What are realistic and unrealistic values for the high energy gamma ray spectral indices of pulsar wind nebulae? This summer, I was working on a project fitting very high energy gamma ray spectra of the Crab Nebula, a pulsar wind nebula. At energies about $\sim$1 TeV, a simple power law suffices, i.e. $\phi(E)\propto E^{\alpha}$. Most power law models of the Crab find a spectral index of $\alpha\simeq-2.47$ (e.g. Hillas 1998, $\alpha=-2.49\pm0.06$). Using a normal analysis method on one dataset, I found $\alpha=-2.469\pm0.02248$, which matches the literature values quite well.

Using the modified analysis method I was testing, however, I found $\alpha=-2.316\pm0.02040$. We believe that method is a poor choice because I showed the calculated flux values were incompatible with previous observations. The spectral index also cast doubts on its viability. However, I was wondering if, for a random pulsar wind nebula, a slightly harder spectral index like $\alpha=-2.316$ is unphysical, or reasonable.

Therefore, I'm wondering: What is the distribution of spectral indices for power-law models of very high energy spectra of pulsar wind nebulae? Do we have a large enough population of pulsar wind nebulae that we can determine what a typical value for $\alpha$ is, and what range we would expect a spectral index to fall into? In other words, could if I had obtained a certain value for $\alpha$, what value would it need to have for me to reject the result as unphysical?
You might find these websites for your answers:


https://www.cv.nrao.edu/course/astr534/SynchrotronSpectrum.html
http://astronomy.swin.edu.au/cosmos/S/Synchrotron+Emission
https://ui.adsabs.harvard.edu/abs/2006MNRAS.371.1441S/abstract


We have a large amount of high energy emission these days, and the third reference shows the distribution from 394 GRBs.