© The Authors 2024

1. Introduction

Galaxy clusters (GCs) are collections of 50 to a few thousand galaxies held together by gravitational forces. These clusters have typical masses around 10¹⁴ M_⊙ and radii of R ≳ 1 Mpc. The majority of the mass in GCs (up to 84−90%) is in the form of dark matter (DM). Only 10−16% is in the form of baryonic matter, which is primarily made up of hot gas and known as the intra-cluster medium (ICM). Stars only account for 1−5% of the total mass (Cimatti et al. 2019).

The ICM is an almost fully ionised plasma at temperatures around T_ICM ∼ 10^7 − 8 K. This plasma extends throughout the entire cluster volume and emits X-rays predominantly through thermal bremsstrahlung (Sarazin 1988; Birkinshaw 1999). Electrons in the ICM are not only scattered by ions (bremsstrahlung), but can themselves scatter photons. Therefore, apart from being extremely luminous X-ray sources, GCs also leave an imprint on the cosmic microwave background (CMB; Penzias & Wilson 1979) through the Sunyaev–Zeldovich effect (SZ; Zeldovich & Sunyaev 1969), arising from the scattering of CMB photons off ICM electrons through the inverse Compton effect (see Rephaeli 1995; Birkinshaw 1999; Carlstrom et al. 2002; Mroczkowski et al. 2019, for reviews).

The SZ effect can be separated in two main components: the thermal (tSZ) and the kinetic (kSZ) effects. The thermal signal is produced as CMB photons travel through the ICM of a GC and get scattered, in any direction, by the random thermal motions of the ICM electrons and, since the ICM is very hot, it shifts the CMB spectrum towards higher frequencies. The change in the black-body CMB temperature (T) due to the tSZ effect (first calculated by Zeldovich & Sunyaev 1969 using the Kompaneets 1957 equation) is given by Mroczkowski et al. (2019):

$$\begin{array}{c}\hfill \frac{\mathrm{\Delta }{T}_{\mathrm{tSZ}}}{T}=f(\nu ,T)y_{\mathrm{tSZ}},\mathrm{with}{y}_{\mathrm{tSZ}}=\frac{{\sigma }_{\mathrm{T}}}{m_{\mathrm{e}}{c}^2}{\displaystyle \int \mathrm{d}}s{k}_{\mathrm{B}}{T}_{\mathrm{e}}{n}_{\mathrm{e}},\end{array}$$(1)

where n_e is the electron number density, T_e the electron gas temperature, m_e the electron mass, σ_T the Thomson scattering cross-section, k_B the Boltzmann constant, and c the speed of light. On the other hand, f(ν, T) is a function of frequency and CMB temperature¹, y_tSZ is the so called Compton y-parameter, which encapsulates the GC temperature and density distributions, and ds is the physical distance travelled by the photon along the line of sight (LoS). We remark that considering the electron gas as an ideal gas (which is accurate enough given the temperature and density regime), the equation of state is P_e = k_BT_en_e and, therefore, y_tSZ measures the electron pressure (P_e) integrated along the LoS.

On the other hand, the kinetic effect arises from the fact that due to the bulk motion of the GC and the internal movements of the ICM (rotation, turbulence, etc.), the scattering medium is moving with respect to the CMB reference frame (Hubble flow). The change in the CMB temperature due to the kSZ effect alone is given by Sunyaev & Zeldovich (1980), Mroczkowski et al. (2019):

$$\begin{array}{c}\hfill \frac{\mathrm{\Delta }{T}_{\mathrm{kSZ}}}{T}=-y_{\mathrm{kSZ}},\mathrm{with}{y}_{\mathrm{kSZ}}={\sigma }_{\mathrm{T}}{\displaystyle \int \mathrm{d}s}\frac{v_{\mathrm{los}}}{c}{n}_{\mathrm{e}},\end{array}$$(2)

where v_los is the velocity along the LoS and y_kSZ is the Compton y-parameter for the kSZ effect. The sign criterium is such that if the gas is moving away from the observer (v_los > 0), it decreases the CMB temperature. In this case, y_kSZ encapsulates the kinematic structure of the cluster and its peculiar motion. In contrast to the tSZ signal, the kinetic effect does not depend on frequency and, therefore, it cannot modify the CMB black-body spectrum. In this sense, the kSZ effect can be understood as a Doppler boost of CMB photons (e.g. Baxter et al. 2019). In fact, as Rephaeli (1995) pointed out, Eq. (2) can be obtained from a simple relativistic transformation.

The kSZ signal can be further split in different contributions. The one that contributes the most is the bulk or monopole component (e.g. Hand et al. 2012; Chen et al. 2022), which is due to the movement of the cluster as a whole with respect to the CMB. Nevertheless, if we subtract this component, we get more contributions attributed to the ICM internal motions such as rotation, which produces a dipole-like signal (e.g. Cooray & Chen 2002; Chluba & Mannheim 2002), or random motions due to turbulence which fill most of the ICM volume (e.g. see Vazza et al. 2017; Vallés-Pérez et al. 2021a,b). While the monopole can be used to measure GC peculiar velocities, the other contributions are useful to understand the internal dynamics of the gas in GCs. The kinetic SZ signal remaining from the subtraction of the monopole has been studied especially for an ideal rotation alignment, that is, when the angular momentum is perpendicular to the LoS. In this particular case, the dipole signal due to rotation maximises and the effect is known as the rotational kSZ signal (rkSZ; Cooray & Chen 2002; Chluba & Mannheim 2002). However, in general, the rotation axes will be randomly aligned, and hence, the dipole signal due to rotation will not necessarily be the dominant contribution after the monopole. Other ICM internal movements, such as turbulence or outflows, could contribute significantly.

While the frequency-dependent tSZ effect is the dominant component and is relatively easy to observe, the kSZ effect, which generates a fainter frequency-independent signal, can be easily confused with primary CMB anisotropies (e.g. Colafrancesco 2007). Nevertheless, on small angular scales, where these anisotropies have minimal influence, the kSZ effect enables the measurement of gas peculiar velocities, particularly velocity gradients within the ICM (see Dupke & Bregman 2002 for a discussion on these measurements). Hence, in combination with X-ray emission and the tSZ signal, the kSZ effect can help in disentangling the complex ICM velocity structure produced by AGN feedback, cosmological structure formation, or turbulent motions (see e.g. Battaglia et al. 2017; Simionescu et al. 2019). Moreover, since the kSZ signal is independent of temperature and linearly proportional to electron density, it is especially convenient for exploring low density and temperature regions such as the outskirts of galaxy clusters and groups of galaxies. Thus, it offers the potential of shedding light on the missing baryon problem (e.g. see Hernández-Monteagudo et al. 2015 for observational evidence or Planelles et al. 2018 for a computational approach). From a different point of view, the SZ effect measurements can be used to disentangle the nature of other components of the Universe such as dark matter or dark energy, as well as a tool to test the feasibility of different cosmological models (e.g. non-flat Universe) or other theories of gravity (see e.g. Alfano et al. 2024).

