Our universe is homogeneous and isotropic, and its perturbations obey
translation and rotation symmetry. In this work we develop Translation and
Rotation Equivariant Normalizing Flow (TRENF), a generative Normalizing Flow
(NF) model which explicitly incorporates these symmetries, defining the data
likelihood via a sequence of Fourier space-based convolutions and pixel-wise
nonlinear transforms. TRENF gives direct access to the high dimensional data
likelihood p(x|y) as a function of the labels y, such as cosmological
parameters. In contrast to traditional analyses based on summary statistics,
the NF approach has no loss of information since it preserves the full
dimensionality of the data. On Gaussian random fields, the TRENF likelihood
agrees well with the analytical expression and saturates the Fisher information
content in the labels y. On nonlinear cosmological overdensity fields from
N-body simulations, TRENF leads to significant improvements in constraining
power over the standard power spectrum summary statistic. TRENF is also a
generative model of the data, and we show that TRENF samples agree well with
the N-body simulations it trained on, and that the inverse mapping of the data
agrees well with a Gaussian white noise both visually and on various summary
statistics: when this is perfectly achieved the resulting p(x|y) likelihood
analysis becomes optimal. Finally, we develop a generalization of this model
that can handle effects that break the symmetry of the data, such as the survey
mask, which enables likelihood analysis on data without periodic boundaries.

本文提出了一种基于 Translation 和 Rotation 的 Equivariant Normalizing Flow (TRENF) 模型，该模型能够作为生成模型并为高维数据提供完整的似然函数，在宇宙学的应用中，通过使用宇宙学参数作为标签， TREFN 可以有效地进行似然分析，并且相对于传统的基于摘要统计的方法，具有更高的信息保留能力，同时还可以用于关于对称性破坏的数据分析。

适用于最优宇宙学分析的平移和旋转等变归一化流 (TRENF)

Translation and Rotation Equivariant Normalizing Flow (TRENF) for  Optimal Cosmological Analysis

We use nine years of gamma-ray data provided by the Fermi Large Area
Telescope (LAT) to systematically study the light curves of more than two
thousand active galactic nuclei (AGN) included in recent Fermi-LAT catalogs.
Ten different techniques are used, which are organized in an automatic
periodicity-search pipeline, in order to search for evidence of periodic
emission in gamma rays. Understanding the processes behind this puzzling
phenomenon will provide a better view about the astrophysical nature of these
extragalactic sources. However, the observation of temporal patterns in
gamma-ray light curves of AGN is still challenging. Despite the fact that there
have been efforts on characterizing the temporal emission of some individual
sources, a systematic search for periodicities by means of a full likelihood
analysis applied to large samples of sources was missing. Our analysis finds 11
AGN, of which 9 are identified for the first time, showing periodicity at more
than 4sigma in at least four algorithms. These findings will help in solving
questions related to the astrophysical origin of this periodic behavior.

本文利用费米大型区域望远镜提供的九年伽马射线数据，应用十种不同的自动周期性搜索技术，对两千多个活动星系核的光变曲线进行全面分析，总共发现了 11 个 AGN 表现出明显的周期性。这些发现可以帮助我们更好地了解超星系源的天体物理特性，解答与该周期性行为起源相关的问题。