Force to fix sigma8 or A_s for Forecasts

[fab13]

TL;DR Summary

I would like to understand why we have to apply a correction on A_s (amplitude of primordial power spectrum) in the context of Forecasts in cosmology.

Hello,

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 $As$ (amplitude of primordial power spectrum), by a simpe relation of proportionality on the 2 $\sigma8$ (fiducial and linear), to get a new $As_{\text{modified}}$ 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 :

$As_{\text{modified}} = As_{\text{fiducial}}\,\bigg(\dfrac{\sigma8_{\text{fiducial}}}{\sigma8_{\text{linear}}}\bigg)^2$

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

My main issue of understanding :

1) Why have we got to do this correction ?, I mean to compute a new $As_{\text{modified}}$ that will give a new $\sigma8_{\text{non_linear}}$ equal to $\sigma8_{\text{fiducial}}$ (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) ?

Thanks in advance for your help, even a succinct answer would be nice.

Regards

 

[AndyW]

Hello,

Thank you for your question. The reason for doing this correction is to ensure that the resulting non-linear power spectrum is consistent with the fiducial (or linear) power spectrum. This is important because any discrepancies between the two power spectra could lead to biased results in the Fisher matrix.

To understand this better, let's consider the definition of $\sigma_8$. This parameter represents the amplitude of matter fluctuations on a scale of 8 Mpc/h. In the linear regime, this parameter is directly related to the amplitude of the primordial power spectrum, $A_s$. However, in the non-linear regime, the relation between $\sigma_8$ and $A_s$ becomes more complicated due to the effects of gravity and other non-linear processes.

So, when we compute the non-linear power spectrum using CAMB, we need to ensure that the resulting $\sigma_8$ is consistent with the fiducial (or linear) value. This is why we use the correction on $A_s$ before launching the non-linear CAMB regime. By doing this, we are essentially adjusting the primordial power spectrum to compensate for the non-linear effects and maintain the desired value for $\sigma_8$.

As for why we don't compute the non-linear regime directly, it is because the linear regime is computationally less expensive and provides a good starting point for the non-linear calculations. Additionally, by running the code twice (once in linear and once in non-linear regime), we can easily compare the results and ensure that the correction has been applied correctly.

I hope this helps clarify your understanding. Please let me know if you have any further questions.