A non-standard Halo Mass Function as a solution to the structure-growth tension, application to KiDS-1000 and DES-y3

Semi-analytical computation of the matter power spectrum often relies on the halo mass function (HMF) as a key component. In this paper, we explore how certain variations of the HMF affect the modelling of the matter power spectra and quantify the impact on the structure growth parameter $S_8 = \sigma_8\sqrt{\Omega_\mathrm{m}/0.3}$. We use the weak gravitational lensing 2-point correlation functions from both Kilo Degree Survey (KiDS-1000) and Dark Energy Survey (DES-y3) to constrain the HMF parameters, which are sensitive to dark matter properties, structure formation, and baryonic feedback. When assuming the Planck cosmology, the canonical HMF parameters are rejected at more than $2\sigma$ level for both KiDS and DES, where reconstructed HMF from these posteriors give a $48.8\%^{+8.3\%}_{-9.4\%}$ lower for KiDS-1000 and a $29.5\%^{+8.5\%}_{-8.5\%}$ lower for DES-y3 in terms of total halo mass larger than $M_\mathrm{Halo} > 10^{14} M_\odot$ compared to a canonical HMF model. In addition, a Planck-like $S_8$ is also preferred if massive haloes were to have a $\sim 20\%$ lower abundance compared to a canonical halo mass function. Under one of these alternative HMFs, we found $S_8 \sim 0.826_{-0.019}^{+0.021}$ for KiDS-1000 and $0.851_{-0.021}^{+0.020}$ for DES-y3. Our work suggests that varying the halo abundance provides an alternative to varying the matter power spectrum when exploring possible solutions to the $S_8$ tension. By comparing the posteriors of both cosmological and HMF parameters between two different surveys (KiDS-1000 and DES-y3), we are also testing the self-consistency of the cosmological interpretation at a level that has corresponding particle and/or astrophysical interpretations.