OFFSETS BETWEEN THE X-RAY AND THE SUNYAEV–ZEL'DOVICH-EFFECT PEAKS IN MERGING GALAXY CLUSTERS AND THEIR COSMOLOGICAL IMPLICATIONS

We assume a spherical symmetric density profile to model the initial mass distributions of the DM and the gas in a cluster. The Navarro–Frenk–White (NFW) profile (Navarro et al. 1997) is used for the DM mass density distribution within the virial radius r_vir,

$$\begin{equation} \rho _{\rm DM}(r) = \frac{\rho _{\rm s}}{r/r_{\rm s}(1+r/r_{\rm s})^{2}}, \quad {\ \rm if\ } r \le r_{\rm vir}, \end{equation} \tag{ 1 }$$

where ρ_s and r_s ≡ r_vir/c_vir are the scale density and the scale radius, respectively, and c_vir is the concentration parameter. The r_vir is calculated from the spherical collapse model (Bryan & Norman 1998), and c_vir is given by a fitting formula obtained from N-body simulations (Duffy et al. 2008; see also Bullock et al. 2001; Wechsler et al. 2002; Zhao et al. 2003; Macciò et al. 2007). The effect of the evolution of r_vir and c_vir with redshift will be discussed in Section 4.2. The DM density distribution outside the virial radius is given by

$$\begin{eqnarray} \rho _{\rm DM}(r) & = & \rho _{\rm DM}(r_{\rm vir}) \left(\frac{r}{r_{\rm vir}}\right) ^{\delta }\exp \left(-\frac{r-r_{\rm vir}}{r_{\rm decay}}\right), \nonumber \\ & & \quad \quad \quad \quad \quad {\ \rm if\ } r > r_{\rm vir}. \end{eqnarray} \tag{ 2 }$$

where we follow Kazantzidis et al. (2004) and implement an exponential cutoff that suppresses the profile on a truncation scale r_decay to avoid a divergent total mass and the parameter δ is set by keeping the first derivative of the DM density profiles continuous at r = r_vir. The parameter ρ_s is set so that the total mass obtained by integrating Equations (1) and (2) over the space is the total DM mass of the cluster. Given the mass density distribution, the distribution function of the DM in the six-dimensional space f(r, v) is assumed to be ergodic and solved via the Eddington's formula (Equation (4.46) in Binney & Tremaine 2008). In both the SPH and the AMR simulations, DM is described by Lagrangian particles. To keep the stability of the DM distribution within the virial radius over cosmological relevant timescales, the truncation scale is set to 0.3r_vir, as done in Zemp et al. (2008), and we have tested that a single cluster with the truncated model is stable within the Hubble time in our simulations.

We choose the Burkert profile (Burkert 1995) to represent the initial gas density profile as follows,

$$\begin{equation} \rho _{\rm gas}(r)=\frac{\rho _{\rm c}}{[1+(r/r_{\rm c})^2](1+r/r_{\rm c})},\quad { \rm if\ } r\le r_{\rm vir}, \end{equation} \tag{ 3 }$$

as done in ZuHone et al. (2009), where r_c = 0.5r_s is the core radius and the normalization density ρ_c is set so that the baryonic mass fraction within r_vir is consistent with the cosmological average value Ω_b/Ω_m = 0.17. For the region outside the virial radius, we assume that the gas density profile traces the DM density profile as follows,

$$\begin{equation} \rho _{\rm gas}(r)=\rho _{\rm DM}(r)\frac{\rho _{\rm gas}(r_{\rm vir})}{\rho _{\rm DM}(r_{\rm vir})},\quad { \rm if\ } r> r_{\rm vir}. \end{equation} \tag{ 4 }$$

The specific internal energy of the gas at radius r is deter-mined by

$$\begin{equation} \mathcal {E}(r)=\frac{1}{\rho _{\rm gas}(r)(\gamma -1)}\int _r^{\infty }\rho _{\rm gas}(r ^{\prime \prime })\frac{GM(r^{\prime \prime })}{r^{\prime \prime 2}}dr^{\prime \prime }, \end{equation} \tag{ 5 }$$

where the gas is assumed to be in hydrostatic equilibrium and ideal monatomic gas state with mean molecular weight per ion μ = 0.592, γ = 5/3 is the ratio of the heat capacity at constant pressure to that at constant volume, and M(r) is the total mass within radius r. The temperature distribution of the gas can be obtained from its internal energy distribution.

We also test different gas density distribution models in our work, e.g., the β-model (Cavaliere & Fusco-Femiano 1978) or a gas distribution tracing the DM density profile at all radii. We find that the different gas models do not affect our results significantly, as we focus on the positions of the maxima of projected X-ray and SZ maps of merging systems.

In our simulations, the centers of the two clusters with masses M₁ and M₂ (M₁ ⩾ M₂) are initially separated by a distance d_ini set to twice the sum of their virial radii, with initial relative velocity V and impact parameter P. The center of mass of the two clusters is initially put at rest at the origin of the coordinate system in the simulations. The initial position of each cluster center is put at the x–y-plane with coordinate z = 0, and their (x, y, z) coordinates are (M₂P/(M₁ + M₂), M₂d_ini/(M₁ + M₂), 0) and (− M₁P/(M₁ + M₂), − M₁d_ini/(M₁ + M₂), 0), respectively. The initial relative velocity is along the y-axis, which are (0, − M₂V/(M₁ + M₂), 0) and (0, M₁V/(M₁ + M₂), 0) for the two clusters, respectively.

We explore a large range of the parameter space of the initial conditions. We perform a series of simulations with M₁ = 1 × 10¹⁴, 2 × 10¹⁴, 5 × 10¹⁴, 1 × 10¹⁵ M_☉. The mass ratio ξ(≡ M₁/M₂) is generally set to be 1, 2, 3, or 5. For some special cases, the values of the mass ratio are set more intensively from 1 to 5. Different from previous work in which a fixed initial relative velocity (1.1 times circular velocity at the virial velocity of the larger cluster) is adopted (McCarthy et al. 2007; ZuHone 2011), our simulations are done for a large range of V from 250 to 4000 km s⁻¹ and show that the SZ–X-ray offset has a strong dependence on the relative velocities (see Section 3.1.1). The impact parameter P spans from 0 to 600 kpc, as head-on or nearly head-on mergers are relevant here (see Section 3.1.2).

We use GADGET-2, an efficient parallel TreeSPH code, to carry out our simulations (Springel et al. 2001). The mass of each gas particle is m_gas = 1.25 × 10⁸ M_☉, and the mass of each dark matter particle is m_DM = 6.25 × 10⁸ M_☉. The gravitational softening length is set to 4 kpc, for which other choices (e.g., 1.5 or 15 kpc) are also tested and our main results are not affected.

As mentioned above, we also use FLASH, a mesh-based Eulerian code, to test whether the discrepancy between Eulerian and Lagrangian methods affects our results significantly (Fryxell et al. 2000). FLASH uses adaptive mesh refinement (AMR), and solves the equations of hydrodynamics by piecewise-parabolic method (PPM) of Colella & Woodward (1984). The gravitational potential is computed by the multigrid solver (Ricker 2008). We simulate the cases with ξ = 2, V = 500 km s⁻¹, P = 0 kpc for M₁ = 2 × 10¹⁴, 5 × 10¹⁴, 1 × 10¹⁵ M_☉. For the box size of 13.0 Mpc of all the FLASH mergers, the finest resolution achieved in our simulations is 12.7 kpc.

Given any time of the merging processes, we can obtain the two-dimensional maps of the mass surface density, the X-ray surface brightness, and the SZ emission of the simulated merging clusters by the following equations.

• 1.  
  The mass surface density Σ is given by an integral of the mass density along the line of sight (LOS), i.e.,

  $$\begin{equation} \Sigma =\int _{\rm LOS}^{}(\rho _{\rm DM}+\rho _{\rm gas})d\ell. \end{equation} \tag{ 6 }$$
• 2.  
  The X-ray surface brightness is obtained by using an approximate expression for the relativistic X-ray thermal bremsstrahlung as follows (see Equation (5.25) in Rybicki & Lightman 1979), considering that the gas temperatures T_gas of some regions heated by shocks can be up to >30 keV,

  $$\begin{eqnarray} I_{\rm X} &\propto & \int _{\rm LOS}^{}\rho _{\rm gas}^{2}{T_{\rm gas}}^{1/2}g_{\rm B} (1+4.4\times 10^{-10}T_{\rm gas})d\ell, \end{eqnarray} \tag{ 7 }$$

  where $g_{\rm B}=(2\sqrt{3}/\pi)[1+0.79(4.95\times 10^{5}{\rm \,K}/T_{\rm gas})^{1/2}]$ is the frequency average of the velocity averaged Gaunt factor (Rephaeli & Yankovitch 1997).
• 3.  
  The SZ surface brightness is given by

  $$\begin{eqnarray} I_{\rm SZ}& \propto & \frac{\sigma _{\rm T}}{m_{\rm e}c^{2}}\int _{\rm LOS}^{}\rho _{\rm gas} T_{\rm gas} \nonumber \\ & & \times (Y_{0}+Y_{1}\Theta +Y_{2}\Theta ^{2}+Y_{3}\Theta ^{3}+Y_{4}\Theta ^{4})d\ell, \end{eqnarray} \tag{ 8 }$$

  where σ_T is the Thomson cross section, m_e is the electron mass, c is the speed of light, Θ ≡ k_BT_gas/(m_ec²), k_B is the Boltzmann constant, and Y₀, Y₁, Y₂, Y₃, Y₄ are the factors given by Equations (2.26)–(2.30) in Itoh et al. (1998).

