Revisiting the Galaxy Shape and Spin Alignments with the Large-Scale Tidal Field: An Effective Practical Model
aa r X i v : . [ a s t r o - ph . C O ] J a n Revisiting the Galaxy Shape and Spin Alignments with theLarge-Scale Tidal Field: An Effective Practical Model
Jounghun Lee
Astronomy Program, Department of Physics and Astronomy, Seoul National University,Seoul 08826, Republic of Korea [email protected]
ABSTRACT
An effective practical model with two characteristic parameters is presentedto describe both of the tidally induced shape and spin alignments of the galactichalos with the large-scale tidal fields. We test this model against the numericalresults obtained from the Small MultiDark Planck simulation on the galactic massscale of 0 . ≤ M/ (10 h − M ⊙ ) ≤
50 at redshift z = 0. Determining empiricallythe parameters from the numerical data, we demonstrate how successfully ourmodel describes simultaneously and consistently the amplitudes and behaviorsof the probability density functions of three coordinates of the shape and spinvectors in the principal frame of the large scale tidal field. Dividing the samplesof the galactic halos into multiple subsamples in four different mass ranges andfour different types of the cosmic web, and also varying the smoothing scale ofthe tidal field from 5 h − Mpc to 10 , , h − Mpc, we perform repeatedly thenumerical tests with each subsample at each scale. Our model is found to matchwell the numerical results for all of the cases of the mass range, smoothing scaleand web type and to properly capture the scale and web dependence of the spinflip phenomenon.
Subject headings: cosmology:theory — large-scale structure of universe
1. Introduction
The physical properties of the observed galaxies in the universe is a reservoir of informa-tion on the conditions under which they formed, the evolutionary processes which they wentthrough, and the interactions in which they are involved. Although the local conditions andprocesses at the galactic scales must have had the most dominant impact on the galaxies, 2 –the non-local effects beyond the galactic scales are also believed to have contributed partlyto their physical properties (e.g., Pandey & Sarkar 2017). Subdominant as its contributionis, the non-local effects on the galaxies are worth investigating, since it may contain valuableindependent information on the galaxy formation and the background cosmology as well.The non-local effects on the galaxies are manifested by the correlations between thegalaxy properties and the large-scale environments. Among various properties of the galax-ies that have been found correlated with the large-scale environments, the shape and spinalignments of the galaxies with the large-scale structures (collectively called the galaxyintrinsic alignments) have lately drawn considerable attentions, inspiring vigorous exten-sive studies (see Joachimi et al. 2015; Kiessling et al. 2015; Kirk et al. 2015, for recent re-views). It is partially because the galaxy intrinsic alignments, if present and significant,could become another systematics in the measurements of the extrinsic counterparts causedby the weak gravitational lensing (see Troxel & Ishak 2015, and references therein). Theother important motivation for the recent flurry of research on this topic is that the ori-gin of the galaxy intrinsic alignments is amenable to the first order perturbation theoryand thus a rather fundamental approach to this topic is feasible (e.g., Heavens et al. 2000;Lee & Pen 2000; Catelan et al. 2001; Crittenden et al. 2001; Lee & Pen 2001; Porciani et al.2002; Hui & Zhang 2008; Blazek et al. 2011, 2015; Tugendhat & Sch¨afer 2018).In the first order Lagrangian perturbation theory (Zel’dovich 1970; Buchert 1992), theminor (major) eigenvectors of the inertia momentum tensors of the proto-galactic regions areperfectly aligned with the major (minor) eigenvectors of the local tidal tensors around theregions. Several N -body simulations have indeed detected the existence of strong correlationsbetween the inertia momentum and local tidal tensors at the proto-galactic sites (Lee & Pen2000; Porciani et al. 2002; Lee et al. 2009). Since the tidal fields smoothed on differentscales are cross correlated, the eigenvectors of the inertia momentum tensors of the proto-galactic regions are expected to be aligned with those of the large-scale tidal fields. Themajor eigenvectors of the inertia momentum tensors of the proto-galaxies correspond to themost elongated axes of their shapes, while the minor eigenvectors of the large-scale tidaltensors correspond to the directions along which the surrounding matter become minimallycompressed. Henceforth, this expectation based on the first order Lagrangian perturbationtheory basically translates into the possible alignments between