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
- 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
- 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)
- The top panel shows the unbinned data and the bottom panel shows the binned data for each survey
- 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)
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
- In order, the parameters in the text are (see equation definition above):
- The top panel shows the unbinned data and the bottom panel shows the binned data for each survey
- 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)