In practice, we obtain all the above quantities at a given position calculated from the SPH method by smoothing N neighbor particles output from each snapshot of the simulations, where N = 50 ± 2 is a neighbor particle number typically used for smoothing.

The values and the spatial positions at the peaks of the projected mass surface density, X-ray and SZ images can then be obtained. Note that all the maps have been smoothed by a Gaussian distribution function with dispersion σ = 70 kpc. We also try other values of σ = 30, 40 kpc, and find that the variations of the positions of the peaks are smaller than 10%. The SZ peak position is not sensitive to the σ value even when σ ⩾ 140 kpc. A similar result is reported in Molnar et al. (2012) although they use a lower SZ resolution. Thus, not only are the obtained peak positions not affected by the smoothing, but also they can be straightforwardly compared with the current SZ observations, though the current observational resolutions are at the sub-arcminute level.

Finally, we clarify the definition of the "SZ–X-ray offset" to be obtained from our simulations, which is not exactly the same as that obtained from observations in the following two points. (1) The offset that we obtain from our simulations is the distance between the positions of the maxima of the whole X-ray and SZ surface brightness maps, which covers a sufficiently large region. However, the offsets obtained from observations are sometimes the displacements between the X-ray and SZ peaks located in a small local region, hence they can be smaller than the offsets obtained in this work. (2) The centroids of the galaxy clusters obtained in the X-ray survey are always obtained by computing the mean emission-weighted positions after removing the extended secondary X-ray maxima, instead of finding the maximum in the surface brightness maps as done here (e.g., Vikhlinin et al. 2009; Andersson et al. 2011). The above points may cause the underestimate of the probability of large SZ–X-ray offsets in the observations to be discussed in Section 4.

Generally, the merging process of two clusters involves the five distinct stages (Poole et al. 2006): pre-interaction stage, primary core–core interaction, apocentric passage, secondary core accretion, and relaxation (see an example illustrated in Figure 2), if the initial relative velocity is not high enough to detach the two clusters from each other at a later time. As mentioned above, the DM and the gas have different physical behavior during the mergers. The DM is collisionless so that the DM halos of the two clusters can go through each other, undergoing only gravity, while gas experiences gas pressure and shocks can be created during collisions of the gas halos. The different behavior can be revealed from the images of the mass surface density, and the X-ray and the SZ emissions of the merging clusters, as DM dominates the entire mass of the merging system and the X-ray and the SZ emissions are merely sensitive to the baryon distribution.

3.1.1. Dependence on Initial Relative Velocities

Figure 1 shows a time sequence of the snapshots of the mass surface density (red contours), the X-ray surface brightness (blue contours), and the SZ emission (green contours) of a head-on merger of two clusters (M₁ = 2 × 10¹⁴ M_☉, ξ = 2) with different initial relative velocities. For simplicity, we show the results of the first three stages (the entire five evolution stages can be viewed in Figure 2). The first column shows a snapshot in the pre-interaction stage for each merging system. The second column represents the time of the first pericentric passage of the two clusters. For simplicity, we set the evolution time t = 0.0 Gyr at the first pericentric passage. As seen from the second column, the blue and the green contours are flattened, i.e., gas is squeezed outward in the direction normal to the collision axis; and the green contours are relatively more flattened, as the outer region of the merging cluster heated by shocks contributes more to the SZ surface brightness than to the X-ray surface brightness. The SZ and the X-ray peaks (labeled by "×" and "+" in the figure) start to separate after the first pericentric passage, when the gas at the inner region of the merging clusters is strongly disturbed. Since the central mass density of the large cluster is lower than that of the small cluster, the larger cluster core is penetrated by the small one, and the offset is gradually stretched (see the third column). The fourth column presents the snapshots with the largest peak offsets occurred during the merging processes. The offset returns to a small value or disappears at a later time as the surrounding gas gradually falls into the gravitational potential of the larger cluster (see the fifth column). After the secondary core–core interaction (though not shown in Figure 1; see the primary core–core interaction in the second column), gas is usually partially relaxed. The SZ–X-ray offsets are then not larger than 100 kpc except for some special massive systems (e.g., bottom panel in Figure 5 below).

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By comparing the first two rows in Figure 1, one can see that there is no obvious qualitative difference between the evolutionary behavior of the offset for the case with relative velocity V = 500 km s⁻¹ and that with V = 1000 km s⁻¹, except that the largest offset produced during the merging process is larger for the case with a higher relative velocity. For the case with V = 2000 km s⁻¹ (the third row of Figure 1), however, the result is distinctly different as the small cluster escapes away and cannot get back to collide with the large cluster again after its first pericentric passage because of the high relative velocity. The center of the small cluster becomes the maximum of the X-ray brightness map, when it passes through the larger one. The offset can be up to 3 Mpc after their interaction.

Evolution of the SZ–X-ray offsets for different relative velocities is shown in Figure 2. The solid lines give the SZ–X-ray offset as a function of time. The dashed lines represent the evolution of the distance between the mass density centers of the two clusters; and a zero distance is used if the two objects cannot be identified as they overlap or merge together. The merging processes are divided into the five different stages as illustrated in the top panel. Initially, the two clusters are in the pre-interaction stage, and then the primary core–core interaction starts around the time of the first pericentric passage at t_1st(= 0). Before t_1st, it is the pre-interaction stage. The time of the first apocentric passage is marked as t_apo. After t_apo, the second core accretion stage starts; and t_2nd marks the secondary pericentric passage and the end of the second core accretion stage. As the small cluster in the bound orbit is dragged back and forth during the gradual relaxation process, the apocentric distances damp with time. The whole system appears visually relaxed around the time t_relax as marked in the top panel.

As seen from Figure 2, the offset is initially zero. A small jump appears before the first pericentric passage, and it then decreases and disappears as the two clusters get closer. The appearance of the small jump in the pre-interaction stage can be understood as follows: as the two clusters approach each other, their outer layers start to be compressed and heated, during which the position of the maximum SZ signal is affected, but the peak of the X-ray remains located at the center of the larger cluster. After the first pericentric passage, the collision of the two clusters destroys the hydrostatic equilibrium and spherical symmetry of the gas halos. As shown in Figure 2, a significantly large SZ–X-ray offset always occurs between the primary and secondary pericentric passages. The occurrence of the large offset can be understood as follows: the gas core of the larger cluster is disrupted, and the X-ray peak jumps to the densest region around the center of the small cluster. However, the SZ peak does not jump (see the peak positions shown in the fourth column of Figure 1), as the temperature at the center of the small cluster is relatively low (see the top right panel in Figure 6 below), which strongly reduces the SZ surface brightness whose emissivity is proportional to the temperature. The SZ–X-ray offset is therefore nearly boosted to the mass density displacement between the two clusters (hereafter we refer to this as the "jump effect"). As seen from the figure, the maximum offset is positively correlated with the initial relative velocity of the two clusters. The X-ray peaks drop back to the center of the larger cluster mostly after t_apo, when the gas falls back to the trough of the gravitational potential well. The end of the large SZ–X-ray offset is indicated by the sharp discontinuity in the solid line. Our results suggest that the jump effect should be the dominant reason to lead to the large offset. Our calculations show that adopting different image smoothing scales σ may result in different time durations of the jump effect if σ > 70 kpc. The duration of the X-ray peaks locating in the center of the small cluster is shorter if σ is larger, as more substructures in the X-ray map is smoothed. For example, for the offset larger than 300 kpc, the duration obtained with σ = 140 kpc is only half of the value obtained with σ = 70 kpc.

As mentioned above, the evolution of the offset shows different behavior for low (e.g., V < 1000 km s⁻¹) and high (e.g., V > 2000 km s⁻¹) relative velocity mergers. Here we define a critical velocity V_crit to distinguish these bound and unbound collision cases, i.e., the two clusters will merge and relax within 13 ± 0.5 Gyr if V < V_crit. The merger is referred to as the "merger mode" if V < V_crit, otherwise as the "flyby mode." We perform a series of low-resolution simulations (with fixed DM and gas particle numbers, 40,000 and 10,000, respectively, for computing efficiency) to identify V_crit as a function of (M₁, P, ξ). The result is shown in Figure 3, and the uncertainty in the obtained V_crit is ±50 km s⁻¹. We approximate the relation between the V_crit and the initial condition parameters by the following fitting form:

$$\begin{eqnarray} \frac{V_{\rm crit}}{10^3\ {\rm \,km\,s^{-1}}} &=& A_V\left(\frac{M_1}{10^{14}{\,M_\odot }}\right)^{\alpha _V}(1+\xi ^{-1})^{\beta _V} \nonumber \\ && \times \left[1+\gamma _V\left(\frac{P/(100 {\rm \,kpc})}{(M_1/10^{14}{\,M_\odot })^{1/3}}\right)^2\right]^{-1/4}, \quad \end{eqnarray} \tag{ 9 }$$

where the best-fit parameters are A_V = 0.93, α_V = 0.30, β_V = 0.46, and γ_V = 0.069. The critical velocity is strongly related with the primary cluster mass and the mass ratio, but not the impact parameter. The fitting form and the best-fit power-law factors are roughly consistent with the expectation from the escape velocity criterion in a simple two-body gravitational interacting system, that is, $V_{\rm crit}^2/2-GM_1(1+\xi ^{-1})/d_{\rm ini}=0$, and thus $V_{\rm crit}\propto M_1^{1/3}(1+\xi ^{-1})^{1/2}(1+f_VP^2/M_1^{2/3})^{-1/4}$, where f_V is a factor.

