SN isotropy

Some mu residual plots: 

• These plots compute the \mu residual by computing the difference between the expected value of \mu (measured from the redshift and Planck cosmological parameters) and the measured value of \mu.  In all cases, errors of mean residual is determined by adding individual errors in quadrature and dividing by number of added Sn.  
• Here is a plot of the sn on the sky, with colors corresponding to different surveys (the colors themselves are chosen to appear distinct, but are otherwise arbitrary) 
  
• Here is a plot of the residuals on the sky, determined by looking at residual of all Sn within 10 degrees (no redshift information considered).  The color scale denotes number of sigma away from 0.0.
    

• Here is a plot of the residual vs redshift (no sky directional information considered). 
  • For all 4 plots:
    • The top panel shows the residuals of all individual Sn from the surveys considered 
    • The bottom panel shows the average residual in twenty bins of either even size or that contain the same number of SN (up to +/- 1) 
  • The first plot shows all surveys.  There are 20 bins of equal redshift size. 
  • The second plot shows all surveys.  There are 20 bins that contain an equal number of SN.
  • The third plot shows only the PS1 survey.  There are 20 bins of equal redshift size. 
  • The fourth plot shows only the PS1 survey.  There are 20 bins that contain an equal number of SN.
    
    
    
• Now, in order to avoid conflating signals in z with those in extinction, let's look at the above plots divided into the 10 PS1 fields 
  • In addition to PS1, the other two relatively large surveys we have data on are: SDSS and SNLS
  • There are 2 fields that have overlap between PS1 and SDSS and 3 fields that have overlap between PS1 and SNLS (no overlap between SDSS and SNLS)
• Each plot represents one field (generally ~2x2 degrees or so), so extinction is (hopefully) relatively consistent across 
  
  
  
  
• Here are two plot for the mu residuals vs extinction (E(B-V) as measured with Finkbeiner et al: https://arxiv.org/pdf/1507.01005.pdf).  The distinct plots are similar to the residual v redshift plots above
  • For all 4 plots:
    • The top panel shows the residuals of all individual Sn from the surveys considered 
    • The bottom panel shows the average residual in twenty bins of either even size or that contain the same number of SN (up to +/- 1) 
  • The first plot shows all surveys.  There are 20 bins of equal extinction size. 
  • The second plot shows all surveys.  There are 20 bins that contain an equal number of SN.
  • The third plot shows only the PS1 survey.  There are 20 bins of equal extinction size. 
  • The fourth plot shows only the PS1 survey.  There are 20 bins that contain an equal number of SN.
    
  • 
  • 
    
• Here is a plot for the mu residuals and the extinction (as measured with Finkbeiner et al: https://arxiv.org/pdf/1507.01005.pdf) on the sky. 
  • Each of the 10 PanStars1 fields are displayed twice: 
    • the top frame is the extinction (E(B-V)) 
    • the bottom is the PS1 SN, 
      • Each point represents a single SN
      • The points themselves are color-coded by normalized mu residual (ie, [mu residual] / [mu uncertainty])   
  • The right colorbar is for the extinction contour plots 
    • All extinction contour plots are on the same logarithmic scale
  • The left colorbar is for the SN scatter plots
    • All SN are on the same linear scale
  • I am not completely happy with this plot, but I am not sure what the best way to improve it is
  

Here is a first-pass at a plot showing the residual as a value of extinction and redshift together (presently for the PS1 data only). 

• The color describes the value of the residuals, divided by the errors 
• Linear interpolating is used to determine the value between points (a cubic interpolation produced some anomalously 'hot' and 'cold' regions)
• The uneven sampling in the space makes the contour plot a bit of a mess
• Another possibility might be to apply some sort of 'smoothing' to determine the value between points
  

Some notes:

• • Almost all of the 1d projected plots suggest that the residuals are systematically negative.  Some quick calculations confirm this:
    • For ALL the SN mu residuals:
      • The raw mean is ~ - 0.059 (0.005)
      • The weighted mean is ~ - 0.067 (0.004) 
    • For just the PS1 mu residuals:
      • The raw mean is ~ -0.065 (0.009) 
      • The weighted mean is ~ -0.068 (0.009) 
  • Fitting seems like a good next step. Probably start with polynomials. 
  • Could also be interesting to see if patterns identified in one survey are consistent across surveys (PS1 and SDSS probably best places to look)  

CWS notes June 26 2017-

1. Perhaps make a fit of residuals vs. extinction for PanSTARRS data? By eye it looks like there is a statistically significant non-zero coherent residual at low extinction values. 
2. A different binning scheme might be equal number of SNe per bin? 
3. Another interesting plot would be contour surface of residual vs extinction and redshift, both.