Relating distance modulus residuals to physical sources

We attempt to relate the observed residual in distance modulus to some physical structure
I have attempted to understand how a perturbation in the energy density would appear in the distance modulus residuals. The derivation is outlined in this document:
Of particular note, perturbations in the energy density as a function of redshift are somewhat suppressed by other redshift dependences (see the bottom of the above document)
I have written a code to run simultaneous fits on the various data sets, given the above derivations
A sample result is here for proposing a modification to the energy density in which we imagine that dark energy is a mildly oscillatory function of time
- We require DE to take on its presently observed value at t = 0
- note that this translates to an oscillatory function of redshift, though the relation between z and t is nontrivial, and is baked into the calculation
The results of fitting all of the data simultaneously with the same DE function, but allowing for different overall mu offsets (ie, assuming that the oscillitory DE is the same across the entire sky) are shown in this plot:
Though it's hard to get a visual sense of exactly how good the fit is with the sheer number of data points, the oscillations don't seem to pick out the oscillations I trace out by eye. So I'm skeptical that this is a good match.

- The reduced chi square for this fit (discussed in page titled 'Estimating the Significance of Our Detections') is 0.9957

I then considered what affect something like a large overdense/underdense matter region could have the residual
As Chris correctly noted during a meeting, since the residual does not appear to undergo any large steps in redshift, any variation must be both positive and negative (otherwise, there would be an overall offset between older and newer SN that we don't see in the data)
Therefore, I considered the effects of a matter perturbation of the form:
- e(z) = (A * \alpha * exp( - (z - z_bar - 2 \sigma) ^ 2 / (2 * \sigma^2)) + A * exp( - (z - z_bar) ^ 2 / (2 * \sigma^2)) + A * \alpha * exp( - (z - z_bar + 2 \sigma) ^ 2 / (2 * \sigma^2))) * (1+z) ^ 3.0
  - (the factor of (1+z) ^ 3 is there since I'm ascribing this perturbation to matter)
- A sample plot with A = 0.1, \alpha = -0.5, z_bar = 0.4, \sigma =0.03 looks like:
- By allowing A and \alpha to have opposite signs, this is a function that can lead to 0 overall difference between the \mu residuals of the low z and higher z SN while still producing interesting residual behavior in between (as shown in the fits below)
- Physically, this function describes a set of 3 peaks/troughs in energy density centered around some redshift, z_bar. With an appropriate choice of A and \alpha, the overall energy density in the universe remains more or less constant, but just has adjacent (in space or time) overdense and underdense regions
- I chose this particular parameterization of e(z) for a few reasons:
  1. It was the best function I could come up with that had only 4 free parameters (A, \alpha, z_bar, \sigma) that I felt could still match the residual structure that we have noticed
  2. I noted that some discussions on the structure of cosmic voids argued the voids would consist of relatively underdense regions surrounded by overdense regions. Thus, if one were looking back in space through a void, the energy density as a function of z would go up at the edge, down within the void, and up again at the far edge
  3. Of course a true parameterization of a void would be far more sophisticated than the simple model I use here, but this basic valley-peak-valley combination seemed a good first pass
  4. Going a bit further beyond the canonical cosmology model, I came across a discussion on how the local group sitting towards the center of a non-spherical void could serve as an alternative to dark energy (see chapter 11 of this set of proceedings; I have a pdf doc if you can't pull it off of the website)
    1. I am not sure how seriously to take such a discussion, but I did note that the model for the local energy density that they propose (see figure 11.1 in the pdf) does look a bit like peaked overdense regions surrounded by large underdense regions
    2. Again, a true model would be far more sophisticated; I just used this simple model as a first pass stand in
- From here, I ran the simultaneous fitter on the \mu residuals based on this energy density (as outlined in the ExploringSNAnistropy.pdf document above), with the following details
  - I allowed the \mu residuals (not the energy density itself) to have an overall constant offset that was fixed for a particular survey
  - I ran the computation only on the PS1 fields
  - I allowed the A, \alpha, and z_bar parameters to vary individually by field.
    - I allowed these 3 variables to float independently because it was relatively clear from the plots which we've previously examined that the feature which I am trying to isolate (the dip around z ~0.35) occurs only for some of the fields. Physically, it's conceivable that this energy density variation is different in different parts of the sky. A good follow up will be forcing the parameters to be the same for different surveys that cover the same sky region
- The results are below:
  - The numbers listed on the figure are the best fit values for, in order, A, \alpha, z_bar, \sigma, and the overall shift for the PS1 survey
  - As usual, the top frame in each figure is the unbinned data set and the bottom frame is the binned data
  - The fit is run on the unbinned data for each field
- When I made a similar fit to all the data simultaneously, the fitting algorithm did not pick out the same value:

Next, I decided to groun
- Clearly, I can't perturb the Omegas in a way that leaves Omega_M + Omega_L unchanged, as that would then leave the value of H(z) (and thus the integral) unchanged
- Thus, we need to change something about the way Omega_M + Omega_L evolves in redshift.
  - We just considered above what the effects of just changing Omega_M might be. Can we think of a way of changing both at once, with minimal free parameters?