EDP Sciences logo
Subscriber Authentication Point
Search
Menu
Home All issues Volume 687 (July 2024) A&A, 687 (2024) A161 Full HTML
Open Access
Issue
A&A
Volume 687, July 2024
Article Number A161
Number of page(s) 8
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/202449195
Published online 05 July 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

During the early 1980s, neutrino particles were considered as dark matter (DM) candidates due to the mounting evidence of non-zero neutrino masses from particle physics experiments (Zeldovich et al. 1982; Bond et al. 1983). This prompted the development of a top-bottom approach to structure formation, where massive “Zel’dovich Pancakes” were theorised to collapse or fragment into halos and galaxies. Such an approach was thought to allow for galaxy formation despite the substantial free-streaming properties of neutrino particles which counteract the gravitational clustering. However, the neutrino hot dark matter (HDM) model was quickly abandoned in favour of the cold dark matter (CDM) paradigm which predicted a more gradual build-up of galaxies in much better agreement with observations (Blumenthal et al. 1984).

While CDM became a main component of the widely accepted ΛCDM theory, small-scale inconsistencies left room for hot and warm sub-components of DM. These inconsistencies included the missing satellite problem, relating to the mismatch between predicted DM sub-haloes and observed satellite galaxies, and the cusp-core problem, which involved the discrepancy between cuspy halo profiles from gravity-only simulations and cored profiles of observed dwarf galaxies (Moore et al. 1999; de Blok 2010). One popular hypothesis to address these issues was the existence of a warm or hot DM sub-components (e.g. Boyanovsky et al. 2008; Anderhalden et al. 2013). However, subsequent observations have shown this solution to be in tension with other data from the Lyman-α forest (Viel et al. 2013; Markovic & Viel 2014) and Milky Way satellite counts (Schneider et al. 2014).

Today it is well established that the missing satellite and cusp-core problems can be alleviated by baryonic feedback effects and do not require additional modifications of the DM sector (e.g. Brooks et al. 2013; Del Popolo & Le Delliou 2021). However, in recent years another clustering tension has appeared between the S8 clustering parameter from the CMB experiment Planck and the stage-III lensing surveys KiDS (Kilo Degree Survey, Heymans et al. 2021), DES (Dark Energy Survey, Amon et al. 2022; DES Collaboration et al. 2022), or HSC (Hyper Suprime Camera, Aihara et al. 2018). The S8 parameter is defined as

S 8 = σ 8 Ω m / 0.3 , $$ \begin{aligned} S_8=\sigma _8\sqrt{\Omega _m/0.3}, \end{aligned} $$(1)

which includes the cosmological parameters Ωm and σ8, describing the matter abundance and the clustering amplitude, respectively.

Attempts to alleviate the S8 tension are often based on extensions to the ΛCDM framework that lead to modifications of the clustering process between the last scattering surface of the CMB and today. A few examples are decaying DM models (Fuß & Garny 2023; Abellán et al. 2021; Bucko et al. 2023, 2024), new couplings between dark energy and DM (Poulin et al. 2023), or DM-baryon scattering (He et al. 2023).

In Das et al. (2022), the effect of a hot sub-component of DM on the S8 tension is studied at the linear level, assuming observations from the CMB together with low-redshift galaxy clustering probes. In this paper, we considered the same physical scenario, but we modeled the full non-linear process of structure formation. This allowed us to predict at the same time the CMB and the weak-lensing signal, obtaining a self-consistent test of the mixed DM scenario from these observations.

Another important goal of the present work was to use data from KiDS and Planck to derive new constraints on the particle mass (mhdm) and the fraction of hot-to-total DM

f hdm = Ω hdm Ω hdm + Ω cdm , $$ \begin{aligned} f_{\text{hdm}}=\frac{\Omega _{\text{hdm}}}{\Omega _{\text{hdm}}+\Omega _{\text{cdm}}}, \end{aligned} $$(2)

where Ωcdm and Ωhdm are the abundances of the cold and hot components, respectively. In particular, we are interested in models with a HDM particle mass in the eV to keV range. In this mass regime, not many constraints currently exist as previous investigations have focused on either the sub-eV (Planck Collaboration V 2020) or the keV mass scales (Boyarsky et al. 2009; Schneider 2015; Baur et al. 2017).

In general, an additional HDM component leads to a suppression of the matter power spectrum at small scales (Viel et al. 2013). This suppression is caused by the free-streaming of the hot component and depends on both the particle mass and the momentum distribution (e.g. Boyarsky et al. 2009; Merle et al. 2016). Depending on the hot-to-total DM fraction (fhdm), the suppression can either have a steep cutoff (if fhdm is close to one) or it can be more gradual and shallow (if fhdm remains close to zero).

The MDM terminology is convenient in cosmological studies as it allows us to group together a wide variety of differently motivated theoretical models into a single phenomenological framework. Some prospective candidates in this mass range are the gravitino (Osato et al. 2016, eg.) and the sterile neutrino (Dodelson & Widrow 1994; Shi & Fuller 1999). The latter has been studied extensively for a variety of different production mechanisms, such as non-resonant mixing, resonant mixing or production via early decays (see Drewes et al. 2017, for a review). Since these particles are never in thermal equilibrium, their momenta cannot be described by a Maxwell-Boltzmann distribution. As a consequence, they exhibit some differences in the way they suppress the matter power spectrum. However, in many cases, a simple re-mapping of the particle mass allows us to interpret the suppression in terms of the standard thermal particles at a sufficient accuracy (Abazajian 2006; Merle & Schneider 2015; Bozek et al. 2016).

