Source and potential field do not correspond

[Lagrang3]

pw-phi-plus.pdf

[Lagrang3]

[Lagrang3]

As seen in the picture. The power spectrum of T00 and k^2 Phi do not match at the small and again at the large scales. Why?

[Lagrang3]

This problem is obtained at this commit 716f5d25ae1, running Gadget.

[Lagrang3]

First idea: produce a minimal working gevolution main, that loads a snapshot (or produces one), computes the potentials, prints the power spectrum and exits.

[Lagrang3]

Next. Let us find the exact conversion factor from T00 to Phi. The first key functions to look at are relativistic_pm::sample and relativistic_pm::compute_potential

[Lagrang3]

Question: if $\tilde A(\vec k) = \tilde B (\vec k) / (|\vec k|)$, then what is the relation between $P(\tilde A, k)$ and $P(\tilde B,k)$?

[Lagrang3]

In gevolution, the T00 component of the energy momentum tensor is substracted the background value of T00 and multiplied by a factor fourpiG dx^2 / a, where fourpiG = 4 pi G in gevolution units, which is numerically equal to 1.5 L^2/c^2, where L is the boxsize in Mpc/h and c is the speed of light in km/s. dx = L/N, where N is the number of grid points per dimension. Out of the previous operation gevolution obtains a new field S00 which becomes the source of the potential Phi. In the Poisson equation solver Phi = 1/k^2 S00 / N^3 / dx^2, where k = 2 pi n being n the integer vector identifying the Fourier mode.

[Lagrang3]

The previous power spectrum corresponds to Pk(T00) and const k^4 Pk(Phi), and they don't match. The proportionality constant is correct, but their shape it just isn't the same. The issue is that Pk( f(k) A ) is different from f(k)^2 Pk(A).

[Lagrang3]

In the PowerI4 documentation by Emiliano Sefussati, we find that the power spectrum of a field A is defined as P_k(A) = kf^3 / |\Omegak| \sum{q \in \Omega_k} |A_k|^2 where kf = 2 pi/L is the fundamental mode, \Omega_k is a set of modes such that q\in \Omega_k if and only if k-dk/2 <= |q| < k+dk/2, and dk is a bin size. With that in mind, given a mode q we can find which set \Omega_k it belongs: k/dk = \floor(1/2 + |q|/dk) where |q| =\sqrt(q_1^2 + q_2^2 + q_3^3)

[Lagrang3]

Then consider f(|q|) a function of the modulo of the mode P_k(f A) = kf^3 / |\Omegak| \sum{q\in \Omega_k} |A_q|^2 |f(|q|)|^2 if |q| had the same value for every q \in \Omega_k, for instance when dk is very small, q\in\Omega_k implies |q|=k, then P_k(f A) = kf^3 / |\Omegak| |f(k)|^2 \sum{q\in\Omega_k} |A_q|^2.

[Lagrang3]

However, if we have dk = kf, we are actually stating that the bins are shells thick enough as the distance between two consecutive nodes, and yet we will show that for the different values of q\in\Omega_k there will be non unique values of |q|.

[Lagrang3]

Let us count for example some modes and their bins for dk = kf	q/kf	degeneracy	q^2/kf^2	k/kf
(0,0,0)	1	0	0
(0,0,1)	3	1	1
(0,1,1)	6	2	1
(1,1,1)	4	3	2
(0,0,2)	3	4	2
(0,1,2)	12	5	2
(1,1,2)	12	6	2

[Lagrang3]

so several values of q^2 will correspond to the same value of k, and we cannot have P_k(f A) = f(k)^2 P_k(A)

[Lagrang3]

One possible definition of the power spectrum that is bilinear could be the following, P_k(A,B) = 1/|Nk| \sum{q \in N_k} A*(q) B(q) where N_k = { q | q^2 = k^2 },

then P_k(A f, B g) = f*(k) g(k) P_k(A,B)

[Lagrang3]

with this new definition, we obtain a more wigling power spectrum because there are less points on each bins, but P_k is bilinear and the power spectra of the source and the potential are related by a factor k^4

[Lagrang3]

Smoothing the power spectra with bins of size 1 we obtain. This is a gevolution (1 step) simulation L=500 Mpc/h, N = 256^3 particles.

[Lagrang3]

My guess is that with a bigger boxsize we will be able to start noticing deviations between T00 and k^2 Phi at large scales due to the Hubble terms in the Einstein equation for Phi. For example, for L=2000 Mpc/h, N=256^3 we find the following