Fast partial-sky spherical harmonic transforms

Abstract

We discuss in some details a novel algorithm for performing partial-sky spherical harmonic transforms (SHT), building on the Fourier-sphere method of Reinecke et al (2023) handling efficiently high numbers of arbitrary locations on the sphere. Our main motivations are Cosmic Microwave Background lensing from the South Pole Telescope, and the South Pole Observatory program targeting primordial gravitational waves from inflation, requiring high-resolution, numerically intensive work on small sky fractions. We achieve speed-up factors ranging from 3 to 10 on SPT-3G main field and BICEP3 deep footprint, and much more on smaller patches. More generally, the algorithm eliminates in our case study the usual disadvantages of arbitrary pixelisations in comparison to isolatitude pixelisations or flat-sky approximations, making it ideal for ambitious workflows that require repeated SHTs on limited sky regions.

Figures (5)

• Figure 1: Sky regions we consider here for our tests and that motivated this work. Black is SPT-3G main field SPT-3G:2025bzu, blue the BICEP3 polarization weight map contour enclosing the central 630 square degrees, and orange the South Pole Telescope observation of the Euclid Deep Field South (EDFS) SPT-3G:2025rxd. Their area is about 1500, 800 and 57 square degrees, respectively.
• Figure 2: Illustration of the synthesis general operation, for a scalar transform. The top panel shows the map to be evaluated on a set of points of sphere, starting from given spherical harmonic coefficients. In this case, the region of interest is the SPT-3G region as defined by the black contours on Fig. \ref{['fig:dfs_ortho']}. This is shown after placing the north pole $\theta=0$ in the center of the patch. Placing the pole in the center stretches the map to a maximum horizontally, reducing the band-limit in that direction (reducing $m_{\text{max}}$). The lower panel shows the Fourier map which is actually evaluated, after stretching and weighting in the vertical direction. The effective Fourier band-limit of the map is now reduced to a minimum in both directions, allowing faster evaluation. In the case of BICEP3, in blue, we can visibly stretch significantly more, improving performance further.
• Figure 3: Polar optimization and equatorial symmetry: Performance comparison of ducc0 spherical harmonic synthesis on regions of fixed co-latitude extent (here 20 degrees), as a function of the center co-latitude. Each region has the same number of pixels, which is set very small for this test. Shown are spin-0 and spin-2 transforms (the latter rescaled by a factor $1/2$) , using $\ell_{\text{max}} = 5000$. Owing to the irrelevance of $m$-values well above $\ell_{\text{max}} \sin \theta$, optimal performance is reached near the poles. The trough at the equator arises when the band crosses the equator: the underlying $\theta$ grid is such that it is symmetric with respect to the equator, and the calculation of a pair of symmetric rings has the same cost than for one member of the pair.
• Figure 4: Upper panel: Construction of our window, green, from the convolution of a top-hat, in blue, with a compactly supported bell-like function, $b$, orange (here the ES kernel, indistinguishable from a Gaussian on this figure), rescaled such that its support is $\Delta \Theta$. The support of the top-hat is $2\pi-\Theta$. In this construction, the rise and decay of the window is given by the cumulative function of the bell. In Fourier space, the window is in all cases the same universal function at the rescaled frequencies $k = n \Delta\Theta /2\pi$, given by the product of the two Fourier windows. The problem of optimizing the window is reduced to that of picking the best bell function. Lower panel: Fourier transform of the Slepian-alike Kaiser-Bessel bell, for two values of shape parameter (main lobe width) $\alpha$, as given by Eq. \ref{['eq:KB']}. Desirable is $\alpha$ as small as possible, reducing the width of the main-lobe and the corresponding apodization length. However, this is only possible as long as the side-lobe of the window is well below the desired tolerance $\epsilon$. This reasoning gives the direct correspondences for the best choice of window $\pi \alpha \sim |\ln \epsilon|$, and $2\alpha \sim \Delta \ell$.
• Figure 5: Top panel: One-dimensional power spectra $c_m$ along great circles for a few special cases of spherical power spectra $C_\ell$. This is for $\ell_{\text{max}} = 4000$, and with arbitrary normalization. Blue has a perfectly white spherical spectrum, and orange a perfectly scale-invariant one. Green shows the case where there is power only at $\ell_{\text{max}}$ and nowhere else. Red results from a $\Lambda$CDM polarization $E$-mode power spectrum. Lower Panel: Predicted (envelope of) the error, Eq.\ref{['eq:st']}, as function of $\theta$ in our baseline apodization procedure, for three spherical caps $\theta^*$ as indicated by the vertical dashed lines. The colored bands indicate the apodization region. The circles indicate the worst error inside the cap. This is for $\ell_{\text{max}} = 4000$ and desired tolerance $\epsilon=10^{-7}$ (dashed black), and for the case of a perfectly white spherical power spectrum.