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3.1.2. Dependence on Impact Parameters

Figure 4 shows the evolution of the SZ–X-ray offsets for different impact parameters, with V = 500 km s⁻¹. For the low initial relative velocities, the large offsets (>100 kpc) occur in both head-on and nearly head-on impacts (i.e., P < 400 kpc). They can reach up to 600–700 kpc. A larger impact parameter results in a smaller offset, which implies that the size of the SZ–X-ray displacement is strongly related with the intensity of the primary core–core interaction. Molnar et al. (2012) studied the high-initial relative velocity case with V = 4800 km s⁻¹ and find that significant displacements (∼300 kpc) between the SZ and X-ray peaks can be produced for nonzero impact parameters (about 100–250 kpc) and they decrease with increasing impact parameters. We also perform simulations with V = 4000 km s⁻¹ similar to those done in Molnar et al. (2012). For the nonzero impact parameter cases, we find that the patterns of the mass surface density, the X-ray, and the SZ emission are all very similar with those shown in Molnar et al. (2012); for the head-on merger, a displacement up to 150 kpc can also be produced, while the distance between the mass centers of the two clusters is 1.5 Mpc.

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As seen from the bottom panel of Figure 4, there is no significant SZ–X-ray offset for a large impact parameter P = 400 kpc. This can be understood through the density and temperature maps of the colliding clusters shown in Figure 6. Figure 6 compares the density and temperature slices of the gas at coordinate z = 0 plane between the head-on (P = 0 kpc) and the off-axis (P = 400 kpc) mergers. For the head-on merger shown in the top panels, as discussed above, the core of the large cluster is disrupted at the collision, and the denser region is near the center of the small cluster; however, the temperature at the center of the small cluster and its surrounding region is relatively low. However, for the off-axis merger with a large impact parameter shown in the bottom panels, the gas cores of the two clusters sideswipe each other at the primary collision, and the center of the large cluster remains dense; thus the X-ray and SZ peaks are both near the center of the larger cluster. It is worthy to note that though shocks can heat the gas at shock fronts to a relatively high temperature (e.g., a few tens of keV), the X-ray and the SZ peaks still locate near the centers of the clusters, as the observed X-ray and the SZ emission is integrated along the line of sight (see Equations (7) and (8)).

Our simulation results demonstrate that only the head-on or nearly head-on mergers are possible to produce offsets larger than 100 kpc, e.g., P < 400 kpc for the simulation with M₁ = 2 × 10¹⁴ M_☉ or P < 600 kpc for M₁ = 5 × 10¹⁴ M_☉(ξ = 2, V = 500 km s⁻¹). In addition, a smaller impact parameter induces a longer time duration of the nonzero offset.

3.1.3. Dependence on Masses

We also investigate the simulations for different masses of the primary cluster. Figure 5 presents the dependence of the SZ–X-ray offset evolution on M₁(= 10¹⁴, 5 × 10¹⁴, 10¹⁵ M_☉). As seen from the figure, a significantly large SZ–X-ray offset (≳ 400 kpc) can occur in the whole mass range of the galaxy clusters. For the more massive systems, the maximum of the spatial separation between the SZ and the X-ray peaks is larger, and the time duration of a nonzero offset is longer. Our simulation results show that the maxima of the SZ–X-ray offsets (denoted by d_max; see Equation (10) below) for different masses are approximately as large as the first apocentric distances and it also does for different velocities (V < V_crit). The fitting form to the dependence of d_max on the initial parameters can be obtained first through the following analysis and then from our numerical simulation results. If approximating dynamics of the merging system by the dynamics of a two-body system, d_max can be approximately obtained through the energy conservation law V²/2 − GM₁(1 + ξ⁻¹)/d_ini ≃ −GM₁(1 + ξ⁻¹)/d_max, where G is the gravitational constant. By setting ξ = 2 and considering the scaling relation between the virial radius and the mass of galaxy clusters, we have $d_{\rm ini}=f_d M_1^{1/3}$ and $d_{\rm max}\simeq d_{\rm ini}(1-V^2d_{\rm ini}/3GM_1)^{-1} \propto M_1^{1/3}+f_d V^2/3GM_1^{1/3}$, where f_d is a factor connecting d_ini and $M_1^{1/3}$. Thus we fit our numerical simulation results of d_max in the following form,

$$\begin{eqnarray} d_{\rm max}& = &A_d\left(\frac{M_1}{10^{14}{\,M_\odot }} \right)^{\alpha _d} \nonumber \\ & & + B_d\left(\frac{V}{10^3 {\rm \,km\,s^{-1}}}\right)^2\left(\frac{M_1}{10^{14} {\,M_\odot }}\right)^{-1/3}, \end{eqnarray} \tag{ 10 }$$

and obtain the best-fit parameters A_d = 366 kpc, B_d = 653 kpc, α_d = 0.42. Equation (10) is used in the integration of Equation (15) to obtain the offset rate. Here we do not consider the dependence of d_max on impact parameters because (1) d_max is not sensitive to the impact parameter (e.g., P < 400 kpc in Figure 4) unless P is too large to suppress the large offset, and (2) the cases with large impact parameters do not contribute much to the integration of Equation (15) below, as their time durations of the nonzero offsets are shorter (or the 〈S_p〉 term in Equation (15) is zero when P is too large).

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As seen from the middle and the bottom panels of Figure 5, for the mergers of high-mass clusters (e.g., M₁ = 5 × 10¹⁴, 1 × 10¹⁵ M_☉), a significantly large offset can still appear after the third pericentric passage (where the SZ peak deviates from the center of the large cluster due to an offset between the positions of the maxima of the gas density and the temperature distributions). However, when doing the statistical analysis in Section 4 below, we consider the offsets (>50 h⁻¹ kpc) triggered merely between the primary and secondary pericentric passages for the following reasons. (1) Massive mergers with cluster masses larger than 5 × 10¹⁴ M_☉ are rare events in the universe, which approximately occupy 5% among all major mergers of galaxy clusters; and (2) the time duration of the offset after the third pericentric passage is nearly five times smaller than that of the offset triggered by the first core–core collision.

3.1.4. Dependence on Mass Ratios

The mass ratio ξ is also an important parameter to affect the values of the offsets. According to our simulations, if the initial relative velocity is below 2000 km s⁻¹ for M₁ = 2 × 10¹⁴ M_☉, we find that only when 1 < ξ < 3 (i.e., major mergers) can the offsets be larger than 100 kpc. The evolution of the SZ–X-ray offsets for different mass ratios with V = 500 km s⁻¹ is shown in Figure 7. As seen from the figure, if the mass ratio is larger, the strength of the collision is weaker and has less power to disturb the large cluster, and thus the offset becomes less significant. For producing the offset larger than 100 kpc with higher ξ, more massive merging systems (e.g., M₁ > 5 × 10¹⁴ M_☉) or higher relative velocities are required. Note that when the mass ratio approaches unity, the offset also turns to be insignificant, as the merging configuration is symmetric.

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The dependence of the SZ–X-ray offset on the mass ratio provides a complementary method to constrain the kinematics of an individual merging cluster (see Section 5 below). For example, the Bullet Cluster is a system with two merging clusters with quite different masses; and by doing the simulations with M₁ = 5 × 10¹⁴ M_☉, ξ = 8, P = 0, we find that only when V > 3000 km s⁻¹ can the maximum of the offset be larger than 150 kpc.

The time duration of the SZ–X-ray offset is important in this work, as it is directly associated with the probability of the offset appearing in the observation. As studied in Section 3.1, the duration has a strong relation with the initial parameters V, P, M₁, and ξ. To quantitatively describe the duration, we define the "offset ratio" R(d > d_c) by

$$\begin{equation} R(d>d_{\rm c})\equiv \frac{\Gamma (d>d_{\rm c})}{D}, \end{equation} \tag{ 11 }$$

where D ≡ t_2nd − t_1st is the time duration between the primary and the secondary pericentric passages. Γ(d > d_c) is the time duration of the SZ–X-ray offset larger than d_c (e.g., see Γ and D marked in the middle panel of Figure 2 where d_c = 200 h⁻¹ kpc)³.

