Cosmic Shear Research Articles

Third-order intrinsic alignment of SDSS BOSS LOWZ galaxies

Cosmic shear is a powerful probe of cosmology, but it is affected by the intrinsic alignment (IA) of galaxy shapes with the large-scale structure. Upcoming surveys such as Euclid and Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) require an accurate understanding of IA, particularly for higher-order cosmic shear statistics that are vital for extracting the most cosmological information. In this paper, we report the first detection of third-order IA correlations using the LOWZ galaxy sample from the Sloan Digital Sky Survey (SDSS) Baryon Oscillation Spectroscopic Survey (BOSS). We compare our measurements with predictions from the MICE cosmological simulation and an analytical model inspired by the Non-linear Linear Alignment (NLA) model and informed by second-order correlations. We also explore the dependence of the third-order correlation on the galaxies' luminosity. We find that the amplitude $A_ IA $ of the IA signal is non-zero at the $4.7 level for scales between Mpc Mpc Mpc Mpc $ the inferred $A_ IA $ agrees both with the prediction from the simulation and estimates from second-order statistics within 1sigma but deviations arise at smaller scales. Our results demonstrate the feasibility of measuring third-order IA correlations and using them for constraining IA models. The agreement between second- and third-order IA constraints also opens the opportunity for a consistent joint analysis and IA self-calibration, promising tighter parameter constraints for upcoming cosmological surveys.

Euclid preparation. LIII. LensMC, weak lensing cosmic shear measurement with forward modelling and Markov Chain Monte Carlo sampling

is a weak lensing shear measurement method developed for and Stage-IV surveys. It is based on forward modelling in order to deal with convolution by a point spread function (PSF) with comparable size to many galaxies, sampling the posterior distribution of galaxy parameters via Markov chain Monte Carlo, and marginalisation over nuisance parameters for each of the 1.5 billion galaxies observed by We quantified the scientific performance through high-fidelity images based on the Flagship simulations and emulation of the VIS images, realistic clustering with a mean surface number density of $ arcmin^ for galaxies, and $ arcmin^ for stars, and a diffraction-limited chromatic PSF with a full width at half maximum of $ $ and spatial variation across the field of view. measured objects with a density of $ arcmin^ in $4500 deg^2 $. The total shear bias was broken down into measurement (our main focus here) and selection effects (which will be addressed in future work). We found measurement multiplicative and additive biases of $m_1=(-3.6 $, $m_2=(-4.3 $, $c_1=(-1.78 $, and $c_2=(0.09 $; a large detection bias with a multiplicative component of $1.2 $ and an additive component of $-3 $; and a measurement PSF leakage of $ $ and $ $. When model bias is suppressed, the obtained measurement biases are close to requirement and largely dominated by undetected faint galaxies ($-5 $). Although significant, model bias will be straightforward to calibrate given its weak sensitivity on galaxy morphology parameters. is publicly available at

Cosmological forecasts from the combination of Stage-IV photometric galaxy surveys and the magnification from forthcoming GW observatories

In this work we have investigated the synergy between Stage-IV galaxy surveys and future GW observatories for constraining the underlying cosmological model of the Universe, focussing on photometric galaxy clustering, cosmic shear and GW magnification as cosmological probes. We have implemented a Fisher matrix approach for the evaluation of the full 6×2pt statistics composed by the angular power spectra of the single probes together with their combination. For our analysis, we have in particular considered dynamical dark energy and massive neutrino scenarios. We have found that the improvement to galaxy survey performance is below 1%, in the case of ℓ GW max=100 and a luminosity distance error of σ dL /dL =10%. However, when extending the analysis to ℓ GW max=1000, we find that the GW magnification improves the galaxy survey performance on all the cosmological parameters, reducing their errors by 3%-5%, when σ dL /dL =10%, and by 10%-18% when σ dL /dL =1%, especially for Mν , w 0 and wa . However, here our analysis is unavoidably optimistic: a much more detailed and realistic approach will be needed, especially by including systematic effects. But we can conclude that, in the case of future gravitational wave observatories, the inclusion of the gravitational wave magnification can improve Stage-IV galaxy surveys performance on constraining the underlying cosmological model of the Universe.

