# Force to fix $\sigma_{8}$ or $A_{s}$ for Forecasts

In the context of Forecasts with Fisher's formalism, I make vary cosmological parameters to compute the elements of the Fisher matrix.

First, I generate with CAMB code a linear power spectrum. Then, from this, I am computing $$\sigma_{8,\text{linear}}$$.

Secondly, Before relaunching the code CAMB in non-linear regime, I apply a correction on $$A_s$$ (amplitude of primordial power spectrum), by a simpe relation of proportionality on the 2 $$\sigma8$$ (fiducial and linear), to get a new $$A_{s,\text{modified}}$$ that will procude the same $$\sigma_{8,\text{fiducial}}$$ after the non-linear regime execution of CAMB code.

For example, if $$\sigma_{8,\text{linear}}$$ is higher than $$\sigma_{8,\text{fiducial}}$$, I do the following correction before launching the non-linear CAMB regime :

$$A_{s,\text{modified}} = A_{s,\text{fiducial}}\,\bigg(\dfrac{\sigma_{8,\text{fiducial}}}{\sigma_{8,\text{linear}}}\bigg)^2$$

So, $$A_{s,\text{modified}}$$ will be smaller than $$A_{s,\text{fiducial}}$$.

My main issue of understanding :

1) Why have we got to do this correction ?, I mean to compute a new $$A_{s,\text{modified}}$$ that will give a new $$\sigma_{8,\text{non_linear}}$$ equal to $$\sigma_{8,\text{fiducial}}$$ (or maybe $$\sigma8_{linear}$$, I am not sure from the relation of proportionality above) ?

For the moment, I think we want to keep a fixed value for $$\sigma_8$$ to be consistent with observations data where $$\sigma_8$$ doesn't change (does $$\sigma_8$$ always correspond implicitly to amplitude of fluctuations at $$z =0$$ ?) : but I am not sure to grasp this subtility.

2) By the way, why we don't compute only once directly the non-linear regime instead of launching 2 times the code CAMB (first in linear regime and second in non-linear regime with this correction between both) ?