropy. Finally, note that our assumption of neglecting the effect of synchrotron cooling on the particle evolution is well justified in PWNe, where the cooling time of radio-emitting electrons is longer than the age of the system. For the Crab Nebula, it is generally quoted that the hard spectral slope persists over six orders of magnitude in frequency (for an extended discussion, see, e.g., Bietenholz et al. 1997 ). Since the hard range of our synchrotron spectra extends over four decades, contributions from turbulent regions with moderately different properties would be required. While a detailed comparison to the Crab Nebula radio emission is left for future work, we mention that in our model we would imply that (i) the total number of radio-emitting electrons is at least one order of magnitude higher than in the case with isotropic electrons. More precisely, the number of radio electrons increases by a factor of (ii) the initial magnetization of the plasma is relatively low and/or the predissipation
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