Improved weak lensing photometric redshift calibration via StratLearn and hierarchical modelling

ABSTRACT Discrepancies between cosmological parameter estimates from cosmic shear surveys and from recent Planck cosmic microwave background measurements challenge the ability of the highly successful $\Lambda$CDM model to describe the nature of the Universe. To rule out systematic biases in cosmic shear survey analyses, accurate redshift calibration within tomographic bins is key. In this paper, we improve photo-z calibration via Bayesian hierarchical modeling of full galaxy photo-z conditional densities, by employing ${\it StratLearn}$, a recently developed statistical methodology, which accounts for systematic differences in the distribution of the spectroscopic training/source set and the photometric target set. Using realistic simulations that were designed to resemble the KiDS + VIKING-450 data set, we show that ${\it StratLearn}$-estimated conditional densities improve the galaxy tomographic bin assignment, and that our ${\it StratLearn}$-Bayesian framework leads to nearly unbiased estimates of the target population means. This leads to a factor of $\sim 2$ improvement upon often used and state-of-the-art photo-z calibration methods. Our approach delivers a maximum bias per tomographic bin of $\Delta \langle z \rangle = 0.0095 \pm 0.0089$, with an average absolute bias of $0.0052 \pm 0.0067$ across the five tomographic bins.

Ellipsoidal Universe and Cosmic Shear

We consider a Bianchi I geometry of the universe. We obtain a cosmic shear expression related to the eccentricity of the universe. In particular, we study the connections among cosmic shear, eccentricity, and CMB. The equations are self-contained, with only two parameters.

The future of cosmological likelihood-based inference: accelerated high-dimensional parameter estimation and model comparison

We advocate for a new paradigm of cosmological likelihood-based inference, leveraging recent developments in machine learning and its underlying technology, to accelerate Bayesian inference in high-dimensional settings. Specifically, we combine (i) emulation, where a machine learning model is trained to mimic cosmological observables, e.g. CosmoPower-JAX; (ii) differentiable and probabilistic programming, e.g. JAX and NumPyro, respectively; (iii) scalable Markov chain Monte Carlo (MCMC) sampling techniques that exploit gradients, e.g. Hamiltonian Monte Carlo; and (iv) decoupled and scalable Bayesian model selection techniques that compute the Bayesian evidence purely from posterior samples, e.g. the learned harmonic mean implemented in harmonic. This paradigm allows us to carry out a complete Bayesian analysis, including both parameter estimation and model selection, in a fraction of the time of traditional approaches. First, we demonstrate the application of this paradigm on a simulated cosmic shear analysis for a Stage IV survey in 37- and 39-dimensional parameter spaces, comparing ΛCDM and a dynamical dark energy model ( w0waCDM). We recover posterior contours and evidence estimates that are in excellent agreement with those computed by the traditional nested sampling approach while reducing the computational cost from 8 months on 48 CPU cores to 2 days on 12 GPUs. Second, we consider a joint analysis between three simulated next-generation surveys, each performing a 3x2pt analysis, resulting in 157- and 159-dimensional parameter spaces. Standard nested sampling techniques are simply unlikely to be feasible in this high-dimensional setting, requiring a projected 12 years of compute time on 48 CPU cores; on the other hand, the proposed approach only requires 8 days of compute time on 24 GPUs. All packages used in our analyses are publicly available.

Constraining Cosmological Parameters Using the Splashback Radius of Galaxy Clusters

Cosmological parameters such as ΩM and σ 8 can be measured indirectly using various methods, including galaxy cluster abundance and cosmic shear. These measurements constrain the composite parameter S 8, leading to degeneracy between ΩM and σ 8. However, some structural properties of galaxy clusters also correlate with cosmological parameters, due to their dependence on a cluster’s accretion history. In this work, we focus on the splashback radius, an observable cluster feature that represents a boundary between a cluster and the surrounding Universe. Using a suite of cosmological simulations with a range of values for ΩM and σ 8, we show that the position of the splashback radius around cluster-mass halos is greater in cosmologies with smaller values of ΩM or larger values of σ 8. This variation breaks the degeneracy between ΩM and σ 8 that comes from measurements of the S 8 parameter. We also show that this variation is, in principle, measurable in observations. As the splashback radius can be determined from the same weak lensing analysis already used to estimate S 8, this new approach can tighten low-redshift constraints on cosmological parameters, either using existing data, or using upcoming data such as that from Euclid and LSST.

