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 
• In each plot, the upper panel shows the unbinned data and the lower panel shows the binned data 
  • binning scheme is 20 bins of equal redshift space, determined for each survey individually 
    • so in a field with multiple surveys, the bins for each survey are different and likely do not cover the whole displayed z-range; only the range covered by the survey in question 
• Qualitatively, I notice the following:
  • In some of the fields (particularly 0, 1, 5, 8, and 9 most obviously), the PS1 data does appear to have a visible dip in residual around z~0.3
  • In other fields (particular 2, 3 and maybe 4) there seems to be a slight 'hole' in the observed SN around this value
  • In the remaining fields (6 and 7) I'd say one could try to convince oneself that the dip is there, but only after one knows to look for it (informed by the other fields).  
  • When the data is broken up like this, it's hard to say if there is evidence for other dips (like those marginally seen in the aggregated PS1MD SN above)
  • Unfortunately, the other surveys with numerous SN that overlap on the sky with the PS1MD fields don't sample this z value very efficiently
    • SNLS generally has higher z SN.  In the 3 frames with both PS1MD and SNLS observations (0, 3, and 6), by eye
      • frame 0 may show evidence that the SNLS data also have a dip around z ~ 0.3
      • frame 3 seems to have a hole around z~0.3 (and maybe a smaller one around z ~ 0.8) 
      • frame 6 don't seem to have much of a drop in the SNLS residuals around z ~ 0.3
      • Note that all of those behaviors (a dip in 0, a hole in 3 and relative flatness in 6) are similar to the behaviors of the PS1MD residuals in their respective frames (stressing again, that all of these observations are qualitative, 'by eye', notes) 
    • In the frames with both PS1MD and SDSS SN (8 and 9), the SDSS SN either don't go to high enough redshift (field 8) or are too few in number (field 9) to allow an effective sampling of the z value of interest
      • By constraining ourselves to only the PS1MD fields, we are ignoring a lot of the SDSS SN.  Perhaps we could now do something similar for the SDSS fields... 

• 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.
    
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• 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.