A Multidimensional Dependence of the Substructure Evolution on the Tidal Coherence
aa r X i v : . [ a s t r o - ph . C O ] J u l A Multidimensional Dependence of the Substructure Evolutionon the Tidal Coherence
Jounghun Lee
Astronomy Program, Department of Physics and Astronomy, Seoul National University,Seoul 08826, Republic of Korea [email protected]
ABSTRACT
We numerically explore how the subhalo mass-loss evolution is affected bythe tidal coherences measured along different eigenvector directions. The meanvirial-to-accretion mass ratios of the subhalos are used to quantify the severity oftheir mass-loss evolutions within the hosts, and the tidal coherence is expressedas an array of three numbers each of which quantifies the alignment between thetidal fields smoothed on the scales of 2 and 30 h − Mpc in each direction of threeprincipal axes. Using a Rockstar halo catalog retrieved from a N-body simulation,we investigate if and how the mass-loss evolutions of the subhalos hosted bydistinct halos at fixed mass scale of [1-3]10 h − M ⊙ are correlated with threecomponents of the tidal coherence. The tides coherent along different eigenvectordirections are found to have different effects on the subhalo mass-loss evolution,which cannot be ascribed to the differences in the densities and ellipticities of thelocal environments. It is shown that the substructures surrounded by the tideshighly coherent along the first eigenvector direction and highly incoherent alongthe third eigenvector direction experience the least severe mass-loss evolution,while the tides highly incoherent only along the first eigenvector direction isresponsible for the most severe mass-loss evolution of the subhalos. Explainingthat the coherent tides have an obstructing effect on the satellite infalls onto theirhosts and that the strength of the obstruction effect depends on which directionsthe tides are coherent or incoherent along, we suggest that the multidimensionaldependence of the substructure evolution on the tidal coherence should be deeplyrelated to the complex nature of the large-scale assembly bias. Subject headings: cosmology:theory — large-scale structure of universe 2 –
1. Introduction
The classical excursion set theory based on the standard ΛCDM (cosmological con-stant Λ and cold dark matter) model provided an analytical framework within which theformation and evolution of DM halos, the building blocks of the large- scale structure inthe universe, can be physically tracked down (Press, & Schechter 1974; Bardeen et al. 1986;Bond et al. 1991; Bond, & Myers 1996; Sheth et al. 2001). According to this theory, thehierarchical accretion and merging events, which are the dominant driver of the halo growth,owe their frequencies solely to the halo masses. N-body simulations that were performed tocomplement the theory with desired accuracy and precision, however, invalidated this simplepicture, discovering a puzzling phenomenon, so called the ”halo assembly bias”: The clus-tering strength of the DM halos affect their formation epochs and growth rates on the samemass scale (Gao & White 2007). Although the discovery of this phenomenon baffled for longthe community of the large-scale structure, it is now generally accepted that the cosmic web,anisotropic large-scale tidal environments surrounding DM halos (Bond et al. 1996), mustbe mainly responsible for the deviation of the simple prediction of the excursion set theoryon the halo growths from the reality (e.g., Sandvik et al. 2007; Hahn et al. 2009; Wang et al.2011; Borzyszkowski et al. 2017; Tojeiro et al. 2017; Yang et al. 2017; Musso et al. 2018;Mansfield, & Kravtsov 2019; Ramakrishnan et al. 2019). Thus, a key to understanding thehalo assembly bias is to figure out what aspect of the anisotropic tidal fields affects the halogrowths.The cosmic web is further classified into four different types each of which has adistinct geometrical shape and dimension: zero dimensional knots, one dimensional fil-aments, two dimensional walls and three dimensional voids (Hahn et al. 2007). Amongthem, the most anisotropic web-type, the filament, turned out to embed the majority ofDM halos (e.g., Ganeshaiah Veena et al. 2019) which were believed to grow via the prefer-ential merging and accretion of satellites along the narrow one-dimensional channels (e.g.,West et al. 1995; Plionis, & Basilakos 2002; Vera-Ciro et al. 2011). A recent numerical workof Borzyszkowski et al. (2017) based on a high-resolution N-body simulation, however, re-vealed that the motions of satellites confined in the filamentary environments could haveopposite effects on the growths of galactic halos, depending on the filament thickness (seealso Gonz´alez, & Padilla 2016). If multiple fine filaments cross one another at some nodes,the radial motions of the satellites along the filaments facilitate their infalls onto the galac-tic halos located at the nodes, enhancing the growths of the hosts. Whereas, in the bulkyfilaments thicker than the sizes of the constituent galactic halos, the satellites preferentiallymove in the tangential directions orthogonal to the filament axes, which lead to the deter-rence of the satellite infalls and the retarded growths of their hosts. Quantifying the filamentthickness in terms of the ellipticity of the surrounding large-scale structure and incorporat- 3 –ing it into the conditions for the halo formation, Borzyszkowski et al. (2017) proposed a newextension of the excursion set theory which could accommodate the opposite effects of thelarge-scale tidal environments on the growths of the