Living with Neighbors. II. Statistical Analysis of Flybys and Mergers of Dark Matter Halos in Cosmological Simulations

The distance between two halos in a pair is the most important factor in identifying the interacting pair sample. Interactions happen when a halo is affected by the gravitational force from another halo (noncontact interactions) and when their virial regions of halos overlap each other (contact interactions). In this paper, we take into account only the contact interaction case, whose distance criterion is expressed by

$$\begin{eqnarray}&&{D}_{12}\lt {R}_{\mathrm{vir},1}+{R}_{\mathrm{vir},2},\end{eqnarray} \tag{ 1 }$$

where D₁₂ is the distance between the center positions of two constituent halos and R_vir is the virial radius of each halo. As mentioned in Section 2, the halo center is defined as the center of mass of the bound particles in the halo's innermost region to avoid misinterpretation by the tidally perturbed structures during the contact interaction.

This study focuses on the major interaction between two halos with similar masses (e.g., Milky Way and Andromeda). We define the major interaction as the mass ratio (M_neigh/M_target) of halo pairs ranging from 1/3 to 3, where M_target is the mass of the target halo and M_neigh is the mass of its neighboring halo. The mass range of the target halos we deal with is 10^10.8–10^13.0 h⁻¹M_⊙. The minimum mass (10^10.8 h⁻¹M_⊙) of a target halo is equivalent to ∼400 member particles, and its neighboring halo with one-third of 10^10.8 h⁻¹M_⊙ has over 130 particles.

There are three main methods for defining the environment (Muldrew et al. 2012): (1) nth-nearest-neighbor, (2) fixed aperture, and (3) fixed annulus. We apply the fixed aperture method to the environment definition because this method is well known to describe the large-scale environment (such as fields, filaments, and clusters) better than other methods (Muldrew et al. 2012). The number count of halos is replaced by the total mass of halos in order to include the effect of massive halos. To obtain the environmental parameter (${{\rm{\Phi }}}_{\mathrm{Env}}$), we first measure the total mass (M_Env) of halos in our halo catalog (M_halo > 10^9.8 h⁻¹M_⊙) within a sphere of a comoving radius of R = 5 h⁻¹ Mpc. Then, Φ_Env is defined as a percentile rank order of M_Env given by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{Env},{\rm{i}}}\equiv \mathrm{rank}({M}_{\mathrm{Env},{\rm{i}}})\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&{M}_{\mathrm{Env},i}=\sum {M}_{\mathrm{halo},j}\ \mathrm{for}\ {D}_{{ij}}\lt 5\,{h}^{-1}\mathrm{Mpc},\end{eqnarray} \tag{ 3 }$$

where D_ij is the distance between the ith target halo and its jth neighboring halo. The range of Φ_Env is from 0 to 100.

Figure 1 shows three snapshots of the projected halo distributions with thickness of 10 h⁻¹ Mpc in the Z-direction at three different redshifts. The percentile rank method has an advantage of tracing given overdense regions from high redshifts to the present. The highly ranked environment at z = 0.0 is largely highly ranked at z = 3.1. This thus makes us reduce the selection effect when we analyze the redshift evolution of interaction fractions. For the analysis of the environmental dependence of the redshift evolution, we classify the environment into low-density (Φ_Env ≤ 35), intermediate-density (35 < Φ_Env ≤ 65), and high-density (Φ_Env > 65) environments. The figure shows that the environmental parameter, ${{\rm{\Phi }}}_{\mathrm{Env}}$, properly represents the cosmic web type, and it is consistent with other environment definitions (e.g., L'Huillier et al. 2015; see Appendix B).