The time duration D obtained from our simulations can be fit as a function of the pairwise velocity, cluster mass, and redshift by the following form,

$$\begin{eqnarray} D& = &A_D\left[1-B_D\left(\frac{V/10^3{\rm \,km\,s^{-1}}}{(M_1/10^{14}{\,M_\odot })^{1/3}}\right)^2\right]^{-3/2} \nonumber \\ & & \times (1+z)^{\alpha _D}, \end{eqnarray} \tag{ 12 }$$

where A_D = 1.53 Gyr, B_D = 0.62, α_D = −0.77. The relaxation timescale of the merging clusters in our simulations (i.e., t_relax − t_1st) is about several Gyr, which is typically two to three times longer than D (e.g., see Figures 2, 4, and 5). The time duration of the mergers listed in Poole et al. (2006) also gives a similar result. In this work, we assume t_relax − t_1st = κD with the factor κ = 2. We will present our detailed studies of the relaxation timescale in a separate paper.

In Figure 8, we present the offset ratio R(d > d_c) measured from mergers with different initial conditions. We select d_c = 50, 100, 150, 200, 300 h⁻¹ kpc, which fall into the typical offset range in observations. We only consider the "merger mode" whose D is shorter than 6 Gyr in our estimation for its dominant contribution to the large offset in the observation (see Section 4.1.3). Except for the massive mergers (see Section 3.1.3), the offsets larger than 50 h⁻¹ kpc appear only between the primary and secondary collisions, consequently we have 0 ⩽ R(d > d_c) < 1.

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The top left panel of Figure 8 shows the offset ratio as a function of the relative velocity with different M₁ for d_c = 200 h⁻¹ kpc, which indicates that mergers of more massive clusters produce higher offset ratios. For each mass, the ratios reveal an anti-correlation with the relative velocity, and the correlation slopes for different masses are close. The top right panel presents the dependence of the ratio on d_c for M₁ = 5 × 10¹⁴ M_☉, where the duration of the larger offset is shorter. However, the difference is not significant at the high-velocity end. The bottom left panel presents the dependence on different impact parameters for M₁ = 5 × 10¹⁴ M_☉. As seen from the panel, the slopes of the curves decrease as the impact parameters increase, and consequently the effective velocity range for a positive offset ratio decreases strongly, e.g., V < 1000 km s⁻¹ for P = 400 kpc. The bottom right panel shows the impact of the different mass ratios on the offset ratio for M₁ = 5 × 10¹⁴ M_☉ and P = 0 kpc. We find that if 1.5 < ξ < 2.5, the amplitudes of the offset ratios are close; while if ξ is larger than 3 or approaches to 1, the offset ratio decreases significantly. The non-monotonic feature of the ξ = 1.5 curve that R becomes smaller with decreasing V at the low-V end can be understood as follows: the merging configuration is close to symmetric and thus the offset becomes insignificant; the effect becomes more significant as the relative velocity is lower due to the weaker collision and the longer interaction time.

According to the dependence behavior of the offset ratio on the initial conditions shown in Figure 8, we find that the following functional form fits the data well,

$$\begin{eqnarray} R(d>d_{\rm c}) &= & A_{\rm l}\cdot D_{\rm scale}^{\alpha _{\rm l}}\cdot \left[\left(\frac{V}{10^3 {\rm \,km\,s^{-1}}}\right) \right. \nonumber \\ & & \cdot\left. \exp \left(\delta _{\rm l}\cdot P_{\rm scale}\right) - \beta _{\rm l}\left(\frac{M_{1}}{10^{14} {\,M_\odot }}\right)^{\gamma _{\rm l}}\right], \nonumber \\ & & {\rm for} \ (d_{\rm c}\ge 50 \,h^{-1}{\rm \,kpc}), {\ \rm and} \nonumber \\ D_{\rm scale} &=& \frac{d_{\rm c}/100\, h^{-1}{\rm \,kpc}}{(M_{1}/10^{14} {\,M_\odot })^{1/3}}, \nonumber \\ P_{\rm scale} &=& \frac{P/100 {\rm \,kpc}}{(M_{1}/10^{14} {\,M_\odot })^{1/3}}, \end{eqnarray} \tag{ 13 }$$

where the offset ratio is linearly correlated with the initial relative velocity. The mass-scaled terms D_scale and P_scale are included to indicate the dependence on d_c and the impact parameter. We fit simultaneously to all of the simulation results but fix ξ = 2. Considering the limited simulation test explored in this study, the mass ratio is not taken as an argument in the above fitting formula. The possible effects caused by different ξ will be discussed at the end of Section 4.2. The best-fit parameters (A_l, α_l, β_l, γ_l, δ_l) and their standard errors are given in Table 1. The average deviation between Equation (13) and the simulated data is 10%. Note that Equation (13) is merely suitable for M₁ > 10¹⁴ M_☉ and d_c = 50–300 h⁻¹ kpc and the ratio R(d > d_c) is constrained to be non-negative. Therefore, the effective velocity range for the positive offset ratio is $V<\beta _{\rm l}(M_1/ 10^{14}{\,M_\odot })^{\gamma _{\rm l}}\times 10^3 {\rm \,km\,s^{-1}}$ for P = 0, and the upper limit of the effective velocity range is a few tens of percent larger than the critical velocity V_crit given in Equation (9). However, when P > 0, the effective velocity could be much smaller than V_crit.

We investigate those cases with offsets smaller than 50 h⁻¹ kpc below, separately, which is necessary especially when we estimate the observational expectation of the SZ–X-ray offset distribution over all ranges of the offset size in Section 4.4. There are at least the following several reasons to separately investigate the large offsets (i.e., d_c ⩾ 50 h⁻¹ kpc) and the small ones (i.e., 0 < d_c < 50 h⁻¹ kpc), respectively. (1) As discussed above, the two ranges of the offsets are caused by different reasons. The large offsets are strongly affected by the "jump effect," mostly appearing between the primary and secondary pericentric passages. On the contrary, the small offsets can be viewed during the whole merging processes. (2) As the uncertainty in our offset estimation is about a few kiloparsecs the time duration of the small offsets (≲ 50 h⁻¹ kpc) has a relatively larger error (see Table 1). (3) Furthermore, the statistics of the offsets smaller than 50 h⁻¹ kpc performs relatively irregular behavior, and is more complex to be described. Note that the criterion to separate the large and the small offsets is at 50 h⁻¹ kpc or a few dozens of kiloparsecs, which is reasonable because the core radius of the initial gas distribution is ∼100 h⁻¹ kpc. While the size of the SZ–X-ray offset is comparable with or larger than the core radius, the SZ and the X-ray peaks actually locate near the centers of the big and the small clusters, respectively. Only the disturbed core region of the big cluster itself could not generate such a large offset.

We count the ratio of the small offsets by using our simulations and setting d_c = 10, 20, 30, 40 h⁻¹ kpc, respectively. We find that the offset ratios also show a linear correlation with the relative velocity as those of d_c ⩾ 50 h⁻¹ kpc, though the correlation has a larger scatter than that shown in Figure 8. In addition, for d_c < 50 h⁻¹ kpc, the offset ratios are not strongly related with the impact parameter, and thus we use a mild dependence on the impact parameter in the fitting function of R(d > d_c) given by

$$\begin{eqnarray} R(d>d_{\rm c}) &=& A_{\rm s}\cdot D_{\rm scale}^{\alpha _{\rm s}}\cdot \left[\left(\frac{V}{10^3 {\rm \,km\,s^{-1}}}\right)\right. \nonumber\\ &&\left. -\beta _{\rm s}\cdot D_{\rm scale}^{\gamma _{\rm s}} \cdot \exp (\delta _{\rm s}\cdot D_{\rm scale}\cdot P_{\rm scale})\right], \nonumber \\ && \quad \quad {\rm for\ } d_{\rm c}<50\,h^{-1}{\rm \,kpc}, \end{eqnarray} \tag{ 14 }$$

where the best-fit parameters (A_s, α_s, β_s, γ_s, δ_s) and their standard errors are given in Table 1. The average deviation between Equation (14) and the simulated data is 20%.

