## Periodic Reporting for period 3 - PUNCA (Preparing for Unveiling the Nature of the Cosmic Acceleration)

Reporting period: 2020-03-01 to 2021-08-31

This project aims to address the question about the origin and nature of the observed accelerated Hubble expansion of our Universe.

The question is one of the most challenging open questions in modern cosmology, and the astronomical community have invested heavily in it, including collective efforts from researchers to tackle the question from different points of view, and the construction and launch of expensive observational missions to collect data for model tests. The ongoing interest and effort from the astronomical community, however, are not the only reason why this is important: because the accelerated Hubble expansion does not seem to fit into the cosmological model that is based on our knowledge in gravity and particle physics, it has been widely considered as an indication that this knowledge may be incomplete, and understanding it will hopefully brings knowledge about new physics.

In this project, the overall objective is to better our understanding of the various theoretical models of gravitation which have been proposed to explain the accelerated Hubble expansion, to identify observational probes which are best suited to test such models, and thereby to prepare for maximally exploiting the next generation of astronomical surveys to understand the nature of it.

The question is one of the most challenging open questions in modern cosmology, and the astronomical community have invested heavily in it, including collective efforts from researchers to tackle the question from different points of view, and the construction and launch of expensive observational missions to collect data for model tests. The ongoing interest and effort from the astronomical community, however, are not the only reason why this is important: because the accelerated Hubble expansion does not seem to fit into the cosmological model that is based on our knowledge in gravity and particle physics, it has been widely considered as an indication that this knowledge may be incomplete, and understanding it will hopefully brings knowledge about new physics.

In this project, the overall objective is to better our understanding of the various theoretical models of gravitation which have been proposed to explain the accelerated Hubble expansion, to identify observational probes which are best suited to test such models, and thereby to prepare for maximally exploiting the next generation of astronomical surveys to understand the nature of it.

The project has 3 working groups (WG), which are respectively devoted to testing model using 3 main cosmological probes: weak gravity lensing (WG1), clustering of galaxies (WG2) and galaxy clusters (WG3).

In WG1, one of the tasks is to run a large number of simulations of various models, to measure one of the main observables of weak lensing: the convergence power spectrum. The original plan was slightly changed as during the preparation we found some other interesting observables that can be extracted from lensing observations. A lot of effort was spent on the study of weak lensing by cosmic voids, and a novel probe developed in this process is cosmic voids detected from 2D weak lensing maps, which were found to have a greater potential for testing models using future lensing data. We also looked at the statistics of weak lensing peaks, which had not been studied in great details before, and discovered some surprisingly simple relations they follow. Consequently, the details of the planned simulations were modified from the original proposal so that they can be suitable for studying these new probes.

In WG2, our main goal is to use measurements of galaxy clustering to test models, an approach that relies on three building blocks: (1) accurately predicting the clustering of dark matter in real space, usually quantified by the matter power spectrum or correlation function, (2) understanding how the clustering of galaxies differs from that of dark matter, given that the former are biased tracers of the latter, and (3) understanding how best to model redshift space distortions (RSD), which can mislead cosmologists into obtaining wrong galaxy positions from observations. We have tackled (1) using three approaches: (1a) a halo model approach to predict the matter power spectrum, by improving our understanding of the different ingredients of the halo model, (1b) a reaction method to predict the matter power spectrum for dark energy and modified gravity models at precent accuracy, and (1c) an emulation approach by running simulations for a selection of models and doing interpolation to get the matter power spectra for models which are not simulated. We have tackled (3) using three approaches: (3a) ay measuring RSD from simulations of modified gravity models, (3b) an improvement of the traditional Gaussian streaming model for RSD based on a more accurate modelling of the probability distribution of galaxy pairwise velocities, and (3c) a novel backward modelling approach, with an iterative reconstruction method to remove the RSD effect from observed galaxy clustering for galaxy pairs separated by 20Mpc or more. The reconstruction approach has also been applied to study galaxies bias for building block (2). Along this process, we have developed the first self-consistent galaxy formation simulations in modified gravity theories. Furthermore, a few papers were published about the use of the so-called marked correlation function to test gravity.

In WG3, we have followed the work plan to develop a novel framework to test gravity using the abundance of galaxy clusters. We have analysed a large number of simulations for a popular modified gravity model (chameleon f(R) gravity), to study the most important properties of dark matter haloes, including their masses, density profiles and abundances. These analyses have led to some surprisingly simple and accurate fitting functions for these properties. Currently, we are working on the planned hydrodynamical cluster simulations, and we have added a new feature to them, namely the simulations will be run using initial conditions for our local Universe (i.e. rather than a random initial condition, as usually done in the literature, which is only `statistically equivalent' to the real Universe) obtained through reconstruction; we expect this will make the simulations more realistic and relevant, and more readily comparable to observational data.

In WG1, one of the tasks is to run a large number of simulations of various models, to measure one of the main observables of weak lensing: the convergence power spectrum. The original plan was slightly changed as during the preparation we found some other interesting observables that can be extracted from lensing observations. A lot of effort was spent on the study of weak lensing by cosmic voids, and a novel probe developed in this process is cosmic voids detected from 2D weak lensing maps, which were found to have a greater potential for testing models using future lensing data. We also looked at the statistics of weak lensing peaks, which had not been studied in great details before, and discovered some surprisingly simple relations they follow. Consequently, the details of the planned simulations were modified from the original proposal so that they can be suitable for studying these new probes.