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The total energy of an interacting pair is required to determine whether or not the halos are bound to each other by mutual gravitational potential (Tonnesen & Cen 2012; Moreno et al. 2013). In this analysis, the halo pair is simply modeled as an isolated two-body system. The total energy is a sum of kinetic and potential energies and calculated by

$$\begin{eqnarray}&&{E}_{12}={M}_{1}{M}_{2}\left\{\displaystyle \frac{1}{2}\displaystyle \frac{| {{\boldsymbol{V}}}_{{\bf{1}}}-{{\boldsymbol{V}}}_{{\bf{2}}}+H(z){{\boldsymbol{R}}}_{{\bf{12}}}{| }^{2}}{{M}_{1}+{M}_{2}}-\displaystyle \frac{G}{| {{\boldsymbol{R}}}_{{\bf{12}}}| }\right\},\end{eqnarray} \tag{ 4 }$$

where M₁ and M₂ are the masses of the two constituent halos, ${{\boldsymbol{V}}}_{{\bf{1}}}$ and ${{\boldsymbol{V}}}_{{\bf{2}}}$ are their peculiar velocities with respect to the center of mass of the system, H(z) is the Hubble parameter at a given redshift, and ${{\boldsymbol{R}}}_{{\bf{12}}}$ is the displacement vector between the centers of the two halos. The Hubble flow, $H(z){{\boldsymbol{R}}}_{{\bf{12}}}$, is usually in the form of the recession velocity when calculating the relative velocity between the halos.

If the total energy of a pair system is positive (negative), the pair would be unbound (bound). However, the total energy of the system is reduced by the dynamical friction during the contact interactions, and even the sign of the total energy can turn from positive to negative. Sinha & Holley-Bockelmann (2012) showed that ∼20% of flybys finally merged after their first passage. For a more robust classification (i.e., flybys or mergers) of the interaction, we employ the capture criterion of Gnedin (2003). The capture criterion is defined as

$$\begin{eqnarray}&&{\rm{\Delta }}E=2.4\,\mu \,\displaystyle \frac{{M}_{\mathrm{neigh}}}{{M}_{\mathrm{target}}}\,\displaystyle \frac{{\sigma }_{\mathrm{target}}^{4}}{{V}_{\mathrm{rel}}^{2}+{\sigma }_{\mathrm{target}}^{2}},\end{eqnarray} \tag{ 5 }$$

where μ is a reduced mass of the pair system, σ_target is a velocity dispersion of the target halo, and V_rel is a magnitude of relative velocity between two constituent halos. Although Equation (5) is analytically derived from the assumption of an isothermal density profile of a halo, the number fraction of pair systems with 0 ≤ E₁₂ < ΔE is consistent with the finding of Sinha & Holley-Bockelmann (2012). Approximately 20% of target halos with interacting neighbors with E₁₂ ≥ 0 have a total energy smaller than the capture criterion (see Appendix C), and they merge with their neighbors despite the positive total energy. We label an interacting halo pair as a flyby or a merger when the total energy is greater than the capture criterion (E₁₂ ≥ ΔE) or smaller than the criterion (E₁₂ < ΔE), respectively. The analysis focuses only on the current status of pairs, so it is not necessary to consider the merging timescale with respect to the Hubble time.

In this study, three interaction fractions are considered in the halo pair catalog: the interaction fraction (F_I), the flyby fraction (F_F), and the merger fraction (F_M). The interaction fraction is defined as

$$\begin{eqnarray}&&{F}_{{\rm{I}}}\equiv \displaystyle \frac{{N}_{{\rm{I}}}}{{N}_{\mathrm{target}}},\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&{N}_{{\rm{I}}}=\displaystyle \sum _{i}^{{N}_{\mathrm{target}}}{N}_{{\rm{I}},i},\end{eqnarray} \tag{ 7 }$$

and the interaction number of the ith halo is N_I,i = 0 or 1. We define ${N}_{{\rm{I}},i}=1$ even in the case that the ith target halo has multiple interacting (flybying or merging) neighbors. Consequently, F_I ranges between 0 and 1. This thus enables us to avoid the number of interactions to be scaled with ${ \mathcal O }({N}^{2})$. Likewise, we apply the same rule to the flyby and merger cases:

$$\begin{eqnarray}&&{N}_{{\rm{F}}}=\displaystyle \sum _{i}^{{N}_{\mathrm{target}}}{N}_{{\rm{F}},i}\ \ \mathrm{and}\ \ {N}_{{\rm{M}}}=\displaystyle \sum _{i}^{{N}_{\mathrm{target}}}{N}_{{\rm{M}},i}.\end{eqnarray} \tag{ 8 }$$

Note that a halo may have flyby and merger events at the same time. In the halo pair sample of 10^10.8 h⁻¹M_⊙ < M_target < 10^13.0 h⁻¹M_⊙ at z = 0 extracted from 11 simulations, 30,392 halos have interacting neighbors, among which 14,693 halos have flybying neighbors and 17,155 halos have merging neighbors. About 4.8% of the halo pair sample have flyby and merger events simultaneously at z = 0. We will discuss the multiple-interaction cases in detail in Section 4.3.

