Y. Rasera , M-A. Breton, P-S. Corasaniti, J. Allingham, F. Roy, V. Reverdy, T.Pellegrin, S. Saga, A. Taruya, S. Agarwal, and S. Anselmi

**General Relativistic effects on the clustering of matter in the universe provide a sensitive probe of cosmology and gravity theories that can be tested with the upcoming generation of galaxy surveys. Here, we present a suite of large volume high-resolution N-body simulations specifically designed to generate light-cone data for the study of relativistic effects on lensing-matter observables. RayGalGroupSims (or in short RayGal) consists of two N-body simulations of $(2625\,h^{-1}\,{\rm Mpc})^3$ volume with $4096^3$ particles of a standard flat $\Lambda$CDM model and a non-standard $w$CDM phantom dark energy model. Light-cone data from the simulations have been generated using a parallel ray-tracing algorithm that has accurately solved billion geodesic equations. Catalogues and maps with relativistic weak-lensing which include post-Born effects, magnification bias (MB) and redshift space distortions (RSD) due to gravitational redshift, Doppler, transverse Doppler, Integrated Sachs-Wolfe/Rees-Sciama effects, are publicly released. Using this dataset, we are able to reproduce the linear and quasi-linear predictions from the Class relativistic code for the 10 (cross-)power spectra (3$\times$2 points) of the matter density fluctuation field and the gravitational convergence at $z=0.7$ and $z=1.8$. We find 1-30\% level contribution from both MB and RSD to the matter power spectrum, while the Fingers-of-God effect is visible at lower redshift in the non-linear regime. MB contributes at the $10-30\%$ level to the convergence power spectrum leading to a deviation between the shear power-spectrum and the convergence power-spectrum. MB also plays a significant role in the galaxy-galaxy lensing by decreasing the density-convergence spectra by $20\%$, while coupling non-trivial configurations (such as the one with the convergence at the same or even lower redshift than the density field).**

- Breton, Michel-Andrès;
- Rasera, Yann;
- Taruya, Atsushi;
- Lacombe, Osmin;
- Saga, Shohei

**The apparent distribution of large-scale structures in the Universe is sensitive to the velocity/potential of the sources as well as the potential along the line of sight through the mapping from real space to redshift space (redshift-space distortions, RSD). Since odd multipoles of the halo cross-correlation function vanish when considering standard Doppler RSD, the dipole is a sensitive probe of relativistic and wide-angle effects. We build a catalogue of ten million haloes (Milky Way size to galaxy-cluster size) from the full-sky light cone of a new `RayGalGroupSims’ N-body simulation which covers a volume of (2.625 h ^{-1} Gpc)^{3} with 4096^{3} particles. Using ray-tracing techniques, we find the null geodesics connecting all the sources to the observer. We then self-consistently derive all the relativistic contributions (in the weak-field approximation) to RSD: Doppler, transverse Doppler, gravitational, lensing and integrated Sachs-Wolfe. It allows us, for the first time, to disentangle all contributions to the dipole from linear to non-linear scales. At large scale, we recover the linear predictions dominated by a contribution from the divergence of neighbouring line of sights. While the linear theory remains a reasonable approximation of the velocity contribution to the dipole at non-linear scales it fails to reproduce the potential contribution below 30-60 h^{-1} Mpc (depending on the halo mass). At scales smaller than ∼10 h^{-1} Mpc, the dipole is dominated by the asymmetry caused by the gravitational redshift. The transition between the two regimes is mass dependent as well. We also identify a new non-trivial contribution from the non-linear coupling between potential and velocity terms.**

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Magrathea-Pathfinder: a 3D adaptive-mesh code for geodesic ray-tracing in N-body simulations