the galaxy shapes and themost elongated axes of the large-scale structures such as the axes of the filaments, the signalsof which have been detected by several numerical and observational studies (e.g., Altay et al.2006; Hahn et al. 2007b; Zhang, Yang & Faltenbacher 2009; Zhang et al. 2013; Chen et al.2016, and references therein).In the linear tidal torque (LTT) theory that Doroshkevich (1970) formulated by com- 3 –bining the first order Lagrangian perturbation theory with the Zel’dovich approximation(Zel’dovich 1970), the anisotropic tidal field of the surrounding matter distribution origi-nates the spin angular momentum of a proto-galaxy provided that its shape departs from aspherical symmetry. The generic and unique prediction of this LTT theory is the inclinationsof the spin vectors of the proto-galaxies toward the intermediate eigenvectors of the large scaletidal field (Lee & Pen 2000), which has also garnered several numerical and observationalsupports (e.g., Navarro et al. 2004; Trujillo et al. 2006; Hahn et al. 2007a; Lee & Erdogdu2007; Wang et al. 2011; Zhang et al. 2015; Chen et al. 2016).The recently available large high-resolution N -body simulations that covered a broadmass range, however, limited the validity of the LTT prediction to the mass scale of M ≥ M t ∼ h − M ⊙ , showing that on the mass scale below M t the spin vectors of dark matterhalos at z = 0 are aligned not with the intermediate but rather with the minor eigenvectorsof the large scale tidal field, similar to the axes of the halo shapes (Arag´on-Calvo et al.2007; Hahn et al. 2007b; Paz et al. 2008; Zhang, Yang & Faltenbacher 2009; Codis et al.2012; Libeskind et al. 2013; Trowland et al. 2013; Dubois et al. 2014; Veena et al. 2018).This difference in the spin alignment tendency between the low and high mass scales werefound most conspicuous in the filament environments: the spin axes of the galactic halos withmasses lower (higher) than M t measured at z = 0 tend to be parallel (perpendicular) to theelongated axes of their host filaments, in contradiction with the LTT prediction. The transi-tion of the spin alignment tendency at M t is often called ”spin flip” phenomenon (Codis et al.2012) and the break-down of the LTT prediction below M t has also been witnessed in recentobservations (Tempel et al. 2013; Tempel & Libeskind 2013; Hirv et al. 2017; Chen et al.2018).The detection of this spin-flip phenomenon puzzled the community and urged it to finda proper answer to the critical question of what the origin of this phenomenon is. Whathas so far been suggested as a possible origin includes the major merging events, mass de-pendence of the merging and accretion processes, assembly bias, vorticity generation insidefilaments, web-dependence of the galaxy formation epochs, nonlinear tidal interactions, geo-metrical properties of the host filaments and etc (Bett & Frenk 2012; Lacerna & Padilla 2012;Codis et al. 2012; Libeskind et al. 2013; Welker et al. 2014; Codis et al. 2015; Laigle et al.2015; Bett & Frenk 2016; Wang & Kang 2017; Veena et al. 2018). Although these previ-ously suggested factors were believed to play some roles for the occurrence of the spin-flipphenomenon, none of them are fully satisfactory in explaining all aspects of the spin-flipphenomenon including the dependence of the transition mass scale M t on the types of thecosmic web, redshifts, and scales of the filaments.The occurrence of the spin-flip phenomenon basically implies that for the case of the 4 –galaxies with masses M ≤ M t , the tendency of the spin alignments with the large scale tidalfield becomes similar to that of the shape alignments. Thus, it is suspected that whatevercaused the spin-flip phenomenon, it should be linked to the shape alignments with the largescale tidal field. To address these remaining issues, what is highly desired is an effectivemodel that can describe consistently and simultaneously both of the galaxy shape and spinalignments. Here, we attempt to construct such a model by modifying the original LTTtheory and to explore if the shapes of the galaxies also show any transition of the alignmenttendency like the spin counterpartsThe organization of this Paper is as follows. A refined analytic model for the galaxyshape alignments is presented in Section 2.1 and tested against the numerical results inSection 2.2. An effective model for the tidally induced spin alignments is presented inSection 3.1 and tested against the numerical results in Section 3.2. A discussion over thepossible application of this model as well as a summary of the results is presented in Section4. Throughout this Paper, we will assume a Planck universe whose total energy density isdominantly contributed by the cosmological constant (Λ) and the cold dark matter (CDM)(Planck Collaboration et al. 2014).
