Figures for the letter

[landmanbester]

I will post em here when they are ready

[landmanbester]

Ok something is not right here. Here is what I get for sigmasq*D**2

Look at the figure on the left (I haven't worked out perturbed FLRW on the inside of the PLC). The black line is the perturbed FLRW on the PLC0. The blue contours shows what the data gives and the magenta line is sigma^2D^2 for an LTB model if I compute it with the CIVP. The LTB model is constrained to tend to EdS at the boundary so I would expect sigma^2 to taper off to zero. Now either I have made a mistake in how I copied the formula from Maple or there is a mistake in the Maple formula. I have copied the formula twice and gotten the same result so I need to ask a favour. @julienlarena pls could you make the following substitutions in the formula and send it to me again?

R{,nu} = S R{,w} = Q A_{,nu} = Z

Its easiest for me to check if you also multiply through by R^2 and expand it. If you don't have time I will only be able to do this when I regain access to Maple (hopefully by the weekend)

[julienlarena]

Will do this first thing tomorrow morning.

On 17 Jan 2017 20:00, "Landman Bester" notifications@github.com wrote:

Ok something is not right here. Here is what I get for sigmasq*D**2

[image: sigmasq] https://cloud.githubusercontent.com/assets/16836765/22032595/a42e85fe-dced-11e6-966d-f36fe25857d7.png

Look at the figure on the left (I haven't worked out perturbed FLRW on the inside of the PLC). The black line is the perturbed FLRW on the PLC0. The blue contours shows what the data gives and the magenta line is sigma^2D^2 for an LTB model if I compute it with the CIVP. The LTB model is constrained to tend to EdS at the boundary so I would expect sigma^2 to taper off to zero. Now either I have made a mistake in how I copied the formula from Maple or there is a mistake in the Maple formula. I have copied the formula twice and gotten the same result so I need to ask a favour. @julienlarena https://github.com/julienlarena pls could you make the following substitutions in the formula and send it to me again?

R{,nu} = S R{,w} = Q A_{,nu} = Z

Its easiest for me to check if you also multiply through by R^2 and expand it. If you don't have time I will only be able to do this when I regain access to Maple (hopefully by the weekend)

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[landmanbester]

Ok so we have to think carefully about what the physical significance of the uv cutoff is. For example in this figure

we can see that the data are compatible with a uv-cut of 20 or 50 but not 10 (I think units are Mpc). Here I didn't even use redshift drift data, only D(z), H(z) and a lower bound on t0 (all simulated, busy doing a run with real data). Here is a plot of the CDF of the marginal likelihood of the Gaussian process over rho as a function of the length scale

The CDF reaches 0.16 at about l = 1.45 Gpc and 0.025 at about l = 1 Gpc. @julienlarena if we set the uv cutoff to 1000 Mpc the perturbed FLRW shear is effectively zero. I am not sure how to interpret this in terms of the physics. Any suggestions?

[julienlarena]

Hi Landman.

In the Maple sheet, I wrote kUV= N/kH0 to get an adimensional integrand. N is the dimensional k in Mpc^{-1}. So N=1/UVCut-off in Mpc.

Is that what you did?

Why is the numerical result diverging at the z=1 boundary? Is it a numerical artefact?

[landmanbester]

Yep that's what I did. See this function

https://github.com/landmanbester/Copernicus/blob/master/genFLRW.py#L74

for how I compute the shear and this one

https://github.com/landmanbester/Copernicus/blob/master/Plotter.py#L163

for how I plot it.

As for the divergence, that is just what the data gives. The LTB shear computed with the CIVP agrees with the shear computed using a parametric specification (at least on the PLC0). If these curves look weird to you its because I don't plot as a function of redshift in these figures, I plot as a function of the normalised affine parameter v/v_max. I could redo them as a function of z for the letter but it won't change the discrepancy between the data and the models. The LTB shear for non-zero bang time models also seems to "diverge". I think if I went to a higher redshift we will see it re-converging. I will test this for simulated data and get back to you.

[julienlarena]

Hi Landman.

Yes, for the paper, we will have to plot as functions of redshift. How do you superimpose the FLRW shear then, as this is a function of z?

I agree that the FLRW shear is strongly dependent on the cut-off and that is a bit of a problem. However, if our stat analysis disregards features below a few 100Mpc, then that's it, that probably means we ought to neglect the FLRW intrinsic shear. That means we are effectively probing very large scale shear, which is a genuine non-FLRW feature.

[landmanbester]

No. To superimpose the shear I convert it to a function of v using the background FLRW v(z) relation. It was just easier to do this for the 3 FLRW shear relations than for the thousands of samples produced by the MCMC. I'll change that on the next run.

I am a bit worried about the comparison we are making. It seems like the better the data get's the larger the cutoff needs to be for the FLRW shear to compatible with observations. But does this have any physical significance? Maybe I am just being slow here but I don't see how we are going to use this to say something meaningful..

[julienlarena]

On 24 January 2017 at 08:50, Landman Bester notifications@github.com wrote:

No. To superimpose the shear I convert it to a function of v using the background FLRW v(z) relation. It was just easier to do this for the 3 FLRW shear relations than for the thousands of samples produced by the MCMC. I'll change that on the next run.

Ok. No problem, as long as we are comparing like with like.