In WG2, our main goal is to use measurements of galaxy clustering to test models, an approach that relies on three building blocks: (1) accurately predicting the clustering of dark matter in real space, usually quantified by the matter power spectrum or correlation function, (2) understanding how the clustering of galaxies differs from that of dark matter, given that the former are biased tracers of the latter, and (3) understanding how best to model redshift space distortions (RSD), which can mislead cosmologists into obtaining wrong galaxy positions from observations. We have tackled (1) using three approaches: (1a) a halo model approach to predict the matter power spectrum, by improving our understanding of the different ingredients of the halo model, (1b) a reaction method to predict the matter power spectrum for dark energy and modified gravity models at precent accuracy, and (1c) an emulation approach by running simulations for a selection of models and doing interpolation to get the matter power spectra for models which are not simulated. We have tackled (3) using three approaches: (3a) ay measuring RSD from simulations of modified gravity models, (3b) an improvement of the traditional Gaussian streaming model for RSD based on a more accurate modelling of the probability distribution of galaxy pairwise velocities, and (3c) a novel backward modelling approach, with an iterative reconstruction method to remove the RSD effect from observed galaxy clustering for galaxy pairs separated by 20Mpc or more. The reconstruction approach has also been applied to study galaxies bias for building block (2). Along this process, we have developed the first self-consistent galaxy formation simulations in modified gravity theories. Furthermore, a few papers were published about the use of the so-called marked correlation function to test gravity.

In WG3, we have followed the work plan to develop a novel framework to test gravity using the abundance of galaxy clusters. We have analysed a large number of simulations for a popular modified gravity model (chameleon f(R) gravity), to study the most important properties of dark matter haloes, including their masses, density profiles and abundances. These analyses have led to some surprisingly simple and accurate fitting functions for these properties. Currently, we are working on the planned hydrodynamical cluster simulations, and we have added a new feature to them, namely the simulations will be run using initial conditions for our local Universe (i.e. rather than a random initial condition, as usually done in the literature, which is only `statistically equivalent' to the real Universe) obtained through reconstruction; we expect this will make the simulations more realistic and relevant, and more readily comparable to observational data.

The project has led to progresses beyond the state of the art in all working groups.

In WG1, we have proposed and tested the uses of weak gravitational lensing by cosmic voids identified from lensing maps, and of the correlation function of peaks of weak lensing maps, to constrain cosmological models, both of which were new ideas which had rarely been touched previously. The new simulations are ongoing now, and we expect that by the end of this grant we will be able to get a much clearer idea about how much additional constraining power these new probes will contribute to the test of models for the accelerated Hubble expansion.

In WG2, there are a number of noteworthy highlights. The reaction approach has been found to be able to predict the matter power spectrum for a wide range of cosmological models at percent accuracy. The improved streaming model for RSD now gives us accurate predictions of the galaxy correlation function down to a galaxy separation of ~10Mpc, while the standard Gaussian streaming model or perturbation methods usually fail at much larger scales. These approaches will be further developed and we expect that by the end of this grant they will be fully ready to be applied to model tests using real data.

Some of the results by WG2 were published in two Nature Astronomy papers: in the first we used a set of high-resolution simulations to study galaxy clustering in f(R) gravity and placed a strong constraint on it; in the second we reported the first self-consistent galaxy formation simulations in f(R) gravity, and studied various galactic properties.

In WG3, we found simple fitting functions to some of the most important halo properties in f(R) gravity, and these greatly simplified our understanding of the nonlinear physical processes involved in halo formation in this popular gravity model. The ongoing works of cluster simulations in modified gravity are again likely to be the first of their kind. These simulations will be used to study the cluster scaling relations in modified gravity, which will then be applied into our newly-developed framework for testing gravity using cluster abundance. We expect this to be finished on time before the end of this grant.

In WG1, we have proposed and tested the uses of weak gravitational lensing by cosmic voids identified from lensing maps, and of the correlation function of peaks of weak lensing maps, to constrain cosmological models, both of which were new ideas which had rarely been touched previously. The new simulations are ongoing now, and we expect that by the end of this grant we will be able to get a much clearer idea about how much additional constraining power these new probes will contribute to the test of models for the accelerated Hubble expansion.

In WG2, there are a number of noteworthy highlights. The reaction approach has been found to be able to predict the matter power spectrum for a wide range of cosmological models at percent accuracy. The improved streaming model for RSD now gives us accurate predictions of the galaxy correlation function down to a galaxy separation of ~10Mpc, while the standard Gaussian streaming model or perturbation methods usually fail at much larger scales. These approaches will be further developed and we expect that by the end of this grant they will be fully ready to be applied to model tests using real data.

Some of the results by WG2 were published in two Nature Astronomy papers: in the first we used a set of high-resolution simulations to study galaxy clustering in f(R) gravity and placed a strong constraint on it; in the second we reported the first self-consistent galaxy formation simulations in f(R) gravity, and studied various galactic properties.

In WG3, we found simple fitting functions to some of the most important halo properties in f(R) gravity, and these greatly simplified our understanding of the nonlinear physical processes involved in halo formation in this popular gravity model. The ongoing works of cluster simulations in modified gravity are again likely to be the first of their kind. These simulations will be used to study the cluster scaling relations in modified gravity, which will then be applied into our newly-developed framework for testing gravity using cluster abundance. We expect this to be finished on time before the end of this grant.