NumCosmoMathFftlogSBesselJ

Logarithm fast fourier transform for a kernel given by the spatial correlation function multipoles.

This object computes the function (see NcmFftlog) $$Y_n = \int_0^\infty t^{\frac{2\pi i n}{L}} K(t) dt,$$ where the kernel are the spherical Bessel function of the first kind multiplied by a power law,

\begin{equation}\label{eq:kerneljl} K(t) = t^q j_{\ell}(t). \end{equation}

Note that the spherical Bessel function’s order, $\ell$ (NcmFftlogSBesselJ:ell), must be an integer number.

The spatial correlation function multipoles, $\xi_{\ell}^{(n)}(r)$, can be defined as (see [Matsubara (2004)][XMatsubara2004a] [arXiv])

\begin{equation}\label{eq:xi_multipoles} \xi_{\ell}^{(n)}(r) = \frac{(-1)^{n+\ell}}{r^{2n-\ell}} \int_{0}^{\infty} \frac{\mathrm{d} k}{2\pi^2} \frac{k^2}{k^{2n-\ell}} j_{\ell}(kr) P(k) \,\, . \end{equation} Where, $P(k)$ is the power spectrum (see NcmPowspec).

The multipoles integral can be written in the following format

\begin{equation} \xi_{\ell}^{(n)}(r) = \frac{(-1)^{n+\ell}}{r^2} \int_{0}^{\infty} \mathrm{d}k \, (kr)^{2-2n+\ell} \, j_{\ell}(kr) P(k) \,\, . \end{equation}

The object NcmFftlogSBesselJ can be used to evaluate the above integral in several ways. For example, the integral can be evaluated by defining the function (see NcmFftlog for more information) \begin{equation} F(k) = k^{2-2n+\ell} \, P(k) \end{equation} and the kernel \begin{equation} K(t) = j_{\ell}(t) \,\, . \end{equation} Where, $t=kr$ and $r^{(2-2n+\ell)}$ was taken out of the integral. Comparing this kernel with the one defined in Eq. \eqref{eq:kerneljl}, we have $q=0$.

But instead, one might choose another format for the function, \begin{equation} F(k) = k^{\ell} \, P(k) \end{equation} and the kernel \begin{equation} K(t) = t^{2-2n} \, j_{\ell}(t) \,\, , \end{equation} which evaluates the same integral, but now with $q=2-2n$, and in this case, the term $r^{\ell}$ was the one taken out of the integral. Therefore, the parameter $q$ is the power of the wavenumber $k$ times the distance $r$, $t=kr$, included to the kernel with the spherical Bessel function. Hereafter, it will be referred to as “spherical Bessel power” (NcmFftlogSBesselJ:q).

In general, $q=0$ is an accurate and fast choice to make, but it is interesting to perform tests to evaluate which kernel format fits best for each type of integral.

The NcmPowspecCorr3d object already evaluates Eq. \eqref{eq:xi_multipoles} for the case of the monopole, $n=\ell=0$, with support for redshift evolution.