The number of the mergers with observational offset⁴ d_phy > d_c per unit redshift (or per unit cosmic time after multiplying by dz/dt) per unit comoving volume at redshift z, which is referred to as the offset rate hereafter, over a given range of mass M₀ ∈ [M_min, M_max] (M₀ ≡ M₁(1 + ξ⁻¹), is the total mass of the merging system), mass ratio ξ ∈ [ξ_min, ξ_max], impact parameter P ∈ [P_min, P_max], and initial relative velocity V ∈ [V_min, V_max] can be given by

$$\begin{eqnarray} \frac{dN(d_{\rm phy} > d_{\rm c}|z)}{dz} & = & \int _{M_{\rm min}}^{M_{\rm max}}dM_0 \int _{\xi _{\rm min}}^{\xi _{\rm max}}d\xi \int _{P_{\rm min}}^{P_{\rm max}} dP \nonumber \\ & & \cdot f_P(P) \int _{V_{\rm min}}^{V_{\rm max}} f_V(V) \cdot H(d_{\rm max}-d_{\rm c}) \nonumber \\ & &\cdot \langle S_{\rm p}(d_{\rm phy}>d_{\rm c}|M_0,\ \xi,\ P,\ V)\rangle \nonumber \\ & & \cdot B(M_0,\ \xi,\ z-\Delta z)\ dV, \end{eqnarray} \tag{ 15 }$$

where f_P(P) and f_V(V) are the initial distribution functions of the impact parameters and the relative velocities of merging cluster pairs with ∫f_P(P)dP = 1 and ∫f_V(V)dV = 1, respectively. In Equation (15), S_p(d_phy > d_c|M₀, ξ, P, V) is defined as the specific probability of a merging system with a given initial condition (M₁, ξ, P, V) showing the observed offset d_phy > d_c in one observational direction, which can be obtained through the ratio of the time duration of the observed offset (>d_c) over the total merging time (from the primary pericentric passage to the complete relaxation; see Equation (16) below), and 〈S_p(d_phy > d_c|M₀, ξ, P, V)〉 is the average specific probability of the observed SZ–X-ray offset d_phy > d_c over all random observational directions. The H(d_max − d_c) is the Heaviside step function, where d_max is the maximum of the SZ–X-ray offset during the merger (i.e., Equation (10)). The merger rate B(M₀, ξ, z) is defined so that B(M₀, ξ, z)dM₀dξdz represents the comoving number density of galaxy cluster mergers completed at redshift z → z + dz, with primary mass in the range M₀ → M₀ + dM₀ and mass ratio in the range ξ → ξ + dξ. The Δz is obtained by the cosmic time difference t(z − Δz) − t(z) = (t_relax − t_1st) − (1/2)(t_2nd − t_1st) for large offsets (e.g., ≳ 50h⁻¹ kpc) and t(z − Δz) − t(z) = (1/2)(t_relax − t_1st) for small offsets (e.g., ≲ 50h⁻¹ kpc), where the large offsets occur mainly at the primary core–core interaction stage as mentioned in Section 3.1.3.

Below we present the detailed forms of the functions 〈S_p〉, B, f_V, and f_P in Sections 4.1.1–4.1.4, respectively.

4.1.1. The Average Specific Probability of the Offset 〈S_p〉

We define the average specific probability of offset in Equation (15) by

$$\begin{equation} \langle S_{\rm p}(d_{\rm phy}>d_{\rm c}|M_0,\ \xi,\ P,\ V)\rangle \equiv \frac{\langle \Gamma (d_{\rm phy}>d_{\rm c})\rangle }{t_{\rm relax}-t_{\rm 1st}}, \end{equation} \tag{ 16 }$$

where 〈Γ(d_phy > d_c)〉 is the average of the time duration of the observed offsets larger than d_c over all possible observational directions. To connect the average duration and the duration observed along the z-axis that was discussed in Section 3, we introduce the following projection factor

$$\begin{equation} A_{\rm p}(d_{\rm phy}>d_{\rm c}|d>d_{\rm c},\ M_{1},\ \xi,\ P,\ V)\equiv \frac{\langle \Gamma (d_{\rm phy}>d_{\rm c})\rangle }{\Gamma (d>d_{\rm c})}. \end{equation} \tag{ 17 }$$

In principle, the projection factor should depend on the initial parameters and d_c. We investigate the dependence by using the Monte Carlo method and randomly selecting the observational directions for the given snapshots, and the result is shown in Figure 9.

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As seen from Figure 9, the values of the points at a given d_c are quite close. The only outliers are the red points in the lower panel for the case with M₁ = 5 × 10¹⁴ M_☉, ξ = 1.5, V = 500 km s⁻¹, and P = 0 kpc, which is due to the near symmetry of the ξ = 1.5 merging system. The factors for the mergers with V = 500 km s⁻¹ is around 10% smaller than that for the corresponding cases with V = 1000 km s⁻¹. That is, the projection factor does not significantly depend on the cluster mass, the mass ratio, the impact parameter, and the initial relative velocity. Thus, we assume a universal factor 〈A_p(d_c)〉, which is only a function of d_c. For simplicity, we set 〈A_p(d_c)〉 to be the mean value of the data shown in the top panel of Figure 9 at each d_c. For the offsets smaller than 50 h⁻¹ kpc, we also find a similar result that the projection factor can be assumed to be only a function of d_c. The results of 〈A_p(d_c)〉 are listed in Table 2.

Table 2. The Projection Factor 〈A_p(d_c)〉 (Section 4.1.1)

dc(h−1 kpc)	10	20	30	40
〈Ap(dc)〉	0.92	0.79	0.79	0.86
dc(h−1 kpc)	50	100	150	200	300
〈Ap(dc)〉	0.95	0.90	0.87	0.84	0.73

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By combining Equations (11), (16), and (17), we have

$$\begin{equation} \langle S_{\rm p}(d_{\rm phy}>d_{\rm c}|M_{0},\ \xi,\ P,\ V)\rangle =\langle A_{\rm p}(d_{\rm c})\rangle \cdot \kappa ^{-1}R(d>d_{\rm c}), \end{equation} \tag{ 18 }$$

where κ ≡ (t_relax − t_1st)/D as defined after Equation (12).

4.1.2. The Merger Rate B

We approximate the merger rate of cluster pairs by using the following universal fitting form of the halo merger rate obtained in Fakhouri et al. (2010, see also Genel et al. 2009):

$$\begin{eqnarray} B(M_0,\ \xi,\ z) & = & A_B \cdot \left(\frac{M_0}{\tilde{M}}\right)^{\alpha }\cdot \xi ^{-\beta }\cdot \exp [(\xi \cdot \tilde{\xi })^{-\gamma }]\nonumber \\ & & \cdot (1+z)^{\eta }\cdot n(M_0,\ z), \end{eqnarray} \tag{ 19 }$$

where A_B = 0.0104, $\tilde{M}=1.2\times 10^{12}{\,M_\odot }$, α = 0.133, β = −1.995, $\tilde{\xi }=0.00972$, γ = 0.263, η = 0.0993. Note that the mass ratio defined in this work is the reciprocal of that in Fakhouri et al. (2010). For the dark matter mass function n(M₀, z), we use the universal form derived from the N-body simulations (Tinker et al. 2008; see also Sheth & Tormen 1999; Jenkins et al. 2001),

$$\begin{equation} n(M_0,\ z)=f(\sigma)\frac{\overline{\rho }_m}{M_0}\frac{d\ln \sigma ^{-1}}{dM_0}, \end{equation} \tag{ 20 }$$

where $\overline{\rho }_m$ is the present mean mass density of the universe, σ is the square root of the variance of mass, and f(σ) gives the fraction of the mass associated with halos in a unit range of ln σ⁻¹ (see Equation (3) in Tinker et al. 2008).

4.1.3. Initial Relative Velocity Distribution f_V(V)

We approximate the probability distribution function of the pairwise velocity obtained from cosmological simulations with halo masses above 10¹⁴ M_☉ (Thompson & Nagamine 2012) as the distribution of initial relative velocity f_V(V) in this work. The 2016 h⁻¹ Mpc box size employed in the simulation of Thompson & Nagamine (2012) guarantees our requirement for the statistic analysis of massive major merging systems. The best-fit of a skewed normal distribution to the simulation results obtained in Thompson & Nagamine (2012) is

$$\begin{eqnarray} f_x(x)& = & \frac{1}{\sqrt{2\pi }w}\cdot \exp \left[-\frac{1}{2}\left(\frac{x-e}{w}\right)^2\right] \nonumber \\ & & \cdot \left[1+{\rm erf}\left(\frac{a}{\sqrt{2}}\cdot \frac{x-e}{w}\right)\right], \end{eqnarray} \tag{ 21 }$$

where x = log₁₀(V/ km s⁻¹), ∫f_x(x)dx = 1, and the best-fit parameters are (a, e, w) = (− 2.19, 2.90, 0.295). The peak of this distribution locates at 500–600  km s⁻¹. However, the velocity distribution could depend on cluster masses, which was not considered in Thompson & Nagamine (2012). Some other works in the literature also discussed the relative motion of cluster pairs. For example, Dolag & Sunyaev (2013) presented the dependence of the relative motions of cluster pairs on their distance, where the median relative velocity can be described by a simple functional form of 〈V〉 = 270 km s⁻¹ + 1000 km s⁻¹ × d⁻¹ (here d represents the separation of the two clusters and is measured in units of the sum of their virial radii). In our simulations, the initial separation of two merging clusters is set to d = 2, and thus correspondingly the median relative velocity is 〈V〉 = 770 km s⁻¹ if adopting the results from Dolag & Sunyaev (2013), which is about 25% larger than the median value of Equation (21). In addition, Wetzel (2011) suggested that the average value of the infalling velocities of the satellite halos shown in cosmological simulations is approximately the circular velocity of the primary halo at the virial radius (v_c), which is smaller than that adopted in our study.