The same occurs in the ultra-light axion scenario with one or several axion-like particles (Marsh & Silk 2014; Hlozek et al. 2015; Giri & Schneider 2022; Vogt et al. 2023; Rogers et al. 2023). In principle, these bosonic DM particles exhibit very different dynamics at the scale of their particle wavelengths. However, for many applications, the results can be brought into reasonable agreement with the fermionic mixed DM scenario (Hui et al. 2017).

Finally, many interacting DM scenarios have a similar effect on structure formation as the simple MDM model. This is especially true for the case where potential interactions of the dark sector are restricted to the early universe as is the case for most of the ETHOS (Cyr-Racine et al. 2016) parameter space, for example. In this particular, simplest non-minimal scenario, the DM particle is allowed to interact with a mediator particle playing the role of dark radiation. The early-universe interactions between the DM and dark radiation particles cause a suppression of the power spectrum at small scales including, in some cases, dark acoustic oscillation features. As these features quickly disappear at nonlinear scales (Schaeffer & Schneider 2021), the ETHOS interaction framework often resembles the mixed DM case (Archidiacono et al. 2019).

In summary, the analysis presented in this paper is, strictly speaking, valid for the case of a mixed DM scenario where the hot particles undergo free streaming caused by a momentum distribution that is identical or close to a Maxwell-Boltzmann distribution. The prime example of such a model is CDM combined with a sterile neutrino produced in the standard way via the Dodelson-Widrow scenario. However, many other DM models mentioned above will yield very similar results and some of the conclusions can be transferred to other scenarios.

In the next section, we will describe our data analysis modeling method. In Sect. 3, we will present our findings, and finally, we will conclude our study in Sect. 4.

2. Methods

In this section, we summarise our pipeline to predict the cosmic shear and the CMB data for both the CDM and MDM models. Note that the description remains at a general level, more details can be found in Schneider et al. (2022) and Bucko et al. (2023).

2.1. Cosmic shear signal

For this analysis, we relied on the band power data from the KiDS-1000 cosmic shear observations published in Asgari et al. (2021). A detailed description of how to obtain predictions for the band power data is given in Joachimi et al. (2021). See also Sect. 3.1. of Schneider et al. (2022) for more details. In general, the band power data is obtained via an integral over the angular power spectrum multiplied by the band filter functions (Joachimi et al. 2021). The angular power spectrum is obtained via the Limber approximation (Limber 1953)

C i , j A , B ( ℓ ) = ∫ 0 χ H W i A ( χ ) W j B ( χ ) f κ ( χ ) 2 P tot ( ℓ χ , χ ) d χ $$ \begin{aligned} C_{i,j}^{A,B}(\ell )=\int _0^{\chi _H}\frac{W_i^A(\chi )W_j^B(\chi )}{f_\kappa (\chi )^2}P_{\rm tot}\left(\frac{\ell }{\chi },\chi \right) \mathrm{d}\chi \end{aligned} $$(3)

where the i, j subscripts refer to the tomographic redshift bins and Ptot is the total matter power spectrum.

The window functions WA and WB describe the lensing (G) and intrinsic alignment (I) weights, [A,B] ∈ [I,G]. In a flat universe, they are given by

W i G ( χ ) = 3 Ω m H 0 2 2 c 2 χ ( 1 + z ) ∫ χ χ H d χ ′ n i , S ( χ ′ ) ( χ ′ − χ ) χ ′ , W i I ( χ ) = − A IA C 1 ρ cr Ω m D ( z ) n i , S ( χ ) , $$ \begin{aligned} W^G_{i}(\chi )&= \frac{3\Omega _m H_0^2}{2c^2} \chi (1+z) \int _{\chi }^{\chi _H} \mathrm{d}\chi^\prime n_{i,S}(\chi^\prime ) \frac{(\chi^\prime -\chi )}{\chi^\prime } \ ,\nonumber \\ W_i^I(\chi )&=-&A_{IA} \frac{C_1 \rho _{\text{cr}}\Omega _m}{D(z)}n_{i,S}(\chi ) \ , \end{aligned} $$(4)

where χ is the comoving distance, H0 the Hubble parameter, ρcr the critical density, and D(z) the growth factor. The redshift distribution of galaxies, ni, S(z), is obtained from Asgari et al. (2021) assuming 5 tomographic bins. The parameter AIA describes the amplitude of the intrinsic alignment effect – assuming the Non-Linear Alignment (NLA) Model described in Hirata & Seljak (2010) and Hildebrandt et al. (2017) – and was kept as a free parameter. The C1 value was calibrated to match SuperCOSMOS Sky Surveys (SSS) observations (Brown et al. 2002), such that the free parameter AIA has value of unity. In accordance with Heymans et al. (2021), any redshift dependence of the intrinsic alignment effect was ignored for this analysis.

The total nonlinear matter power spectrum is given by

P tot ( k , z ) = S baryon ( k , z ) × S mdm ( k , z ) × P NL ( k , z ) , $$ \begin{aligned} P_{\rm tot} (k,z) = S_{\rm baryon}(k,z)\times S_{\rm mdm}(k,z)\times P_{\rm NL}(k,z) \ , \end{aligned} $$(5)

where PNL(k, z) is the nonlinear, gravity-only, matter power spectrum obtained with the revised_halofit method of Takahashi et al. (2012). Note that PNL(k, z) only depends on the cosmological, but not on any astrophysical or DM parameters. The function Sbaryon(k, z) refers to the suppression due to the baryonic feedback effect. We modeled this effect using the emulator BCemu (Giri & Schneider 2021), which is based on the baryonification method (Schneider & Teyssier 2015; Schneider et al. 2019). This emulator depends on the cosmic baryon fraction (fb = Ωb/Ωm) plus seven free baryonic parameters. However, instead of varying all of them (as in Schneider et al. 2022), we only vary the three baryonic parameters {log10Mc, ηd, θj} along with fb. This reduced setup of parameters has been shown in Giri & Schneider (2021) to provide accurate fits to all investigated hydrodynamical simulations. We show posteriors of baryonic feedback parameters in Appendix A

