Compressed ‘CMB-lite’ Likelihoods Using Automatic Differentiation

The compression of multi-frequency cosmic microwave background (CMB) power spectrum measurements into a series of foreground-marginalised CMB-only band powers allows for the construction of faster and more easily interpretable ‘lite’ likelihoods. However, obtaining the compressed data vector is computationally expensive and yields a covariance matrix with sampling noise. In this work, we present an implementation of the CMB-lite framework relying on automatic differentiation. The technique presented reduces the computational cost of the lite likelihood construction to one minimisation and one Hessian evaluation, which run on a personal computer in about a minute. We demonstrate the efficiency and accuracy of this procedure by applying it to the differentiable SPT-3G 2018 TT/TE/EE likelihood from the candl library. We find good agreement between the marginalised posteriors of cosmological parameters yielded by the resulting lite likelihood and the reference multi-frequency version for all cosmological models tested; the best-fit values shift by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mn>0.1</mml:mn> <mml:mspace width="0.167em"/> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>σ</mml:mi> </mml:math> is the width of the multi-frequency posterior, and the inferred parameter error bars match to within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mn>10</mml:mn> <mml:mi>%</mml:mi> </mml:mrow> </mml:math> . We publicly release the SPT-3G 2018 TT/TE/EE lite likelihood and a python notebook showing its construction at https://github.com/Lbalkenhol/candl .