Cosmic shear with small scales: DES-Y3, KiDS-1000 and HSC-DR1

We present a cosmological analysis of the combination of the DES-Y3, KiDS-1000 and HSC-DR1 weak lensing samples under a joint harmonic-space pipeline making use of angular scales down to ℓmax=4500, corresponding to significantly smaller scales (δθ ~ 2.4') than those commonly used in cosmological weak lensing studies. We are able to do so by accurately modelling non-linearities and the impact of baryonic effects using Baccoemu. We find S 8 ≡ σ 8√(Ωm/0.3) = 0.795+0.015 -0.017, in relatively good agreement with CMB constraints from Planck (less than ~1.8σ tension), although we obtain a low value of Ωm =0.212+0.017 -0.032, in tension with Planck at the ~3σ level. We show that this can be recast as an H0 tension if one parametrises the amplitude of fluctuations and matter abundance in terms of variables without hidden dependence on H0. Furthermore, we find that this tension reduces significantly after including a prior on the distance-redshift relationship from BAO data, without worsening the fit. In terms of baryonic effects, we show that failing to model and marginalise over them on scales ℓ ≲ 2000 does not significantly affect the posterior constraints for DES-Y3 and KiDS-1000, but has a mild effect on deeper samples, such as HSC-DR1. This is in agreement with our ability to only mildly constrain the parameters of the Baryon Correction Model with these data.

Reconstructing the matter power spectrum with future cosmic shear surveys

ABSTRACT Analyses of cosmic shear typically condense weak lensing information over a range of scales to a single cosmological parameter, $S_8$. This paper presents a method to extract more information from Stage IV cosmic shear measurements by directly reconstructing the matter power spectrum from linear to non-linear scales. We demonstrate that cosmic shear surveys will be sensitive to the shape of the matter power spectrum on non-linear scales. We show that it should be possible to distinguish between different models of baryonic feedback and we investigate the impact of intrinsic alignments and observational systematics on forecasted constraints. In addition to providing important information on galaxy formation, power spectrum reconstruction should provide a definitive answer to the question of whether weak lensing measurements of $S_8$ on linear scales are consistent with the Planck Lambda cold dark matter cosmology. In addition, power spectrum reconstruction may lead to new discoveries on the composition of the dark sector.

Dark scattering: accelerated constraints from KiDS-1000 with ReACT and CosmoPower

ABSTRACT We present constraints on the dark scattering model through cosmic shear measurements from the Kilo Degree Survey (KiDS-1000), using an accelerated pipeline with novel emulators produced with CosmoPower. Our main emulator, for the dark scattering non-linear matter power spectrum, is trained on predictions from the halo model reaction framework, previously validated against simulations. Additionally, we include the effects of baryonic feedback from HMCode2016, whose contribution is also emulated. We analyse the complete set of statistics of KiDS-1000, namely band powers, COSEBIs, and correlation functions, for dark scattering in two distinct cases. In the first case, taking into account only KiDS cosmic shear data, we constrain the amplitude of the dark energy–dark matter interaction to be $\vert A_{\rm ds} \vert \lesssim 20$$\rm b\,GeV^{-1}$ at 68 per cent C.L. Furthermore, we add information from the cosmic microwave background (CMB) from Planck, along with baryon acoustic oscillations (BAO) from 6dFGS, SDSS, and BOSS, approximating a combined weak lensing+CMB+BAO analysis. From this combination, we constrain $A_{\rm ds} = 10.6^{+4.5}_{-7.3}$$\rm b\,GeV^{-1}$ at 68 per cent C.L. We confirm that with this estimated value of $A_{\rm ds}$ the interacting model considered in this work offers a promising alternative to solve the $S_8$ tension.

