Linear vs non linear matter power spectrum

[Jip]

Hi,

I have troubles understanding the difference between linear and non linear matter power spectrum. These words are commonly used in the litterature, but I have found no definitions yet.

My understanding is that there is one definition of the power spectrum for matter distribution, basically as the Fourier transform of the 2-point correlation function of the density field.

It covers both very large scales where linear approximation for the dynamics should be accurate, while at small scales the non linear gravitational collapse heavily impacts the dynamics.
Still, we have only one power spectrum.

So, what does linear or nonlinear power spectrum refers to? (in terms of maths, I mean)

Thanks for your help!

 

[Mordred]

If you graph a linear relation the graph will show a straight line whether or not its flat, diagonal or vertical. This is described as value a is directly or indirectly proportional to value b.
A non linear relation will be a curve, examples include bell curves, sinusoidal wave forms
etc. Hence non linear.
value a is exponentially proportional to b. This type of relation will give a bell curve

 

[Jip]

Yes, I'am aware of this, but I'm rather talking about mathematical description of large scale structure formation! I have found that details can be found in peeble's book, 1980. But I don't have it. Any other ref is welcome! :)

 

[Mordred]

Ah ok wasn't sure on what you were unclear on. I may have an article that will help but I'll have to dig around in my archives to find it.

 

[Chalnoth]

Hi,

I have troubles understanding the difference between linear and non linear matter power spectrum. These words are commonly used in the litterature, but I have found no definitions yet.

My understanding is that there is one definition of the power spectrum for matter distribution, basically as the Fourier transform of the 2-point correlation function of the density field.

It covers both very large scales where linear approximation for the dynamics should be accurate, while at small scales the non linear gravitational collapse heavily impacts the dynamics.
Still, we have only one power spectrum.

So, what does linear or nonlinear power spectrum refers to? (in terms of maths, I mean)

Thanks for your help!

Well, I think the general idea is to start using the non-linear power spectrum at the scales in which there is a noticeable difference between the linear power spectrum and the non-linear approximations. I doubt there is a firm limit on precisely where this occurs, but the best way to investigate it is probably to start investigating some nonlinear power spectrum approximations.

 

[Mordred]

This link covers the linear power spectrum vs the nonlinear in regards to LSS and peebles. Hope this helps

http://ned.ipac.caltech.edu/level5/March01/Coles/Coles2.html