Based on the relative velocity distribution obtained from Thompson & Nagamine (2012), we estimate the probability of the flyby mode defined in Section 3.1.1. It is smaller than 1% of the whole merging events, which is the reason why we only use the merger rate B(M₁, ξ, z) in Equation (15) but ignore the contribution from the rare flyby mode. We ignore the redshift evolution of the pairwise velocity distribution for clusters, which seems insignificant at redshift z ∼ 0–0.5 (Thompson & Nagamine 2012). The peak positions of the pairwise velocity distribution in Bouillot et al. (2014) show a slightly increasing trend from z = 0 to 0.5, which however has no significant effect on the result (more discussions seen in Section 4.3).

4.1.4. The Impact Parameter Distribution f_P(P)

The distribution of impact parameters for major mergers of massive clusters was investigated in the literature, and it is not easy to give a universal quantitative description for this distribution. We list some works below. Sarazin (2001) suggested that most mergers are expected to involve fairly small impact parameters comparable to the sizes of the gas cores in clusters, which may be also biased to a lower value if most mergers occur along large-scale structure filaments. Wetzel (2011; see also Vitvitska et al. 2002; Benson 2005) presented the distributions of the radial (v_r) and the tangential (v_t) velocities of DM substructures at the time of crossing within the virial radius of a larger host halo, where the peak of the distribution is centered on v_r ≈ 0.89v_c and v_t ≈ 0.64v_c and the impact parameters are implied to be a few hundred kiloparsecs. However, Vitvitska et al. (2002) reported that the tangential velocity decreases with the increase of the secondary mass, and major mergers are significantly more radial than minor mergers. In addition, Benson (2005) found the evidence for a mass dependence of the distributions of orbital parameters, i.e., the orbits of more massive merging systems are more radial and less tangential, though their small sample size limits an accurate determination of this dependence. According to the argument in Poole et al. (2006), the statistics including the secondary or tertiary encounters with the primary one tends to overestimate the tangential velocity, and the average impact parameter is likely to be smaller than what v_t/v_r implies in Benson (2005). Khochfar & Burkert (2006) also reported a distribution of the impact parameters, which shows a peak around 350 h⁻¹ kpc; however, the application of that result to the analysis of major mergers of galaxy clusters would be limited as the simulation box size is not sufficiently large enough.

In this work, we construct the distribution of the impact parameters by using the following form,

$$\begin{equation} f_P(P)=A_P \cdot \left(\frac{P}{\lambda }\right)^{\mu }\cdot \exp \left(-\mu \frac{P}{\lambda }\right), \end{equation} \tag{ 22 }$$

where λ and μ are the free parameters to control the position and the width of the distribution peak, and A_P is the normalization. When the parameters are (A_P, λ, μ) = (7.620 × 10⁻³h kpc⁻¹, 327.1 h⁻¹ kpc, 1.636), Equation (22) reduces to the best-fit distribution (Equation (11)) obtained in Khochfar & Burkert (2006). In this work, we use (A_P, λ, μ) = (5.0 × 10⁻³h kpc⁻¹, 200.0 h⁻¹ kpc, 1.0) as the fiducial model, which gives the distribution peak at 200 h⁻¹ kpc. In Section 4.2 below, we also try various choices of the parameters to test the dependence of the probability of the SZ–X-ray offsets on the distribution of the impact parameters.

We use Equation (15) to estimate the offset rate. In our calculation, the integration limits in Equation (15) are set as follows. Current SZ surveys have discovered clusters in a large mass range from a few 10¹⁴ to a few 10¹⁵ M_☉ (Williamson et al. 2011; Reichardt et al. 2013). We set M_min = 2  ×  10¹⁴ h⁻¹ M_☉ and M_max = 3  ×  10¹⁵ h⁻¹ M_☉ in our calculation. We also investigate the results with different M_min in Figure 11. We set ξ_min = 1 and ξ_max = 3, as only major mergers could produce the obvious offsets in the simulations. We set V_max = V_crit (see Equation (9)), as we ignore the flyby mode in this work; and V_min is 10 km s⁻¹. We set P_min = 0 and P_max = 600 h⁻¹ kpc. We test for a larger value of P_max and find no much difference in the result, as mergers with P > 400 kpc only induce small offset ratio.

We define the cumulative offset probability of the SZ–X-ray offset as follows,

$$\begin{eqnarray} && \mathbb {P}_{\rm cumul}(d_{\rm phy}>d_{\rm c}|z) \nonumber \\ && \quad\quad = \frac{dN(d_{\rm phy}>d_{\rm c}|z)/dz\cdot |dz/dt|\cdot (t_{\rm relax}-t_{\rm 1st})}{\int ^{M_{\rm max}}_{M_{\rm min}} n(M,\ z)dM}. \quad\quad \end{eqnarray} \tag{ 23 }$$

In Figure 10, we show the cumulative probability as a function of d_c at different redshifts. As seen from the figure, the cumulative probability is smaller while d_c > 50 h⁻¹ kpc at lower redshift. The probability for significant offsets observed at higher redshift is higher because the factor of |dz/dt|(t_relax − t_1st) in Equation (23) is larger at higher redshift. For all redshift cases, the cumulative probability is flat over the range of 50 < d_c < 300 h⁻¹ kpc. This is consistent with the "jump effect" shown in Section 3 that the offset rapidly increases to a few hundred kiloparsecs after the small cluster passes through the larger one, which roughly equals to the displacement between the mass density centers of the two clusters. Forero-Romero et al. (2010) investigated the distribution of displacements between the peaks of the DM and the gas mass densities in clusters from a large nonradiative SPH ΛCDM cosmological simulation. They found that about 10% of the massive clusters (with masses ranging from 2.0  ×  10¹⁴ h⁻¹ M_☉ to 2.5  ×  10¹⁵ h⁻¹ M_☉) at redshift z = 0.5 have displacements larger than 100 kpc, and about 3% are larger than 200 kpc. Their results are roughly consistent with ours for d_c ∼ 50–100 h⁻¹ kpc; however, when d_c > 100 h⁻¹ kpc, their probability is several times smaller than our results. The difference is caused by the jump effect of the X-ray position as discussed above. The period when the X-ray position locates around the center of the small cluster is commonly 0.5 Gyr or longer. This significantly enhances the probability of the offset larger than 100 h⁻¹ kpc, which however might be omitted by the used hierarchical friends-of-friends algorithm.

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We calculate the cumulative probability for d_c > 300 h⁻¹ kpc by extrapolating Equation (13). As shown in Figure 10, the probabilities reveal an obvious decay when d_c ≳ 350 h⁻¹ kpc, mostly because M_min in Equation (15) is larger when V is close to the peak of the velocity distribution (see Equation (10)).

Specifically, we discuss the probability at z = 0.7 in detail for two reasons, (1) the median redshift of the SZ sample in observations is nearly 0.5 (Marriage et al. 2011; Reichardt et al. 2013); (2) the average redshift of the observed SZ clusters to be compared with our estimation is 0.7 (see Section 4.4). As seen from Figure 10, for the significant offsets with d_c = 50, 100, 150, 200, 300 h⁻¹ kpc, the cumulative probabilities of the unrelaxed clusters with SZ–X-ray offset larger than d_c are 11.1%, 9.5%, 8.6%, 8.0%, and 6.5%, respectively. The best fit to these probabilities by a power-law form ($\propto d_{\rm c}^{b_{\rm l}}$) is shown as the dotted line, with b_l = −0.27. As discussed above that significant offsets might be omitted in Forero-Romero et al. (2010), our best-fit power index (b_l = −0.27) is flatter than their result (b_l = −1.0).

In addition, the probability of d_c = 500 h⁻¹ kpc is approximately one third of that of d_c = 300 h⁻¹ kpc at z = 0.7. We assume an exponential decay to describe this behavior and to extend the best-fit power-law form. Consequently, we fit the cumulative probability for d_c > 50 h⁻¹ kpc as follows,

$$\begin{eqnarray} && \mathbb {P}_{\rm cumul}(d_{\rm phy}>d_{\rm c}) \nonumber \\ && \quad = \left\lbrace \begin{array}{@{}l} a_{\rm l}\left(\frac{d_{\rm c}}{d_{\ast }}\right)^{b_{\rm l}}, \quad {\ \rm if \ }50\,h^{-1}{\rm \,kpc}\le d_{\rm c} \le x_d,\\ a_{\rm l}\left(\frac{d_{\rm c}}{d_{\ast }}\right)^{b_{\rm l}}\exp \left(\frac{x_d - d_{\rm c}}{x_{\rm s}}\right),\quad {\ \rm if\ } d_{\rm c}>x_d, \end{array} \right. \end{eqnarray} \tag{ 24 }$$

where a_l = 0.085, b_l = −0.27 are the best-fit parameters, and d_* is set to 140 h⁻¹ kpc. The best fits of x_d and x_s are 369 and 115 h⁻¹ kpc, respectively.