The function Smdm(k, z) characterizes power spectrum changes due to hot/warm DM using an improved emulator derived from Parimbelli et al. (2021). This new version is trained on a larger set of simulations (100 vs. 74) across a broader parameter range (fhdm, mhdm), encompassing masses as low as 0.03 keV. Initial conditions are established through the fixed-and-paired technique (Angulo & Pontzen 2016), with simulations conducted within a 120 Mpc h−1 box. This ensures precision convergence of better than 1% for k ≲ 10 h Mpc−1, effectively maintaining a link with the linear regime.

The emulator obviates the need for computing a separate linear MDM spectrum, as it yields the non-linear MDM power spectrum by adjusting a non-linear ΛCDM spectrum. In the parameter space accepted by the KiDS-1000 data, σ8 discrepancies between ΛCDM and MDM calculations are negligible. Furthermore, Parimbelli et al. (2021) demonstrated that the suppression induced by MDM remains cosmology-independent within a 2% range across various Ωm and σ8 values.

Note that in our modelling we implicitly assumed that the baryonic feedback and MDM suppression effects are independent from each other. As a consequence, the functions Sbaryon(k, z) and Smdm(k, z) can be multiplied as shown in Eq. (5) greatly simplifying the analysis. The validity of this approximation has been confirmed in Parimbelli et al. (2021) with the help of baryonified CDM and MDM simulations. It turns out that multiplying the two suppression functions only adds a small, sub-percent error to the full power spectrum. This is significantly smaller than the estimated error of power spectrum estimator itself.

In Fig. 1 we show the band power spectrum from the KiDS-1000 analysis together with predictions for MDM models with fixed particle mass (mhdm = 50 eV) and a varying fraction (fhdm = 0.1,  0.5,  1) using the pipeline summarised above. All other cosmological parameters are kept at the Planck 18 values. The plot shows that the higher the ratio of hot-to-cold DM, the more the band power is suppressed. Furthermore, the suppression is more pronounced at higher l-modes, which correspond to smaller physical scales, as well as at higher redshift bin numbers indicating a stronger suppression at larger redshift. Note that although the effect of mixed DM is very well visible, potential degeneracies with cosmology and baryonic physics may strongly reduce the effect.

thumbnail Fig. 1.

Auto and cross angular band power spectra of the KiDS-1000 data set separated in five tomographic redshift bins (black data points). The data is obtained from Asgari et al. (2021). The lines correspond to predictions assuming a ΛCDM model (orange dashed) and three ΛMDM models with fixed thermal mass (mhdm = 50 keV) and increasing hot-to-total DM fraction (fhdm = 0.1, 0.5, 1.0) in green, purple, and blue. All plotted models are run assuming the Planck-18 best-fitting cosmology.

The angular power spectra were calculated using the PyCosmo package (Refregier et al. 2018; Tarsitano et al. 2020). The prior ranges are summarised in Table 1. More details about the selection of the priors are provided in Bucko et al. (2023). Note that for comparison, we also ran an analysis assuming the standard ΛCDM model. We thereby find good agreement with the KiDS-1000 results from Asgari et al. (2021) and Schneider et al. (2022) validating our pipeline.

Table 1.

Prior ranges used in the Planck-18 TTTEEE and KiDS-1000 analysis.

2.2. CMB data

We extended our exploration beyond cosmic shear and delved into the impact of MDM on the CMB temperature and polarization data. For this analysis, we employed the Planck-lite-py module from Prince & Dunkley (2019), using the Planck-18 TTTEEE likelihood provided in the package, including low-ℓ bins. The Planck-18 likelihood was coupled with the Boltzmann solver CLASS (Blas et al. 2011). CLASS features an integrated option for additional hot and warm DM sub-species (Lesgourgues & Tram 2011). While this study was conducted with the thermal relics mass mtherm, wdm, a conversion is possible to sterile neutrino mass mνs through following expression (provided in Bozek et al. 2016):

m ν , s = 3.90 keV ( m thermal 1 keV ) 1.294 ( f wdm Ω DM h 2 0.1225 ) − 1 / 3 . $$ \begin{aligned} m_{\nu _,s}=3.90 \ \mathrm{keV} \ \left(\frac{m_{\mathrm{thermal}}}{1\,\mathrm{keV}}\right)^{1.294}\left(\frac{f_{\mathrm{wdm}}\Omega _{\mathrm{DM}}h^2}{0.1225}\right)^{-1/3} \ . \end{aligned} $$(6)

The CLASS Boltzmann solver takes the sterile neutrino mass as an input for the m_ncdm parameter.

In Fig. 2, we visualize the influence of an extra HDM sub-species across diverse particle masses and hot-to-total DM fractions. The three panels depict models for particle masses of mhdm = {10, 30, 100} eV, demonstrating the diminishing effects on the C(ℓ) as the mass rises to 100 eV. While the apparent increase in C(l)TT, ΛMDM seems in conflict with the objective of lowering S8, the larger peaks are due to a shift in the energy budget from matter to radiation when increasing fhdm for low masses. This decreases ωm/ωR therefore delaying the matter-radiation zeq which leaves a shorter time for the BAO peaks to decrease until recombination.

thumbnail Fig. 2.

