free parameter creates instabilities in the fits to the other parameters, due to their high degree of cross-correlation (as noted in, e.g., @KanekoCat). Finally, the sum of several exponentials evolving under under certain conditions can approximate a power law, as we shall show below. Conversely, the sum of many power laws with different spectral indicies is guaranteed not to result in a power law. It is one of the outstanding questions of GRB spectroscopy why the data should ever be consistent with the Band function.

As mentioned above, assuming it represents a characteristic energy, [$E_{\rm peak}$]{}is the unique spectral parameter that encodes relative motion between the source and the observer. Given that GRB redshifts have been measured for many events, the distribution of these currently falls between $z \approx 0.01$ [@galama98] and $z \approx 9.4 $ [@Cucchiara], with a peak between $z = 1$ and 2. Without any other consideration, we might expect the distribution of [$E_{\rm peak}$]{} to follow the redshift distribution, with the smallest values corresponding with the largest redshifts. However, there is good reason to suspect that the GRB emission arises within a relativistic blast wave [@reesmesz]. Any characteristic energy, as seen in the observer’s frame, should 