2. Tidally Induced Shape Alignments2.1. An Analytic Model
Suppose a galactic halo located in a region where a tidal tensor T has its major, inter-mediate and minor eigenvectors (ˆ u , ˆ u and ˆ u , respectively), corresponding to the largest,second to the largest, smallest eigenvalues ( λ , λ and λ , respectively). The tidal tensor T depends on the smoothing scale, R f , as T ( x ) ∝ ∂ i ∂ j R d x ′ Φ( x ′ ) W ( | x − x ′ | ; R f ), where Φ( x )is the perturbation potential field and W ( | x − x ′ | ; R f ) is a window function with a filteringradius R f . In the current analysis, we adopt a Gaussian window function.As mentioned in Section 1, the first order Lagrangian perturbation theory (Zel’dovich1970; Buchert 1992) predicts a strong anti-correlation between the principal axes of theinertia momentum tensor of a galactic halo and the local tidal tensor in the Lagrangianregime. According to this theory, the correlation between the two tensors is strongest if thetwo tensors are defined on the same scale (i.e., the virial radius of the halo, R g ), becomesweaker if R f is larger than R g . If the shape of this galactic halo can be approximatedby an ellipsoid, then the direction of the coordinate vector of the largest shape ellipsoid, e = ( e , e , e ) (i.e., the major principal axis of the inertia momentum tensor) is expected tobe aligned with ˆ u (i.e., the minor principal axis of the local tidal tensor) along which the 5 –surrounding matter is least compressed, provided that λ = λ = λ .This alignment tendency can be statistically quantified by the conditional joint proba-bility density function of three coordinates of the largest shape ellipsoid axis, p ( e , e , e | ˆ T ),where ˆ T is a unit traceless tidal tensor defined as ˆ T ≡ ( T − Tr( T ) / / | T − Tr( T ) / | withTr( T ) denoting the trace of T . As Lee & Pen (2001) and Lee (2004) did, we assume herethat p ( e , e , e | ˆ T ) follows a multivariate Gaussian distribution of p ( e , e , e | ˆ T ) = 1[(2 π ) det( Σ )] / exp (cid:20) − (cid:0) e · Σ − · e (cid:1)(cid:21) , (1)where the components of the covariance matrix, Σ = (Σ ij ), are defined as the conditionalensemble averages, Σ ij ≡ h e i e j | ˆ T i . Here, we suggest the following practical formula for h e i e j | ˆ T i : h e i e j | ˆ T i = 1 + d t δ ij − d t ˆ T ij , (2)where d t is the shape correlation parameter that measures the alignment strength between e and ˆ u . Note that this formula describes a linear dependence of the covariance, h e i e j | ˆ T i ,on T (Catelan et al. 2001; Lee & Pen 2008; Hui & Zhang 2008), unlike a spin vector whosecovariance has a quadratic dependence on T (White 1984; Lee & Pen 2000, 2001).Focusing only on the direction of e , we marginalize p ( e , e , e | T ) over e ≡ | e | to have p (ˆ e | ˆ T ) = Z p ( e | ˆ T ) e de = 14 π det( Σ ) / (cid:0) ˆ e · Σ − · ˆ e (cid:1) − / , (3)where ˆ e ≡ e /e denotes the unit vector in the direction of the largest shape ellipsoid axis.While ˆ T shares the same orthonormal eigenvectors with T , its eigenvalues, ˆ λ , ˆ λ , ˆ λ , aresubject to two additional constraints of P i =1 ˆ λ i = 0 and P i =1 ˆ λ i = 1 (Lee & Pen 2001).Putting Equation (2) into Equation (3) leads to the following analytic expression p (ˆ e | ˆ T ) = = 12 π " Y n =1 (cid:16) d t − d