Residual angular power spectra of the CMB temperature fluctuations from Planck-18 assuming a mixed DM model with varying hot-to-total DM fractions (coloured lines) and a thermal particle mass of mhdm = 10 eV (left), mhdm = 30 eV (middle), and mhdm = 100 eV (right). Note that the blue line (fhdm = 0) corresponds to the ΛCDM model. The grey error bars denote the 68% confidence interval of the Planck-18 TT data.

2.3. Bayesian inference analysis

We performed a Bayesian inference analysis with a Monte-Carlo Markov chain (MCMC) method implemented in the emcee package (Foreman-Mackey et al. 2013). This method determines the posterior probability p(θ|d) on the interested parameters θ, given the observation d. To explore the vast multidimensional parameter space, our analysis was performed on the supercomputer facilities at the Swiss National Supercomputing Centre (CSCS), running in parallel on 128 CPU cores. Our priors p(θ) are mostly flat and wide as described in Table 1. Note that we have verified our analysis pipeline by reproducing the fiducial ΛCDM analysis from Planck.

3. Results

Here we present our new constraints on the MDM model (Sect. 3.1) and its implications on the S8 tension (Sect. 3.3).

3.1. Constraints on MDM model

The constraints on the MDM parameters were obtained from the full Bayesian inference chains by marginalising over all parameters except mhdm and fhdm. We show the results of this analysis in Fig. 3. All constraints are provided at the 95 % confidence level. For a pure ΛHDM scenario with fhdm = 1, the cosmic shear and CMB analysis yield constraints of mth < 0.2 keV and mth < 0.14 keV, respectively. We should note that these limits are significantly weaker than other constraints from e.g. Milky-Way satellite counts or the Lyman-α forest.

thumbnail Fig. 3.

Constraints on the MDM (ΛMDM) model from weak lensing (KiDS-1000) in green, the CMB (Planck-18 TTTEEE) in blue, and the combined WL+CMB in red (dotted line). TheΛMDM model is parametrised by the thermal mass of the hot species (mhdm) and the fraction of hot-to-total DM (fhdm). The top-left corner of the plot corresponds to the excluded region. All contours are shown at the 95% confidence level.

Both the cosmic shear and CMB data are much more powerful in constraining models with small hot-to-total DM fractions. For a particle mass of mhdm ≤ 20 eV we obtain limits of fhdm < 0.09 and < 0.08 from the WL and the CMB analysis. This means that a HDM particle can not make up more than 8-9% of the total DM budget.

The constraints on fhdm become weaker when going to larger particle masses. For the cosmic shear (and CMB) analysis we obtain the limits fhdm < 0.16 (< 0.24) for mhdm ≤ 50 eV and fhdm < 0.3 (< 0.55) for mhdm ≤ 100 eV.

In Fig. 3 we also show the constraints of the combined KiDS-1000 and Planck-18 analysis. They are slightly weaker than the limits obtained with the runs based on the individual weak-lensing cosmic shear and CMB data. This surprising result is caused by the existing clustering (or S8) tension between the two data-sets. It turns out that the tension is compensated in the combined chain by allowing for slightly higher values of the hot-to-total DM fraction.

Parts of the MDM parameter ranges investigated here were explored in the past using data from the Lyman-α forest (Boyarsky et al. 2009; Baur et al. 2017) and Milky-Way satellite counts (Schneider 2015). These studies found constraints of fhdm ∼ 0.1 − 0.2 for small particle masses which is only slightly weaker than our constraints. However, they only focused on the regime above mhdm ∼ 0.2 keV and they did not perform a Bayesian inferences analysis including cosmological parameters. Furthermore, the Lyman-α studies assumed a IGM temperature-evolution that follows a power law which is known to yield very constraining results (Garzilli et al. 2021).

We have focused this work on the full exploitation of small scale information in large scale structure. While Lyman-α probe the structure formation at high redshift, CMB lensing probes the universe at a similar redshift range on larger scales. Recent data as published by ACT (Madhavacheril et al. 2024) and SPT (Pan et al. 2023) could therefore help to constrain low-mass particles with large free-streaming scales.

3.2. Comparison to other work

In a recent study by Das et al. (2022) a MDM model with additional neutrino-like sub-species was investigated at the linear level. The paper focused on Planck data combined with BAO measurements from the BOSS survey as well as the Pantheon SNIa catalogue data. An additional prior mimicking the S8 measurement from KiDS-1000 (Heymans et al. 2021) was included in the analysis. Note that their approach included results obtained under the premises of ΛCDM and can therefore only be used as an approximation.

Converting the model of Das et al. (2022) into our parametrisation, we find that their best-fit model (which was claimed to alleviate the S8 tension) corresponds to mtherm, hdm = 11.36 eV and fhdm = 0.08 (see Table III in Das et al. 2022). Interestingly, this point is right on the 95% confidence level of our cosmic shear analysis (see Fig. 3), which means that our results neither confirm nor strongly disfavour the best fitting model found by Das et al. (2022). For a description of the different ways to parameterize light relics, we refer the reader to Acero & Lesgourgues (2009).

A recent forecast study by Schneider et al. (2020b,a) investigated the potential of the Euclid weak-lensing survey to constrain the MDM model. They found that Euclid will provide much stronger constraints on the mass and fraction of a potential HDM particle. Assuming a ΛCDM universe, the reported limits are fhdm < 0.01 for mhdm ≲ 30 eV (at the 95% confidence level). This confirms that future, stage-IV lensing surveys will be able to detect hot particle sub-species even if they only make up a percent of the total DM budget.