galactic halos (see also Garaldi et al.2017).Motivated by the insightful work of Borzyszkowski et al. (2017), several attempts weremade to improve their model by incorporating more realistic conditions from the halo growthsor by extending the model to the larger scales or to the other web types (Lazeyras et al. 2017;Musso et al. 2018; Lee 2019). For instance, Lee (2019) introduced a new concept of the ”tidalcoherence” for a quantitative explicit description of the filament thickness, suggesting thatbulky thick (multiple fine) filaments should be outcomes of the highly coherent ( incoherent )tides defined as the strong (weak) alignments between the first eigenvectors corresponding tothe larges eigenvalues of the tidal fields smoothed on two widely separated scales. With thenumerical analysis on the cluster scales, Lee (2019) indeed found that the radial (tangential)motions of the infall-zone satellites around host clusters are obstructed (facilitated) by thehighly coherent tides, which implies that the halo growth sensitively depends on the degreeof the tidal coherence.Yet, the prime focus of Lee (2019) was the future evolution of the cluster halos ratherthan their past evolutions, dealing with the infall-zone satellites which have yet to fall into thehalos. It is necessary to treat the real satellites for the investigation of the effect of the tidalcoherence on the past growths of the DM halos. Besides, the original definition of the tidalcoherence in terms only of the first eigenvector direction may neglect the possibilities thatthe coherence in the second and third eigenvector directions corresponding to the secondlargest and smallest eigenvalues are not evinced by the coherence in the first eigenvectordirection and that the simultaneous coherence of the tides in multiple eigenvector directionsmay have different effects on the halo growths.In this Paper, we attempt to incorporate the multi-dimensional aspect of the tidalcoherence into the idea of Lee (2019) and to explore how it affects the halo growths bymeasuring a correlation between the mass-loss evolution of the halo satellites and the multi-dimensional tidal coherence. In Section 2.1 the definition of the multi-dimensional tidalcoherence as well as the description of the numerical data sets utilized for this analysis arepresented. In Sections 2.2-2.4, the effects of the simultaneous coherence of the tides alongone, two and three eigenvector directions on the subhalo mass-loss evolutions are presented.In Section 3 the final results are summarized and its implication on the halo assembly biasis discussed. Throughout this analysis, we will assume a concordance cosmology with initialconditions prescribed by the Planck result (Planck Collaboration et al. 2014). 4 –
2. Dependence of the Satellite Mass-Loss on the Tidal Coherence2.1. Tidal Coherence as a Multi-Component Array
For this analysis, we utilize the catalog of the Rockstar halos (Behroozi et al. 2013)and density field at z = 0 retrieved from the website of the Small MultiDark Plancksimulation (SMDPL, Klypin et al. 2016), a DM-only N-body simulation performed on aperiodic box of linear size 400 h − Mpc, containing 3840 DM particles of individual mass m p = 9 . × h − M ⊙ for the Planck cosmology (Planck Collaboration et al. 2014). Thecatalog contains both of the distinct halos and the subhalos, which can be distinguishedby their parent ID (pId): The former has pId= − − h − M ⊙ , we identify their subhalos whose pId’s match their ID’s.For each subhalo belonging to each host, we determine the ratio, ξ m ≡ M vir /M acc , of itsvirial mass, M vir , to its accretion mass, M acc , defined as the subhalo mass at the momentof its accretion to its host. The majority of the subhalos are to lose their masses aftertheir infalls via various processes like the tidal stripping/heating and dynamical frictions(van den Bosch et al. 2005), for which cases we expect ξ m <
1. The lower value of ξ m below unity indicates that the given subhalo must have experienced the severe mass-lossprocesses for longer time after the infall. Yet, in some rare occasions, the subhalos can gainmasses through merging inside the hosts for which case ξ m can exceed unity. From here on,two terms, subhalos and satellites , will be interchangeablly used to refer to the non-distinctRockstar halos gravitationally bound to some larger distinct halos.As done in Lee (2019), we compute the tidal field, T ij ( x ), from the density field definedon the 512 grid points, ρ ( x ), by taking the following steps: (i) Calculating the densitycontrast field as δ ( x ) ≡ ( ρ ( x ) − ¯ ρ ) / ¯ ρ where ¯ ρ is the mean density averaged over the gridpoints. (ii) Performing the Fourier transformation of δ ( x ) into ˜ δ ( k ). (iii) Smoothing thedensity field in the Fourier space with a Gaussian filter on the scale of R f = 30 h − Mpcas ˜ δ s ( k ) ≡ ˜ δ ( k ) exp( − k R f / T ij ≡ k i k j ˜ δ s ( k ) /k . (v) Performing the inverse Fourier transformation of ˜ T ij ( k ) into T ij ( x ).At the grid point, x h , where each of the selected hosts is located, we diagonalize T ij ( x h )to find a set of three eigenvalues { λ i } i =1 (with a decreasing order) and the correspondingeigenvectors { e i } i =1 . Then, we repeat the whole process but with a smaller filtering scale of R ′ f = 2 h − Mpc to obtain a new set of { λ ′ i } i =1 and { e ′ i } i =1 . q , was originally defined as q ≡ | e · e ′ | (Lee 2019). In the current work, we redefine q as a multi-component array as q i = | e i · e ′ i | for each i ∈ { , , } . (1)If q i is equal to or higher than 0 . .
