2025/10/08 by Nickolas Kokron, Shi-Fan Chen, Kokron, Nickolas +1 · 1 voice
Physics and Astronomy · Mathematics · #Galaxies: Formation, Evolution, Phenomena #Cosmology and Gravitation Theories #Markov Chains and Monte Carlo Methods
paper · doi:10.33232/001c.162533
openalex created_date 2025/10/11 · openalex publication_date 2026/05/26 · openalex updated_date 2026/07/28
Control variates have seen recent interest as a powerful technique to reduce the variance of summary statistics measured from costly cosmological <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -body simulations. Of particular interest are the class of control variates which are analytically calculable, such as the recently introduced ‘Zeldovich control variates’ for the power spectrum of matter and biased tracers. In this work we present the construction of perturbative control variates in Eulerian and Lagrangian perturbation theory, and adopt the matter bispectrum as a case study. Eulerian control variates are analytically tractable for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> -point functions, but we show that their correlation with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -body <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> -point function decays at a rate proportional to the sum-of-squared wavenumbers, hampering their utility. We show that the Zeldovich approximation, while possessing an analytically calculable bispectrum, is less correlated at low- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>k</mml:mi> </mml:math> than its Eulerian counterpart. We introduce an alternative – the ‘shifted control variate’ – which can be constructed to have the correct tree-level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> -point function, is Zeldovich-resummed, and in principle has an analytically tractable bispectrum. We find that applying this shifted control variate to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math> matter bispectrum is equivalent to averaging over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>4</mml:mn> </mml:msup> </mml:math> simulations for the lowest- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>k</mml:mi> </mml:math> triangles considered. With a single <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -body simulation, for a binning scheme with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>1400</mml:mn> </mml:mrow> </mml:math> triangles from to , we obtain sub-2% precision for every triangle configuration measured. This work enables the development of accurate bispectrum emulators – a probe of cosmology well-suited to simulation-based modeling – and lays the theoretical groundwork to extend control variates for the entire <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> -point hierarchy.