Currently, thanks to measurements carried out by observational facilities such as Planck (Planck Collaboration XXIX 2014), South Pole Telescope (SPT; Reichardt et al. 2013), Atacama Cosmology Telescope (ACT; Hasselfield et al. 2013), or Combined Array for Research in Millimeter-wave Astronomy (CARMA; Plagge et al. 2013), the number of clusters studied using the SZ signal has increased significantly. Moreover, in the upcoming decades future X-ray missions, such as Xrism² and Athena³, are expected to provide high precision measurements of ICM motions. In a similar way, future mm/sub-mm facilities (see e.g. AtLAST⁴ or SKA⁵) are expected to have a high-enough resolution to be able to measure the ICM velocity field by means of the kSZ effect. This synergy between future X-ray and mm/sub-mm observations will offer us unprecedented insights on the ICM thermodynamic properties.

Many successful attempts to measure bulk motions with the kSZ effect have been carried out so far (e.g. Hand et al. 2012; Soergel et al. 2016; Hill et al. 2016; Chen et al. 2022). Concerning the internal movements contribution to the kSZ effect, an early attempt to produce a spatially resolved map of the kSZ signal in a galaxy cluster was presented by Sayers et al. (2013). An improved measurement was provided by Adam et al. (2017), who detected a dipolar structure in the kSZ signal of the merging cluster MACS J0717.5+3745. The signal was consistent with two sub-clusters, moving with velocities of several thousand km s⁻¹: one moving away and the other approaching the observer. In Matilla & Haiman (2020), the authors study the viability of statistically detecting a rotational kSZ signal through the combination of CMB data across numerous galaxies, where the spin orientation can be spectroscopically estimated.

Other recent observational efforts to further study gas thermodynamics with both thermal and kinetic SZ include Schaan et al. (2016, 2021), where the authors use both effects to study the baryon density and pressure profiles of galactic halos, finding that they are shallower than their DM counterparts. In a similar direction, Amodeo et al. (2021) constrain the amount of non-thermal pressure inside the virial radius of GCs and Mallaby-Kay et al. (2023) presented a novel hybrid estimator to measure kSZ, putting constraints on the thermodynamic properties of galaxy halos. Furthermore, regarding the cluster dynamical state and SZ connection, Adam et al. (2024) confirm that the observed disturbed clusters have less concentrated and shallower electron pressure profiles compared to what would be expected for a relaxed cluster.

Early works using numerical simulations, such as Springel et al. (2001), already put constraints on the mean Comptonisation due to tSZ. More recent works (e.g. Planelles et al. 2017, 2018; Lokken et al. 2023) have further expanded our knowledge studying the tSZ effect on clusters and their surroundings producing mock y maps and scaling relations. Moreover, other studies such as Baldi et al. (2018) or Altamura et al. (2023) have focused on the kSZ effect, trying to disentangle the different components (namely the cluster peculiar velocity and rotation) producing the signal.

In view of upcoming observational efforts and as a continuation of the computational studies aforementioned, in this paper, we aim to study the tSZ and kSZ effects on a sample of simulated galaxy clusters (N ∼ 10²) at different redshifts and dynamical states to provide results to interpret and lead future observations. For each simulated cluster in our sample we produce maps of the tSZ and kSZ signals. With these results, we study the mean features of both SZ effects and look for possible correlations with the dynamical state of the clusters in our sample. We present the individual results for several prototypical clusters, together with a statistical approach considering the whole sample. The redshift evolution is also considered.

The structure of the manuscript is a follows. In Sect. 2, we give the details of the simulation, the halo finder and the cluster sample and we describe how the dynamical state of the clusters is defined. In Sect. 3, we describe the results obtained for the analysis of single clusters and the whole sample. Finally, we present our conclusions and discuss our results in Sect. 4.

2. Methods

2.1. The simulation

In this work we examine a numerical simulation performed with the cosmological code MASCLET (Quilis 2004; Quilis et al. 2020). This code, designed primarily for cosmological purposes, employs (magneto-)hydrodynamics and N-body techniques. The gaseous (collisional) component is evolved using high-resolution shock-capturing methods, while the DM component is evolved using an N-body multilevel Particle-Mesh (PM) scheme (Hockney & Eastwood 1988). The gravity solver ensures the coupled evolution of both components. Also, to enhance spatial and temporal resolution, MASCLET incorporates an adaptive mesh refinement (AMR) scheme.

The simulation encompasses a periodic, cubic domain with a size of L = 147.5 Mpc, discretised into 256³ cubical cells. It assumes a flat ΛCDM cosmology characterised by a matter density parameter Ω_m = 0.31 (Ω_Λ = 1 − Ω_m), a baryon density parameter Ω_b = 0.048, and a Hubble parameter h ≡ H₀/(100 km s⁻¹ Mpc⁻¹) = 0.678. The initial conditions originate from a realisation of the primordial Gaussian random field, assuming a spectral index of n_s = 0.96 and an amplitude resulting in σ₈ = 0.82. These conditions are set up at redshift z_ini = 100 using a CDM transfer function (Eisenstein & Hu 1998). The chosen cosmological parameter values are in line with the latest findings reported by Planck Collaboration VI (2020).

Starting from the initial conditions and evolving them until the present time, using a low-resolution run with equal-mass particles, we selected regions that meet specific refinement criteria to establish three levels of refinement (ℓ = 1, 2, and 3) within the AMR scheme. In these initially refined levels, the DM component is sampled with particles that are 8, 64, and 512 times lighter than those employed for regions in ℓ = 0. As the evolution progresses, local baryonic and DM densities are utilised to create finer grids, ultimately reaching a maximum refinement level of ℓ = 6. The ratio between cell sizes for a given level (ℓ + 1) and its parent level (ℓ) is Δx_ℓ + 1/Δx_ℓ = 1/2, striking a balance between the prevention of numerical instabilities and the increase in resolution. This enables us to achieve a peak physical resolution of approximately 9 kpc at z = 0. With four different DM particle species, the mass resolution amounts to approximately 2 × 10⁶ M_⊙, equivalent to using 2048³ particles in the entire computational domain.

In addition to gravity, the simulation takes into account cooling mechanisms (free-free, inverse Compton, and atomic and molecular cooling for a primordial gas) and heating mechanisms (UV background radiation; Haardt & Madau 1996).

2.2. Cluster sample and dynamical state

In order to identify the GCs produced in the simulation, we used the publicly available spherical overdensity halo finder ASOHF⁶ (see Planelles & Quilis 2010; Knebe et al. 2011; Vallés-Pérez et al. 2022, for more details). It finds DM bound structures (halos) and substructures (subhalos) in each snapshot of the simulation. In the simulation analysed in this work, ASOHF identifies N = 86 halos with virial mass⁷, M_vir, greater than 5 × 10¹³ M_⊙ at z = 0. In Fig. 1, we plot the halo mass function of this simulation in the last iteration against the fit provided by Tinker et al. (2008). In the M_vir/M_⊙ ∈ [10¹³,  5 × 10¹⁴] range, the fit and simulation are in fair agreement.

Fig. 1. Cumulative halo mass function at z = 0 of the simulation (points) and the fit (black line) described in Tinker et al. (2008) for the cosmology given in Sect. 2.1 using the COLOSSUS package (Diemer 2018). The errorbars and shaded region correspond to 1σ Poissonian error.

