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### Explanations on making correction on A_s between linear and non-linear regime

Posted: December 31 2019
Hello,

I take as reference the following link : viewtopic.php?t=475

I would like to understand why one has to to a correction on between the linear and non-linear regime of CAMB.

Actually, in my code, I launch a first time CAMB in linear regime in order to produce 100 files of matter power spectrum ( one 's file for each of the 100 redshifts considered).

Then, I compute from these matter power spectrum : I call this .

So, I make the following computation to on and so calculate its new value :

and represent the fiducial/reference values (As far I know, and

After, I launch again CAMB but now, in non-linear regime (with Takabird Halofit model) by using .

Finally my question is : why have we got to make this correction on between linear ans non-linear model launching.

Is this for keeping like a parameter of calibration for linear regime ? But this is not which is constant, like for , I mean I can't see which quantity remains constant between the 2 CAMB executions (linear and non linear).

If asomeone could explain me what is the goal of this correction on between the 2 runs of CAMB. This would allow me to grasp better the subtilities of the linear and non-linear model.

Kind regards.

### Re: Explanations on making correction on A_s between linear and non-linear regime

Posted: January 02 2020
$\sigma_8$ should be definition be linear and hence independent of the non-linear model as long as you are only changing $A_s$ (keeping $H_0$ etc fixed)

### Re: Explanations on making correction on A_s between linear and non-linear regime

Posted: January 04 2020
Thanks for your quick answer. The current issue on all this post is about the non-linear Takabird Halofit model (option 4 in params.ini).

I have doubts about the current relation I am using, i.e :

Indeed, it is indicated on this link viewtopic.php?t=475 that :
if you set get_transfer = T, \sigma_8 will be computed for whatever your input scalar_amp(1). Since \sigma_8^2 scales like scalar_amp(1)
So, if scalar_amp(1) is proportional to , shouldn't we have rather the relation :

and finally, we would have instead :

Question 1) If this is not a right relation, could you tell me why one must have :

Question 2) Does it mean that if is higher than , then one has to make decrease the next with equation ?

Question 3)

Does it mean into my code : After a first run (linear) by taking , I compute, from , the value.

After, I re-run CAMB in non-linear regime, by using the same initial value.

Then, into a loop on different little steps, I make vary with this step the cosmological parameter on which I want to compute the corresponding (in my case, I study ).

I re-run a third time CAMB (the first time into the loop), by setting again the input value : I compute from output the value and Now, I make the following computation (see equation above) :

Finally, always on this loop on the steps making vary , I launch CAMB a last time (the second time into the loop) with , i.e with the corrected previous value of as input value.

And I store all coming from this last run and associated to the current step of variation on value.

I repeat the two last runs of CAMB for each different step iteration.

Question 4) Does all this procedure seem to be correct (first 2 steps with run linear and non-linear regime, and correction achieved after the first linear run into loop) ?

I hope having been cleared on this issue since with Takabird HALOFIT non-linear model, the derivative of as respect of presents, at little scales (see attachment for k=0.1 and k=0.2), a cut-off (a sudden peak) at roughly 1e-3, for a chosen redshift . At large scales, it is perfectly stable (a large flatness from 1e-6 to 1e-1) (see attachment for k = 0.001). I would like to find the origin of this peak for k > 0.1.
dPk_dOm_0.001.png (7.88 KiB) Viewed 1703 times
dPk_dOm_0.1.png (8.78 KiB) Viewed 1703 times
dPk_dOm_0.2.png (9.66 KiB) Viewed 1703 times
If you want further informations or more detailled explanations, feel free to let me know it.

Regards