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rules for photon polarization modes, one-photon pair pro- duction into low Landau levels for the e $^{±}$, and resonant in- verse Compton scattering from polar cap hot spots [with T = (0 . 3–3) × 10 6 K]. Both cascades initiated by a single electron (representing the entire cascade process described in Section 1) and by a single photon (representing one branch of the cascade) are simulated, for a variety of initial ener- gies of the primary particle ( γ$_{0}$m$_{e}$c 2 and ϵ$_{0}$ ). We have made an effort to present our numerical results systematically, in- cluding empirical relations for the pair multiplicity (i.e., the number of electrons and positrons produced per primary particle). 

Locally the cascade behaves very differently above and below the critical QED field strength B$_{Q}$ ≡ 4 . 414 × 10 13 G (e.g., pair production is ∼ 10 times more efficient for B > B$_{Q}$ but synchrotron emission is highly suppressed; see Sec- tion 4). Globally, however, the cascade followed from the surface to the light cylinder behaves similarly regardless of surface field strength B$_{p}$ . For example, we find that the to- tal number of pairs produced in electron-initiated cascades, as well as their energy spectrum, depends on the polar cap voltage B$_{p}$P − 2 but is weakly dependent on B$_{p}$ alone. Addi- tionally, the total photon spectra for B > B$_{Q}$ and B < B$_{Q}$ have similar shapes over the energy range we consider in our simulation (10 keV-1 TeV); this is because curvature radia- tion, which dominates the photon spectrum over most of this energy range, is independent of magnetic field strength. Pho- ton splitting, a process that is only active for B > ∼ B$_{Q}$ , could potentially distinguish between neutron stars with high and low surface fields, by lowering the multiplicity of electrons and positrons N$_{E}$ produced in the cascade to such a de- gree that the radio emission mechanism can not function (Baring & Harding 2001). However, we find that even if both photons polarized parallel to and perpendicular to the mag- netic field are allowed to split, photon splitting lowers N$_{E}$ by at most 50% for B$_{p,}$$_{12}$ = 1000 and by < 10% for B$_{p,}$$_{12}$ ≤ 100. With the correct selection rule ( ⊥→‖ ‖ ), the effect of photon splitting is even smaller. 

Our results show that a strongly non-dipole magnetic field with radius of curvature ∼ 10 6 cm near the stellar sur- face ( r less than a few stellar radii) can account for pulsars which lie below the standard death line for dipole fields (see Section 2). Whether or not such a strongly-curved geome- try could form is another question altogether. It is doubtful that cascades initiated by resonant inverse Compton scat- tering in the open field region of the magnetosphere can ac- count for the “missing” pulsars, as the multiplicity of pairs produced in such cascades ( < ∼ 1 per primary electron) are too low for the current models of the pulsar radio emission. High-multiplicity cascades due to resonant ICS may still oc- cur along twisted lines in the closed field region (where the primary electrons never reach Lorentz factors larger than γ$_{0}$ ∼ 10 $^{3}$), as they do in magnetars (e.g., Thompson et al. 2002; Beloborodov & Thompson 2007; Thompson 2008a,b; Beloborodov 2009). 

At any altitude in the magnetosphere, the photon spec- trum at low energies ( < ∼ 1 MeV) has a power-law shape with spectral index Γ = 2 / 3 (where dN/dϵ ∝ ϵ − $^{Γ}$); this is true regardless of whether synchrotron radiation or cur- vature radiation dominates the spectrum, as both pro- cesses have the same low-energy shape. The photon spec- trum dN/dϵ ∝ ϵ − 2 / 3 is harder than the hard X-ray spec- tra observed in several pulsars, which typically have pho- ton indices Γ ∼ 1–2 (see, e.g., Kuiper & Hermsen 2009; Bogdanov & Grindlay 2009). One way to reconcile our re- sults with observations is to include an additional radiative process, cyclotron resonant absorption, in the simulation. Such an approach is taken by Harding et al. (2005) (see also Harding et al. 2008), who find that in addition to the syn- chrotron radiation emitted immediately upon creation, the e ± particles emit an additional component of synchrotron radiation at large altitudes that dominates the low-energy spectrum. The efficiency of this high-altitude synchrotron emission is due to the large pitch angles of the e $^{±}$, which they obtain through resonant absorption of radio photons that are beamed from some intermediate height in the magneto- 