Since our aim is to study if and how the SZ effect changes depending on the dynamical state of the clusters, we need a robust estimation of this property. For this study, we employed the procedure described in Vallés-Pérez et al. (2023), in which a redshift-dependent combination of five different dynamical and morphological indicators is proposed to estimate the cluster dynamical state. We briefly summarize the main procedure and we refer to the previous work for a detailed description of the methodology employed to classify clusters according to their dynamical state. We employed the following dynamical indicators of the DM halo:

• Sparsity: s_200c, 500c = M_200c/M_500c

• Ellipticity: ϵ = 1 − c/a

• Centre offset: Δ_r = |r_peak − r_CM|/R_vir

• Mean radial velocity: ${⟨\stackrel{\sim }{v_{\mathrm{r}}}⟩}_{\mathrm{DM}}\equiv \frac{\left|{⟨v_{\mathrm{r}}⟩}_{\mathrm{DM}}\right|}{V_{\mathrm{circ},\mathrm{vir}}}$

• Virial ratio: $\eta \equiv \frac{2E_{\mathrm{k}}}{|E_{\mathrm{p}}|}$

where M_200c and M_500c are the masses enclosed within the radius with an average overdensity of 200ρ_c and 500ρ_c, respectively, c and a are the smallest and largest semi-axes of the best fitting ellipsoid describing the DM mass distribution, r_peak and r_CM are the position of the density peak and centre of mass with respect to the simulation box centre, |⟨v_r⟩_DM| and V_circ, vir are the mean radial velocity of DM particles and the circular velocity at R_vir, and E_k, E_p are the total kinetic and potential energy within the same aperture.

Each indicator, X_i, has a redshift dependent threshold, $X_i^{\mathrm{thr}}(z)$, below which the cluster is classified as relaxed. Therefore, for a particular cluster, if every indicator X_i satisfies $X_i<X_i^{\mathrm{thr}}(z)$ the cluster is regarded as “totally relaxed”. Otherwise, the five dynamical indicators are combined to define a new parameter, χ, representing the degree of relaxedness of the considered halo (see Vallés-Pérez et al. 2023, for details on the definition of this parameter). Following this procedure, a given cluster which does not fall into the totally relaxed category will be classified as “marginally relaxed” if χ ≥ 1 or “unrelaxed” whenever χ < 1.

Applying this procedure to our cluster sample, we ended up with N_relaxed = 11, N_unrelaxed = 41, and N_marginally = 34 clusters at z = 0. However, in order to avoid contamination of our results due to poorly resolved halos, for every cluster in our sample we quantified the fraction of the cluster mass inside R_vir that is refined at ℓ = 6, ℓ ≥ 5 and ℓ ≥ 4 (see Fig. 2). According to these results, we excluded from our sample all those halos that fulfill the mass fraction at each level, f_i, of: f_ℓ = 6 < 15%, f_{ℓ ≥ 5} < 50%, or f_{ℓ ≥ 4} < 90%. After pruning these poorly resolved halos, we ended up with a sample of N_relaxed = 11, N_unrelaxed = 20, and N_marginally = 32; hence, we have a total of N = 63 clusters at z = 0, reducing our initial sample by 25%.

Fig. 2. Number of clusters in our sample as a function of the fraction of their mass, within Rvir, that is refined at ℓ = 6 (upper panel), ℓ ≥ 5 (middle panel) and ℓ ≥ 4 (lower panel). To avoid poorly resolved halos from contaminating our results, in our final sample we have removed halos with fℓ = 6 < 15%, fℓ ≥ 5 < 50% and fℓ ≥ 5 < 90%, where fi is the corresponding mass fraction.

On the other hand, we are also interested in studying how the SZ effect in a particular cluster evolves with z. For that purpose, we produced the merger tree from the ASOHF outputs and we traced the cluster population at z = 0 backwards in cosmic time, so that we could analyse the evolution of each of those N = 63 clusters individually.

2.3. SZ mock y maps

There are available packages for producing mock y maps for the kinetic and thermal SZ effects, such as pyMSZ (Cui et al. 2018; Baldi et al. 2018). Nevertheless, they are made for particle data in SPH simulations; hence, we would need to properly transform our AMR data in order to use them. For this reason, we have designed our own package in order to produce mock SZ observations from block-based AMR simulations⁸ (in particular, for the MASCLET code).

The algorithm works as follows. For each cluster in our sample, we define a map of physical size 2 R_vir × 2 R_vir centred on the DM density peak, with the maximum resolution of the simulation. In the current version of our algorithm, the map’s LoS has to be parallel to one of the three coordinate axes of the computational domain. Once the map size is defined, we can establish the integration width or depth, w, within which we integrate Eqs. (1) and (2). In this case, we choose w = 2 R_vir and we centre the integration in the density peak so that the integration interval is ( − R_vir, R_vir). Then, in order to calculate the y-value in each pixel, we integrate, using the rectangle rule, the former equations along the photon path with an integration step ds equal to the maximum resolution of our AMR grid. The integration is done at fixed z, that is, using just one snapshot for each case. This is because the photon crossing time, Δt_γ ∼ 1 − 10 Myr, is much smaller than the timescale for typical ICM motions, Δt_ICM, which we define as the timescale on which the ICM could change noticeably its structure at the considered scales due to its internal dynamics (see Fig. 3 of Vallés-Pérez et al. 2021a). In this regard, since the gas moves at v_ICM ≲ 10³ km s⁻¹ (e.g. Simionescu et al. 2019) and R_vir ∼ 1 Mpc, we have Δt_ICM ∼ 1 Gyr. Thus, we should not expect the density, temperature and velocity fields to change significantly during the photon crossing time.

Since our simulation does not take into account stellar feedback nor AGNs, overcooling affects to small volumes of the clusters, leading to unphysical values of density and temperature, and therefore, of y parameters. To clear up this unwanted effect, we set up an upper comoving baryon density threshold for all cells involved in the integrated quantities along the LoS. After some experimentation, we chose the value ρ_thres = 5 × 10⁻²⁷ g cm⁻³. Finally, since the Compton scattering is significantly produced only in the hot-ICM phase, we eliminate from our calculation those cells with T < 10⁶ K. With both thresholds we assure that all the cells contributing to the calculation have typical ICM values. In practice, for all clusters, only < 2% of the volume is dismissed.

To overcome the contained size of our sample (and only to increase the statistics when needed), we have considered the projections along the x, y and z axes, that is, projections on the YZ, XZ, and XY planes, for each individual cluster as three different clusters in our extended sample. In this manner, the number of objects in the sample can be increased and get closer to the observational conditions in which no privileged direction is considered.