The infall region as a complementary probe to cluster abundance

ABSTRACT Galaxy cluster abundance measurements provide a classic test of cosmology. They are most sensitive to the evolved amplitude of fluctuations, usually expressed as $S_8 = \sigma _8\sqrt{\Omega _{\rm m}/0.3}$. Thus, abundance constraints exhibit a strong degeneracy between $\sigma _8$ and $\Omega _{\rm m}$, as do other similar low-redshift tests such as cosmic shear. The mass distribution in the infall region around galaxy clusters, where material is being accreted from the surrounding field, also exhibits a cosmological dependence, but in this case it is nearly orthogonal to the $S_8$ direction in the $\Omega _{\rm m}$–$\sigma _8$ plane, making it highly complementary to halo abundance or cosmic shear studies. We explore how weak-lensing measurements of the infall region might be used to complement abundance studies, considering three different tests. The splashback radius is a prominent feature of the infall region; we show that detection of this feature in lensing data from the Euclid survey could independently constrain $\Omega _{\rm m}$ and $\sigma _8$ to $\pm 0.05$. Another feature, the depletion radius where the bias reaches a minimum, also shows cosmological dependence, though it is challenging to observe in practice. The strongest constraints come from direct measurements of the shear profile in the infall region at 2–$4\, r_{200{\rm c}}$. Combining the latter with abundance constraints such as those reported from SRG$/$eROSITA should reduce the area of the error contours by an estimated factor of 1.2 using a sample of clusters observed by the UNIONS survey, or a factor of 3 using clusters observed by the Euclid Wide survey over a broader range of redshift.

Forecast of Joint Analysis of Cosmic Shear and Supernovae Magnification from the CSST and LSST

Cosmic shear and cosmic magnification reflect the same gravitational lensing field. Each of these two probes are affected by different systematics. We study the auto- and cross-correlations of cosmic shear from the China Space Survey Telescope and cosmic magnification of supernovae from the Large Synoptic Survey Telescope. We want to determine to what extent, by adding the magnification data, we can remove the systematic bias in cosmic shear measurements. We generate mock shear/magnification maps based on the correlation between different tomographic bins. After obtaining the corrected power spectra, we adopt the Markov Chain Monte Carlo technique to fit the theoretical models and investigate the constraints on the cosmological and nuisance parameters. We find that, with only the cosmic shear data, there are 1σ biases in the σ 8 and intrinsic alignment model parameters. By adding the magnification data, we are able to remove these biases perfectly.

Deciphering baryonic feedback with galaxy clusters

Upcoming cosmic shear analyses will precisely measure the cosmic matter distribution at low redshifts. At these redshifts, the matter distribution is affected by galaxy formation physics, primarily baryonic feedback from star formation and active galactic nuclei. Employing measurements from the Magneticum and IllustrisTNG simulations and a dark matter + baryon (DMB) halo model, this paper demonstrates that Sunyaev-Zel'dovich (SZ) effect observations of galaxy clusters, whose masses have been calibrated using weak gravitational lensing, can constrain the baryonic impact on cosmic shear with statistical and systematic errors subdominant to the measurement errors of DES-Y3 and LSST-Y1, with systematic errors on S8 and Ω m reaching 10% and 50% of the statistical errors, respectively. For LSST-Y6 and Roman surveys, these systematic errors increase to 150% and 100% of the statistical errors, indicating the necessity for further model developments for future surveys. We further dissect the contributions from different scales and halos with different masses to cosmic shear, highlighting the dominant role of SZ clusters at scales critical for cosmic shear analyses. These findings suggest a promising avenue for future joint analyses of Cosmic Microwave Background (CMB) and lensing surveys.

The SRG/eROSITA All-Sky Survey

Context. Number counts of galaxy clusters across redshift are a powerful cosmological probe if a precise and accurate reconstruction of the underlying mass distribution is performed – a challenge called mass calibration. With the advent of wide and deep photometric surveys, weak gravitational lensing (WL) by clusters has become the method of choice for this measurement. Aims. We measured and validated the WL signature in the shape of galaxies observed in the first three years of the Dark Energy Survey (DES Y3) caused by galaxy clusters and groups selected in the first all-sky survey performed by SRG (Spectrum Roentgen Gamma)/eROSITA (eRASS1). These data were then used to determine the scaling between the X-ray photon count rate of the clusters and their halo mass and redshift. Methods. We empirically determined the degree of cluster member contamination in our background source sample. The individual cluster shear profiles were then analyzed with a Bayesian population model that self-consistently accounts for the lens sample selection and contamination and includes marginalization over a host of instrumental and astrophysical systematics. To quantify the accuracy of the mass extraction of that model, we performed mass measurements on mock cluster catalogs with realistic synthetic shear profiles. This allowed us to establish that hydrodynamical modeling uncertainties at low lens redshifts (z < 0.6) are the dominant systematic limitation. At high lens redshift, the uncertainties of the sources’ photometric redshift calibration dominate. Results. With regard to the X-ray count rate to halo mass relation, we determined its amplitude, its mass trend, the redshift evolution of the mass trend, the deviation from self-similar redshift evolution, and the intrinsic scatter around this relation. Conclusions. The mass calibration analysis performed here sets the stage for a joint analysis with the number counts of eRASS1 clusters to constrain a host of cosmological parameters. We demonstrate that WL mass calibration of galaxy clusters can be performed successfully with source galaxies whose calibration was performed primarily for cosmic shear experiments, opening the way for the cluster cosmological exploitation of future optical and NIR surveys like Euclid and LSST.