M-A. Breton, V. Reverdy

We present \textsc{Magrathea-Pathfinder}, a ray-tracing framework which accurately reconstructs the past light-cone of an observer in numerical simulations. Our code directly computes the 3D trajectory of light rays through the null geodesic equations, with the weak-field limit as its only approximation. Therefore, it does not rely on any other standard ray-tracing approximations such as plane-parallel, Born or multiple-lens. \textsc{Magrathea-Pathfinder} fully takes advantage of the small-scale clustering of matter by using adaptive integration steps and interpolation within an Adaptive-Mesh Refinement (AMR) structure to accurately account for the non-linear regime of structure formation. It uses MPI parallelization, C\texttt{++}11 \texttt{std::thread} multithreading, and is optimised for High-Performance Computing (HPC) as a post-processing tool for very large $N$-body simulations. In this paper, we describe how to produce realistic cosmological observables from numerical simulation using ray-tracing techniques, in particular the production of simulated catalogues and maps which accounts for all the effects at first order in metric perturbations (such as peculiar velocities, gravitational potential, Integrated Sachs-Wolfe, time delay, gravitational lensing, etc\ldots). We perform convergence tests of our gravitational lensing algorithms and conduct performance tests of the null geodesic integration procedures. \textsc{Magrathea-Pathfinder} provides sophisticated ray-tracing tools to make the link between real space ($N$-body simulations) and light-cone observables. This should be useful to refine existing cosmological probes and to build new ones beyond standard assumptions in order to prepare for next-generation large-scale structure surveys.

### Forecasting cosmological parameter constraints using multiple sparsity measurements as tracers of the mass profiles of dark matter haloes

Corasaniti, P. S.; Le Brun, A. M. C.; Richardson, T. R. G.; Rasera, Y.; Ettori, S.; Arnaud, M.; Pratt, G. W.

The dark matter halo sparsity, i.e. the ratio between spherical halo masses enclosing two different overdensities, provides a non-parametric proxy of the halo mass distribution that has been shown to be a sensitive probe of the cosmological imprint encoded in the mass profile of haloes hosting galaxy clusters. Mass estimations at several overdensities would allow for multiple sparsity measurements, which can potentially retrieve the entirety of the cosmological information imprinted on the halo profile. Here, we investigate the impact of multiple sparsity measurements on the cosmological model parameter inference. For this purpose, we analyse N-body halo catalogues from the Raygal and M2Csims simulations and evaluate the correlations among six different sparsities from spherical overdensity halo masses at Δ = 200, 500, 1000, and 2500 (in units of the critical density). Remarkably, sparsities associated to distinct halo mass shells are not highly correlated. This is not the case for sparsities obtained using halo masses estimated from the Navarro-Frenk-White (NFW) best-fitting profile, which artificially correlates different sparsities to order one. This implies that there is additional information in the mass profile beyond the NFW parametrization and that it can be exploited with multiple sparsities. In particular, from a likelihood analysis of synthetic average sparsity data, we show that cosmological parameter constraints significantly improve when increasing the number of sparsity combinations, though the constraints saturate beyond four sparsity estimates. We forecast constraints for the CHEX-MATE cluster sample and find that systematic mass bias errors mildly impact the parameter inference, though more studies are needed in this direction.

### Galaxy cluster number counts with individual lensing mass estimates: Forecasts for Euclid

Artis, E.; Melin, J. -B.; Bartlett, J. G.; Murray, Calum; Euclid Consortium

The ΛCDM model is gradually becoming challenged by observational data. Upcoming cosmological surveys will increase the number of detected galaxy clusters by several orders of magnitude. Therefore, clusters will shortly provide precise cosmological constraints and improve our understanding of structure formation in the Universe. In the following, we present a cluster likelihood based on individual weak lensing mass estimates and forecast Euclid’s performances within this framework. We use a matched filter for weak lensing mass estimation and model its characteristics with a set of simulations. We use the Flagship N-body simulation to emulate the expected cluster mass distribution of a Euclid-like sample and test our statistical framework against it. Finally, we simultaneously constrain the observable-mass relation and the cosmological parameters.

### Accurate modelling of extragalactic microlensing by compact objects

Bosca, V., Fleury, P., & Garcia-Bellido, J.

Microlensing of extragalactic sources, in particular the probability of significant amplifications, is a potentially powerful probe of the abundance of compact objects outside the halo of the Milky Way. Accurate experimental constraints require an equally accurate theoretical model for the amplification statistics produced by such a population. In this article, we argue that the simplest (strongest-lens) model does not meet this demanding requirement. We thus propose an elaborate practical modelling scheme for extragalactic microlensing. We derive from first principles an expression for the amplification probability that consistently allows for: (i) the coupling between microlenses; (ii) realistic perturbations from the cosmic large-scale structure; (iii) extended-source corrections. An important conclusion is that the external shear applied on the dominant microlens, both by the other lenses and by the large-scale structure, is practically negligible. Yet, the predictions of our approach can still differ by a factor of a few with respect to existing models of the literature. Updated constraints on the abundance of compact objects accounting for such discrepancies may be required.

