I have color coded the rejects to correspond to the k bin that is responsible for exceeding $1 \sigma$ and making the tested model non-degenerate with WMAP7. The color coding is actually a spectrum, but one gets the same answer for wide bins in $k$, because it is always one end of $k$ or the other that is causing the model to go out of range. It's pretty clear that this is the reason that the edges of the accepted range are so straight, yielding a parallelogram shape.

Here are a few choices of parameter spaces. The story is the same in each.







The degenerate range does not depend strongly on the range in $k$ that is being probed. The same outside values are the ones going out of range, regardless of what the cutoff value of $k$ is. For example, extending $k$ to larger scales we get:



And, extending $k$ to smaller scales into the non-linear regime we get:


And oh ho ho! What have we here?! It looks like a different $k$ bin is responsible for the rejection, not the absolute outside bin, and furthermore, it looks like the top is a slightly different $k$ bin than the bottom. I will have to look into this more tomorrow!

Sorted out contour bug

Just a quick post to demonstrate, I have sorted out the contour bug. Thanks to some help from Adrian.

Changing the threshold

Just for completeness, here is a plot of the 2-parameter case, where I change the size of the error bars (or alternatively the threshold that triggers a point to be labeled degenerate). Below are the $1\sigma$ degenerate regions for three different choices of error on $\Delta^2(k)$. Don't forget you can click on the plots.

Changing the k range (preliminary)

I am so frustrated, I cannot get the gridding routine to work. Of course I could grid it up myself, but I really would prefer to use the scipy version, with the pretty cubic splining instead of just a 2D histogram. But for now, I have made some k range studies using points instead. Below I have varied only the two parameters, and I have zoomed in on the region in question:



Looks here like the k range has little to no effect on the degeneracy region. The hypothesis about the edge effects dominating the threshold must therefore be wrong. There must be some other explanation for these super straight edges.

Below I am just showing that extending the k range in the other direction seems to have more of an effect (I forgot to zoom in that time):



Here I have varied all the parameters but $\Omega_b h^2$, which we saw in previous plots has little to no effect on the degeneracy region.



Not the colors on the two ranges have been swapped (oops). We see that qualitatively this seems to be the same type of effect as in the two parameter case in the bottom right. Hard to tell in the upper left. All this will look much clearer once I can start drawing contours.

Varying all the parameters

Just for completeness, Here is a run where all the parameters are varied. This was 40k points, run in two separate chains. I found the bug. The minimum value of $\sigma_8$ that I was using was wrong, so every once in a while the chain would run out of range and the code would crash. I can now, in principle, run arbitrarily long chains (which is more accurate than running a short chain 15 times).



So, what we see here is that this looks esentially like the plot where $\Omega_b h^2$ is fixed, which is consistent with what we were seeing before, $\Omega_b h^2$ is less important than the other parameters in terms of it's effect on the $\Omega_m h^2 - \sigma_8$ plane. I think this is because the major driver for the boundaries of the degeneracy range are the cutoff values of k, whereas $\Omega_b h^2$ is controlling the relative values of the peak heights. It is probably true that the points which first exceed the $1\sigma$ threshold are either at one end or the other of the allowed range in k. I think this is also the reason that the degenerate region has straight edges, but we will have to see.

varying more parameters

Here I am showing how allowing other parameters to vary opens up the degenerate range in the $\Omega_m h^2$ - $\sigma_8$ parameter space. I am no longer plotting the reject points, because they overlap with the accepted points, due to projection. Also, I have only plotted points whose deviation is less than 1-sigma (for every k, the degenerate model is less than one sigma away from the fiducial WMAP7 model), because it's easier to see how the region grows.

Here is the plot where only two parameters are allowed to vary, the other three are fixed, as noted in the title.



Next, I show plots where a third parameter is allowed to change. What we see is that some of the parameters can compensate for changes in $\Omega_m h^2$ and $\sigma_8$, so the degenerate region opens up. More points fail, which is why the density seems to have gone down. If I were a more patient person, I'd up the density to compensate for the increase in the dimensionality of the parameter space, but these are just preliminary plots.






Finally, here are some plots where only one parameter is fixed. I ran some of these a couple of times to get a higher density of keepers...:







Don't have time right now to run the one with no parameters fixed, I'll do it tomorrow. I have a bug, I think the cosmic emulator dies sometimes, probably because the point goes out of range, which kills my code. I've written a check on the parameter values but it's not working properly. I need to track it down so the chain can run longer (like overnight).

Degeneracy space with the emulator

I have downloaded the cosmic emulator, and have begun exploring the parameter space, looking for degeneracies. I have specified the fiducial model to agree with the WMAP 7 parameters. These are

omhh = 0.1344
obhh = 0.02246
ns = 0.961
sig8 = 0.807
w = -1.0

I have written a Monte-Carlo flavored code, that explores the parameter space. I've designed a criterion to decide if a set of parameters yeilds a "degenerate" power spectrum. For a specified range in k, a degenerate model does not depart from the fiducial model by more than $2\sigma$ at any value of k. The value of $\sigma$ could in principle be taken from the WMAP error bars, but for the time being, I have just used 5% error bars on $\Delta^2$. The code has the ability to fix any of the 5 parameters. Below are some plots showing the degenerate models in blue, and the non-degenerate models in red, where only the two parameters indicated have been varied. These show the degeneracy of $\Omega_m h^2$ with all the other parameters being considered. Note: you can click on the plots to make them larger.








These results surprised me: they were wider ranges than I would have expected, and also the very straight edges seemed a little strange to me, though of course I cannot prove that they are non-physical.

The next steps: I will make plots where more than two of the parameters have been varied, I'll look at how the swath changes as I change the range in k, and also the value of $\sigma$ (3% rather than 5%, for example).

To consider: I did not use a metropolis-hastings algorithm, because I wanted to specify precisely, either the model is degenerate within two sigma, or it is not. However, the M-H algorithm might have better properties in terms of exploring the parameter space, which might become more important once I start opening it up to 3,4,5 dimensions.

Talked with Salman

Before leaving IAS I talked with David Weinberg about a project I had in mind to try and leverage the stringyness or filamentarity of LSS to get an independent set of constraints on the cosmological parameters, most notably $\sigma_8$. He told me lots of people have gone down this dark dark path, but he also said my proposed observable was different than any he'd heard of and it was worth checking to see if it helps. He said to keep the study focussed and just try and see if I can use it to break the $\sigma_8$ degeneracy with $\Omega_m$. Most importantly, don't get lost in trying to define what a filament is, or characterize its profile and that sort of thing, because those things do not seem to be observationally tractable as yet.

I talk to Salman, to see if they have any halo catalogs from sims with different values of these two parameters, with degenerate power spectra. He suspected not. They did do parameter scans, but did not specifically try and generate degenerate power spectra. He suggested I use the emulator (newest version is HERE) they have developed to find two cosmologies that I want, and they would do the sims and give me some halo catalogs. I need to think about what would be best regarding the size versus the resolution of the sims. Maybe the best thing would be to use one of their existingsims, and just have them run the degenerate twin. For this, however, I would need the CPs for one of the ones they used. Maybe they use WMAP as one of the fiducial runs??