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Clusters of galaxies possess the capability to accelerate cosmic rays (CRs)to very high energy up to $\sim10^{18}$~eV due to their large size and magneticfield strength which favor CR confinement for cosmological times. During theirconfinement, they can produce neutrinos and $\gamma-$rays out of interactionswith the background gas and photon fields. In recent work,\cite{hussain2021high, hussain2023diffuse} have conducted three-dimensionalcosmological magnetohydrodynamical (MHD) simulations of the turbulentintracluster medium (ICM) combined with multi-dimensional Monte Carlosimulations of CR propagation for redshifts ranging from $z \sim 5$ to $z = 0$to study the multi-messenger emission from these sources. They found that whenCRs with a spectral index in the range $1.5 - 2.5$ and cutoff energy$E_\mathrm{max} = 10^{16} - 10^{17}$~eV are injected into the system, they makesignificant contributions to the diffuse background emission of both neutrinosand gamma-rays. In this work, we have revisited this model and undertakenfurther constraints on the parametric space. This was achieved by incorporatingthe recently established upper limits on neutrino emission from galaxyclusters, as obtained by the IceCube experiment. We find that for CRs injectedwith spectral indices in the range $2.0 - 2.5$, cutoff energy $E_\mathrm{max} =10^{16} - 10^{17}$~eV, and power corresponding to $(0.1-1)\%$ of the clusterluminosity, our neutrino flux aligns with the upper limits estimated byIceCube. Additionally, the resulting contribution from clusters to the diffuse$\gamma$-ray background (DGRB) remains significant with values of the order of$ \sim 10^{-5}\, \mathrm{MeV} \, \mathrm{cm}^{-2} \,\mathrm{s}^{-1} \,\mathrm{sr}^{-1}$ at energies above $500$ GeV.

Neutrinos and gamma-rays from Galaxy Clusters constrained by the upper limits of IceCube