Figure 2 illustrates the interaction fraction (F_I), flyby fraction (F_F), merger fraction (F_M), and ratio between F_F and F_M (F_F/F_M) as functions of the halo mass and environment at three different redshifts: z ≃ 0 (left columns), z ≃ 1 (center), and z ≃ 3 (right). In the top row, the interaction fraction shows a marginal correlation with the halo mass and environment at all redshifts. The F_I values slightly decrease with increasing halo masses and with decreasing Φ_Env. The trend is consistent qualitatively with that of L'Huillier et al. (2015). Whereas L'Huillier et al. (2015) used a mass ratio criterion of M_neigh/M_target > 0.4 with no upper boundary, our criterion (1/3 < M_neigh/M_target < 3) sets an upper boundary, by which we do not count interactions of less massive target halos with more massive neighbors, which occur more frequently in the denser environment. As a result, our mass and environmental dependence is weaker than that of L'Huillier et al. (2015).

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In the second row, the flyby fraction shows a clear dependence on the halo mass and environment at all redshifts of interest. At a given environment (and redshift), ${F}_{{\rm{F}}}$ for the least massive halos is higher by up to a factor of 50 than for the most massive counterparts. The mass dependence tends to diminish in denser environments. While the anticorrelation between the flyby fraction and the halo mass is consistent with the interaction fraction and L'Huillier et al. (2015), it is opposite to Sinha & Holley-Bockelmann (2012), who showed a positive correlation. We attribute the opposite behavior to Sinha & Holley-Bockelmann (2012) having a mass ratio criterion that is different from ours. The flyby sample of Sinha & Holley-Bockelmann (2012) has no lower boundary in the mass ratio (i.e., M_neigh/M_target < 1.0). With no lower boundary, the flyby fraction is boosted by numerous interactions of massive halos with smaller satellites and thus correlated with the halo mass. At a given mass (and redshift), F_F increases by up to a factor of 400 with the increasing environmental parameter. This environmental dependence is qualitatively consistent with observational surveys (e.g., Lin et al. 2010) and simulations (e.g., Jian et al. 2012).

In the third row, the merger fraction shows a quite different trend from the flyby fraction in that the mass and environmental dependencies of F_M are very weak over all redshifts analyzed in this study. The apparent mass independence of F_M is consistent with observations (Xu et al. 2012). However, the environmental independence is at odds with the findings of Lin et al. (2010) and Jian et al. (2012). The origin of the mass and environmental dependence of ${F}_{{\rm{F}}}$ and the independence of F_M will be discussed in Section 5.1.

In the bottom row, we plot the F_F/F_M ratio. The ratio depends on the halo mass and environment, analogously to the flyby fraction. Obviously, this is because the flyby fraction only shows the clear dependence on these parameters, while the merger fraction does not. The F_F/F_M ratio increases with decreasing halo mass (by up to a factor of 100) and with increasing Φ_Env (by up to a factor of 300). The maximum values of F_F/F_M are 4.97, 2.49, and 1.12 at z ≃ 0, 1, and 3, respectively. In other words, the fractional contributions of flybys to the pair fraction are 0.83, 0.71, and 0.53 at z ≃ 0, 1, and 3, respectively. The flyby, as a source of tidal perturbations, starts to outnumber the merger (F_F/F_M > 1) at high redshift (z ∼ 3) from the bins with the least halo mass in the highest density. At z ≃ 0, the flyby outnumbers the merger in 58 bins among the total 220 bins (∼26%) on the mass–environment plane.