On the impact of lensing magnification on redshift-space galaxy clustering analysis

- Breton, Michel-Andrès;
- de la Torre, Sylvain;
- Piat, Jade

We study the impact of lensing magnification on the observed redshift-space three-dimensional galaxy clustering. For this, we use the RayGal suite of *N*-body simulations, from which we extract samples of dark matter particles and haloes in the redshift regime of interest for future large redshift surveys. Several magnitude-limited samples are built that reproduce various levels of magnification bias ranging from *s*=0 to *s*=1.2, where *s* is the logarithmic slope of the cumulative magnitude number counts, in three redshift intervals within 1<*z*<1.95. We study the two-point correlation function multipole moments in the different cases, in the same fashion as we would do on real data, and investigate how well one can recover the growth rate of structure parameter. In the analysis, we use an hybrid model that combines non-linear redshift-space distortions and linear curved-sky lensing magnification. We find that the growth rate is underestimated when magnification bias is not accounted for in the modelling. This bias becomes non-negligible for *z*>1.3 and can reach 10\% at *z*≃1.8, depending on the properties of the target sample. However, adding the lensing linear correction allows us to recover an unbiased estimation of the growth rate in most cases, even when the fiducial cosmology is not that of the data. Our results show the importance of knowing in advance *s* instead of letting this parameter free with flat priors, since in this case the error bars significantly increase. For large values of *s*, we find that one has to be careful with the weak-lensing limit, as it may not be a good approximation at high redshift.

Detectability of the gravitational redshift effect from the asymmetric galaxy clustering

- Saga, Shohei;
- Taruya, Atsushi;
- Breton, Michel-Andrès;
- Rasera, Yann

It has been recently recognized that the observational relativistic effects, mainly arising from the light propagation in an inhomogeneous universe, induce the dipole asymmetry in the cross-correlation function of galaxies. In particular, the dipole asymmetry at small scales is shown to be dominated by the gravitational redshift effects. In this paper, we exploit a simple analytical description for the dipole asymmetry in the cross-correlation function valid at quasi-linear regime. In contrast to the previous model, a new prescription involves only one dimensional integrals, providing a faster way to reproduce the results obtained by Saga et al. (2020). Using the analytical model, we discuss the detectability of the dipole signal induced by the gravitational redshift effect from upcoming galaxy surveys. The gravitational redshift effect at small scales enhances the signal-to-noise ratio (S/N) of the dipole, and in most of the cases considered, the S/N is found to reach a maximum at *z*≈0.5. We show that current and future surveys such as DESI and SKA provide an idealistic data set, giving a large S/N of 10∼20. Two potential systematics arising from off-centered galaxies are also discussed (transverse Doppler effect and diminution of the gravitational redshift effect), and their impacts are found to be mitigated by a partial cancellation between two competitive effects. Thus, the detection of the dipole signal at small scales is directly linked to the gravitational redshift effect, and should provide an alternative route to test gravity.

- Corasaniti, Pier-Stefano;
- Sereno, Mauro;
- Ettori, Stefano

In recent years, the availability of large, complete cluster samples has enabled numerous cosmological parameter inference analyses using cluster number counts. These have provided constraints on the cosmic matter density Ω_{m} and the amplitude of matter density fluctuations σ_{8} alternative to that obtained from other standard probes. However, systematics uncertainties, such as the mass calibration bias and selection effects, may still significantly affect these data analyses. Hence, it is timely to explore other proxies of galaxy cluster cosmology that can provide cosmological constraints complementary to those obtained from cluster number counts. Here we use measurements of the cluster sparsity from weak-lensing mass estimates of the LC^{2}-single and HSC-XXL cluster catalogs to infer constraints on a flat ΛCDM model. The cluster sparsity has the advantage of being insensitive to selection and mass calibration bias. On the other hand, it primarily constrains a degenerate combination of Ω_{m} and σ_{8} (along approximately constant curves of *S*8=*σ*8Ω*m*/0.3−−−−−−√ and, to a lesser extent, the reduced Hubble parameter h. Hence, in order to break the internal parameter degeneracies, we perform a combined likelihood analysis of the cluster sparsity estimates with cluster gas mass fraction measurements and BAO data. We find marginal constraints that are competitive with those from other standard cosmic probes: Ω_{m} = 0.316 ± 0.013, σ_{8} = 0.757 ± 0.067 (corresponding to S_{8} = 0.776 ± 0.064), and h = 0.696 ± 0.017 at 1σ. Moreover, assuming a conservative Gaussian prior on the mass bias of gas mass fraction data, we find a lower limit on the gas depletion factor Y_{b,500c} ≳ 0.89.