2.4. Mass normalisation

In order to analyze the SZ signal dependence on the clusters’ dynamical state, we should get rid of implicit dependencies on other quantities, such as mass. To do this, we need to find a mass dependence for each effect. For the thermal signal we have:

$$\begin{array}{c}\hfill y_{\mathrm{tSZ}}\propto {\displaystyle \int \mathrm{d}s}{n}_{\mathrm{e}}{T}_{\mathrm{e}}\propto {\overline{\mathrm{\Sigma }}}_{\mathrm{gas}}{\overline{T}}_{\mathrm{gas}}\propto f(z)\frac{M_{\mathrm{vir}}^{5/3}}{R_{\mathrm{vir}}^2}\propto f(z)M_{\mathrm{vir}},\end{array}$$(3)

where we have used that ${\overline{\mathrm{\Sigma }}}_{\mathrm{gas}}\propto M_{\mathrm{vir}}/R_{\mathrm{vir}}^2$ is the mean surface density, the self-similar temperature-mass relation ${\overline{T}}_{\mathrm{gas}}\propto M_{\mathrm{vir}}^{2/3}f(z)$ for the mean ICM temperature (e.g. Voit 2005, noting that f(z) is a function of redshift that is not relevant, since the normalisation is done at fixed z) and the fact that, by definition, $R_{\mathrm{vir}}\propto M_{\mathrm{vir}}^{1/3}$. On the other hand, for the kinetic part we have:

$$\begin{array}{c}\hfill y_{\mathrm{kSZ}}\propto {\displaystyle \int \mathrm{d}s}{n}_{\mathrm{e}}{v}_{\mathrm{los}}\propto {\overline{\mathrm{\Sigma }}}_{\mathrm{gas}}{\overline{v}}_{\mathrm{los}}\propto g(z)\frac{M_{\mathrm{vir}}^{3/2}}{R_{\mathrm{vir}}^{5/2}}\propto g(z)M_{\mathrm{vir}}^{2/3},\end{array}$$(4)

where we consider that the average ICM velocity v_los scales as the circular velocity at the virial radius and, hence, ${\overline{v}}_{\mathrm{los}}\propto g(z)\sqrt{G{M}_{\mathrm{vir}}/R_{\mathrm{vir}}}$, being g(z) a function of redshift, which will not contribute to our normalisation, since it is done at fixed z.

Although based on simple arguments, the obtained scaling relations, Eqs. (3) and (4), allow us to purge the mass dependence. As shown in the results presented in the next section, the clusters in our sample fit well to these scaling relations within the errors, except for the kSZ signal of relaxed clusters; as discussed in Sect. 3.1.1, it should not follow this mass dependence.

3. Results

In this section, we present the results obtained from the analysis of the mock SZ y maps produced for every cluster in the sample. First, in Sect. 3.1, we focus on the results at z ∼ 0, and in Sect. 3.2, we extend the analysis to the effect of cosmic evolution on SZ effects. Throughout this work, unless otherwise specified, we use the term “kinetic SZ effect” (kSZ) to refer to the effect produced when the monopole is subtracted, that is, the kinetic effect produced by the gas movements within the ICM in the cluster’s reference rest frame.

3.1. Analysis of the SZ effect at z = 0

Taking into account the dynamical state of the halos in our sample, we analyse their associated SZ signals at z = 0 by means of scaling relations (Sect. 3.1.1), radial profiles (Sect. 3.1.3), and projected maps (Sect. 3.1.4) of the most relevant quantities in order to find trends between different dynamical classes. We also try to disentangle the relevance of rotation on the kinetic effect (Sects. 3.1.1 and 3.1.4). Additionally, in Sect. 3.1.2, we discuss how varying the line-of-sight direction can affect the signal (the projection effect), depending on cluster dynamical state.

3.1.1. Scaling relations

A well-known scaling relation is the one between the tSZ y-parameter (e.g. Planck Collaboration XX 2014), integrated within some specific radius (for example, R_500c) and the mass enclosed within that aperture. This relation has also been confirmed from a numerical point of view (e.g. Planelles et al. 2017). In our case, we used this relation as a test to validate our results.

One of the goals of the present paper is to look for similar correlations and scaling relations for the kSZ effect. However, the analysis of this effect requires some caution to be taken. Given that the kSZ signal can have positive and negative values, mean values are not a well suited choice for this analysis. So, to make a comparison with tSZ, we defined the root mean square (RMS) of the y_kSZ pixel values of the mock kSZ maps⁹. In this way, the kSZ is described with a positive quantity and is directly comparable to the mean of the y_tSZ pixel values of the tSZ map. Hence, we used the mean y-parameter instead of the integrated quantity for the tSZ effect in order to be consistent¹⁰.

The results are presented in Fig. 3, where we show the scaling relation between M_vir and the mean y_tSZ (upper panel, denoted by ${\overline{\mathit{y}}}_{\mathrm{tSZ}}$) and the RMS of the y_kSZ values (lower panel, denoted by RMS_kSZ), both integrated in a R_vir aperture. The shown errobars correspond to the difference between the maximum and minimum values of ${\overline{\mathit{y}}}_{\mathrm{tSZ}}$ and RMS_kSZ found in the three projections (XY, XZ, and YZ) of each cluster. Furthermore, in order to capture the difference between the trends of different cluster dynamical states, we perform a fit of the type y = A(M_vir/M₀)^α where M₀ = 10¹⁴ M_⊙ is the pivot mass. The best fitting parameters are presented in Table 1. No mass normalisation is carried out here.

Fig. 3. Scaling relation between the virial mass, Mvir, and the mean y-parameter for the tSZ effect (top panel) and for the RMS associated with the kSZ effect (bottom panel). The integration aperture is Rvir. The represented points correspond, for each cluster, to the mean across the three spatial projections. In both panels, the data have been fitted according to y = A(Mvir/M0)α, where M0 = 1014 M⊙ is the pivot mass and A is a normalisation. The best fitting parameters are presented in Table 1. The shaded regions correspond to the 16 − 84 confidence interval around the fit, whereas the error bars represent the variation (maximum and minimum values) across the three spatial projections for each cluster.

As expected, we obtain a tight correlation between ${\overline{\mathit{y}}}_{\mathrm{tSZ}}$ and M_vir, without any significant difference due to dynamical state. The slopes are similar and consistent within the 16 − 84 percentile errors shown, being also compatible with the theoretical α = 1 in Eq. (3). Regarding the kSZ part, we observe more scatter, especially at lower mass, and we also obtain different trends depending on the cluster dynamical state, as shown by data fittings. The unrelaxed population shows a steeper (α = 0.77) trend compared to the relaxed clusters (α = 0.35), but, more importantly, it shows a higher RMS_kSZ value at all masses compared to the relaxed population. As expected, the marginally relaxed population is halfway of the other two. We note how the fits for unrelaxed and all clusters are compatible, within their errors, with the theoretical value α = 2/3 given in Eq. (4), while the relaxed fit is incompatible (within 1σ). Also note how the marginally relaxed population, which is the largest, follows almost perfectly the fit obtained for the whole sample (dashed line), which is the most consistent with the theoretical α = 2/3.

A plausible explanation for this separation in the kinetic effect, due to dynamical state, is the fact that the recent mergers and high accretion suffered by the unrelaxed clusters can trigger high velocity motions in the ICM (turbulence, outflows, etc.); hence, the overall kSZ signal of the cluster is increased. To justify this finding, in Fig. 4, we plot the kSZ signal against the gas velocity dispersion with an analogous fit. We show that the higher the velocity dispersion is, the more intense the kinetic effect gets. On the other hand, the unrelaxed population is shifted towards higher velocity dispersion, while the relaxed sample is shifted towards lower velocity dispersion. Thus, the unrelaxed population has an overall higher kSZ signal, on average. Moreover, at fixed σ_v, the value of RMS_kSZ for disturbed clusters is higher than for the relaxed ones. More precisely, we obtain, approximately, RMS${}_{\text{kSZ}}\propto {\sigma }_v^{3/2}$ and RMS_kSZ ∝ σ_v from the disturbed and relaxed fittings, respectively.

