One sign of a «good» chi^2 is that there is no big scale patterns in the maps, but rather white noise fluctuations. There will almost always be big scale patterns along the galaxy plane, so this is why we mask this out. I usually smooth the chi^2 maps to easier see if there are any large scale patterns.
When there are big scale patterns in the chi^2 maps, I try to recognize which foreground might have caused this pattern to see which foreground that needs to be fitted better. In the case of the real data from Pico, we would start by fitting the known foregrounds and then look at the chi^2 and the residuals if there might be signals that we’re not taken into account.
We have 21 input maps, one pr frequency. When I’m fitting for 6 parameters (synchrotron amplitude and beta, dust amplitude, beta and temperature and cmb) as for model 90.91, I will have 21-6=15 degrees of freedom and thus expect the chi^2 for a well fitted model to have an average value of 15. When I’m fitting for 12 parameters (synchrotron amplitude, curvature and beta, dust amplitude, beta and temperature for two dust models, AME amplitude and peak frequency and cmb) as for model 90.92, I will have 21-12=9 degrees of freedom and expect the average chi^2 value to be 9. However, counting degrees of freedom is complicated when informative priors are used since a free parameter then doesn't really count as a full degree of freedom. But it's an OK rule-of-thumb. The chi^2 value for high latitudes in model 90.91 is approximately 15, but the level is not as low as 9 for 90.92, so this can not be used as the only criterion
The numerical value for the chi^2 is difficult to see in the unsmoothed chi^2 maps, I will try to make this clearer in future presentations.