Theoretical and numerical perspectives on cosmic distance averages

- Breton, Michel-Andrès;
- Fleury, Pierre

The interpretation of cosmological observations relies on a notion of an average Universe, which is usually considered as the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) model. However, inhomogeneities may statistically bias the observational averages with respect to FLRW, notably for distance measurements, due to a number of effects such as gravitational lensing and redshift perturbations. In this article, we review the main known theoretical results on average distance measures in cosmology, based on second-order perturbation theory, and we fill in some of their gaps. We then comprehensively test these theoretical predictions against ray tracing in a high-resolution dark-matter *N*-body simulation. This method allows us to describe the effect of small-scale inhomogeneities deep into the non-linear regime of structure formation on light propagation up to *z*=10. We find that numerical results are in remarkably good agreement with theoretical predictions in the limit of super-sample variance. No unexpectedly large bias originates from very small scales, whose effect is fully encoded in the non-linear power spectrum. Specifically, the directional average of the inverse amplification and the source-averaged amplification are compatible with unity; the change in area of surfaces of constant cosmic time is compatible with zero; the biases on other distance measures, which can reach slightly less than 1% at high redshift, are well understood. As a side product, we also confront the predictions of the recent finite-beam formalism with numerical data and find excellent agreement.

- Saga, Shohei;
- Taruya, Atsushi;
- Breton, Michel-Andrès;
- Rasera, Yann

The observed galaxy distribution via galaxy redshift surveys appears distorted due to redshift-space distortions (RSD). While one dominant contribution to RSD comes from the Doppler effect induced by the peculiar velocity of galaxies, the relativistic effects, including the gravitational redshift effect, are recently recognized to give small but important contributions. Such contributions lead to an asymmetric galaxy clustering along the line of sight, and produce non-vanishing odd multipoles when cross-correlating between different biased objects. However, non-zero odd multipoles are also generated by the Doppler effect beyond the distant-observer approximation, known as the wide-angle effect, and at quasi-linear scales, the interplay between wide-angle and relativistic effects becomes significant. In this paper, based on the formalism developed by Taruya et al., we present a quasi-linear model of the cross-correlation function taking a proper account of both the wide-angle and gravitational redshift effects, as one of the major relativistic effects. Our quasi-linear predictions of the dipole agree well with simulations even at the scales below 20*h*−1 Mpc, where non-perturbative contributions from the halo potential play an important role, flipping the sign of the dipole amplitude. When increasing the bias difference and redshift, the scale where the sign flip happens is shifted to a larger scale. We derive a simple approximate formula to quantitatively account for the behaviours of the sign flip.

The relativistic galaxy number counts in the weak field approximation

- Di Dio, Enea;
- Beutler, Florian

We present a novel approach to compute systematically the relativistic projection effects at any order in perturbation theory within the weak field approximation. In this derivation the galaxy number counts is written completely in terms of the redshift perturbation. The relativistic effects break the symmetry along the line-of-sight and they source, contrarily to the standard perturbation theory, the odd multipoles of the matter power spectrum or 2-point correlation function, providing a unique signature for their detection in Large Scale Structure surveys. We show that our approach agrees with previous derivations (up to third order) of relativistic effects and, for the first time, we derive a model for the transverse Doppler effect. Moreover, we {show} that in the Newtonian limit this approach is consistent with standard perturbation theory at any order.

Modeling relativistic contributions to the halo power spectrum dipole

- Beutler, Florian;
- Di Dio, Enea

We study the power spectrum dipole of an N-body simulation which includes relativistic effects through ray-tracing and covers the low redshift Universe up to z_{max} = 0.465 (RayGalGroup simulation). We model relativistic corrections as well as wide-angle, evolution, window and lightcone effects. Our model includes all relativistic corrections up to third-order including third-order bias expansion. We consider all terms which depend linearly on H/k (weak field approximation). We also study the impact of 1-loop corrections to the matter power spectrum for the gravitational redshift and transverse Doppler effect. We found wide-angle and window function effects to significantly contribute to the dipole signal. When accounting for all contributions, our dipole model can accurately capture the gravitational redshift and Doppler terms up to the smallest scales included in our comparison (k=0.48 h Mpc^{-1}), while our model for the transverse Doppler term is less accurate. We find the Doppler term to be the dominant signal for this low redshift sample. We use Fisher matrix forecasts to study the potential for the future Dark Energy Spectroscopic Instrument (DESI) to detect relativistic contributions to the power spectrum dipole. A conservative estimate suggests that the DESI-BGS sample should be able to have a detection of at least 4.4σ, while more optimistic estimates find detections of up to 10σ. Detecting these effects in the galaxy distribution allows new tests of gravity on the largest scales, providing an interesting additional science case for galaxy survey experiments.

