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• 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
  • There is no particular order to the grid set up (it would also not be hard to rearrange the grid, if there is a reason a new layout would be helpful)
  • 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

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• • For ease of reference, the best fit constants line by survey are:
    • CFA1: -0.13096 (0.00242) 
    • HST: -0.09145 (0.00222) 
    • PS1MD: -0.06807 (0.00007)
    • CFA4p1: -0.10461(0.00083)
    • CSP: -0.0567(0.00073) 
    • SDSS: -0.06075(0.00005) 
    • CFA3S: -0.10639(0.0011) 
    • CFA4p2: -0.05857(0.0021)
    • SNLS: -0.04682(0.00008) 
    • CFA3K: -0.09714 (0.00038) 
    • CFA2: -0.0983 (0.00153) 
• Notably, all the offsets are definitively negative, with values ranging from about -0.05 to -0.1
  • According to the uncertainties provided by the fit (the only element in the fit covariance matrix), the numbers  are also in some statistical tension between surveys.  I am unsure how much relevance should be given to those results, however.  
  • Also, I will note that the 3 surveys with (by far) the largest data sets (PS1MD, SDSS, SNLS) have constants on the low side (of the other 8 surveys, only 2 (CSP and CFA2p2) have constant residual offsets as small or smaller. The remaining 6 surveys (CFA1, HST, CFA4p1, CFA3S, CFA3K, CFA2) have MUCH larger offsets (all < - 0.09) 

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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
  • There is no particular order to the grid set up (it would also not be hard to rearrange the grid, if there is a reason a new layout would be helpful)
  • 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)
    View filenameshifted_sine_fits_all_surveys_1.pdfheight250
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  Fitting to a single mode gaussian:

  • Noting that, by eye, many of the fields for the PS SN show an apparent dip in the residual around z~0.3, I tried fitting that data to a single mode Gaussian binned to the SN from each survey in each field individually 
  • Basics of the fit include:
    • The fitting function was of the form: Res(z) = C + A * e ^ (-(z - \mu)^2 / (2 w^2)) where C, A, \mu, and w 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
    • All fits were primed with the following values to try to guide them to the object of interest: (C, A, \mu, w) = (-0.05, -0.1, 0.3, 0.05)
  • In the below plots:
    • 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 
    • The best fit parameters (rounded to 5 decimal points) are shown in the text underneath.  
      • In order, the parameters in the text are (see equation definition above):
        • C: the constant, overall residual shift
        • A: the constant scale
        • \mu the center
        • w the width
    • The colors of the fit curves and text match the colors of the points
  • Some of the fits did pick out the dip I've been looking at 
    • For the PS1MD data, Fields 0, 1, 3(?), 5 and 6 identify the dip 
    • For the SNLS data, Field 0 identified the dip 
    • For the SDSS data, field 9 identified the dip
  • I think some of the other fields could identify the dip as well if we apply some constraints on the range of the fit, or make it a bit more flexible (maybe allow for a polynomial addition instead of a constant shift?)
    • However, we should think about how much a fit could be made to identify a dip anywhere, with sufficient prompting... 

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• To compare the PS1MD fields (those above) to the SDSS fields, I generated a plot for just the SDSS SN and fit a single mode Gaussian to them 
  • The fitting information is like that above, with similar constraints
  • The fit did select a dip at a similar position to that noted above (though somewhat narrower).  This data mays also show evidence of an earlier dip (around z ~0.18) that I do not note in the PS1MD fields above (perhaps because of insufficient sampling) 
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• In light of the above, I have attempted to fit the PS1MD, SDSS, and SNLS data (the three sets with numerous SN) simultaneously with a single function
  • I started with a sine function of the form: res(z) = A * sin(omega * x + phi) + shift 
    • With A, omega, phi allowed to vary, but forced to be the same between all data sets
    • shift is allowed to vary independently 
  • I ran the fit 7 times for each of the different possible combinations of data (each data set alone, the 3 2-set combinations, and all three simultaneously). 
  • The plots show both unbinned data (top panel) and binned data (bottom panel).  The fits are always run on the unbinned data.  
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