Fig. 4. Scaling relation between the gas velocity dispersion inside the aperture and the root mean square for the kSZ. The integration aperture is Rvir. The fit corresponds to RMSkSZ = A(σv/σ0)B, where σ0 = 300 km s−1 is the pivot velocity dispersion and the shown error corresponds to a 16 − 84 confidence interval. The fitting slopes (α) and errors are shown below the figure legend.

As mentioned above, the slope of the fit of the relaxed population for the RMS_kSZ is incompatible within 1σ error with the theoretical value (α = 2/3). For a cluster of our sample to be relaxed, five different thresholds have to be fulfilled (see Sect. 2.2). This means that our criteria for considering a cluster as relaxed are quite restrictive; hence, their dynamics needs to be really calm. In fact, relaxed clusters are mostly supported by thermal pressure (they are close to be in hydrostatic equilibrium) and the kinetic pressure produced by ICM bulk motions does not play an important role in its support against gravitational collapse. Thus, we can expect the scaling derived for the kSZ mass dependence in Eq. (4) to be overestimating the value of v_los which, for our relaxed population, is likely to be mostly contributed by residual motions, causing the disagreement. In principle, this difference in the slopes between relaxed and disturbed clusters for the kSZ signal could be used to infer the cluster dynamical state depending on which of both trends is followed.

It is also interesting to study the connection between the kinetic effect and the cluster angular momentum. In order to mitigate the mass dependence for studying this correlation, we have to normalize the kinetic effect according to Eq. (4)¹¹. Besides, it would be desirable to study an angular momentum related quantity which is independent of mass. Thus, we have used the spin parameter presented in Bullock et al. (2001):

$$\begin{array}{c}\hfill \lambda =\frac{L}{\sqrt{2}MVR},\end{array}$$(5)

where L is the magnitude of the total angular momentum inside a given radius, R, M is the mass enclosed within that radius, and V is the circular velocity at radius, R. Since we are working with projections, we try to correlate to the kinetic signal only the component of the total angular momentum that lies in the projection plane. For instance, in the XY projection we only consider $L=\sqrt{L_x^2+L_{\mathit{y}}^2}$, since the L_z component of the rotation will not contribute to the kinetic signal, as it does not have velocity component along the LoS.

The results are presented in Fig. 5, where we apply a similar fit to the data to disentangle the differences between populations. We note that since we have three projections for each cluster, and the effect and angular momentum are different in each projection, we can treat each projection as an independent cluster signal; thus we are able to enhance our statistics by a factor of three. Unrelaxed clusters hardly show any degree of correlation since the scatter is considerable and the fit is compatible with an horizontal line (α = 0.03 ± 0.10, with p-value = 0.75). On the other hand, for the relaxed case, we observe less scatter and a clear tendency to have a higher kSZ signal with increasing angular momentum (α = 0.29 ± 0.06), which is translated into a small p-value (that is, a correlation does exist). Therefore, from this figure, we can infer that in the disturbed case, post-merger motions dominate over rotation and, as a result, blur any possible correlation with angular momentum; on the other hand, relaxed clusters are dominated by rotation, showing a clear correlation with λ.

Fig. 5. Scaling relation between the Bullock spin parameter (λprojection) of the angular momentum contained in each projection against the RMS of the kSZ signal of each projection normalised to a Mvir = 1014 M⊙ mass. Since neither of the two quantities depend on mass the relationship shown correlates the cluster spin with the kSZ effect. The fitting slopes (α) and errors are shown in the upper left part.

3.1.2. Projection effects

In Fig. 3, we show for each cluster the average value of the RMS_kSZ when considering three different lines of sight aligned with the Cartesian axes of the computational domain, whereas the error bars show the range between the maximum and minimum values over the three different projections. It is noticeable that the variation is, generally, significantly higher in disturbed clusters. To quantify this trend, in Fig. 6 we present an histogram of the variation of the RMS of the kSZ signal across the three different projections (lines of sight) for the relaxed and disturbed dynamical state classes. It is clear that low variations are dominated by relaxed clusters and broad variations are mainly given in the unrelaxed cases. Hence, the impact of having the information projected in a certain plane is more important for disturbed clusters, since they tend to be aspherical and present significant substructure that can be erased due to projection effects. The opposite happens for relaxed clusters, as they generally have a rounder shape, minimising the impact of projection.

Fig. 6. Histogram of the number of clusters of each dynamical state class against the logarithmic difference between the maximum and minimum values of the RMS of the kSZ effect obtained with the three projections XY, XZ and YZ.

For the tSZ signal, the projection effect is minimum compared to the kinetic part, to the point where the variation range is smaller than the data points in Fig. 3. A plausible explanation comes from the fact that the kSZ signal depends on v_los, which has a different value depending on the projection; meanwhile, the tSZ effect involves scalar quantities that do not vary with the direction of observation.

3.1.3. Radial profiles

Another approach to assessing whether the cluster dynamical state affects the SZ signal would be to represent the radial profiles of this effect for each cluster population. In Fig. 7, we show the median profile at z = 0 of the y-parameter for the tSZ effect (top panel) and for the kSZ effect (bottom panel) for different cluster populations in terms of dynamical state. The thermal profile is obtained averaging over radial shells, whilst the kinetic profile is obtained computing its root mean square (RMS) in each shell. Since all clusters have been normalised to the same mass, M_vir = 10¹⁴ M_⊙, we employ the symbols $\stackrel{\sim }{\mathit{y}}$ and $\stackrel{\sim }{\mathrm{RMS}}$, respectively. This normalisation ensures that the differences between populations are solely due to differences in the dynamical state. The shadowed regions represent the error corresponding to the 68th percentile. The small panels below show the same quantity divided by the mean across the three dynamical state populations, thus highlighting the differences between them.

Fig. 7. Profiles at z = 0 of the y-parameter for the tSZ (top panel) and the kSZ (bottom panel) effects for different cluster populations. The thermal profile is obtained averaging over radial shells, whilst the kinetic profile is obtained computing its root mean square (RMS) in each shell. The solid lines represent the median of the profiles for each cluster population, whereas the shadowed areas correspond to a 68th percentile. All clusters are normalised to the same mass, Mvir = 1014 M⊙. Lower panels show the same quantity divided by the mean across the three dynamical state populations, thus highlighting the differences between them.

