# Robustness to redshift variation (Revised):

Now, the $\delta_F$-vs-$\rho_{DM}$ relation for different redshifts makes sense. z=2.4 does look like a transition from z=2.3 to z=2.6.


The estimators are :

$\rm \left(\frac{\rho_{\rm DM}}{\langle \rho_{\rm DM} \rangle}\right)^{sm} = a_2 \ {\delta^{sm}_F}\ {}^2 \ + \ a_1 \ \delta^{sm}_F + a_0$

| | $a_2$ | $a_1$ | $a_2$ |
|----|---|---| --- |
| z=2.3 | 14.6 $\pm$ 0.6| -4.97$\pm$ 0.06 | 0.97 $\pm$ 0.00|
| z=2.45 | 11.4 $\pm$ 0.5| -4.59$\pm$ 0.05 | 0.97 $\pm$ 0.01|
| z=2.6 | 10.2 $\pm$ 0.6| -4.37$\pm$ 0.05 | 0.99 $\pm$ 0.00|

However, I show this evolution makes a very small difference in our final results:

## $M_{DM}$-vs-$M_{tomo}$ relation :

If we use the $\delta_F$-vs-$\rho_{DM}$ realtion at each redshift or use the z=2.45 estimator for all redshifts, we would get the same estimator for $M_{DM}$-vs-$M_{tomo}$. The differneces are smaller than the uncertanties on the slope and intercept by factor of a few.

Now, assume we use the same $\delta_F$-vs-$\rho_{DM}$ estimator for all redshfits to get $M_{tomo}$. Now, we should be asking if therese is any extra redhift evolution in $M_{DM}$-vs-$M_{tomo}$ ?

|z | slope | intercept |
|--|--| --|
|2.3| 0.43+-0.09 | 14.54+-0.06|
|2.4| 0.39+-0.07 | 14.54+-0.04|
|2.6|0.35+-0.05 | 14.58+-0.03|

- Also the scatter around these estimators do seem unchanged (The plots are [here](https://github.com/mahdiqezlou/LyTomo-Watershed/blob/main/MDM_Mtomo.ipynb))

- I think, it does not change either, what do you think ?

## $M_{desc}$-vs-$M_{tomo}$ relation :

I checked again if it makes any difference to use $\delta_F$-vs-$\rho_{DM}$ realtion at the mid-redshift for the entire box.

I found it the difference is so minuscule.

Now, We can check if the $M_{desc}$-vs-$M_{tomo}$ estimator is redshift dependant :

|z | # Watersheds | slope | intercept |
|--|--|--| --|
|2.3| -- | 0.76 +-0.09 | 13.74+-0.06|
|2.4| -- | 0.72+-0.09 | 13.76+-0.05|
|2.6| -- | 0.77+-0.09 | 13.76+-0.05|

- The errors shown on the bottom panel of figure 10 stay the same for these redshifs (I have the plots [here](https://github.com/mahdiqezlou/LyTomo-Watershed/blob/main/M0_Mtomo.ipynb))

## Conclusion :

- None of the estimators or the estimated errors in the draft are not varying with chaning redshift within 2.3-2.6.

- So, we should be able to safely use the same $\kappa$ and $\nu$ and all the estimators to the entire volume of LATIS.