For the offsets with d_c < 50 h⁻¹ kpc, the jump effect has little influence on the cumulative probability. Our results show that the probabilities at z = 0.7 are 62.9%, 34.0%, 21.4%, and 14.6%, when d_c = 10, 20, 30, and 40 h⁻¹ kpc, respectively. We follow Equation (1) in Forero-Romero et al. (2010) to fit the results as

$$\begin{eqnarray} \mathbb {P}_{\rm cumul}(d_{\rm phy}>d_{\rm c})& = & a_{\rm s}\left(\frac{d_{\rm c}}{d_{\ast }}\right)^{b_{\rm s}}\exp \left(-\frac{d_{\rm c}}{d_{\ast }}\right), \nonumber \\ & & {\ \rm if\ }d_{\rm c}<50h^{-1}{\rm \,kpc}. \end{eqnarray} \tag{ 25 }$$

The best-fit gives a_s = 0.072, b_s = −0.85, approximately consistent with those in Forero-Romero et al. (2010). Since the fitting form is divergent when d_c approaches zero, we set an cutoff at the offset where the cumulative probability is 1.0 in Equation (25).

In Figure 11, we show the cumulative probabilities of the SZ–X-ray offset with different M_min (Equation (23)) in panel (a) and the contribution of mergers in different cluster mass ranges to the cumulative probability of those clusters with M_min = 2 × 10¹⁴ h⁻¹ M_☉ in panel (b), respectively. Mergers of the massive systems (>5 × 10¹⁴ h⁻¹ M_☉) tend to produce a larger offset. The cumulative probability of d_c = 100 h⁻¹ kpc is 5% for the M_min = 1 × 10¹⁴ h⁻¹ M_☉ case, while it is about 30% for the M_min = 5 × 10¹⁵ h⁻¹ M_☉ case (see panel (a)). The results in panel (b) shows that the clusters in the low mass range (2 × 10¹⁴–5 × 10¹⁴ h⁻¹ M_☉) dominate the contribution of the offset. In Table 3, we list the best-fit parameters of the cumulative probabilities for Equations (24) and (25) for different redshift and mass range (see Figures 10 and 11(a)). We find that the parameters of a_s = 0.015(1 + z)^3.1, b_s = −0.85, a_l = 0.018(1 + z)^3.1, b_l = −0.27, x_d = 369 h⁻¹ kpc, and x_s = 151(1 + z)^−0.5 h⁻¹ kpc may fit the redshift dependence of the cumulative probability shown in Figure 10 well (M_min = 2 × 10¹⁴ h⁻¹ M_☉).

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The SZ surveys with large sky coverage may be able to determine both the probabilities of the clusters with offsets larger than a given d_c and then the offset rate dN(d_phy > d_c|z)/dz, which depends on the cluster merger rate B in Equation (15) and thus help to constrain it.

Finally, we discuss some possible uncertainties in the above estimation of the cumulative offset probability.

• 1.  
  In the fitting form of the offset ratio in Equations (13) and (14), we do not consider the mass ratio as an argument. The bottom right panel of Figure 8 reveals that the offset ratio with ξ = 3 is about 30%–50% lower than that with ξ = 2. Thus, we might overestimate the probability by a factor of ≲1.5–2. If we select a tighter constraint of the mass ratio range in Equation (15) with (ξ_min, ξ_max) = (1.5, 2.5), the cumulative probability shown in Figure 10 (solid line) becomes 5.4%, 4.6%, 4.2%, 3.9%, and 3.2% for d_c =50, 100, 150, 200, and 300 h⁻¹ kpc, respectively, which are nearly half of the values reported above.
• 2.  
  In this work, the offset ratio R(d > d_c) is assumed to be redshift independent. The redshift evolution shown in Figure 10 is purely introduced by the halo merger rate B(M₀, ξ, z − Δz) in Equation (15). However, the virial radius of the galaxy cluster is proportional to (1 + z)⁻¹; the concentration parameter is anti-correlated with redshift; and the physical mass densities of the dark matter and gas halos also depend on redshift. We use Figure 12 to indicate the effect on the maximum and the duration of the SZ–X-ray offset from the redshift-dependent physical size of the galaxy cluster. In Figure 12, we show the results of the ratio of R(d > d_c) at redshift z = 0.5, 1.0, and  2.0 to its value at redshift z = 0, by performing a series of cluster merger simulations with M₁ = 2 × 10¹⁴, 5 × 10¹⁴, 1 × 10¹⁵ M_☉, P = 0 kpc, V = 500 km s⁻¹, ξ = 2. As seen from the figure, the deviation of the ratio from unity is smaller than 10% (or 20%) at redshift z = 0.5 (or z = 1) for most of the cases. For d_c ⩽ 100 h⁻¹ kpc, the ratios are possibly larger than 1 at higher redshift due to the higher central density of the cluster and thus a strong collision intensity. For larger d_c, the ratios shift to a lower level at high redshift. At redshift z = 2, the redshift evolution of the offset ratio becomes more significant, especially for d_c ⩾ 200 h⁻¹ kpc. The maximum of the SZ–X-ray offset in the M₁ = 2  ×  10¹⁴ M_☉ case is smaller than 200 h⁻¹ kpc at z = 2, which implies that the redshift-dependence of the cluster size (∝(1 + z)⁻¹) plays a more important role on the SZ–X-ray offset size with increasing redshift, although a higher central density enhances the strength of the collision. We conclude that the redshift evolution effect is an important factor for estimating the offset probability only at z ≳ 1. Considering both the smaller offset ratio R(d > d_c) and the higher mass limit of M_min in Equation (15) at high redshift, the cumulative probability of the offset for d_c ⩾ 50 h⁻¹ kpc may be suppressed.

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In Section 4.2, we estimate the cumulative probability of the SZ–X-ray offset by using the fiducial model of the distributions of relative velocities and impact parameters presented in Sections 4.1.3 and 4.1.4. The forms and the parameters of the fiducial model are motivated or obtained from cosmological simulations, but there is no observational constraint on them so far. In this section, we discuss the impacts of different distribution parameters on the cumulative offset probability, and the realistic observational results may put constraints on them.

In Figure 13, we explore the influence of the pairwise velocity distribution on the probability. In the left panel, in addition to e = 2.9 in the fiducial model fitted from the cosmological simulation, we also obtain the results with different values of e(=2.5, 3.2, 3.5) in Equation (21) to test the effects of different peak positions of the velocity distribution, where w is fixed to be 0.295. The peaks locate in the range of ∼200–2200 km s⁻¹. In the right panel, we test the effect of different FWHMs in the velocity distribution, where we set different values of w(=0.10, 0.15, 0.29, 0.50) and the value of e is selected to keep the peak positions of the distributions the same as that of the fiducial model. According to the results shown in Figure 13, we find that the amplitude of the probability is sensitive to the peak position of the velocity distribution, but not to the width. The probability obtained in the fiducial model is approximately larger than that obtained with e = 3.2 and 3.5 by a factor of 1.6 and 5.0, respectively. As seen from the figure, the shapes of the cumulative probabilities of d_c ⩽ 300 h⁻¹ kpc with different velocity distributions are approximately similar. That is because the shape of the probability is mainly determined by the dependence of the offset ratios and the projection factor on d_c (see Equation (18)), and the offset ratios are proportional to $d_{\rm c}^{\alpha _{\rm l}}$ in Equation (13), where the best-fit index α_l is approximately a constant and not dependent on the parameters V and P. The same feature also appears in Figure 14 for the effects of different impact parameter distributions below. However, the cumulative probabilities of d_c > 300 h⁻¹ kpc are sensitive to the peak position of the velocity distribution. The reason is that the mergers with high relative velocities produce large offsets. In the e = 2.5 case, the upper boundary of the velocity distribution is about 600 km s⁻¹, and only the massive clusters could produce offsets larger than 400 h⁻¹ kpc (see Equation (10)). Due to the relatively small number of the massive clusters, the cumulative probability decays rapidly with d_c > 400 h⁻¹ kpc. When the peak position of the relative velocity distribution is higher (e.g., e = 3.2), the cumulative probability becomes flatter at d_c > 300 h⁻¹ kpc, as the relatively less massive cluster mergers can also produce relatively large offsets.

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Note that while the velocity distributions shift to the high-velocity end, the flyby mode should not be ignored any more. From our simulations, we find that the time durations of the significant SZ–X-ray offset of the flyby mode are usually 1–1.5 Gyr when V ∼ 2000 km s⁻¹ for different cluster masses (e.g., see the bottom panel of Figure 2). The durations are also inversely proportional to the relative velocity and very sensitive to the impact parameter (because off-axis mergers with high relative velocity, e.g., V > 2000 km s⁻¹, are relatively ineffective in destroying the gas cores in the large clusters, the offsets are strongly suppressed when the impact parameter gets larger). We roughly estimate the effects of the flyby mode on our calculation. We find that when e = 3.2, the effect is smaller than 10%; and when e = 3.5, the probability will increase by ∼50% after including the flyby case. The flyby mode, however, does not significantly weaken the tendency that a higher peak position of the relative velocity distribution results in a smaller probability. Here we stress that the sensitivity of the probability to the velocity is the crux of the Bullet Cluster problem discussed in Lee & Komatsu (2010) and Thompson & Nagamine (2012). The parameter space search of the Bullet Cluster (1E0657-56) in the literature suggested that such a system requires a high relative velocity during the merger (e.g., Mastropietro & Burkert 2008), which possibly challenges the standard ΛCDM model. We suggest that the cumulative probability of the observed SZ–X-ray offset could provide an opportunity to examine the incompatibility existing between observed bullet clusters and cosmological simulations.

