---
# System prepended metadata

title: 'Redshift sensitivity :'

---

# Redshift sensitivity :

We have selected the $\nu$ and $\kappa$ for each individual redshift like the table below. 

| z  | $\nu$ | $\kappa$ |
|----|---|---|
|2.3 | -2.35 | -2.75 |
|2.45 | -2.0 | -2.35 |
|2.6  | -1.90 |  -2.4|

They are not terribly different though. 

However, We have already shown we get roughly the same estimators both for $M_{DM}$ amd $M_{z=0}$ this way. (Tables below).
This shows we can use the same estimator for the entire of the observed volume, but it doesn't say we can use a single set of  ($\nu$ and $\kappa$) and $\delta_F$-$\rho_{DM}$ realtion for all rdshifts. In next session I show what happens if we use same thing for all redshifts.


For $M_{DM}$-$M_{tomo}$

|z | # Watersheds | slope | intercept |
|--|--|--| --|
|2.3|  | 0.41+-0.10 | 14.59+-0.05|
|2.4|  | 0.39+-0.06 | 14.54+-0.04|
|2.6|  | 0.32+-0.05 | 14.67+-0.03|

For $M_{DM}$-$M_{z=0}$:

|z | # Watersheds | slope | intercept |
|--|--|--| --|
|2.3| | 0.668+-0.117 | 13.90+-0.075|
|2.4| | 0.720+-0.090 | 13.76+-0.051|
|2.6| | 0.766+-0.106 | 13.70+-0.067|


## Fixing $\nu=-2.0$, $\kappa=-2.35$ and the $\delta_F$-$\rho_{DM}$ relation :

It turns out the  $\delta_F$-vs-$\rho_{DM}$  relation has noticeable redhsift evolution. The table below shows the polynomial coefficients for each redshift. 

|z| a$x^2$ | bx | c |
|--|--|--| --|
|2.3 | 18.9 +- 0.7 | 5.7 +- 0.1 | 0.96 +- 0.01 |
|2.4 | 11.5 +- 0.5 | 4.60 +- 0.05 | 0.97 +- 0.01 |
|2.6 | 13.0 +- 0.8 | 4.98 +- 0.06 | 0.99 +- 0.0  |




So, if we assume the relation at z=2.45 holds for all other redshifts, we would underestimate the $M_tomo, raw$ and therefore a larger offset is required in $M_{tomo,raw}$-$M_{tomo}$ relation. The plot below shows that :


![](https://i.imgur.com/TNNZjoN.png)



After corrcting this offset we would get the same estimators again :

For $M_{DM}$-$M_{tomo}$


|z | # Watersheds | slope | intercept |
|--|--|--| --|
|2.3|  | 0.45+-0.10 | 14.62+-0.03|
|2.4|  | 0.39+-0.06 | 14.54+-0.04|
|2.6|  | 0.34+-0.05 | 14.62+-0.03|

For $M_{DM}$-$M_{z=0}$:

|z | # Watersheds | slope | intercept |
|--|--|--| --|
|2.3| | 0.80+-0.1 | 14.69+-0.06|
|2.4|  | 0.72+-0.09 | 13.76+-0.05|
|2.6|  | 0.80+-0.09 | 13.72+-0.05|


I should say the scatter the estimators in mass-bins are also not changing.

## Conclusion : 
To me, it seems the redshift evolution of the  $\delta_F$-$\rho_{DM}$ relation is more important than the evolution of $\nu$ and $\kappa$ with redshift. 


I have basically equivalent to each plot on the draft for these two other redshifts on the noebooks [here](https://github.com/mahdiqezlou/LyTomo-Watershed). 