I am a bit worried about the comparison we are making. It seems like the better the data get's the larger the cutoff needs to be for the FLRW shear to compatible with observations. But does this have any physical significance? Maybe I am just being slow here but I don't see how we are going to use this to say something meaningful..

Yep, that is the problem of perturbative calculations. At small scales, pretty much anything can happen, which increases the variance dramatically. The thing is that we are assessing the background structure, so in FLRW, it is not unreasonable to assume that the shear is negligible. At least, that was my first impression, but Roy confused everything with his suggestion... :) If in effect, when building our LTB models statistically from the data we implicitly smooth out any feature below a very large scale (typically the width of the voids), then at those scales, there is no sizeable FLRW shear and that's it. If it makes no difference, that is even better, no?

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[landmanbester]

Well we always knew that we can't use this code to test for small scale effects, for that you would have to be fully 3D (and very careful about how you reduce your data). At least this time we can say that Lambda=0 void models (for arbitrary bang time) seem to be are ruled out. Well at least for the parametrisation we consider...

Where to from here? I can do a run with current data and superimpose the improvement that would result from using future D(z) and dz/dw(z) data? Which figures should I produce though?

[julienlarena]

Okay. Not sure about the figure. Why is this D^{2}Shear not good?

On 24 January 2017 at 15:17, Landman Bester notifications@github.com wrote:

Well we always knew that we can't use this code to test for small scale effects, for that you would have to be fully 3D (and very careful about how you reduce your data). At least this time we can say that Lambda=0 void models (for arbitrary bang time) seem to be are ruled out. Well at least for the parametrisation we consider...

Where to from here? I can do a run with current data and superimpose the improvement that would result from using future D(z) and dz/dw(z) data? Which figures should I produce though?

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[landmanbester]

Ok so this is what the shear looks like for D and dzdw data

And these are the Om0 v OL0 contours for the same data

Ignore the figure on the right I put it there to check if my contour plots were coming out right. Here is the PLC0

The T1 and T2 contours

The MCMC reports that we haven't quite converged so I am running it for a bit longer. I doubt the figures will change by much though. I thought it would be cool to plot the current data on the same figures since that would show the improvement we would get from future data. The current data is a nightmare though, I keep running into issues with the multiprocessing that I didn't have before. Hoping that I have dealt with all of them for the latest run. Will keep you posted

[julienlarena]

It looks brilliant! I have to have a closer look but Zoe is keeping me busy...

Chat later.

On 3 February 2017 at 11:19, Landman Bester notifications@github.com wrote:

Ok so this is what the shear looks like for D and dzdw data

[image: sigmasq] https://cloud.githubusercontent.com/assets/16836765/22585697/bd9c3362-ea01-11e6-978a-4e1d81bb4c40.png

And these are the Om0 v OL0 contours for the same data

[image: contours] https://cloud.githubusercontent.com/assets/16836765/22585708/cdd80878-ea01-11e6-818f-48a445bbaa0d.png

Ignore the figure on the right I put it there to check if my contour plots were coming out right. Here is the PLC0

[image: plc0] https://cloud.githubusercontent.com/assets/16836765/22585756/0c94554e-ea02-11e6-909a-2080d05b105f.png

The T1 and T2 contours

[image: cp] https://cloud.githubusercontent.com/assets/16836765/22585766/1c77cfcc-ea02-11e6-949d-7ed1b59d9bf0.png

The MCMC reports that we haven't quite converged so I am running it for a bit longer. I doubt the figures will change by much though. I thought it would be cool to plot the current data on the same figures since that would show the improvement we would get from future data. The current data is a nightmare though, I keep running into issues with the multiprocessing that I didn't have before. Hoping that I have dealt with all of them for the latest run. Will keep you posted

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[julienlarena]

Ok. It looks quite nice. The FoM seems to say that one could rule out LTB+tbb=0 at 2-sigma if FLRW is correct. That is nice. Yes, including current data is a very good idea to show the improvement. I thought about it the other day, and then, I forgot to wrote to you...

Cheers, J.

On 3 February 2017 at 11:19, Landman Bester notifications@github.com wrote:

Ok so this is what the shear looks like for D and dzdw data

[image: sigmasq] https://cloud.githubusercontent.com/assets/16836765/22585697/bd9c3362-ea01-11e6-978a-4e1d81bb4c40.png

And these are the Om0 v OL0 contours for the same data

[image: contours] https://cloud.githubusercontent.com/assets/16836765/22585708/cdd80878-ea01-11e6-818f-48a445bbaa0d.png

Ignore the figure on the right I put it there to check if my contour plots were coming out right. Here is the PLC0

[image: plc0] https://cloud.githubusercontent.com/assets/16836765/22585756/0c94554e-ea02-11e6-909a-2080d05b105f.png

The T1 and T2 contours

[image: cp] https://cloud.githubusercontent.com/assets/16836765/22585766/1c77cfcc-ea02-11e6-949d-7ed1b59d9bf0.png

The MCMC reports that we haven't quite converged so I am running it for a bit longer. I doubt the figures will change by much though. I thought it would be cool to plot the current data on the same figures since that would show the improvement we would get from future data. The current data is a nightmare though, I keep running into issues with the multiprocessing that I didn't have before. Hoping that I have dealt with all of them for the latest run. Will keep you posted

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