Similarly to Das et al. (2022), a recent study by Rogers & Poulin (2024) found preference for a MDM type suppression in the eBOSS Lyman-α forest data. The parameters they find to alleviate the tension of 4.9σ to 1.34σ is log ( m wdm ) = 1 . 01 − 0.44 + 0.30 $ \mathrm{log}(m_{\mathrm{wdm}})=1.01_{-0.44}^{+0.30} $ and f wdm = 0 . 0219 − 0.0042 + 0.0030 $ f_{\mathrm{wdm}}=0.0219_{-0.0042}^{+0.0030} $. The 10eV mass which was found to alleviate the tension is right at the edge of our prior, and the fwdm fraction of 0.02 is comfortably inside our constraints. While the results presented in Rogers & Poulin (2024) seem coherent with what we have found, more stringent weak lensing data would be required to probe these low fractions of MDM.

3.3. Impact on the S8 tension

A further goal of this work was to investigate the effects of the ΛMDM model on the S8 tension. The gradual power suppression towards small scales caused by a subdominant HDM species could affect the nonlinear clustering by pushing the weak-lensing estimates for S8 further up. This could then lead to a better agreement with the measurements from the CMB.

In Fig. 4 we plot the two-dimensional posterior contours of S8 and Ωm for both the ΛCDM and ΛMDM scenarios. The weak-lensing posteriors (green) are indeed pushed to somewhat higher values of S8 in the case of ΛMDM, but the shift is not very significant. At the same time the posterior from the CMB analysis (blue) moves slightly down, helping to alleviate the S8 tension. For completeness in Fig. 4, we also provide the results from the combined KiDS-1000 and Planck-18 TTTEEE analysis. The corresponding posteriors are shown in red. Not surprisingly, they lie between the results from the individual weak lensing and CMB analyses. The posterior means for KiDS-1000 are S 8 = 0 . 749 − 0.029 + 0.034 $ S_8=0.749_{-0.029}^{+0.034} $ and S 8 = 0 . 754 − 0.030 + 0.034 $ S_8=0.754_{-0.030}^{+0.034} $ for ΛCDM and ΛMDM respectively. For the Planck 18 data the values are S8 = 0.841 ± 0.017 and S8 = 0.832 ± 0.019 for ΛCDM and ΛMDM respectively. We find that the remaining cosmological parameters are not modified by the change of model. We used two metrics to estimate the tension, which we refer to as 2D and 1D.

thumbnail Fig. 4.

Posterior contours in the S8 − Ωm plane for the ΛCDM (left) and ΛMDM models (right) at 68 and 95 % confidence level. The green, blue, and red contours correspond to the cosmic shear (KiDS-1000), the CMB (Planck-18 TTTEEE) and the combined analysis.

To calculate the 2D tension, we used the lmtttensiometer package (Raveri & Doux 2021). The package first computes the difference between the parameters in the chains and then estimates the probability of the shift using a Kernel Density Estimate algorithm (KDE) described in Raveri et al. (2020). Formally, we obtained a reduction of the tension from 2.9σ in ΛCDM to 1.6σ in the ΛMDM model (see also Table 2).

Table 2.

2D tension metrics between Planck-18 and KiDS-1000 datasets for ΛCDM and ΛMDM models computed using parameter shifts in the Ωm − S8 plane with the tensiometer package.

When computing the gaussian 1D tension in S8, we used the metric from Asgari et al. (2021) that is given as

τ S 8 = S 8 CMB − S 8 WL Var [ S 8 CMB ] + Var [ S 8 WL ] , $$ \begin{aligned} \tau _{S_8} = \frac{S_8^\mathrm{CMB} - S_8^\mathrm{WL}}{\sqrt{\mathrm{Var}\left[S_8^\mathrm{CMB}\right] + \mathrm{Var}\left[S_8^\mathrm{WL}\right]}}, \end{aligned} $$(7)

where S 8 CMB $ S_8^{\rm CMB} $ and S 8 WL $ S_8^{\rm WL} $ are the most probable values derived from CMB and WL observations respectively. In the denominator, the variance of the same quantities are used. In this 1D framework, the tension is reduced from 2.4σ to 2.0σ only. While the tension values computed using the 1D and 2D estimates differ quite strongly, the non Gaussian shapes of the weak lensing contours tend to make us favour the 2D tension estimate based on parameter shifts.

4. Conclusion

The presence of a sub-dominant HDM species, such as an additional neutrino particle, is a straight-forward extension of the ΛCDM scenario. In this paper, we explored the power of cosmic shear together with CMB temperature and polarisation data to constrain the mixed DM model (ΛMDM) which is parametrised by the (thermal) particle mass of the hot species (mhdm) and the hot-to-total DM fraction (fhdm). As observations we use the KiDS-1000 band power data from Asgari et al. (2021) as well as the Planck-18 TTTEEE data from Planck Collaboration V (2020).

We find new constraints on the ΛMDM parameters that are summarised in Fig. 3. At the 95% confidence level, the hot-to-total DM fraction is limited to fhdm < 0.08 for a hot species with mhdm ≤ 20 eV. This limit is weakened to fhdm < 0.16 for mhdm ≤ 50 eV and fhdm < 0.30 for mhdm ≤ 100 eV. Scenarios with (thermal) particle masses beyond mhdm ∼ 200 eV remain unconstrained by the weak lensing and CMB data.

Next to providing constraints, we also investigated the S8 (or lensing) tension describing the clustering mismatch between cosmic shear probes and the CMB data. We find that the S8 tension is decreased from 2.9σ in ΛCDM to 1.6σ in the ΛMDM model. This improvement is caused by both a slight upper shift of the KiDS-1000 as well as a small downward shift of Planck-18 TTTEEE contours. The two dimensional Ωm − S8 posterior contours are illustrated in Fig. 4. Using 1D estimates the reduction in tension is milder, from 2.4σ to 2.0σ. While these estimates are quite different we tend to favour the 2D parameter shift methods as it was designed to capture non-Gaussian features in the posteriors.

