Issue about the percentage of falling of height of Likelihood when I project the edge 1 C.L sigma joint distribution on the 1D Likelihood

1. I have currently an issue about the height at which the projection of 1 sigma edges in 2D contour should intersect the associated Likelihood.

Here a figure (called "triplot") to illustrate my issue :

At bottom left is represented the joint distribution (shaded blue = contours at 2 sigma (95% C.L) and classic blue = contours at 1 sigma (68% C.L) of the 2 parameters considered (w0 and wa).

On the top is represented the normalized Likelihood of w0 parameter.

In all contours (with all triplot representing other parameters) and in all tripltot of thesis documents I have seen, the projection from the edge of 1 sigma contours on the likelihood intersects the likelihood at a height relatively low (on my scheme, roughly at 25%-30%, at first sight, of the maximum height of the likelihood).

However, a colleague told me that Likelihood should be intersected by the 1 sigma edge of joint distribution at roughly 70% of the maximum height of Likelihood (green bar and text on my figure)

For this, he justifies like this :

Concerning \Delta\chi2, distribution function is a \chi2 law with 2 freedom degrees ; pdf is written as :

f(delta(chi2))=1/2 exp^(-delta(chi2)/2)

So for a fixed confidence level C.L, we have :

1-CL=\int_{delta(chi2)_{CL}}^{+\infty}1/2 exp^(-delta(chi2)/2) d chi2

=exp^{-delta(chi2)_{CL}/2}`

and taking CL=0.68, we get :

delta(chi2)_{CL}=-2\ln(1-CL)

delta(chi2)_{CL}=2.28

And Finally, he concludes by saying that Maximum of Likelihood shoud fall from about 30% , i.e :

exp(-2.3/2) = 0.31 

2. So I don't know why I get a falling of about 70% ~ (1-0.31) and not only of 31% ~ 0.3 like one says on my figure (red line on my figure above).

MOST RECENT EDIT IMPORTANT : I think my colleague is wrong since the 1 sigma on Gaussian Likelihood covers a larger 1 sigma than the projection of 1 C.L edges of 2D contours, so I think we can expect a max height for the intersection of exp(-2.28/2) ~ 0.32 ~ 30% and not a height of 70% : what do you think about this ?

ADDING FURTHER OLD DCUMENTATION : I tried below to give documentation and links to justify an intersection at 70% of the max height of Likelihood Gaussian but this statement seems to be wrong as indicated above in MOSST RECENT EDIT IMPORTANT.

ps1 : I give you a small tutorial about chi2 distribution and Likelihood for Fisher formalism Area of ellipse

ps2 : I have seen an ineresting remark on https://docs.scipy.org/doc//numpy-1.10.4/reference/generated/numpy.random.normal.html which suggests a maximum at 60.7% of the max, which is not really what I expect (~ 70%).

ps3 : I have also found another interesting page, maybe more important since it talks about multivariate distribution :

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.multivariate_normal.html

3. Here too, I give you what it seeems to me to be a justification of my reasoning :

Any help is welcome.

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