For the tSZ effect we appreciate how the unrelaxed clusters have a flatter profile than the relaxed ones, which are more concentrated towards the cluster centre. Indeed, it can be seen that for R > 0.2 R_vir, the unrelaxed clusters show a greater signal, whereas for R < 0.2 R_vir the relaxed clusters have a higher profile. This result is in agreement with recent observations carried out by Adam et al. (2024), where they confirm that disturbed clusters have a relatively flatter core and a shallow outer slope of the pressure profile (note that y_tSZ is an indirect measure of the electron gas pressure). On the other hand, for the kSZ effect we observe that the unrelaxed population has a higher value for all radii, with the shape of the profile equivalent in all dynamical states. Marginally relaxed clusters show a behaviour that is halfway between the other two. The explanation for the described facts is similar to the one given in Sect. 3.1.1: the thermal profile for unrelaxed clusters is less concentrated towards the centre primarily due to recent merger(s) that move matter to the outer regions; hence, the rising SZ signal observed there (both thermal and kinetic). Furthermore, turbulence and other motions caused by recent mergers suffered by the unrelaxed population produce high velocities that increase the overall kSZ signal; hence, the overall higher profile for the unrelaxed population. Also, since marginally relaxed clusters are transitioning from unrelaxed to relaxed or vice versa, they exhibit a middle term behaviour.

An important result given by theoretical studies (e.g. Rephaeli 1995) and observations (e.g. Adam et al. 2017) is the fact that the tSZ signal is approximately an order of magnitude larger than the kSZ signal, making the first one much easier to measure. This result is in agreement with the trend shown in Fig. 7, where the maximum values of the profiles satisfy that ${\stackrel{\sim }{\mathit{y}}}_{\mathrm{tSZ}}/{\stackrel{\sim }{\mathrm{RMS}}}_{\mathrm{kSZ}}\sim 10$ at the innermost region. Nevertheless, we see that in cluster outskirts, at R ≥ R_vir, both effects become equally important, due to the higher importance of non-thermal motions in these regions.

3.1.4. Kinetic SZ maps and multipole expansion

In Fig. 8 we present the kSZ map for all clusters in our sample with M_vir > 10¹⁴ M_⊙ at z = 0, separated by dynamical state. The size of the maps is 2 R_vir × 2 R_vir with the highest spatial resolution being 9 kpc and they are centred on the density peak. It can be easily seen that the major difference between relaxed and unrelaxed maps is the fact that the first have the signal significantly concentrated towards the inner regions whilst the second have more features and present a more extended signal across the cluster volume. This explains, for example, the radial profiles shown in Fig. 7, where the unrelaxed clusters clearly have a higher kSZ effect (and particularly towards the outer regions) or the scaling relation in Fig. 3; here, the unrelaxed population shows an overall higher kSZ signal for all masses.

Fig. 8. 2 Rvir × 2 Rvir maps of the kSZ effect for every cluster with Mvir > 1014 M⊙ at z = 0. The pixel size in the images is 9 kpc, coinciding with the highest numerical resolution in the simulation. The maps are centred on the cluster density peak. The relaxed clusters are on top, the unrelaxed in the middle and the marginally relaxed are below. Colours display temperature variations according with the palette on the right.

On the other hand, because of cluster rotation, one should expect a dipole-like map once we have subtracted its bulk velocity and, hence, only internal motions remain (e.g. Baldi et al. 2018; Altamura et al. 2023). Nevertheless, this is only appreciated in some maps, since the orientation of the angular momentum does not necessarily rest on the plane of the map; in some cases, it could be even perpendicular, thereby leaving the kSZ signal only to ICM turbulence, other bulk flows, and so on.

With respect to quantifying the importance of the rotation imprint in the kinetic SZ maps, we performed a multipole expansion, similar to that done in Gouin et al. (2020), which is based on the methodology presented in Schneider & Bartelmann (1997). In these studies, the multipole expansion is applied to matter surface density, while in our particular case, it is applied to the kSZ map. The goal is to obtain general angular symmetries for each dynamical state class. In particular, we are interested in discerning the importance of the rotation on the maps, whose imprint is a dipole-like signal (e.g. Altamura et al. 2023). The multipole moments of a certain quantity (Σ) defined on a polar map (R, ϕ) are given by:

$$\begin{array}{c}\hfill Q_{\mathrm{m}}(\mathrm{\Delta }R)={\displaystyle {\int }_{\mathrm{\Delta }R}}R\mathrm{d}R{\displaystyle {\int }_0^{2\pi }}\mathrm{d}\varphi e^{im\varphi }\mathrm{\Sigma }(R,\varphi ),\end{array}$$(6)

where Q_m are the multipole moments and ΔR is the radial aperture over which we integrate. In our case, we performed the integrals with ΔR = (0.1,  0.5) R_vir, where the dominant part of the effect is produced (see Fig. 8).

The choice of the multipole expansion centre is crucial, as the centre of angular symmetries may not necessarily align with the image centre. Our choice for the new centre involves selecting the cell that corresponds to the arithmetic mean weighted by the absolute value of the signal. Through this straightforward recentring algorithm, we aim to minimise the introduction of a dipole signal in the expansion caused by the offset in the symmetry centre.

The results for the multipole expansion performed to all the sample kSZ maps are presented in Fig. 9. Here, we show the mean for each of the dynamical state classes of the module of the multipole expansion coefficients (|Q_m|) normalised by the sum of the complete multipolar expansion. On average, the trends of the three populations are similar, being the dipole coefficient (m = 1) clearly the most relevant, accounting for roughly 50% of the sum over all coefficients. This implies that, in general, the dipole (and, in principle, rotation) is the most important contribution to the effect. Nevertheless, the 68% confidence intervals shown here indicate that the weight of Q₁ highly varies across all clusters, due to projection effects. Moreover, as it can be seen, the m = 2, 3 coefficients also contribute significantly to the signal and cannot be neglected. In fact, this is the main difference in the multipole expansion between disturbed and relaxed clusters: the first present a more prominent quadrupole behaviour and less dipole component. Considering the kSZ effect to be produced solely by rotation would result in a biased interpretation of the observations.

Fig. 9. Mean across dynamical state classes of the module of the multipole expansion coefficients of the kinetic SZ maps (|Qm|) normalised by their sum against the coefficient order (m), for each projection. The integration aperture is (0.1,  0.5) Rvir and the error shows the 68% confidence interval.

Figure 9 shows an important feature of the kSZ signal that could be exploited by observations. Every kSZ map can show particular attributes, however, when averaged over a significant number of clusters, the mean multipole expansion is dominated by the dipole and all the other coefficients become almost negligible, especially if only relaxed clusters are used. Therefore, if they are properly recentred and oriented, many kSZ maps could be stacked. As a result, a dipole signal would emerge, which would be correlated to the global rotation properties of the sample. A similar procedure to the one described is applied in Xia et al. (2021) for intergalactic filaments.

For a more in-depth exploration of the rotational influence on the kinetic signal, it is crucial to know the extent to which the dipole is attributable to rotation versus other contributing factors. In Fig. 10, we present a scaling relation for the whole sample between the module of the dipole coefficient (|Q₁|) of the kinetic SZ map (normalised by the sum of all the multipole expansion coefficients) against the Bullock spin parameter (λ) of the angular momentum for the three considered projections along the Cartesian axes. It can be seen that the dipole weight generally increases with increasing angular momentum in the corresponding projection. However, the scatter is considerable, therefore, apart from rotation, there must be other ICM movements contributing to the dipole.

