Fitting Residuals v Redshi

Here, I will provide the results of my various attempts to fit the SN residual data as a function of redshift for various surveys.

Fitting to a constant

• Canonically, the residuals should be consistent with 0.0.  However, it has become clear that there appears to be a small negative offset in.
• It is not entirely unreasonable that such an offset could arise.  However, we should expect that (assuming all SN are calibrated consistently across surveys) that the shift should be uniform across surveys.
• Here, I show the best fit constant line associated with all surveys for which we have >2 SN.  Results are shown below.  
• In the below plots:
  • The top panel shows the unbinned data and the bottom panel shows the binned data for each survey
    • There were 20 bins of equal redshift space over the redshift range covered by the particular survey in question
  • The fit lines are shown in red
  • The best fit constant (rounded to 5 decimal points) is shown in the red text underneath each line 

Fitting to a basic sine function:

• Noting that, by eye, the binned residual data for some of the surveys with a relatively wide redshift range had apparent oscillations in redshift (probably most apparent in PS1MD), I tried fitting a shifted sine function to all of the surveys for which we have >2 SN.  The results are shown below.  Basics of the fit include:
  • The fitting function was of the form: Res(z) = C + A * sin(k * z + \phi) where C, A, k, and \phi are the fitted parameters
  • The domain of the fitting parameters was largely unconstrained
  • The fit was performed on the unbinned data for each survey 
  • The fits for each survey were entirely independent
• In the below plots:
  • The top panel shows the unbinned data and the bottom panel shows the binned data for each survey
    • There were 20 bins of equal redshift space over the redshift range covered by the particular survey in question
  • The fit lines are shown in red
  • The best fit parameters (rounded to 5 decimal points) are shown in the red text underneath each line
    • In order, the parameters in the text are (see equation definition above):
      • C: the constant, overall residual shift
      • A: the amplitude 
      • k: the wavenumber 
      • \phi: the phase shift
• I see no consistent signal in the below plots.  However, I am not yet convinced that there is nothing here.  I think we just need to:
  • (a) be smarter about our approach to fitting, perhaps starting with a better function
  • (b) recognize that most of the surveys cover too small a range of redshifts to give much information about larger variations
  • (c) Try to focus on SN in matching regions of sky to be sure we aren't entangling extinction artifacts with redshift signals (though I do note that we saw now obvious extinction dependence in our previous analysis)