Cosmic-Ray Electron Excess from Pulsars is Spiky or Smooth?: Continuous and Multiple Electron/Positron injections

We investigate the observed spectrum of cosmic-ray electrons and positrons from astrophysical sources, especially pulsars, and the physical processes for making the spectrum spiky or smooth via continuous and multiple electron/positron injections. We find that (1) the average electron spectrum predicted from nearby pulsars are consistent with PAMELA, Fermi and H.E.S.S. data. However, the ATIC/PPB-BETS peak around 500GeV is hard to produce by the sum of multiple pulsar contributions and requires a single (or a few) energetic pulsar(s). (2) A continuous injection produces a broad peak and a high energy tail above the peak, which can constrain the source duration ($\lesssim 10^5$yr with the current data). (3) The H.E.S.S. data in the TeV range suggest that young sources with age less than $\sim 6 \times 10^4$yr are less energetic than $\sim 10^{48}{\rm erg}$. (4) We also expect a large dispersion in the TeV spectrum due to the small number of sources, that may cause the high energy cutoff inferred by H.E.S.S. and potentially provide a smoking-gun for the astrophysical origin. These spectral diagnostics can be refined in the near future by the CALET experiments to discriminate different astrophysical and dark matter origins.