2) at a given region, the tides is said tobe highly coherent ( incoherent ) along the i th eigenvector direction at the region. A criticalquestion to which we would like to find an answer in the following Subsections is whether ornot the subhalos located in the regions where the tides are highly coherent or incoherent indifferent eigenvector directions exhibit different mass-loss evolutions. In this Subsection, we are going to study how the mean value of the subhalo virial-to-accretion mass ratios depends on each of the three components of the tidal coherence, { q i } i =1 , calling it one-dimensional (1D) dependence of the subhalo mass-loss evolution onthe tidal coherence. We first divide the sample of the selected host halos into two subsamples:One contains those hosts surrounded by the tides highly coherent along the first eigenvectordirection, satisfying the condition of q ≥ .
9. The other consists of those surrounded by thetides not so strongly coherent along the first eigenvector direction with q < .
9. Table 1 liststhe mean masses ( h M h i ) and numbers ( N h ) of the hosts contained in each subsample. As canbe seen, although the latter subsample (i.e., q < .
9) contains three times larger number ofhosts, no significant difference in h M h i between the two subsamples is noted, which assuresthat if the values of h ξ m i from the two subsamples are significantly different from each other,then it should not be ascribed to the mass difference.For each host contained in each subsample, we select only those subhalos which expe-rienced the mass-loss process, i.e., ξ m <
1, excluding those few subhalos which experiencedthe mass-gain process, ξ m ≥
1. Then, we calculate the mean virial-to-accretion mass ra-tio, h ξ m i , averaged over the selected subhalos of the hosts contained in each subsample.The errors, σ ξ m , in the measurement of h ξ m i , is calculated as its standard deviation as σ ξ m ≡ [ h ( ξ m − h ξ m i ) i / ( N sub − / where N sub is the total number of the subhalos of thehosts contained in each subsample.Figure 1 plots the values of h ξ m i from the two subsamples with q ≥ . q < . σ ξ m in its left panel,explicitly demonstrating that the former yields a significantly higher value of h ξ m i thanthe latter. This trend implies that the satellites located in the regions surrounded by thetides highly coherent along the first eigenvector direction experience less severe mass-loss 6 –evolution after their infalls onto their hosts than the other counterparts with q < .
9. Basedon the insights from Lee (2019), we put forth the following explanation to understand thisphenomenon: As the satellites surrounded by highly coherent tides along the first eigenvectordirection develop velocities in the tangential direction, which deter their infalls onto the hosts,reducing the amount of time during which the subhalos are exposed to the effects of the tidalstripping/heating or dynamical fraction inside their hosts.Repeating the above procedure but with the subsamples obtained by contraining thevalue of q ( q ) instead of q with the same threshold of 0 .
9, we also investigate how h ξ m i differs between the cases of q ≥ . q < . q ≥ . q < . q ( q ). As can be seen, thesubhalos of the hosts located in the regions with q ≥ . q ≥ .
