PREPRINT

# $i\left(cm\right)z$, a semi-analytic model for the thermodynamic properties in galaxy clusters: calibrations with mass and redshift, and implication for the hydrostatic bias

S. Ettori, L. Lovisari, D. Eckert

Submitted on 6 November 2022, last revised on 18 November 2022

## Abstract

In the self-similar scenario for galaxy cluster formation and evolution, the thermodynamic properties of the X-ray emitting plasma can be predicted in their dependencies on the halo mass and redshift only. However, several departures from this simple self-similar scenario have been observed. We show how our semi-analytic model $i\left(cm\right)z$, which modifies the self-similar predictions through two temperature-dependent quantities, the gas mass fraction ${f}_{g}={f}_{0}{T}^{{f}_{1}}{E}_{z}^{{f}_{z}}$ and the temperature variation ${f}_{T}={t}_{0}{T}^{{t}_{1}}{E}_{z}^{{t}_{z}}$, can be calibrated to incorporate the mass and redshift dependencies. We used a published set of 17 scaling relations to constrain the parameters of the model. We were subsequently able to make predictions as to the slope of any observed scaling relation within a few percent of the central value and about one $\sigma$ of the nominal error. Contextually, the evolution of these scaling laws was also determined, with predictions within $1.5\sigma$ and within 10 percent of the observational constraints. Relying on this calibration, we have also evaluated the consistency of the predictions on the radial profiles with some observational datasets. For a sample of high-quality data (X-COP), we were able to constrain a further parameter of the model, the hydrostatic bias $b$. By calibrating the model, we have determined that (i) the slopes of the temperature dependence are ${f}_{1}=0.403\left(±0.009\right)$ and ${t}_{1}=0.144\left(±0.017\right)$; and that (ii) the dependence upon ${E}_{z}$ are constrained to be ${f}_{z}=-0.004\left(±0.023\right)$ and ${t}_{z}=0.349\left(±0.059\right)$. These values permit one to estimate directly how the normalizations of a given quantity ${Q}_{\mathrm{\Delta }}$ changes as a function of the mass (or temperature) and redshift halo in the form ${Q}_{\mathrm{\Delta }}\sim {M}^{{a}_{M}}{E}_{z}^{{a}_{z}}\sim {T}^{{a}_{T}}{E}_{z}^{{a}_{Tz}}$, in very good agreement with the current observational constraints.

## Preprint

Comment: 15 pages; A&A in press. Updated to match the published version

Subject: Astrophysics - Cosmology and Nongalactic Astrophysics