Fig. 10. Module of the dipole coefficient (|Q1|) of the kinetic SZ map normalised by the sum of all the multipole expansion coefficients against the Bullock spin parameter (λprojection) of the angular momentum contained in each projection. The representation corresponds to a kernel density estimate (KDE) using all clusters in the sample.

3.2. SZ signal evolution with redshift

In this section, we study how the kinetic and thermal SZ effects evolve with cosmic time, looking for a correlation between the SZ signal variations and the clusters’ dynamical state. First, we discuss the redshift evolution of the effect. Then, we analyse two particular cluster histories to get a deeper description of this connection.

3.2.1. Redshift evolution

The SZ effect is essentially redshift independent (e.g. Carlstrom et al. 2002; Mroczkowski et al. 2019). Nevertheless, our clusters evolve with redshift due to the expansion of the Universe and, hence, this introduces an “artificial” redshift dependence that we have to take into account (see e.g. Schaan et al. 2021 for similar arguments). We define the halos of our simulations according to the over-density $\mathrm{\Delta }=\frac{\rho }{{\rho }_B}$, where ρ is the matter density and ρ_B = Ω_mρ_c is the background density, with ρ_c being the critical density of the Universe. Thus, since Δ is co-moving (independent of z), and ρ_B = ρ_B, 0(1 + z)³, necessarily ρ ∝ Δ(1 + z)³.

On the other hand, the physical size of the cluster is bigger at lower redshift due to the Universe expansion. If ds and dx are the physical and co-moving distance elements, respectively, then ds = a(z)dx, where a(z) = a₀/(1 + z) is the scale factor at a given z. All in all, we have that the redshift evolution of the thermal effect is given by:

$$\begin{array}{c}\hfill y_{\mathrm{tSZ}}\propto {\displaystyle \int n_{\mathrm{e}}}{T}_{\mathrm{e}}\mathrm{d}s\propto {\displaystyle \int \rho }{T}_{\mathrm{e}}\frac{\mathrm{d}x}{1+z}\propto {(1+z)}^2{\displaystyle \int T_{\mathrm{e}}}\mathrm{\Delta }\mathrm{d}x,\end{array}$$(7)

where x is the co-moving line-of-sight distance. Thus, in the last step all the quantities inside the integral are comoving and, hence, redshift independent.

Similarly, the same procedure can be applied to the kinetic effect:

$$\begin{array}{c}\hfill y_{\mathrm{kSZ}}\propto {(1+z)}^2{\displaystyle \int v_{\mathrm{los}}}\mathrm{\Delta }\mathrm{d}x.\end{array}$$(8)

We want to emphasise that this redshift dependence is introduced in our calculations only because of the manner in which we define these objects; that is, the factor (1 + z)² is only taking into account how our clusters are evolving with cosmic time (density and size) due to the expansion of the Universe. Nevertheless, the same object with the same physical size R and physical density n at z = 0 and at arbitrary z would produce the same effect at both redshifts, since there is no explicit z dependence in Eq. (1). Therefore, in order to analyse the redshift evolution of the effect due to dynamical state changes, we removed the (1 + z)² factor due to the intrinsic evolution of the cluster with the Universe expansion.

3.2.2. Individual cluster evolution

Although it is difficult to elucidate, due to the complexity of the mass assembly histories of GCs (e.g. Vallés-Pérez et al. 2020), it is clear from the previous results that there is a connection between the cluster dynamical state and the SZ signal (especially the kinetic part). To make this even clearer, we have chosen two prototypical clusters from our population to analyse how their dynamical state and both tSZ and kSZ signals change with cosmic time. We have taken into account whether the clusters suffer minor or major mergers. The results are presented in Fig. 11, where we depict the cluster dynamical state, the kSZ (green) and tSZ (purple) signals, and whether it has suffered a minor (major) merger in light (dark) grey bars.

Fig. 11. Individual evolution of two prototypical clusters in the sample, CL01 and CL02 (top and bottom panel, respectively). In green (purple) we plot the integrated RMS of the kSZ signal (tSZ signal) as a function of redshift. Different symbols differentiate between the three stages of dynamical state and we also show in light (dark) grey the minor (major) mergers. Major (minor) mergers are highlighted as dark (light) grey regions.

In the first case, CL01 suffers a major merger at z ≈ 0.7, which rises the kSZ and tSZ signals. Once the merger phase ends at z ≈ 0.2 it becomes marginally relaxed and, in the end, relaxed, decreasing its kSZ signal but maintaining the tSZ nearly constant or even increasing it, due to the mass growth. In the second case, CL02 starts with a major merger at z ≈ 1.1, which demonstrates it is an unrelaxed cluster. Once this period ends, it becomes marginally relaxed, reducing the kSZ signal until z ≈ 0.5, where it enters a minor merger period. It then becomes unrelaxed and shows increases both in terms of the thermal and kinetic signals. Finally, when it becomes again marginally relaxed, the kSZ effect decreases but the tSZ remains constant.

Both study cases reassert the correlation between the cluster dynamical state (including mergers) and the kinetic part of the SZ signal. During merging periods (unrelaxed), the kSZ signal increases, while in quiescent periods (relaxed), it decreases. Simultaneously, the tSZ signal rises during merging periods due to mass gain and heating by compression and merger shocks, but does not decrease in relaxed periods. Therefore (in line with Fig. 3), we can anticipate that for the same mass, unrelaxed clusters should generally exhibit a more pronounced kSZ signal than the relaxed population, while the tSZ effect is minimally affected by this classification.

We can also highlight the thin line that can exist between our marginally relaxed an unrelaxed population with these particular clusters. Although CL01 only presents a marginally relaxed dynamical state once it comes to the end of the merging period, CL02 exhibits this state even at the peak of the kSZ signal at z ≈ 0.2 – from which it starts to decrease. Hence, despite the fact that the marginally relaxed state appears in quiescent periods and at the end of merger or high accretion epochs, clusters can be significantly disturbed in this state. That is one of the main reasons why we predominantly highlight the differences between relaxed and unrelaxed populations in all the previous results, putting the marginally relaxed systems in a second plane, since they are halfway of the other dynamical states and can blur the main differences between them.

4. Summary and conclusions

The main goal of this work is to assess the correlation between GCs dynamical state and their associated SZ signals, both thermal and kinetic. To do so, a sample of cluster-like halos extracted from a cosmological AMR simulation is used. The SZ effect strongly depends on density, temperature, and velocity distributions of the baryonic component of such halos. Therefore, those events able to substantially alter these quantities and, thus, changing the cluster dynamical state (e.g. mergers, accretion, etc.), would be crucial to establish the previously mentioned correlation.

We have defined three main dynamical state classes, according to Vallés-Pérez et al. (2023). We have primarily focused on the clearly differentiate cases of relaxed and unrelaxed (or disturbed) clusters to maximise the main differences. Hence, at fixed redshift, we obtain the following results.

• The kinetic part of the effect shows a strong correlation with cluster dynamical state. Disturbed clusters present an overall higher kSZ signal for all masses than the relaxed population and a steeper trend with mass (α = 0.77 for disturbed vs. α = 0.35 for relaxed systems). This would be mostly explained by the fact that the unrelaxed sample has an overall higher velocity dispersion (σ_v) and, therefore, kinetic pressure plays an important role in the cluster dynamics. On the other hand, the relaxed population would be mostly supported by thermal pressure and thus uncorrelated to the kinetic signal.