In the near future, stage-IV lensing surveys such as Euclid will allow us to further probe the ΛMDM model. In particular, it will be possible to constrain models where only about 1% of the DM sector is made of a hot particle. This is about a factor of ten improvement with respect to current weak-lensing observations.

Acknowledgments

For the plotting routines we used the lmttGetDist (Lewis 2019) and lmttmatplotlib (Hunter 2007) packages. We also thank Giovanni Aricò and Andrej Obuljen for informing discussions. This work is supported by the Swiss National Science Foundation under grant number PCEFP2_181157. Nordita is supported in part by NordForsk. The KiDS-1000 data is based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples). We used the gold sample of weak lensing and photometric redshift measurements from the fourth data release of the Kilo-Degree Survey (Kuijken et al. 2019, Wright et al. 2020, Hildebrandt et al. 2021, Giblin et al. 2021), hereafter referred to as KiDS-1000. Cosmological parameter constraints from KiDS-1000 have been presented in Asgari et al. (2021) (cosmic shear), Heymans et al. (2021) (3×2pt) and Tröster et al. (2021) (beyond ΛCDM), with the methodology presented in Joachimi et al. (2021). Fabian Hervas Peters acknowledges support from the Centre National dÉtudes Spatiales.

References

  1. Abazajian, K. 2006, Phys. Rev. D, 73, 063506 [NASA ADS] [CrossRef] [Google Scholar]
  2. Abellán, G. F., Murgia, R., & Poulin, V. 2021, Phys. Rev. D, 104, 123533 [CrossRef] [Google Scholar]
  3. Acero, M. A., & Lesgourgues, J. 2009, Phys. Rev. D, 79, 045026 [NASA ADS] [CrossRef] [Google Scholar]
  4. Aihara, H., Arimoto, N., Armstrong, R., et al. 2018, PASJ, 70, S4 [NASA ADS] [Google Scholar]
  5. Amon, A., Robertson, N. C., Miyatake, H., et al. 2022, MNRAS, 518, 477 [CrossRef] [Google Scholar]
  6. Anderhalden, D., Schneider, A., Maccio, A. V., Diemand, J., & Bertone, G. 2013, JCAP, 03, 014 [CrossRef] [Google Scholar]
  7. Angulo, R. E., & Pontzen, A. 2016, MNRAS, 462, L1 [NASA ADS] [CrossRef] [Google Scholar]
  8. Archidiacono, M., Hooper, D. C., Murgia, R., et al. 2019, JCAP, 10, 055 [CrossRef] [Google Scholar]
  9. Asgari, M., Lin, C.-A., Joachimi, B., et al. 2021, A&A, 645, A104 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  10. Baur, J., Palanque-Delabrouille, N., Yéche, C., et al. 2017, JCAP, 2017, 013 [CrossRef] [Google Scholar]
  11. Blas, D., Lesgourgues, J., & Tram, T. 2011, JCAP, 2011, 034 [Google Scholar]
  12. Blumenthal, G. R., Faber, S. M., Primack, J. R., & Rees, M. J. 1984, Nature, 311, 517 [Google Scholar]
  13. Bond, J. R., Szalay, A. S., & White, S. D. M. 1983, Nature, 301, 584 [NASA ADS] [CrossRef] [Google Scholar]
  14. Boyanovsky, D., de Vega, H. J., & Sanchez, N. 2008, Phys. Rev. D, 78, 063546 [CrossRef] [Google Scholar]
  15. Boyarsky, A., Lesgourgues, J., Ruchayskiy, O., & Viel, M. 2009, JCAP, 2009, 012 [CrossRef] [Google Scholar]
  16. Bozek, B., Boylan-Kolchin, M., Horiuchi, S., et al. 2016, MNRAS, 459, 1489 [NASA ADS] [CrossRef] [Google Scholar]
  17. Brooks, A. M., Kuhlen, M., Zolotov, A., & Hooper, D. 2013, ApJ, 765, 22 [NASA ADS] [CrossRef] [Google Scholar]
  18. Brown, M. L., Taylor, A. N., Hambly, N. C., & Dye, S. 2002, MNRAS, 333, 501 [NASA ADS] [CrossRef] [Google Scholar]
  19. Bucko, J., Giri, S. K., & Schneider, A. 2023, A&A, 672, A157 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  20. Bucko, J., Giri, S. K., Hervas Peters, F., & Schneider, A. 2024, A&A, 683, A152 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  21. Cyr-Racine, F.-Y., Sigurdson, K., Zavala, J., et al. 2016, Phys. Rev. D, 93, 123527 [NASA ADS] [CrossRef] [Google Scholar]
  22. Das, S., Maharana, A., Poulin, V., & Sharma, R. K. 2022, Phys. Rev. D, 105, 103503 [NASA ADS] [CrossRef] [Google Scholar]
  23. de Blok, W. J. G. 2010, Adv. Astron., 2010, 1 [CrossRef] [Google Scholar]
  24. Del Popolo, A., & Le Delliou, M. 2021, Galaxies, 9, 123 [NASA ADS] [CrossRef] [Google Scholar]
  25. DES Collaboration (Abbott, T. M. C., et al.) 2022, Phys. Rev. D, 105, 023520 [CrossRef] [Google Scholar]
  26. Dodelson, S., & Widrow, L. M. 1994, Phys. Rev. Lett., 72, 17 [CrossRef] [Google Scholar]
  27. Drewes, M., Lasserre, T., Merle, A., et al. 2017, JCAP, 01, 025 [NASA ADS] [Google Scholar]
  28. Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 [Google Scholar]
  29. Fuß, L., & Garny, M. 2023, JCAP, 2023, 020 [Google Scholar]
  30. Garzilli, A., Magalich, A., Ruchayskiy, O., & Boyarsky, A. 2021, MNRAS, 502, 2356 [NASA ADS] [CrossRef] [Google Scholar]
  31. Giblin, B., Heymans, C., Asgari, M., et al. 2021, A&A, 645, A105 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  32. Giri, S. K., & Schneider, A. 2021, JCAP, 2021, 046 [Google Scholar]
  33. Giri, S. K., & Schneider, A. 2022, Phys. Rev. D, 105, 083011 [NASA ADS] [CrossRef] [Google Scholar]
  34. Grandis, S., Aricò, G., Schneider, A., & Linke, L. 2024, MNRAS, 528, 4379 [NASA ADS] [CrossRef] [Google Scholar]
