High-Resolution Simulations of Cosmic Microwave Background non-Gaussian Maps in Spherical Coordinates

TL;DR

The paper tackles the challenge of simulating high-resolution CMB maps with primordial non-Gaussianity of the local type. It introduces a real-space, spherical-coordinate pipeline that directly generates correlated linear potential multipoles $\Phi^{\rm L}_{\ell m}(r)$ from white-noise inputs using precomputed filters $W_\ell(r,r_1)$, then combines them with a nonlinear term controlled by $f_{\rm NL}$ and contracts through the real-space transfer function $\Delta_\ell(r)$ to obtain $a_{\ell m}$ up to $\ell_{\max} \approx 3000$, all while maintaining a modest memory footprint. This approach reduces computational demands by avoiding a massive Fourier grid and by employing adaptive radial sampling around the last-scattering surface, enabling accurate treatment of the transfer function. The method is demonstrated with high-resolution simulations and is positioned as a flexible test-bed for non-Gaussian estimators, with potential extensions to include instrumental, foreground, and other secondary effects.

Abstract

We describe a new numerical algorithm to obtain high-resolution simulated maps of the Cosmic Microwave Background (CMB), for a broad class of non-Gaussian models. The kind of non-Gaussianity we account for is based on the simple idea that the primordial gravitational potential is obtained by a non-linear but local mapping from an underlying Gaussian random field, as resulting from a variety of inflationary models. Our technique, which is based on a direct realization of the potential in spherical coordinates and fully accounts for the radiation transfer function, allows to simulate non-Gaussian CMB maps down to the Planck resolution ($\ell_{\rm max} \sim 3,000$), with reasonable memory storage and computational time.

Figures (5)

• Figure 1: The coefficients $\Delta_\ell(r)$ evaluated for different values of $\ell$, assuming a $\Lambda$CDM cosmology ($h=0.65$, $\Omega_b = 0.05$, $\Omega_c=0.25$, $\Omega_\Lambda=0.7$). We put $r=c\eta$, where $\eta$ is the conformal time. In the chosen model, $c\eta_0 \sim 14.9 Gpc$ and $c\eta_* \sim 289$ Mpc, where $\eta_0$ and $\eta_*$ are the present day and the recombination conformal time respectively.
• Figure 2: The top panel shows the spherical Bessel function of order $150$. The bottom panel shows the filters $W_\ell(r,r_1)$ as functions of $r_1$, at fixed $r$, and three different values of $\ell$. We choose the spectral index $n=1$. Note that the filters become more and more peaked around $r_1 = r$ as $\ell$ increases. The Bessel functions oscillate very fast and have to be sampled in many more points than $W_\ell$ to have an accurate numerical integration. For this reason the use of $W_\ell$ instead of $j_\ell$ drastically reduces the needed CPU time, as explained in the text.
• Figure 3: One-point PDF of temperature fluctuations obtained from a single simulated map with $l_{max} = 3000$. Different panels show a comparison between the Gaussian realization ($f_{\rm NL}=0$) and the corresponding non-Gaussian one, for different values of $f_{\rm NL}$. Beam smearing has not been included here.
• Figure 4: The left-hand side of this panel shows the same Gaussian realization, smoothed by three different beams. From top to bottom, the FWHM of the beams is $7^{\circ}$ (COBE), 13 arcmin (WMAP) and 5 arcmin (Planck). The right-hand side shows three corresponding non-Gaussian realizations, obtained from the Gaussian one by adding a non-linear coupling parameter $f_{\rm NL} = 3000$ (such a high value of $f_{\rm NL}$ is chosen to make the non-Gaussian effects visible by eye). The model is a $\Lambda$CDM with primordial spectral index $n=1$. The primordial fluctuations have been COBE-normalized by CMBfast.
• Figure 5: In this panel we show the same Gaussian maps of Fig. \ref{['fig:maps']} on the left-hand side, while on the right-hand side we take $f_{NL}=-3000$. The remaining parameters are the same as in Fig. \ref{['fig:maps']}