Figure 3 presents the redshift evolution of the interaction fractions (top row), flyby fractions (second row), merger fractions (third row), and ratio of the flyby fraction to the merger fraction (bottom row). Three halo mass subsamples are compared in the range of 10^11.4 h⁻¹M_⊙ < M_target(z = 0) < 10^13.0 h⁻¹M_⊙ in three different (low-, intermediate-, and high-density) environments. To take into account the progenitor bias (e.g., van Dokkum & Franx 2001), we set the halo mass range as a function of redshift by adopting the halo mass growth function from Correa et al. (2015):

$$\begin{eqnarray}&&{M}_{\mathrm{target}}(z)={M}_{\mathrm{target}}(z=0){\left(1+z\right)}^{0.24}\,{e}^{-0.75z}.\end{eqnarray} \tag{ 9 }$$

The progenitor halos of M_target(z = 0) have a mass of M_target(z). By using this method, we can estimate the past interaction fraction of halos with M_target(z = 0). An additional mass cut at each redshift should be considered to have its corresponding halo mass equal to 10^10.8 h⁻¹M_⊙ at the starting redshift because the minimum halo mass is set to be M_min = 10^10.8 h⁻¹M_⊙. According to Equation (9), we divide the halo pair sample into three subsamples: high-mass halos (log₁₀(hM_target(z = 0)/M_⊙) ∈ [12.2, 13.0]), intermediate-mass halos (∈[11.7, 12.2]), and low-mass halos (∈[11.4, 11.7]). The lower boundary masses of the subsamples become greater than the minimum mass at z = 2.2, 3.1, and 5.0, respectively.

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In the top row of Figure 3, the interaction fractions increase with redshift, regardless of the halo mass and environment. On average, the F_I(z = 0) values are 0.06, 0.10, and 0.16 and the F_I(z = 2) values are 0.13, 0.18, and 0.25, for the low-, intermediate-, and high-density environments, respectively. The slope seems slightly different depending on the environment. The slopes for the low-, intermediate-, and high-mass halos are steeper by a factor of ∼1.2, ∼1.7, and ∼2.6 in the low-density environment than in the high-density environment.

In the second row, the flyby fraction evolves depending on the environment. In the low-density environment, the flyby fraction increases with redshift. The increasing trend gets stronger for more massive halos: F_F at z = 2 for the high-mass (low-mass) halos is ∼15 (∼2.5) times higher than that at z = 0. In the intermediate-density environment, the redshift evolution of F_F is qualitatively consistent with that in the low-density environment. Note that F_F evolves weakly with redshift at z < 1, but remains constant at z > 1. In the high-density environment, F_F peaks at z ∼ 0.35 (∼0.65), and the maximum value of F_F is 0.13 (0.09) for the high-mass halos (intermediate-mass halos). Then it slightly decreases with redshift. In the third row, the merger fraction monotonically increases with redshift. The value of F_M is roughly 0.06 at z = 0 and 0.15 at z = 2.

The evolutionary trends of F_I and F_M are well described by a form of (1 + z)^m (e.g., Lin et al. 2010; Xu et al. 2012) at all redshifts. The value of m is roughly 1.0 in three interaction fractions in spite of a transition in the slope of F_F. This weak redshift evolution with small m-value was recently reported for the extension of the galaxy pair sample to high redshift (z > 1; e.g., Keenan et al. 2014; Mundy et al. 2017), in contrast to the strong trend at low redshift (z < 1; e.g., Conselice et al. 2009; Genel et al. 2009).

In the bottom row of Figure 3, the redshift evolution of F_F/F_M behaves in different ways depending on the environment. In the low-density environment, F_F/F_M increases with redshift, and the slope tends to be steeper for more massive halos (by up to 10), similarly to F_F. The increasing trend diminishes in the intermediate-density environment and even gets reversed in the high-density environment. This indicates that, as we approach the present epoch, the relative effect of the flyby gets stronger (weaker) in the high-density (low-density) environments. Interestingly enough, in the high-density environment, F_F/F_M for the low-mass halos exceeds one for the majority of the Hubble time. The value converges to 1.6 at z = 0, indicating that 62% of pairs are flybys. The F_F/F_M ratio for the intermediate-mass halos has also been over one since z ∼ 1.3. For the high-mass halos, F_F/F_M touches unity at the present epoch. In the high-density environment, flybys are as frequent as or more frequent than mergers at z ≲ 2. We emphasize that the flyby fraction can affect the amplitude and slope of the observed pair fraction, varying with redshift. Snyder et al. (2017) found that the number fraction of real mergers among galaxy pairs decreases from 0.8 at z = 3 to 0.5 at z = 1. This is in line with our result that F_F/F_M in the high-density environment becomes greater approaching z = 0, and it suggests that the flyby fraction contributes more to the pair fraction at lower redshift. We will discuss how significantly the flyby fraction contaminates the observed pair fraction in Section 5.3.

