Computing High Accuracy Power Spectra with Pico

TL;DR

The paper addresses the bottleneck of rapid, precise CMB power-spectrum and likelihood computations for high-dimensional cosmological models. It introduces Pico's second release, featuring Karhunen–Loève compression, region-based clustering, and per-cluster polynomial fits, augmented by high-accuracy CAMB training data and a hierarchical training scheme to focus accuracy near the likelihood peak. Pico delivers sub-cosmic-variance accuracy across a broad parameter space and achieves dramatic speedups over CAMB (up to $150\times$–$250{,}000\times$ depending on settings), enabling efficient exploration of posteriors for nonflat models with $w_{DE}$ allowed to vary. The work also extends Pico to tensor spectra and the matter transfer function, and leverages distributed computing (Cosmology@Home) to train on large datasets, with regression coefficients and training code released for broader adoption and future cosmological analyses.

Abstract

This paper presents the second release of Pico (Parameters for the Impatient COsmologist). Pico is a general purpose machine learning code which we have applied to computing the CMB power spectra and the WMAP likelihood. For this release, we have made improvements to the algorithm as well as the data sets used to train Pico, leading to a significant improvement in accuracy. For the 9 parameter nonflat case presented here Pico can on average compute the TT, TE and EE spectra to better than 1% of cosmic standard deviation for nearly all $\ell$ values over a large region of parameter space. Performing a cosmological parameter analysis of current CMB and large scale structure data, we show that these power spectra give very accurate 1 and 2 dimensional parameter posteriors. We have extended Pico to allow computation of the tensor power spectrum and the matter transfer function. Pico runs about 1500 times faster than CAMB at the default accuracy and about 250,000 times faster at high accuracy. Training Pico can be done using massively parallel computing resources, including distributed computing projects such as Cosmology@Home. On the homepage for Pico, located at http://cosmos.astro.uiuc.edu/pico, we provide new sets of regression coefficients and make the training code available for public use.

Figures (7)

• Figure 1: The plot shows the value of the temperature spectrum as a function of $\Omega_{\mathrm{b}}h^2$ at various $\ell$-values for a nonflat cosmology. The red ($+$) points correspond to to the default CAMB accuracies ($1$,$1$,$1$) and the green ($\Box$) points correspond to higher accuracy settings ($3$,$3$,$1$). While at low $\ell$ the power spectrum is smooth, the default accuracy becomes numerically noisy at higher $\ell$. This is one reason adding $\Omega_{\mathrm{k}}$ as a free parameter increases the difficulty in fitting the power spectrum. Also plotted, as a blue line, is the power spectrum computed by Pico trained on the $9$ parameter nonflat model discussed in section \ref{['sec:results']}.
• Figure 2: The plots show the percent error between the TT (left) and EE (right) power spectrum computed by CAMB at the default accuracy compared to those computed at high accuracy . Also shown is the percent error between the power spectra computed by Pico and the high accuracy CAMB spectra. This test was done on $25$ models all located within $25$ log-likelihoods of the WMAP peak.
• Figure 3: Value of $-\ln L_{\mathrm{WMAP}}$ as a function of $\Omega_{\mathrm{b}}h^2$ for a nonflat cosmology. Note that this is near the peak of the likelihood in the full space. The red ($+$) points correspond to the default CAMB accuracies ($1$,$1$,$1$) and the green ($\Box$) points corresponding to using higher accuracy settings ($3$,$3$,$1$). The blue line is the value computed by Pico trained over the $9$ parameter nonflat models discussed in section \ref{['sec:results']}. Note that using the default accuracy in CAMB gives a numerically noisy function, which can lead to variations of $1$ or more log-likelihoods, but Pico gives a smooth function through the high accuracy values.
• Figure 4: The above plots compare the performance of Pico with CAMB at high accuracy settings for 9 parameter nonflat models with $w_{\mathrm{DE}}\ne 1$. Pico was trained using the hierarchy method described in section \ref{['subsec:hierarchy']} and the test set consists of $2000$ points within $25$ log-likelihoods of the WMAP peak. The top two rows show the error compared with CAMB in units of cosmic standard deviation for the TT, TE and EE power spectra at high $\ell$ (top) and low $\ell$ (center). The bottom row shows the error in the BB spectra in units of the cosmic standard deviation and the percent error in the matter transfer function. The two lines on each plot denote the mean error and the error bar that bounds $99\%$ of the test set. We note that much of the error at low $\ell$ in the EE spectra is due to the $1\%$ of models with extremely low power over this range. These spectra are too small to detect even with Planck.
• Figure 5: The plots are same as those in Figure \ref{['fig:openw']} except here Pico was trained and tested over models sampled uniformly from the box defined by Table \ref{['tbl:param_bounds']}. Even over this larger region Pico can compute the power spectrum in $99\%$ of the test cases to better than $5\%$ of cosmic standard deviation over most $\ell$ and is never worse than $0.7$ cosmic standard deviation.