9) yield a larger value of h ξ m i than those with q < . q < . h ξ m i between the twosubsamples are found for the case that the threshold condition is imposed on the value of q ( q ) rather than on the value of q .To see whether or not this difference in h ξ m i witnessed in Figure 1 is a secondaryeffect induced by any differences in the local density ( δ ) or ellipticity ( e ) between the twosubsamples, we determine the values δ ,and e at the grid point of each host. The three tidaleigenvalues, { λ ′ i } i =1 on the scale of 2 h − Mpc obtained in Subsection 2.1 is used to calculate δ and e : δ = P i =1 λ ′ i , and e ≡ [(1 + δ ) − P i 9, no significantdifferences are found in h δ i and h e i between the two subsamples. Whereas, the subsamplewith q ≥ . h δ i and h e i than the othersubsample with q < . 9. That is, the regions surrounded by the tides highly coherent alongthe first eigenvectors tend to be more overdense and more anisotropic due to the simultaneouscompression of matter along the coherent first eigenvector direction. This result brings out asuspicion that the higher value of h ξ m i found in the subsample with q ≥ . h δ i and h e i .Now that the tides highly coherent along the eigenvector direction are found to have anobstruction effect on the satellite infalls, the next quest is to investigate whether the tideshighly incoherent along any eigenvector direction have the opposite effect or not. For thisquest, we use two thresholds: an upper-bound threshold of 0 . . q i ≥ . q i < . i ∈ { , , } ) andthen conduct the same analysis. Figures 3-4 plot the same as Figures 1-2, respectively, butwith the conditions of q i ≥ . q i < . q i ≥ . q i < . 9. The leftpanel of Figure 3 reveals that the difference in h ξ m i between the two subsamples obtainedby putting two thresholds of 0 . . q is larger than that by puttingone threshold of 0 . 9. This result indicates that the tides highly incoherent along the firsteigenvector direction indeed have the opposite effect on the satellite infalls: it facilitates thesatellite infalls onto the hosts, leading them to undergo the more severe mass-loss evolutionafter the infalls. Meanwhile, the left panel of Figure 4 shows that the difference in h δ i and h e i between the two subsamples obtained by putting two thresholds of 0 . . q is smaller than that by using one threshold of 0 . 9, which proves that the largervalues of h δ i and h e i are not mainly responsible for the more severe mass-loss evolution ofthe subhalos found from the subsample with q ≥ . incoherent along the third eigenvector direction does not have the expected opposite effect,compared to that coherent along the same direction. The difference in h ξ m i between thesubsamples obtained by putting two thresholds of 0 . . q is smaller than thatbetween the subsamples obtained by putting one threshold of 0 . q . This result indicatesthat the tides highly incoherent along the third eigenvector direction have an obstructingeffect on the satellite infalls rather than facilitating it unlike the tides highly incoherent alongthe first eigenvector direction. This phenomenon may be closely linked with the larger meanellipticity, h e i , found in the subsample with q < . q < . incoherent along the third eigenvectordirection can increase the tidal anisotropy of a region, which in turn makes it harder forthe satellites in the region to fall onto their hosts. As the satellite infalls are deterred, theymust go through less severe mass-loss evolution after the infalls till the present epochs. Notealso in the middle panels of Figures 1-4 that the tidal coherence measured along the secondeigenvector direction have the weakest effect on the subhalo mass-loss evolution, showing nosignificant differences in h ξ m i , h δ i , and h e i among three samples with q ≥ . q < . q < . Now that the surrounding tides coherent along different eigenvector directions are foundto have different effects on the mass-loss evolution of the subhalos, we would like to explorethe effects of the tides coherent simultaneously along two eigenvector directions. Since the 8 –tidal coherence measured along the second eigenvector direction is found to have the weak-est effect on the subhalo mass-loss evolution in Subsection 2.2, we will focus on the tidalcoherence measured simultaneously along the first and third eigenvector directions (i.e., q and q ) in this Subsection.We first separate the selected host halos into four subsamples by simultaneously con-straining the values of q and q with a single threshold of 0 . h ξ m i , σ ξ m ), ( h δ i , σ δ ) and ( h e i , σ e ) by taking the same steps described in Subsec-tion 2.2 for each of the four subsamples, the results of which are displayed in Figures 5-6. Ascan be seen in Figure 5, the two subsamples satisfying the conditions of ( q ≥ . , q < . q < . , q ≥ . 9) yield significantly higher values of h ξ m i than the other two sub-samples. A crucial implication of this result is that the tides highly coherent along the first(third) eigenvector direction but not along the third (first) eigenvector directions have astronger obstructing effect on the satellite infalls than the tides highly coherent along bothof the first and third eigenvector directions.It is interesting to see that while the two subsamples with of ( q ≥ . , q < . 