A target halo can have two or more interacting neighbors at the same time. In our simulations, the number of target halos with two or more comparable-mass (i.e., 1/3–3) companions is 4557 (∼15% of N_I, the total number of target halos with interacting neighbors) at z = 0. About 5% of N_I possess both flybying and merging neighbors (see Section 3.4), and the remaining ∼10% have multiple flybying neighbors only or multiple merging neighbors only. In this subsection, we measure the multiple interaction (MI) fraction, defined by

$$\begin{eqnarray}&&{F}_{\mathrm{MI}}\equiv \displaystyle \frac{{N}_{\mathrm{MI}}}{{N}_{\mathrm{target}}},\end{eqnarray} \tag{ 10 }$$

where N_MI is the number of target halos with multiple interacting neighbors. The multiple interactions are classified into three subsamples: (a) the flyby-dominated multiple interaction (FMI), in which the cumulative number of flybying neighbors is larger than that of merging neighbors (n_F,i > n_M,i); (b) the merger-dominated multiple interaction (MMI; n_F,i < n_M,i), which is opposite to the FMI; and (c) the equally contributed multiple interaction (EMI), in which the numbers of flybying neighbors and merging neighbors are equivalent (n_F,i = n_M,i). Note that ${n}_{{\rm{F}},i}\ne {N}_{{\rm{F}},i};$ n_F,i is a cumulative number, but N_F,i is a noncumulative number (0 or 1).

Figure 4 presents the multiple interaction (MI) fraction (F_MI; top row) and the contribution of the three types of multiple interactions to the MI fraction (second, third, and bottom rows) on the halo mass–environment planes at three different redshifts. In the top row, the MI fraction strongly depends on both the halo mass and environmental parameter at all redshifts, and the trend is similar to that of the flyby fraction (see second row of Figure 2). As the halo mass increases and Φ_Env decreases, F_MI becomes lower by up to 40 and 100, respectively. On average, F_MI increases with redshift. The maximum values of F_MI are 0.078, 0.109, and 0.128 at z ≃ 0, 1, and 3, respectively. The MI fraction derived from our simulations tends to be higher than the observational result of Darg et al. (2011). They identified the multimerger system using the Galaxy Zoo catalog and found that the ratio of triple systems to binary systems is smaller than 2% (multiple/binary ≤ 2.5%). In our simulations, the ratio of F_MI to F_I at z ≃ 0 is on average 0.15 (multiple/binary ≃ 18%), which is over five times higher than the observational upper limit. This is because of our larger distance criterion (approximately 10 times higher than that of Darg et al. 2011). If we set the distance criterion to 30 h⁻¹ kpc, the ratio F_MI/F_I is reduced to 0.045 (multiple/binary ≃ 4.7%) and is still two times higher than the result of Darg et al. (2011). This is attributed to the underestimation of binary systems (binary/single ∼ 0.5% in the case of 30 h⁻¹ kpc) by the low force resolution of our simulations ( = 12.5 h⁻¹ kpc).