• The correlation between gas angular momentum and the kSZ effect is only evident among relaxed clusters, as the signal in disturbed objects would be dominated by turbulence and non-rotational motions. By means of a multipole expansion on the kSZ maps, it becomes apparent that (overall) the dipole constitutes the most significant component of the effect, particularly within the relaxed sample. This characteristic can be leveraged by stacking multiple, appropriately oriented, and centred kSZ maps, allowing the retrieval of a dipole-like signal that is correlated with the global rotation properties of the cluster sample.

• The thermal component lacks a robust correlation with cluster dynamical state, primarily because it is unaffected by the baryonic velocity distribution. However, the radial profiles presented seem to indicate that unrelaxed clusters exhibit a more flattened trend, while the relaxed sample displays greater concentration towards the cluster centre. This has been confirmed by observations in Adam et al. (2024).

Our halos at z = 0 can be traced backwards in cosmic time using the ASOHF merger tree. Therefore, for each simulation snapshot, we have recovered the cluster dynamical state history, together with the minor and major mergers suffered. Joining this information with the thermal and kinetic SZ signals of every cluster at all cosmic epochs we can infer that, during merging periods (when the cluster is disturbed), the kinetic part of the SZ effect rises; meanwhile, in quiescent periods it decreases. The tSZ signal, however, does not decrease once the disturbed phase is over and thus, it seems to be almost independent of the cluster dynamical state.

Based on our findings, the kSZ effect opens a new window of opportunity to assess the dynamical state of clusters, particularly in distinguishing clusters undergoing a quiescent evolution from those marked by a history of significant merging events. Consequently, our results suggest that upcoming observational data focused on measuring the SZ effect in clusters could serve as a valuable complementary approach to describe the evolution of galaxy clusters. Additionally, and especially in cases where clusters exhibit a relaxed state, our analysis suggests that the kSZ effect holds potential for estimating dynamical quantities such as the gas rotation.

Acknowledgments

This work has been supported by the Agencia Estatal de Investigación Española (AEI; grant PID2022-138855NB-C33), by the Ministerio de Ciencia e Innovación (MCIN) within the Plan de Recuperación, Transformación y Resiliencia del Gobierno de España through the project ASFAE/2022/001, with funding from European Union NextGenerationEU (PRTR-C17.I1), and by the Generalitat Valenciana (grant PROMETEO CIPROM/2022/49). O.M. and D.V. acknowledge support from Universitat de València through Atracció de Talent fellowships. Simulations have been carried out using the supercomputer Lluís Vives at the Servei d’Informàtica of the Universitat de València.

References

All Tables

All Figures

	Fig. 1. Cumulative halo mass function at z = 0 of the simulation (points) and the fit (black line) described in Tinker et al. (2008) for the cosmology given in Sect. 2.1 using the COLOSSUS package (Diemer 2018). The errorbars and shaded region correspond to 1σ Poissonian error.
In the text

	Fig. 2. Number of clusters in our sample as a function of the fraction of their mass, within Rvir, that is refined at ℓ = 6 (upper panel), ℓ ≥ 5 (middle panel) and ℓ ≥ 4 (lower panel). To avoid poorly resolved halos from contaminating our results, in our final sample we have removed halos with fℓ = 6 < 15%, fℓ ≥ 5 < 50% and fℓ ≥ 5 < 90%, where fi is the corresponding mass fraction.
In the text

Fig. 3. Scaling relation between the virial mass, Mvir, and the mean y-parameter for the tSZ effect (top panel) and for the RMS associated with the kSZ effect (bottom panel). The integration aperture is Rvir. The represented points correspond, for each cluster, to the mean across the three spatial projections. In both panels, the data have been fitted according to y = A(Mvir/M0)α, where M0 = 1014 M⊙ is the pivot mass and A is a normalisation. The best fitting parameters are presented in Table 1. The shaded regions correspond to the 16 − 84 confidence interval around the fit, whereas the error bars represent the variation (maximum and minimum values) across the three spatial projections for each cluster.

In the text

	Fig. 4. Scaling relation between the gas velocity dispersion inside the aperture and the root mean square for the kSZ. The integration aperture is Rvir. The fit corresponds to RMSkSZ = A(σv/σ0)B, where σ0 = 300 km s−1 is the pivot velocity dispersion and the shown error corresponds to a 16 − 84 confidence interval. The fitting slopes (α) and errors are shown below the figure legend.
In the text

Fig. 5. Scaling relation between the Bullock spin parameter (λprojection) of the angular momentum contained in each projection against the RMS of the kSZ signal of each projection normalised to a Mvir = 1014 M⊙ mass. Since neither of the two quantities depend on mass the relationship shown correlates the cluster spin with the kSZ effect. The fitting slopes (α) and errors are shown in the upper left part.

In the text

	Fig. 6. Histogram of the number of clusters of each dynamical state class against the logarithmic difference between the maximum and minimum values of the RMS of the kSZ effect obtained with the three projections XY, XZ and YZ.
In the text

Fig. 7. Profiles at z = 0 of the y-parameter for the tSZ (top panel) and the kSZ (bottom panel) effects for different cluster populations. The thermal profile is obtained averaging over radial shells, whilst the kinetic profile is obtained computing its root mean square (RMS) in each shell. The solid lines represent the median of the profiles for each cluster population, whereas the shadowed areas correspond to a 68th percentile. All clusters are normalised to the same mass, Mvir = 1014 M⊙. Lower panels show the same quantity divided by the mean across the three dynamical state populations, thus highlighting the differences between them.

In the text

Fig. 8. 2 Rvir × 2 Rvir maps of the kSZ effect for every cluster with Mvir > 1014 M⊙ at z = 0. The pixel size in the images is 9 kpc, coinciding with the highest numerical resolution in the simulation. The maps are centred on the cluster density peak. The relaxed clusters are on top, the unrelaxed in the middle and the marginally relaxed are below. Colours display temperature variations according with the palette on the right.

In the text

	Fig. 9. Mean across dynamical state classes of the module of the multipole expansion coefficients of the kinetic SZ maps (|Qm|) normalised by their sum against the coefficient order (m), for each projection. The integration aperture is (0.1,  0.5) Rvir and the error shows the 68% confidence interval.
In the text

	Fig. 10. Module of the dipole coefficient (|Q1|) of the kinetic SZ map normalised by the sum of all the multipole expansion coefficients against the Bullock spin parameter (λprojection) of the angular momentum contained in each projection. The representation corresponds to a kernel density estimate (KDE) using all clusters in the sample.
In the text

Fig. 11. Individual evolution of two prototypical clusters in the sample, CL01 and CL02 (top and bottom panel, respectively). In green (purple) we plot the integrated RMS of the kSZ signal (tSZ signal) as a function of redshift. Different symbols differentiate between the three stages of dynamical state and we also show in light (dark) grey the minor (major) mergers. Major (minor) mergers are highlighted as dark (light) grey regions.

In the text