9) and( q < . , q ≥ . 9) show no significant difference in the values of h ξ m i and h δ i from eachother, a substantial difference in the value of h e i is found between them (see Figure 6): theregions surrounded by the tides highly coherent along the first eigenvector direction butnot along the third ones are more anisotropic than those surrounded by the tides highlycoherent along the third eigenvector direction but not along the first ones. Given that thetidal anisotropy can also have an effect of obstructing the satellite infalls, the larger valueof h ξ m i found from the subsample with q ≥ . q < . h e i than that from the subsample with q < . q ≥ . 9. The lowestvalue of h ξ m i is found from the subsample with q < . q < . 9, which indicates thatthe tides highly coherent along none of the first nor third eigenvector directions have theweakest obstructing and/or strongest facilitating effects of the satellite infalls.We also investigate the effect of the highly incoherent tides on the subhalo mass-lossevolution and on the local density and ellipticity as well by constraining the value of q and q with double thresholds of 0 . . 2, the results of which are shown in Figures 7-8. Ascan be seen in Figure 7, the subsample with q ≥ . q < . h ξ m i among the four, while its lowest value is found in the subsample with q < . q ≥ . 9. This result indicates that the tides highly coherent along the first eigenvectordirection and highly incoherent along the third eigenvector direction are most effective inobstructing the satellite infalls, while the tides highly coherent along the third eigenvectordirection and incoherent along the first eigenvector direction are most effective in facilitatingthe infalls among the four. Given that the subsample with ( q ≥ . , q < . 2) yields 9 –the highest value of h e i among the four, the largest value of h ξ m i from the subsample with q ≥ . q < . h e i .It is worth recalling that in Subsection 2.2 the tides highly coherent only along the thirdeigenvector direction have been already found to obstruct the satellite infalls rather than fa-cilitate them (see the right panel of Figure 1). Nevertheless, if the tides are simultaneouslyincoherent along the first eigenvector direction, then the facilitating effect of the tidal in-coherence along the first eigenvector direction seem to overwhelm the obstructing effect ofthe tidal coherence along the third eigenvector direction, according to the result shown inFigure 3. In other words, it is the tidal incoherence along the first eigenvector direction thatplays the most decisive dominant role of facilitating the satellite infalls, driving the largestamount of mass-loss of the subhalos in the post-infall stages.Meanwhile, the high coherence of the tides along the first eigenvector direction seemsto be synergetic with its simultaneous incoherence along the third eigenvector direction (seeFigure 7). The subsample with ( q ≥ . , q < . 2) yields the lowest value of h ξ m i notonly among the subsamples obtained by simultaneously constraining both of q and q butalso among the subsamples obtained by constraining only one of three components of { q i } (see Figure 3). Our interpretation is that the high tidal anisotropy associated with the tideshighly incoherent along the third eigenvector direction tends to magnify the obstructingeffect of the high tidal coherence along the first eigenvector direction. Now that the simultaneous constraints of q and q uncovers the complex two-dimensionaldependence of the subhalo mass-loss evolution on the tidal coherence, it should be legitimateto investigate how h ξ m i depends on all of the three components of { q i } i =1 , calling it threedimensional (3D) dependence of the subhalo mass-loss evolution on the tidal coherence. Wefirst separate the host halos into eight subsamples by constraining simultaneously the valuesof ( q , q , q ) with a single threshold of 0 . h ξ m i , σ ξ m ), ( h δ i , σ δ ) and ( h e i , σ e ), for each of the eight subsamples, which are plotted inFigures 9-10. As can be seen in Figure 9, we find the highest and lowest values of h ξ m i from the subsamples with ( q ≥ . , q ≥ . , q < . 9) and ( q < . , q ≥ . , q ≥ . h δ i but substantial difference in h e i , we find the highest and lowest values of h e i from the same two subsamples, which implies that the large difference in h e i among the 10 –two subsamples should be linked with the large difference in h ξ m i .For the case that q ≥ . q < . 9, the simultaneous constraint of q ≥ . h ξ m i (scarlet bar). Whereas, for the case that q < . q ≥ . 9, thesame constraint of q ≥ . h ξ m i (greenbar). Note also that the subsample with ( q < . , q ≥ . , q < . 