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The three types of multiple interactions contribute differently to the MI fraction. In the lower three rows of Figure 4, we plot the number fractions of multiple interaction types with respect to the number of MI target halos: (1) the number fraction of FMI target halos (μ_FMI ≡ N_FMI/N_MI; second row), (2) the number fraction of MMI target halos (μ_MMI ≡ N_MMI/N_MI; third row), and (3) the number fraction of EMI target halos (μ_EMI ≡ N_EMI/N_MI; bottom row). The contribution of FMIs is both mass dependent and environment dependent (second row), similarly to the flyby fraction and MI fraction. For the low-mass halos in the high-density environment, the majority of the MIs are FMIs. Moreover, μ_FMI tends to decrease with redshift. The maximum contributions of FMIs are 1.0, 0.79, and 0.69 at z ≃ 0, 1, and 3, resepctively. This is in line with the trend of F_F/F_M in the high-density environment (see bottom row of Figure 3), where flybys happen the most frequently. The contribution of MMI (${\mu }_{\mathrm{MMI}}$) shows an opposite trend to that of FMI (third row). As the halo mass increases and Φ_Env decreases, μ_MMI becomes higher, and these conditions cause the merger fraction to highly exceed the flyby fraction (see bottom row of Figure 3). In the bottom row, the EMI contribution (μ_EMI) has weaker mass and environmental dependence, compared to μ_FMI and μ_MMI. The mean value of μ_EMI is about 0.23. Only for halos with M_target > 10¹² h⁻¹M_⊙ does the mass dependence emerge; μ_EMI tends to decrease with increasing halo masses.

To further investigate differences in F_MI, μ_FMI, μ_MMI, and μ_EMI, we plot the redshift evolution of four variables for nine subsamples in Figure 5. The top row shows that the MI fraction rises with redshift. For instance, for the intermediate-mass halos, the values of F_MI at z = 0 (z = 2) are 0.003, 0.009, and 0.029 (0.012, 0.028, and 0.062) in the low-, intermediate-, and high-density environments, respectively. In the low- and intermediate-density environments, the evolutionary trend is steeper for the higher mass halos, while it is reversed in the high-density environment. Compared to the interaction fraction, the MI fraction on average evolves two times more rapidly with redshift.

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In the lower three rows, the contributions of the three types of multiple interactions evolve differently with redshift depending on the environment. In the low-density environment, three contributions are almost independent of redshift. In most ranges of redshift, μ_MMI keeps higher values than μ_FMI and μ_EMI. The values of μ_MMI are approximately 0.39, 0.60, and 0.85 for the low-, intermediate-, and high-mass halos. In the intermediate-density environment, μ_MMI slightly increases with redshift while μ_FMI and μ_EMI marginally decrease. The MI type that contributes the most depends on the halo mass: μ_FMI for the low-mass halos and μ_MMI for the high-mass halos. For the intermediate-mass halos, μ_FMI is the highest at z < 1.5, but μ_MMI at z > 1.5. In the high-density environment, three contributions show different evolutionary trends. At all redshifts, μ_FMI decreases, μ_MMI increases, and μ_EMI remains constant. The FMI is mostly the MI type with the highest contribution, except for the high-mass halos at z > 2, where μ_MMI is higher than μ_FMI. At z = 0, μ_FMI is roughly 0.7.

Table 1 gives the number of target halos with flybying and merging neighbors. The number of target halos with flybying neighbors is in general smaller than that with merging neighbors except for the high-density environment. However, in the case of the multiple interaction, the n_F > n_M cases always outnumber the n_M > n_F cases. Even in the high-density environment, ∼20% of target halos with flybying neighbors have multiple flybying neighbors, and thus the frequency of the flyby is about two times greater than that of the merger.

In this subsection, we delve into the physical causes of the mass and environmental dependence of the flyby fraction and the independence of the merger fraction. We consider the following three factors: (1) the possibility of running into a halo with a comparable mass (i.e., an encounter with a mass ratio of 1:3–3:1), which is higher for lower mass halos; (2) the relative velocity of two halos in a pair, which depends on their large-scale environment; and (3) the depth of the gravitational potential well of a halo, which depends on the mass of the halo.

First, the flyby fraction has an anticorrelation with the halo mass. According to the halo mass function, the number density of less massive halos is higher than that of more massive halos (e.g., Press & Schechter 1974). Less massive halos have a higher possibility of an encounter with halos of comparable mass and thus have the higher flyby fraction. For the same reason, the multiple interaction fraction is also higher for the less massive halos. The merger fraction, on the other hand, is rather independent of the halo mass. This is because there are two competing factors regulating the merger fractions: less massive halos tend to encounter more frequently comparable-mass halos but have difficulty at the same time in holding nearby halos, due to their shallower potential well. Many studies have shown the diverse results of the mass dependence of the merger fraction (Genel et al. 2009; Xu et al. 2012; Casteels et al. 2014; Keenan et al. 2014; Rodriguez-Gomez et al. 2015). The diversity can be accounted for by the difference in the mass ratios at the infall moment and right before the merger (e.g., Rodriguez-Gomez et al. 2015) and the diverse definitions of the major merger (Casteels et al. 2014, and references therein).