  35. He, A., Ivanov, M. M., An, R., & Gluscevic, V. 2023, ApJ, 954, L8 [NASA ADS] [CrossRef] [Google Scholar]
  36. Heymans, C., Tröster, T., Asgari, M., et al. 2021, A&A, 646, A140 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  37. Hildebrandt, H., Viola, M., Heymans, C., et al. 2017, MNRAS, 465, 1454 [Google Scholar]
  38. Hildebrandt, H., Busch, J. L. v. d., Wright, A. H., et al. 2021, A&A, 647, A124 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  39. Hirata, C. M., & Seljak, U. 2010, Phys. Rev. D, 82, 049901 [NASA ADS] [CrossRef] [Google Scholar]
  40. Hlozek, R., Grin, D., Marsh, D. J. E., & Ferreira, P. G. 2015, Phys. Rev. D, 91, 103512 [NASA ADS] [CrossRef] [Google Scholar]
  41. Hui, L., Ostriker, J. P., Tremaine, S., & Witten, E. 2017, Phys. Rev. D, 95, 043541 [NASA ADS] [CrossRef] [Google Scholar]
  42. Hunter, J. D. 2007, Comput. Sci. Eng., 9, 90 [Google Scholar]
  43. Joachimi, B., Lin, C.-A., Asgari, M., et al. 2021, A&A, 646, A129 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  44. Kuijken, K., Heymans, C., Dvornik, A., et al. 2019, A&A, 625, A2 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  45. Lesgourgues, J., & Tram, T. 2011, JCAP, 2011, 032 [CrossRef] [Google Scholar]
  46. Lewis, A. 2019, arXiv e-prints [arXiv:1910.13970] [Google Scholar]
  47. Limber, D. N. 1953, ApJ, 117, 134 [NASA ADS] [CrossRef] [Google Scholar]
  48. Madhavacheril, M. S., Qu, F. J., Sherwin, B. D., et al. 2024, ApJ, 962, 113 [CrossRef] [Google Scholar]
  49. Markovic, K., & Viel, M. 2014, PASA, 31, e006 [NASA ADS] [CrossRef] [Google Scholar]
  50. Marsh, D. J. E., & Silk, J. 2014, MNRAS, 437, 2652 [NASA ADS] [CrossRef] [Google Scholar]
  51. Merle, A., & Schneider, A. 2015, Phys. Lett. B, 749, 283 [NASA ADS] [CrossRef] [Google Scholar]
  52. Merle, A., Schneider, A., & Totzauer, M. 2016, JCAP, 04, 003 [CrossRef] [Google Scholar]
  53. Moore, B., Ghigna, S., Governato, F., et al. 1999, ApJ, 524, L19 [Google Scholar]
  54. Osato, K., Sekiguchi, T., Shirasaki, M., Kamada, A., & Yoshida, N. 2016, JCAP, 2016, 004 [CrossRef] [Google Scholar]
  55. Pan, Z., Bianchini, F., Wu, W., et al. 2023, Phys. Rev. D, 108, 122005 [CrossRef] [Google Scholar]
  56. Pandey, S., Lehman, K., Baxter, E. J., et al. 2023, MNRAS, 525, 1779 [CrossRef] [Google Scholar]
  57. Parimbelli, G., Scelfo, G., Giri, S. K., et al. 2021, JCAP, 2021, 044 [CrossRef] [Google Scholar]
  58. Planck Collaboration V. 2020, A&A, 641, A5 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  59. Poulin, V., Bernal, J. L., Kovetz, E. D., & Kamionkowski, M. 2023, Phys. Rev. D, 107, 123538 [NASA ADS] [CrossRef] [Google Scholar]
  60. Prince, H., & Dunkley, J. 2019, Phys. Rev. D, 100, 083502 [NASA ADS] [CrossRef] [Google Scholar]
  61. Raveri, M., & Doux, C. 2021, Phys. Rev. D, 104, 043504 [NASA ADS] [CrossRef] [Google Scholar]
  62. Raveri, M., Zacharegkas, G., & Hu, W. 2020, Phys. Rev. D, 101, 103527 [Google Scholar]
  63. Refregier, A., Gamper, L., Amara, A., & Heisenberg, L. 2018, Astron. Comput., 25, 38 [NASA ADS] [CrossRef] [Google Scholar]
  64. Rogers, K. K., Hložek, R., Laguë, A., et al. 2023, JCAP, 2023, 023 [CrossRef] [Google Scholar]
  65. Rogers, K. K., & Poulin, V. 2024, arXiv e-prints [arXiv:2311.16377] [Google Scholar]
  66. Schaeffer, T., & Schneider, A. 2021, MNRAS, 504, 3773 [NASA ADS] [CrossRef] [Google Scholar]
  67. Schneider, A. 2015, MNRAS, 451, 3117 [NASA ADS] [CrossRef] [Google Scholar]
  68. Schneider, A., & Teyssier, R. 2015, JCAP, 2015, 049 [Google Scholar]
  69. Schneider, A., Anderhalden, D., Maccio, A. V., & Diemand, J. 2014, MNRAS, 441, L6 [NASA ADS] [CrossRef] [Google Scholar]
  70. Schneider, A., Teyssier, R., Stadel, J., et al. 2019, JCAP, 03, 020 [CrossRef] [Google Scholar]
  71. Schneider, A., Refregier, A., Grandis, S., et al. 2020a, JCAP, 2020, 020 [CrossRef] [Google Scholar]
  72. Schneider, A., Stoira, N., Refregier, A., et al. 2020b, JCAP, 04, 019 [CrossRef] [Google Scholar]
  73. Schneider, A., Giri, S. K., Amodeo, S., & Refregier, A. 2022, MNRAS, 514, 3802 [NASA ADS] [CrossRef] [Google Scholar]
  74. Shi, X., & Fuller, G. M. 1999, Phys. Rev. Lett., 82, 2832 [NASA ADS] [CrossRef] [Google Scholar]
  75. Takahashi, R., Sato, M., Nishimichi, T., Taruya, A., & Oguri, M. 2012, ApJ, 761, 152 [Google Scholar]
  76. Tarsitano, F., Schmitt, U., Refregier, A., et al. 2020, arXiv e-prints [arXiv:2005.00543] [Google Scholar]
  77. Tröster, T., Asgari, M., Blake, C., et al. 2021, A&A, 649, A88 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  78. Viel, M., Becker, G. D., Bolton, J. S., & Haehnelt, M. G. 2013, Phys. Rev. D, 88, 043502 [NASA ADS] [CrossRef] [Google Scholar]
  79. Vogt, S. M. L., Marsh, D. J. E., & Laguë, A. 2023, Phys. Rev. D, 107, 063526 [NASA ADS] [CrossRef] [Google Scholar]
  80. Wright, A. H., Hildebrandt, H., Busch, J. L. V. D., & Heymans, C., 2020, A&A, 637, A100 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  81. Zeldovich, Y. B., Einasto, J., & Shandarin, S. F. 1982, Nature, 300, 407 [NASA ADS] [CrossRef] [Google Scholar]