9) corresponding tothe tides highly coherent only along the second eigenvector direction but not along the firstand third ones yields relatively low value of h ξ m i . This result indicates that the effect of thehigh tidal coherence along the second eigenvector direction shifts from the obstruction to thefacilitation of the satellite infalls, depending on which eigenvector direction between the firstand third the tides are simultaneously coherent. If the tides are highly coherent along noneof the first and third eigenvector direction, then the high tidal coherence along the secondeigenvector direction does not have a strong effect on the satellite infalls.It is interesting to see that the tides highly coherent along all of the three eigenvectordirections (red bar) are less effective in obstructing the satellite infalls than the tides highlycoherent along the first and second eigenvector directions but not highly coherent alongthe third eigenvector direction (scarlet bar). It is even not so effective in obstructing thesatellite infalls as the tides highly coherent only along the third eigenvector direction but notalong the first and second eigenvector direction (thick violet bar). Note also that the secondhighest value of h ξ m i is found from the subsample with q < . , q < . , q ≥ . incoherence measured along all of three eigenvector directions is linked with the subhalomass-loss evolution, creating seven new subsamples by constraining simultaneously all of thethree components, ( q , q , q ) with double thresholds of 0 . . q ≥ . , q < . , q < . q < . , q ≥ . , q ≥ . 9) and ( q < . , q ≥ . , q < . h ξ m i , σ ξ m ), ( h δ i , σ δ ) and ( h e i , σ e ) obtained from the rest four non-empty subsamples aswell as from the subsample with ( q ≥ . , q ≥ . , q ≥ . 9) are shown in Figures 11-12.The subsample with ( q ≥ . , q < . , q < . 2) yields the highest value of h ξ m i (olive green bar), while the lowest value (violet bar) is found from the subsample with( q < . , q < . , q ≥ . h e i between the two subsamples is not solarge enough to explain their difference in h ξ m i (see Figure 12), the different mean ellipticities 11 –between the two subsamples should not be the main cause of the significant difference in themean virial-to-accretion mass ratios between them. The tides highly coherent along the firsteigenvector direction but highly incoherent along the second and third eigenvector directionsare much more effective in obstructing the satellite infalls than the tides highly coherent alongthe third eigenvector direction but highly incoherent along the first and second eigenvectordirections.The comparison of the result shown in Figures 9 and 11 reveals that the subsamplewith ( q < . , q ≥ . , q ≥ . 9) yield a lower value of h ξ m i than the subsample with( q < . , q < . , q ≥ . incoherent along the first and second eigenvector direction are less effective infacilitating the satellite infalls than the tides highly coherent along the second and thirdeigenvector direction but not so highly coherent along the first eigenvector direction.Another interesting fact revealed by the comparison between the two Figures is thatthe value h ξ m i from the subsample with ( q < . , q ≥ . , q < . 2) is as high as thatfrom the subsample with ( q ≥ . , q ≥ . , q ≥ . incoherent alongthe first and third eigenvector directions are as effective in obstructing the satellite infallsas the tides highly coherent along all of the three eigenvector directions. It is a rathersurprising unexpected result since we have already found in Subsection 2.2 that the tideshighly coherent along the first eigenvector direction have an obstructing effect on the satelliteinfalls and that the tides highly incoherent along the same direction have the opposite effect,i.e., facilitating the satellite infalls. The slightly larger value of h e i from the subsamplewith ( q < . , q ≥ . , q < . 2) than that from the subsample with ( q ≥ . , q ≥ . , q ≥ . 9) should be related to this puzzling phenomenon (see Figures 10-12). The tidal incoherence along the third eigenvector direction tends to increases the tidal anisotropy (i.e.,mean ellipticity) which plays a role in increasing the value of h ξ m i , as shown in Subsection2.2. The obstructing effect of the high tidal anisotropy caused by the tidal incoherence alongthe third eigenvector direction compensates the facilitating effect of the high incoherence ofthe tides along the first eigenvector direction. 3. Summary and Discussion We have systematically studied the dependence of the subhalo mass-loss evolution onthe multi-dimensional aspect of the tidal coherence by using the numerical datasets retrievedfrom the SMPDL (Klypin et al. 2016). For this study, we have quantified the subhalo mass-loss evolution in terms of the mean virial-to-accretion mass ratios averaged over the subahlos, 12 –and expressed the tidal coherence as an array of three numbers, { q i } i =1 , where q i representsthe alignments between the i th eigenvectors of the tidal fields smoothed on two widelyseparated scales of 2 h − Mpc and 30 h − Mpc. To eliminate the well known strong dependenceof the subhalo mass-loss evolution on the masses of their hosts (van den Bosch et al. 2005),we select only those subhalos belonging to the hosts whose masses lie in the narrow range of1 ≤ M h / (10 h − M ⊙ ) ≤ q i ≥ . , ∀ i ∈ { , , } ) tend to have higher mean values of the virial-to-accretion mass ratios than their counterparts ( q i < . incoherent along a different eigenvector direction, however, has turnedout to have a different effect. The tides highly incoherent along the first eigenvector direction( q < . 2) have an effect opposite to the tides coherent along the same direction ( q ≥ . incoherent along the first eigenvector direction have been found to yieldthe lowest mean virial-to-accretion mass ratios. Whereas, the tides highly incoherent alongthe third eigenvector direction ( q < . 