profiles (e.g., the widely separated double peaks) of the six pulsars detected by EGRET already suggested that models of high-altitude gamma-ray emission were required. More recently, observations of the Crab pulsar by the Ma- jor Atmospheric Gamma-ray Imaging Cherenkov Telescope (MAGIC) and around 50 pulsars with “above average” spin- down powers by the Fermi Gamma-ray Space Telescope have revealed that the high-energy tails of the photon spec- tra fall off exponentially or more slowly than exponential (Aliu et al. 2008; Abdo et al. 2009a,b,c, 2010), while inner gap models predict that the tails fall off faster than ex- ponential (see, e.g., Fig. 12). The outer gap model and the high-altitude version of the slot gap model have been successful in explaining the γ -ray pulsar light curves (e.g., Watters et al. 2009; Venter, Harding, & Guillemot 2009; but see Bai & Spitkovsky 2009). sphere. The high-altitude synchrotron emission will have a hard X-ray part with photon index Γ$_{10}$$_{−}$$_{1000 keV}$ generally different from 2 / 3, because the hard X-ray band lies near the peak of the emission (rather than in the low-energy tail as is the case for the low-altitude emission). Harding et al. (2005) find that for typical millisecond parameters the pho- ton index is in the observed range, 1 < Γ$_{10}$$_{−}$$_{1000 keV}$ < 2. 24 Zach Medin and Dong Lai To fully incorporate the effects of cyclotron absorption into our simulation, we would need a model of the radio beam structure and spectrum, since the evolution in the pitch angle of each e ± particle depends on the angle at which radio photons of the resonant frequency are incident on the particle and their density. Empirical models of radio beams exist (e.g., Rankin 1993; Arzoumanian, Chernoff, & Cordes 2002; Kijak & Gil 2003); however, inclusion of such a model is beyond the scope of this paper. This means that we can not say anything in detail about the hard X-ray spectra of strongly-magnetized neutron stars, such as whether the high-altitude synchrotron component dominates and what its low-energy cutoff is (but see Harding et al. 2005 for ex- amples of these spectra at several different cascade param- eters). Nevertheless, we can use the e ± spectra from our simulation to estimate how the Γ$_{10}$$_{−}$$_{1000 keV}$ photon index varies as a function of the cascade parameters. For syn- chrotron emission from a distribution of e ± particles, the hardness of the e ± spectrum and the hardness of resulting photon spectrum are correlated [for a dN/dE ∝ E − p power- law distribution of e ± the photon spectrum has an index given by the familiar expression Γ = ( p + 1) / 2]. We find that the e ± spectrum is harder, and therefore Γ$_{10}$$_{−}$$_{1000 keV}$ is lower, for larger B$_{p}$ or γ$_{0}$ or shorter P (i.e., anything that increases the energy of the cascade); additionally, for two neutron stars with the same polar cap voltage B$_{p}$P − $^{2}$, Γ$_{10}$$_{−}$$_{1000 keV}$ is lower in the neutron star with the stronger surface field. Therefore, in pulsars where the cyclotron ab- sorption – synchrotron emission mechanism dominates the hard X-ray spectrum, the most-strongly magnetized pulsars will have the hardest spectra (lowest photon indices), all other parameters being equal. Although an outer gap or slot gap accelerator model is required to explain gamma-ray observations, our results based on the inner gap model still have general applica- bility. First, even when the acceleration region is located in the outer magnetosphere a significant fraction of the pair creation must occur in the inner magnetosphere (e.g., Cheng et al. 2000). Indeed, pulsar radio emission is strongly constrained to originate from well inside the light cylinder radius; the only way to generate highly coherent radio emis- sion is to have copious e ± plasma produced by vigorous pair cascades (e.g., Melrose 2004; Lyubarsky 2008). Second, both inner gap and outer gap accelerators may exist in a neutron star simultaneously, whether as one extended acceleration region (e.g., Hirotani 2006) or on different field lines. Ra- dio observations suggest that radio emission can come from intermediate field lines, neither along the pole of the star nor at the edge of the open field region (Gangadhara 2004); if this is true, it provides further support for an inner gap origin of the e ± plasma, since the plasma generated by an outer gap model is formed on field lines close to the last open field line. In addition, as discussed above, hard X-ray (10- 1000 keV) observations of both pulsars and magnetars sup- port a magnetosphere model where the hard X-ray emission is dominated by radiation from the e $^{+}$e − pairs generated by inner gap cascades. Third, it is unlikely that the death lines for the inner gap and outer gap mechanisms overlap completely. There should therefore be regions of the P - ˙ P di- agram where the outer gap mechanism is excluded but the inner gap mechanism still functions; these regions, if they exist, will be located near the inner gap death line, where pulsars have very low spindown powers. Although observa- tions show that the gamma-ray spectra is dominated by slot or outer gap emission for pulsars with moderate to large spindown power, they have not yet ruled out an inner gap origin for pulsars with low spindown power. Luminous hard X-ray (from 20 keV to several hundreds of keV) emission has also be detected from a number of Anomalous X-ray Pulsars by INTEGRAL and RXTE (e.g., Kuiper et al. 2006). Possible mechanisms for this emission were discussed by Thompson & Beloborodov (2005) and Be- loborodov & Thompson (2007). Since the observed hard X- ray luminosity exceeds the spin-down power by several or- ders of magnitude, it must be fed by an alternative source of energy such as the dissipation of a superstrong magnetic field. We note, however, that the observed photon indices, Γ ∼ 0 . 8 − 1 . 8, are similar to (but slightly harder than) those of radio pulsars. Synchrotron radiation by secondary e $^{+}$e − pairs produced in a cascade similar to those studied in this paper could plausibly explain the observations. In constructing our simulations, we have attempted to rely as little as possible on the precise nature of the ac- celeration region. However, there are several key assump- tions that we have made that are only valid for inner gap accelerators (e.g., that the cascade begins at the neu- tron star surface). It has become apparent that models where particle acceleration occurs only in the inner gap can not account the observed high-energy gamma ray emis- sion from young pulsars. For example, the gamma-ray pulse ZM has been supported in part by the Lorne Trottier Chair in Astrophysics and Cosmology and an NSERC Discovery Grant awarded to Andrew Cumming. This work forms part of the Ph.D. thesis of ZM at Cornell University. This work has been supported in part by NASA grant NNX07AG81G and NSF grants AST 0707628. ACKNOWLEDGMENTS