Strong gravitational lensing’s ‘external shear’ is not shear

ABSTRACT The distribution of mass in galaxy-scale strong gravitational lenses is often modelled as an elliptical power-law plus ‘external shear’, which notionally accounts for neighbouring galaxies and cosmic shear along our line of sight. A small amount of external shear could come from these sources, but we show that the vast majority does not. Except in a handful of rare systems, the best-fitting values do not correlate with independent measurements of line-of-sight shear: from weak lensing in 45 Hubble Space Telescope images, or in 50 mock images of lenses with complex distributions of mass. Instead, the best-fit external shear is aligned with the major or minor axis of 88 per cent of lens galaxies; and the amplitude of the external shear increases if that galaxy is discy. We conclude that ‘external shear’ attached to a power-law model is not physically meaningful, but a fudge to compensate for lack of model complexity. Since it biases other model parameters that are interpreted as physically meaningful in several science analyses (e.g. measuring galaxy evolution, dark matter physics or cosmological parameters), we recommend that future studies of galaxy-scale strong lensing should employ more flexible mass models.

Constraints on Bianchi-I type universe with SH0ES anchored Pantheon+ SNIa data

We study the Bianchi-I cosmological model motivated by signals of statistical isotropy violation seen in cosmic microwave background (CMB) observations and others. To that end, we consider various kinds of anisotropic matter that source anisotropy in our model, specifically Cosmic strings, Magnetic fields, Domain walls and Lorentz violation generated magnetic fields. These anisotropic matter sources, taking one at a time, are studied for their co-evolution with standard model (isotropic) sources viz., dust-like (dark/normal) matter, and dark energy modelled as cosmological constant. We constrain the Hubble parameter, density fractions of anisotropic matter, cold dark matter (CDM), and dark energy (Λ) in a Bianchi-I universe with planar symmetry i.e., which has a global ellipsoidal geometry, and try to find signatures of a cosmic preferred axis if any.The latest compilation of Type Ia Supernova (SNIa) data from Pantheon+SH0ES collaboration is used in our analysis to obtain constraints on cosmological parameters and any preferred axis for our universe.In our analysis, we found mild evidence for a cosmic preferred axis. It is interesting to note that this preferred axis lies broadly in the vicinity of other prominent cosmic anisotropy axes reported in the literature from diverse data sets.Also we find some evidence for non-zero (negative) cosmic shear and eccentricity that characterize different expansion rates in different directions and deviation from an isotropic scale factor respectively.The energy density fractions of two of the sources considered are found to be non-zero at a2σ confidence level.To be more conclusive, we require more SNIa host galaxy data for tighter constraints on distance and absolute magnitude calibration which are expected to be available from the future JWST observations and others.

Joint constraints from cosmic shear, galaxy-galaxy lensing and galaxy clustering: internal tension as an indicator of intrinsic alignment modelling error

In cosmological analyses it is common to combine different types of measurement from the same survey. In this paper we use simulated DES Y3 and LSST Y1 data to explore differences in sensitivity to intrinsic alignments (IA) between cosmic shear and galaxy-galaxy lensing. We generate mock shear, galaxy-galaxy lensing and galaxy clustering data, contaminated with a range of IA scenarios. Using a simple 2-parameter IA model (NLA) in a DES Y3 like analysis, we show that the galaxy-galaxy lensing + galaxy clustering combination ( 2×2pt) is significantly more robust to IA mismodelling than cosmic shear. IA scenarios that produce up to 5σ biases for shear are seen to be unbiased at the level of ∼1σ for 2×2pt. We demonstrate that this robustness can be largely attributed to the redshift separation in galaxy-galaxy lensing, which provides a cleaner separation of lensing and IA contributions. We identify secondary factors which may also contribute, including the possibility of cancellation of higher-order IA terms in 2×2pt and differences in sensitivity to physical scales. Unfortunately this does not typically correspond to equally effective self-calibration in a 3×2pt analysis of the same data, which can show significant biases driven by the cosmic shear part of the data vector. If we increase the precision of our mock analyses to a level roughly equivalent to LSST Y1, we find a similar pattern, with considerably more bias in a cosmic shear analysis than a 2×2pt one, and significant bias in a joint analysis of the two. Our findings suggest that IA model error can manifest itself as internal tension between ξ± and γt+w data vectors. We thus propose that such tension (or the lack thereof) can be employed as a test of model sufficiency or insufficiency when choosing a fiducial IA model, alongside other data-driven methods.