2) have an effect of obstructing rather than facilitatingthe satellite infalls, similar to the tides highly coherent along the same direction ( q ≥ . incoherence of the tides along two orthree eigenvector directions have more complex effects on the subhalo mass-loss evolution.The high tidal coherence along the first eigenvector direction has been found to be synergicwith the high tidal incoherence along the minor eigenvector direction ( q ≥ . q < . incoherence along the first eigenvector direction hasturned out to be discordant with both of the high tidal coherence and incoherence along thethird eigenvector direction in facilitating the satellite infalls. The high tidal coherence alongthe second eigenvector direction have turned out to be synergic with the high tidal coherencealong the first eigenvector direction in obstructing the satellite infalls, provided that the tidesare not so coherent along the third eigenvector direction. Meanwhile, provided that the tidesare not so coherent along the first eigenvector direction, the high tidal coherence along the 13 –second eigenvector direction has been found synergic with the high tidal coherence along thethird eigenvector direction in facilitating the satellite infalls.Although the tides highly coherent along one of the three eigenvector direction have anobstructing effect on the satellite infalls, the simultaneous coherence of the tides along all ofthe three eigenvector directions have been found not to reinforce the obstructing effect. Thetides highly coherent along the first eigenvector direction and incoherent along the second andthird eigenvector directions have been found more effective in obstructing the satellite infallsthan the tides simultaneously coherent along all of the three eigenvector directions. Thesame is true for the simultaneous incoherence of the tides along all of the three eigenvectordirections, which have been found not to reinforce the effect of facilitating the satellite infalls.The tides highly coherent along the third eigenvector direction and simultaneously incoherent along the first and second eigenvector direction have been found more effective in facilitatingthe satellite infalls than the tides simultaneously incoherent along all of the three eigenvectordirections.Determining the mean values of the local density contrasts, h δ i , and tidal anisotropies, h e i , averaged over the regions with different tidal coherences, we have found negligible dif-ferences in h δ i and substantial differences in h e i among the regions. Noting that the si-multaneous coherence along all of the three eigenvector directions plays a significant role ofreducing the tidal anisotropy, and recalling that the high tidal anisotropy has been knownto obstruct the satellite infalls (e.g., Borzyszkowski et al. 2017), we have explained that thehigher tidal anisotropy should be contributed to the stronger obstructing effect of the tideshighly coherent along the first eigenvector direction but highly incoherent along the secondand third eigenvector directions than the tides highly coherent along all of the three eigenvec-tor directions. Yet, we have also shown that the multi-dimensional tidal coherence have anindependent net effect on the subhalo-mass loss evolution, which cannot be ascribed simplyto the differences in the tidal anisotropy.Given that the mean virial-to-accretion mass ratios of the subhalos reflect not onlytheir mass-loss evolutions but also how fast their host clusters have grown as well as in whatdynamical states they are (van den Bosch et al. 2005), the bottom line of our work is asfollows: The formation and evolution of the cluster halos at fixed mass scales located in theenvironments with similar densities and tidal anisotropies still show variations with the multi-dimensional effects of the tidal coherence. We suspect that this result may be responsible forthe large scatters around the spherical critical density contrast of δ c ≡ . 68 required for theformation of a cluster halo, which could not be entirely explained by the scale-dependence ofthe non-spherical counter-part, δ ec (e.g., Maggiore, & Riotto 2010; Corasaniti, & Achitouv2011). Our result may be also closely related to the elusive nature of the large-scale as- 14 –sembly bias, whose existence have so far gained no observational confirmations (e.g., seeSunayama, & More 2019). It is not only the density and tidal strengths but also the multi-dimensional tidal coherence that we must take into account to detect the large-scale assemblybias. We plan to work on finding a direct link between the tidal coherence and the large-scale assembly bias as well as on extending the excursion set model by incorporating thetidal coherence, hoping to report the results elsewhere in the near future.I acknowledge the support of the Basic Science Research Program through the Na-tional Research Foundation (NRF) of Korea funded by the Ministry of Education (NO.2016R1D1A1A09918491). I was also partially supported by a research grant from the NRFof Korea to the Center for Galaxy Evolution Research (No.2017R1A5A1070354). 