Next, the flyby fraction has a positive correlation with the environmental parameter. We attribute this to the high velocity dispersion in denser environments (e.g., Evrard et al. 2008). Figure 6 shows the relative velocity distributions of flybying neighbors (upper) and merging neighbors (lower). For the flybying neighbors in the upper panels, the median relative velocity for the higher mass (lower mass) halos is ∼420 (∼270) km s⁻¹ and ∼850 (∼480) km s⁻¹ in the low- and high-density environments, respectively. Intuitively, in the high-density environment, the relative velocity exceeds more easily the escape velocity, and thus the number of flybying neighbors is enhanced. In addition to the high relative velocity, the flyby fraction is higher because the possibility of encountering another halo is higher in denser environments. As the high relative velocity and the high possibility arise at the same time, the flyby fraction becomes more strongly dependent on the large-scale environment. For the merging neighbors in the lower panels, the relative velocity distributions do not depend on the environment but on the halo mass. The median velocity for the higher mass (lower mass) halos is ∼260 (∼110) km s⁻¹ regardless of the environment. The environmental independence of the median relative velocity explains why the merger fraction shows little or no environmental dependence. Despite the higher encounter possibility in denser environments, most relative velocities of interacting neighbors exceed the escape velocity of their target halos, lowering the merger fraction. This is in line with the notion that it is hard for a merger to happen in the cluster environment (Ostriker 1980; Wetzel et al. 2008).

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In this subsection, we interpret the redshift evolution of the flyby fraction, the merger fraction, and their ratio in the context of the formation history of the cosmic large-scale structure. Figure 7 shows the median relative velocities of all interacting pairs (both flybys and mergers) as a function of redshift. In the low- and intermediate-density environments, the median relative velocity slightly increases from z = 4 to ∼1. The enhanced relative velocity hinders the flyby fraction from decreasing, similarly to the merger fraction. But after z ≃ 1 the median relative velocity is roughly constant, and at the same time the flyby fraction starts to decrease (see second row of Figure 3). In contrast, in the high-density environment, the median relative velocity dramatically increases toward z = 0. We attribute the enhancement of the flyby fraction from z = 4 to ∼0.3 in the high-density environment to the dramatic rise in the median relative velocity. This effect also leads to both the enhanced contribution of flyby-dominated multiple interactions and the reduced contribution of merger-dominated multiple interactions with respect to the multiple interaction fraction. Furthermore, the gap of the median values between the low-density and high-density environments gets wider toward z = 0, indicating that the environmental dependence of the ratio F_F/F_M becomes stronger when approaching z = 0. The boosted environmental dependence is in line with the evolution of the cosmic web (from voids to nodes; Bond et al. 1996; Cautun et al. 2014). According to Cautun et al. (2014), as the total volume of voids expands over cosmic time, the relative velocity between two halos gets relatively lower, and accordingly F_F/F_M gets reduced. By contrast, as the total volume of nodes contracts over cosmic time, F_F/F_M gets enhanced.

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Figure 8 is a simple schematic diagram illustrating our scenario, which links the redshift evolution of two fractions to the hierarchical structure formation. We divide the formation history into two phases based on Knebe et al. (2004, 2006), who found a correlation between the dynamical age of simulated clusters and the frequency of flybys: the younger, the higher. First, in Phase I, protoclusters and protogroups formed mainly via major mergers in the early universe. At this epoch, because the relative velocity between halos was still low, the merger fraction was higher than the flyby fraction. During this phase, as the large-scale structure was gradually growing, infalls of halos and groups took place more frequently with the increasing relative velocity. Hence, the flyby fraction remains constant or slightly increases. By contrast, the merger fraction decreased with the decreasing redshift because the hierarchical merging leads to decline in the halo accretion rate over cosmic time (e.g., McBride et al. 2009). For the same reason, the multiple interaction fraction gets lower toward low redshift.