B-modes from galaxy cluster alignments in future surveys

Intrinsic alignment (IA) of source galaxies represents an important contaminant for upcoming cosmic shear surveys. In particular, it is expected on general grounds that IA contains a B-mode while the weak lensing signal does not. Thus, a detection of B-modes offers the possibility to study directly the IA signal of the sources. Galaxy clusters exhibit strong IA and are therefore a natural candidate to look for a B-mode signal. We forecast the signal-to-noise ratio (SNR) for B-modes from IA of galaxy clusters in the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST). We use a perturbative model for the IA multipoles based on the Effective Field Theory of Intrinsic Alignments (EFT of IA), which has recently been validated against N-body simulations. We forecast SNR ≈12 and find that this detectability is not significantly impacted by different analysis choices. Lastly, we also apply our forecast to clusters in the redMaPPer SDSS and DESY1 samples. We find SNR ≈5 and SNR ≈3, respectively, suggesting a detection is within reach, provided accurate redshift information is available.

Accuracy requirements on intrinsic alignments for Stage-IV cosmic shear

In the context of cosmological weak lensing studies, intrinsic alignments (IAs) are one of the most In the context of cosmological weak lensing studies, intrinsic alignments (IAs) are one of the most complicated astrophysical systematic to model, given the poor understanding of the physical processes that cause them. A number of modelling frameworks for IAs have been proposed in the literature, both purely phenomenological or grounded on a perturbative treatment of symmetry-based arguments. However, the accuracy with which any of these approaches is able to describe the impact of IAs on cosmic shear data, particularly on the comparatively small scales () to which this observable is sensitive, is not clear. Here we quantify the level of disagreement between the true underlying intrinsic alignments and the theoretical model used to describe them that can be allowed in the context of cosmic shear analyses with future Stage-IV surveys. We consider various models describing this “IA residual’’, covering both physics-based approaches, as well as completely agnostic prescriptions. The same qualitative results are recovered in all cases explored: for a Stage-IV cosmic shear survey, a mis-modelling of the IA contribution at the ∼10% level produces shifts of ≲0.5σ on the final cosmological parameter constraints. Current and future IA models should therefore aim to achieve this level of accuracy, a prospect that is not unfeasible for models with sufficient flexibility.

Constraining hot dark matter sub-species with weak lensing and the cosmic microwave background radiation

Although it is well known that the bulk of dark matter (DM) has to be cold, the existence of an additional sub-dominant, hot species remains a valid possibility. In this paper we investigate the potential of the cosmic shear power spectrum to constrain such a mixed (hot plus cold) DM scenario with two additional free parameters, the hot-to-total DM fraction ($f_ hdm $) and the thermal mass of the hot component ($m_ hdm $). Running a Bayesian inference analysis for both the Kilo-Degree Survey cosmic shear data ( as well as the cosmic microwave background (CMB) temperature and polarisation data from Planck we derive new constraints for the mixed DM scenario. We find a 95 per cent confidence limit of $f_ hdm <0.08$ for a very hot species of $m_ hdm eV. This constraint is weakened to hdm <0.25$ for $m_ hdm eV. Scenarios with masses above hdm eV remain unconstrained by the data. Next to providing limits, we investigate the potential of mixed DM to address the clustering (or $S_8$) tension between lensing and the CMB. We find a reduction of the 2D ($ - S_8$) tension from 2.9sigma to 1.6sigma when going from a pure cold DM to a mixed DM scenario. When computing the 1D Gaussian tension on $S_8$ the improvement is milder, from 2.4sigma to 2.0sigma