15 – REFERENCES Bardeen, J. M., Bond, J. R., Kaiser, N., et al. 1986, ApJ, 304, 15Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109Bond, J. R., Cole, S., Efstathiou, G., et al. 1991, ApJ, 379, 440Bond, J. R., & Myers, S. T. 1996, ApJS, 103, 1Bond, J. R., Kofman, L., & Pogosyan, D. 1996, Nature, 380, 603Borzyszkowski, M., Porciani, C., Romano-D´ıaz, E., & Garaldi, E. 2017, MNRAS, 469, 594Corasaniti, P. 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V., Helmi, A., et al. 2011, MNRAS, 416, 1377Yang, X., Zhang, Y., Lu, T., et al. 2017, ApJ, 848, 60 This preprint was prepared with the AAS L A TEX macros v5.2. 17 –Fig. 1.— Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts sur-rounded by the tides highly coherent along one of three eigenvector directions as red bar(major, intermediate and third eigenvector directions in the left, middle and right panels,respectively.) In each panel, the complement case of the tides not so highly coherent alongthe same direction is plotted as blue bar. 18 –Fig. 2.— Mean values of the density contrast and ellipticity averaged over the regionssurrounded by the tides highly coherent along one of three eigenvector directions as red barin the top and bottom panels, respectively. In each panel, the blue bar correspond to thecomplement case. 19 –Fig. 3.— Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts sur-rounded by the tides highly coherent ( incoherent ) along one of three eigenvector directionsas red (blue) bar. 20 –Fig. 4.— Mean values of the density contrast and ellipticity averaged over the regionssurrounded by the tides highly coherent ( incoherent ) along one of three eigenvector directionsas red (blue) bar in the top and left panels, respectively. 21 –Fig. 5.— Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts sur-rounded by the tides highly coherent along both of the first and third (thick red bar), alongthe first but not along the third (thick ocher bar), highly coherent along the third butnot along the first (thick green bar), and highly coherent along none of the first and thirdeigenvector directions (thick blue bar). 22 –Fig. 6.— Mean values of the density contrast and ellipticity averaged over the regionssurrounded by the tides for the four different cases described in the caption of Figure 5 inthe top and bottom panels, respectively. 23 –Fig. 7.— Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts sur-rounded by the tides highly coherent along both of the first and third (red bar), highlycoherent along the first but highly incoherent along the third (ocher bar), highly coherentalong the third but highly incoherent along the first (green bar), and highly incoherent alongthe first and third eigenvector directions (blue bar). 24 –Fig. 8.— Mean values of the density contrast and ellipticity averaged over the regionssurrounded by the tides for the four different cases described in the caption of Figure 7 inthe top and bottom panels, respectively. 25 –Fig. 9.— Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts sur-rounded by the tides highly coherent along all of the three eigenvector directions are plottedas carmine bar. The seven complement cases corresponding to the tides coherent along notall of the three eigenvector directions are plotted as different color bars. 26 –Fig. 10.— Mean values of the density contrast and ellipticity averaged over the regionssurrounded by the tides for the eight different cases described in the caption of Figure 9 inthe top and bottom panels, respectively. 27 –Fig. 11.— Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts sur-rounded by the tides highly incoherent along all of the three eigenvector directions are plottedas black bar. For comparison, the case of the tides coherent along all of the three directionsare also plotted as carmine bar. The results from the three cases of the tides highly in-coherent along two of the three eigenvector directions but highly coherent along the otherdirections are plotted as different color bars. 28 –Fig. 12.— Mean values of the density contrast and ellipticity averaged over the regionssurrounded by the tides for the five different cases described in the caption of Figure 11 inthe top and bottom panels, respectively. 29 –Table 1. 1D Tidal Coherence, Mean Mass and Number of the Hosts.condition h M h i N h (10 h − M ⊙ ) q ≥ . . ± . 03 306 q < . . ± . 02 1096 q < . . ± . 01 179 q ≥ . . ± . 04 174 q < . . ± . 01 1228 q < . . ± . 01 275 q ≥ . . ± . 03 254 q < . . ± . 02 1148 q < . . ± . 01 203 30 –Table 2. 2D Tidal Coherence, Mean Mass and Number of the Hosts.condition h M h i N h (10 h − M ⊙ ) q ≥ . , q ≥ . . ± . 06 92 q ≥ . , q < . . ± . 04 214 q < . , q ≥ . . ± . 04 162 q < . , q < . . ± . 02 934 q ≥ . , q < . . ± . 08 26 q < . , q ≥ . . ± . 19 9 q < . , q < . . ± . 06 78 31 –Table 3. 3D Tidal Coherence, Mean Mass and Number of the Hosts.condition h M h i N h (10 h − M ⊙ ) q ≥ . , q ≥ . , q ≥ . . ± . 06 82 q ≥ . , q ≥ . , q < . . ± . 17 11 q ≥ . , q < . , q ≥ . . ± . 19 10 q ≥ . , q < . , q < . . ± . 04 203 q < . , q ≥ . , q ≥ . . ± . 16 10 q < . , q ≥ . , q < . . ± . 06 71 q < . , q < . , q ≥ . . ± . 04 152 q < . , q < . , q < . . ± . 02 863 q ≥ . , q < . , q < . . ± . 09 22 q < . , q ≥ . , q < . . ± . 19 9 q < . , q < . , q ≥ . . ± . 12 19 q < . , q < . , q < . . ± ..