- Taruya, Atsushi;
- Saga, Shohei;
- Breton, Michel-Andrès;
- Rasera, Yann;
- Fujita, Tomohiro

Redshift-space distortions (RSD) in galaxy redshift surveys generally break both the isotropy and homogeneity of galaxy distribution. While the former aspect is particularly highlighted as a probe of growth of structure induced by gravity, the latter aspect, often quoted as wide-angle RSD but ignored in most of the cases, will become important and critical to account for as increasing the statistical precision in next-generation surveys. However, the impact of wide-angle RSD has been mostly studied using linear perturbation theory. In this paper, employing the Zel’dovich approximation, I.e. first-order Lagrangian perturbation theory for gravitational evolution of matter fluctuations, we present a quasi-linear treatment of wide-angle RSD, and compute the cross-correlation function. The present formalism consistently reproduces linear theory results, and can be easily extended to incorporate relativistic corrections (e.g. gravitational redshift).

- Corasaniti, Pier Stefano;
- Rasera, Yann

The dark matter halo sparsity provides a direct observational proxy of the halo mass profile, characterizing halos in terms of the ratio of masses within radii which enclose two different overdensities. Previous numerical simulation analyses have shown that at a given redshift the halo sparsity carries cosmological information encoded in the halo mass profile. Moreover, its ensemble-averaged value can be inferred from prior knowledge of the halo mass function at the overdensities of interest. Here, we present a detailed study of the ensemble average properties of the halo sparsity. In particular, using halo catalogues from high-resolution N-body simulations, we show that its ensemble average value can be estimated from the ratio of the averages of the inverse halo masses as well as the ratio of the averages of the halo masses at the overdensity of interests. This can be relevant for galaxy clusters data analyses. As an example, we have estimated the average sparsity properties of galaxy clusters from the LoCuSS and HIFLUGCS data sets, respectively. The results suggest that the expected consistency of the different average sparsity estimates can provide a test of the robustness of mass measurements in galaxy cluster samples.

Probing Cosmology with Dark Matter Halo Sparsity Using X-Ray Cluster Mass Measurements

- Corasaniti, P. S.;
- Ettori, S.;
- Rasera, Y.;
- Sereno, M.;
- Amodeo, S.;
- Breton, M. -A.;
- Ghirardini, V.;
- Eckert, D.

We present a new cosmological probe for galaxy clusters, the halo sparsity. This characterizes halos in terms of the ratio of halo masses measured at two different radii and carries cosmological information encoded in the halo mass profile. Building on the work of Balmes et al., we test the properties of the sparsity using halo catalogs from a numerical N-body simulation of (2.6 Gpc h ^{-1})^{3} volume with 4096^{3} particles. We show that at a given redshift the average sparsity can be predicted from prior knowledge of the halo mass function. This provides a quantitative framework to infer cosmological parameter constraints using measurements of the sparsity of galaxy clusters. We show this point by performing a likelihood analysis of synthetic data sets with no systematics, from which we recover the input fiducial cosmology. We also perform a preliminary analysis of potential systematic errors and provide an estimate of the impact of baryonic effects on sparsity measurements. We evaluate the sparsity for a sample of 104 clusters with hydrostatic masses from X-ray observations and derive constraints on the cosmic matter density Ω_{ m} and the normalization amplitude of density fluctuations at the 8 Mpc h ^{-1} scale, σ _{8}. Assuming no systematics, we find Ω_{ m} = 0.42 ± 0.17 and σ _{8} = 0.80 ± 0.31 at 1σ, corresponding to {S}_{8}\equiv {σ }_{8}\sqrt{{{{Ω }}}_{m}}=0.48+/- 0.11. Future cluster surveys may provide opportunities for precise measurements of the sparsity. A sample of a few hundred clusters with mass estimate errors at the few percent level can provide competitive cosmological parameter constraints complementary to those inferred from other cosmic probes.