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Next, in Phase II, groups and clusters became dynamically virialized and relaxed, and the number of comparable nearby halos decreased. From the beginning of Phase II, the flyby fraction started to decline. The turnover of the redshift evolution for the flyby fraction is logically consistent with Knebe et al. (2004, 2006). In addition, the diversity of the turnover redshift of the flyby fraction is attributed to the fact that the assembly and virialization of halo systems are mass dependent (e.g., Power et al. 2012) and environment dependent (e.g., Sheth & Tormen 2004). During this phase, the continuous mergers and the resultant reduction of the number of comparable neighbors decrease monotonically the merger fraction. This is more so in the accelerating universe where halo systems become more isolated with time (Busha et al. 2003; Behroozi et al. 2013a).

Given that the observed pair (flyby + merger) fraction is used to derive the merger rate, it is important to examine to what extent the pair fraction is contaminated by flybys that are observationally indistinguishable from merging pairs. Back in Figure 6, even up to ∼90% of the flybys in the low-density region have relative velocities smaller than 500 km s⁻¹, suggesting that it is hard to determine, by using only a certain velocity criterion, whether a pair is a flyby or a merger.

Among the three types of dependence analyzed in this paper, the mass and environmental dependence of the flyby fraction should affect the behavior of the observed pair fraction. Survey observations revealed that the pair fraction does not depend on the stellar mass (e.g., Xu et al. 2012) but strongly depends on the environment (e.g., Lin et al. 2010). The absence of the mass dependence seems to be due to the strict merger definition (D_12,projected ≲ 30 h⁻¹ kpc and V_12,radial ≲ 200 km s⁻¹) in an attempt to reduce the effect of spurious mergers, and thus the mass dependence of the flyby fraction barely contributes to the pair fraction. The interaction fraction also shows little mass dependence because the merger fraction is dominated by the flyby fraction (F_F/F_M ≲ 0.1) in the low-density environment, where the mass dependence of F_F is the most evident. On the other hand, the environmental dependence of the pair fraction has been studied with a rather looser definition for mergers (D_12,projected ≲ 200 h⁻¹ kpc and V_12,radial ≲ 1000 km s⁻¹) in order to search for paired companions in cluster environments. Given that in dense environments, F_F/F_M ≳ 1, the looser merger definition allows the pair fraction to be more contaminated by flybys, resulting in the observed dependence of the pair fraction on the environment.

Flybys also contaminate the redshift evolution of the observational pair fraction. Depending on the definition of galaxy pairs, the redshift evolution is observed to be strong (e.g., Conselice et al. 2009) or weak (e.g., Keenan et al. 2014) at z ≲ 1 (see also Williams et al. 2011; Newman et al. 2012; Man et al. 2016; Mundy et al. 2017). The diverse results are ascribed to the contribution of flybys that are observationally indistinguishable from pairs, depending on the merger definition. The pair fraction is expected to be more flattened if the flyby contaminates the pair fraction more significantly. Hence, in order to derive the real merger rate from the pair fraction, the ratio of flybys to pairs (from 0.004 to 0.62; see bottom panels of Figure 3) should be considered.

Assessing the contribution of interactions to galactic evolution is further complicated by the following two factors. Some flybying companions will be turned into merging companions by dynamical friction. We have minimized the effect of dynamical friction by applying the capture criterion to the classification of flybys. Despite this consideration, the real number of flybys that a galaxy experiences during its lifetime requires the real flyby rate (e.g., Sinha & Holley-Bockelmann 2012) and the flyby timescale (between the beginning and end of a contact interaction). In addition to dynamical friction, some halos experience multiple interactions (Darg et al. 2011). Although we found that the multiple interaction is mainly governed by the flyby-dominated MI, it is uncertain whether the multiple interaction raises the flyby effect (e.g., the restricted three-body problem; Sales et al. 2007) or lowers it (e.g., more mergers due to dynamical friction; Lacey & Cole 1993; Jiang et al. 2008). According to our results, multiple interactions in denser environments are expected to enhance the flyby effect, due to the high relative velocity and numerous concurrent flybys (∼20% of the flyby fraction).