In Figure 14, we show the cumulative probabilities obtained from different distributions of the impact parameters. The left panel displays the results of the distributions with different peak positions (i.e., 50, 200, 500 h⁻¹ kpc), but with the same FWHM. The distribution reported in Khochfar & Burkert (2006) is also tested. We find that if the peak positions are smaller than 200 h⁻¹ kpc, the results show little difference. However, if the peak of the distribution shifts to a larger value, the probability is obviously suppressed as shown by the λ = 500 h⁻¹ kpc case in the panel. Compared with our fiducial model, the cumulative probability obtained from Khochfar & Burkert's (2006) distribution is smaller by a factor of 1.7. In the right panel, we change the width of the distribution, but keep the same peak position. We find that a wider distribution gives smaller probability, which is reasonable as the larger impact parameter components contribute more to the distribution in the wider case but less to the large offsets. For the two extreme cases shown in the panel, the probability obtained with μ = 3.0 is approximately two times larger than that obtained with μ = 0.5. As a result, we find that the amplitude of the cumulative probability (d_c > 50 h⁻¹ kpc) depends both on the peak position and the width of the impact parameter distributions. However, the shape of the cumulative probability does not, which is different from the dependence on the velocity distributions.

In this section, we compare our estimation of the cumulative probability of the SZ–X-ray offset with observations in the real universe. This comparison is motivated mainly by the following reasons: (1) in Section 4.2, we estimate the probability of the offset based on the fiducial model, and find that galaxy clusters with significant offsets are not rare, for example, approximately 10% of the clusters have offsets larger than 50 h⁻¹ kpc; (2) the past several years have seen rapid progress in the SZ cluster observation, in terms of the total numbers and the parameter ranges (precision and redshift); and (3) the comparison would potentially provide constraints to the model used in the estimation of the offset probability, e.g., the pairwise velocity distribution. The majority of the observed SZ clusters to be compared with are at redshift z ∼ 0.7 (Andersson et al. 2011) and 1'' corresponds to 5 h⁻¹ kpc at redshift z = 0.7. In the following comparison, we assume that 1'' corresponds to 5 h⁻¹ kpc, for simplicity.

The spatial offset between the X-ray and the SZ peaks shown in observations can be modeled to comprise the two following components,

$$\begin{equation} d_{\rm obs}=d_{\rm phy}+d_{\rm err}, \end{equation} \tag{ 26 }$$

where d_phy is the physical one produced by the energetic merger defined in Equation (15) and d_err is the observational error. The (differential) distribution of the physical offset d_phy follows the derivative of the offset cumulative probability (i.e., Equations (24) and (25)) with respect to d_c. In this work, we assume that both of the observational errors in the spatial positions of the X-ray and the SZ peaks follow a Gaussian distribution. The standard deviation of d_err for the X-ray peak (σ_X-ray) is set to 1'', according to the current capability of the X-ray instrument (e.g., Chandra X-ray Observatory). For the SZ effect, the typical position uncertainty of the SPT SZ cluster centroid is approximately σ_SZ = 1'2/SNR ∼ 15'', which dominates the uncertainty d_err. In addition, we also test the results by assuming two higher resolutions of the SZ effect (σ_SZ = 2'', and 8'') in this work.

Figure 15 shows the statistical distribution of the SZ–X-ray offsets d_obs by using the Monte-Carlo method to simulate both d_phy and d_err in Equation (26). The bin size of the offsets is 5'' in both panels of Figure 15, and the histogram represents the percentage of the number fraction of the simulated sample in each bin. To illustrate the effect of different distributions of d_phy, we show the observational expectations obtained by assuming that d_phy = 0 and d_phy follows the derivatives of Equations (24) and (25) in the left and the right panels, respectively; for example, the values of the differential distribution of d_phy at 2'', 4'', 6'', 8'', 10'', 20'', 40'', 60'', 80'', and 100'' in the right panel are 2.91 × 10⁻¹, 8.1 × 10⁻², 3.8 × 10⁻², 2.2 × 10⁻², 1.5 × 10⁻², 1.3 × 10⁻³, 5.2 × 10⁻⁴, 3.1 × 10⁻⁴, 2.3 × 10⁻³, and 8.9 × 10⁻⁴ arcsec⁻¹, respectively. In the left panel of Figure 15, the offsets are contributed purely by the observational errors; while those SZ–X-ray offsets in the right panel are contributed by both the observational errors and the underlying physical ones. The distribution of d_phy is displaced in the inset of the right panel of Figure 15 as the solid line, and the discontinuities appear at d_c = 107 and 738, corresponding to the transitions of the different behavior of the cumulative probability in three different regions of d_c as discussed in Section 4.2 (see Equations (24) and 25). As seen from the right panel, the distribution of d_obs is bimodal once the underlying physical distribution d_phy is considered: some clusters peak around d_obs ∼ 0 and the others peak at d_obs ∼ 70''–90''. The lack of clusters with the SZ–X-ray offsets around 50''–60'' is mainly due to the "jump effect" as shown in Figures 2, 4, and 5, respectively. The location of the right peak (∼ 70''–90'') is determined by the combination of the following factors: (1) the maximum offset caused by a merger increases with the cluster mass and the initial relative velocity, as seen from Equation (10); (2) the cluster mass function decreases with increasing mass; and (3) the underlying velocity distribution. A much larger offset is more likely to be contributed by mergers of clusters with relatively large masses and high relative velocities (e.g., see Figure 16).

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The distribution of d_obs may be significantly biased away from the distribution of d_phy, especially when the accuracy in determining the SZ centroid (σ_SZ) is not sufficient. For example, the current SPT SZ survey (σ_SZ = 15'') may obtain mis-estimates of the SZ–X-ray offsets ∼10''–20'' for many clusters simply because of the observational errors. However, these mis-estimates may be modeled and the underlying physical SZ–X-ray offset distribution can still be extracted even if σ_SZ is large. As seen from the right panel of Figure 15, the right peak of the SZ–X-ray offset distribution is hard to be affected by the uncertainties in determining the SZ centroids. If the current SZ surveys can detect many more clusters, the right peak should be able to be revealed. If future SZ surveys can achieve to higher resolution and higher accuracy in determining the SZ centroids (e.g., σ_SZ = 8'' or 2''), the underlying physical distribution of the SZ–X-ray offset may be better determined, and thus can be used to constrain the physics involved in the mergers of clusters.

We show the observational offset distribution obtained from the sample of Andersson et al. (2011) in Figure 15 (plus signs). The observational sample in Andersson et al. (2011) consists of 15 clusters, obtained from observations of 178 deg² of the sky surveyed by the SPT. The average redshift of the sample is z = 0.68. The observational data shown in Figure 15 are the values after transferring the observed displacement between SPT detection and X-ray centroid to the distance at redshift z = 0.7 where 1'' ∼ 5 h⁻¹ kpc. As seen from Figure 15, the uncertainties in determining the SZ cluster centroids probably play a dominant role for those clusters with the SZ–X-ray offsets ≲ 20''. It appears there are a few clusters with the SZ–X-ray offsets ≳ 30'', which cannot be due to the observational errors and must be due to large physical offsets. Our model shows that the cumulative probability of existing clusters with the SZ–X-ray offsets ≳ 30''–40'' is roughly 10%, which is roughly consistent with observations. Note that the current observational sample is still small (15), which may lead to large uncertainties in estimating the distribution of the SZ–X-ray offsets; for example, there is only one of the 15 clusters locating in the offset bin 45''–50'', where the Poissonian error is significant and comparison of that bin point with our model results is not reliable. If SZ surveys can detect many more SZ clusters, it would be possible to accurately estimate the observational SZ–X-ray offset distribution and use it to constrain the underlying cluster merging model and the related physics involved in. In addition, we show that the jump effect plays a dominant role in generating significant offsets, but here we do not consider the secondary X-ray maxima in the data analysis, which would suppress the probability to discover the clusters with offsets larger than 10''.

Figure 16(a) presents the model results on the SZ–X-ray offset distribution of clusters within different mass ranges (i.e., different M_min in Equation (23)). As seen from the figure, more massive clusters contribute larger offsets in the observation, as the right peak of the distribution resulted from the high mass case locates at a larger spatial scale in the high mass range (e.g., 100''–110'' when M_min = 5 × 10¹⁴ h⁻¹ M_☉). In Figure 16(b), we show the distributions obtained with different pairwise velocity distributions. As seen from the figure, the bimodal distribution of the offsets still exist for different velocity distributions. As the mean value of the relative velocity increases, the "jump effect" as shown in Figure 2 is more significant in causing the scarce of the clusters with intermediate offsets. The higher relative velocity case results in more significant offsets (>120''). The dependence of the distribution of the SZ–X-ray offsets on the pairwise relative velocity distribution of clusters suggests that the observed SZ–X-ray offset distribution can be used to probe the cosmological velocity field at the cluster scale.