Appendix A: Degeneracy with baryonic feedback parameters

It was shown in Parimbelli et al. (2021) that the baryonification scheme can be considered independently from the MDM parameters up to k = 5h/Mpc at the percent level precision. As described in Section 2.1 the baryonic feedback parameters were treated as separate from MDM effects. Effectively, broad priors on our baryonic feedback parameters allowed for a large range of feedback suppressions. This broad prior was necessary to provide conservative constraints on the MDM parameters. We show in Fig. A.1 that the baryonic feedback parameters are mainly prior dominated. Recent works use external datasets to constraint baryonic feedback suppresions, such as X-ray and kSZ data (Schneider et al. 2022), tSZ data (Pandey et al. 2023) or gas density profiles from deep X-ray observations (Grandis et al. 2024). This kind of study should improve the constraints obtained on the MDM parameters, as it would allow to disentangle baryonic processes from free-streaming DM effects. We leave this investigation to future work.

thumbnail Fig. A.1.

Posterior contours of the baryonic feedback and MDM Parameters at the 68 and 95 % confidence level. The baryonic feedback parameters are prior-dominated.

All Tables

Table 1.

Prior ranges used in the Planck-18 TTTEEE and KiDS-1000 analysis.

In the text
Table 2.

2D tension metrics between Planck-18 and KiDS-1000 datasets for ΛCDM and ΛMDM models computed using parameter shifts in the Ωm − S8 plane with the tensiometer package.

In the text

All Figures

thumbnail Fig. 1.

Auto and cross angular band power spectra of the KiDS-1000 data set separated in five tomographic redshift bins (black data points). The data is obtained from Asgari et al. (2021). The lines correspond to predictions assuming a ΛCDM model (orange dashed) and three ΛMDM models with fixed thermal mass (mhdm = 50 keV) and increasing hot-to-total DM fraction (fhdm = 0.1, 0.5, 1.0) in green, purple, and blue. All plotted models are run assuming the Planck-18 best-fitting cosmology.
In the text
thumbnail Fig. 2.

Residual angular power spectra of the CMB temperature fluctuations from Planck-18 assuming a mixed DM model with varying hot-to-total DM fractions (coloured lines) and a thermal particle mass of mhdm = 10 eV (left), mhdm = 30 eV (middle), and mhdm = 100 eV (right). Note that the blue line (fhdm = 0) corresponds to the ΛCDM model. The grey error bars denote the 68% confidence interval of the Planck-18 TT data.
In the text
thumbnail Fig. 3.

Constraints on the MDM (ΛMDM) model from weak lensing (KiDS-1000) in green, the CMB (Planck-18 TTTEEE) in blue, and the combined WL+CMB in red (dotted line). TheΛMDM model is parametrised by the thermal mass of the hot species (mhdm) and the fraction of hot-to-total DM (fhdm). The top-left corner of the plot corresponds to the excluded region. All contours are shown at the 95% confidence level.
In the text
thumbnail Fig. 4.

Posterior contours in the S8 − Ωm plane for the ΛCDM (left) and ΛMDM models (right) at 68 and 95 % confidence level. The green, blue, and red contours correspond to the cosmic shear (KiDS-1000), the CMB (Planck-18 TTTEEE) and the combined analysis.
In the text
thumbnail Fig. A.1.

Posterior contours of the baryonic feedback and MDM Parameters at the 68 and 95 % confidence level. The baryonic feedback parameters are prior-dominated.
In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.