t ˆ λ n (cid:17) − X l =1 | ˆ u l · ˆ e | d t − d t ˆ λ l ! − , (4)since ˆ T ij = ˆ λ i δ ij in the principal axis frame of ˆ T . Now, the conditional probability densityfunction, p ( | ˆ u i · ˆ e | ), for i ∈ { , , } , can be obtained as p ( | ˆ u i · ˆ e | ) = Z π p (ˆ e | ˆ T ) dφ jk , (5) 6 –where φ jk is the azimuthal angle of ˆ e in the plane spanned by ˆ u j and ˆ u k perpendicular toˆ u i . Equation (4) indicates that the completion of this analytic model requires us to deter-mine the value of d t . If ˆ e is perfectly aligned with ˆ u , then d t would be unity. Whereas, thezero value of d t would correspond to the case that ˆ e is completely random having no corre-lation with ˆ u . As done in Lee & Pen (2001), for the determination of d t , we first evaluatethe conditional ensemble average, h ˆ e i ˆ e j | ˆ T i , under the assumption of d t ≪ h ˆ e i ˆ e j | ˆ T i = Z ˆ e i ˆ e j p (ˆ e | ˆ T ) d ˆ e , (6) ≈ (cid:20)(cid:18)
13 + 35 d t (cid:19) − d t ˆ λ i (cid:21) δ ij , (7)Note that the off-diagonal elements vanish in Equation (7) since ˆ T ij = ˆ λ i δ ij in its principalframe. Multiplying Equation (7) by ˆ λ i and summing over the three components, we finallyderive a simple analytic formula for d t : d t = − X i =1 ˆ λ i h ˆ e i | ˆ T i . (8)The constraints of P i =1 ˆ λ i = 0 and P i =1 ˆ λ i = 1 are used to derive the above formula.Equation (8) implies that once the values of ˆ λ i and h ˆ e i | ˆ T i are measured, the shape correlationparameter, d t , can be empirically determined.It is worth recalling that the shape correlation parameter, d t , depends on the smoothingscale, R f . It is expected to have the highest value when R f = R g , as mentioned in the above.In the Eulerian regime, however, the approximation of p ( e | T ) as a multivariate Gaussiandistribution and T as a Gaussian random field used for Equation (3) are not valid on thescale of R g due to the nonlinear evolution of ˆ T on the galactic scale. Thus, we consider thescales R f much larger than R g where these approximations still hold true. Since ˆ T on twodifferent scales of R g and R f are cross-correlated, it is expected that ˆ e is still correlated withˆ T smoothed on the scale of R f ≫ R g . The larger the difference between R f and R g is, thelower the value of d t is. 7 – Our numerical analysis is based on the data set from the Small MultiDark Plancksimulation (SMDPL), a DM only N -body simulation conducted in a periodic box with aside length of 400 h − Mpc (Klypin et al. 2016) as a part of the MultiDark simulation project(Riebe et al. 2013) for a Planck universe (Planck Collaboration et al. 2014). The SMDPLtracks down the gravitational evolution of 3840 DM particles each of which has individualmass of 9 . × h − M ⊙ , starting from z = 120 down to z = 0 (Klypin et al. 2016). Thevirialized DM halos were identified via the Rockstar halo-finding algorithm (Behroozi et al.2013) from the spatial distributions of the DM particles at various snapshots of the SMDPL.Through the CosmoSim database that stores all the experimental results from the Mul-tiDark simulations, we first extract the Rockstar catalog, which provides information on adiverse set of the physical properties of the DM halos. For the current analysis, we use suchinformation as the parent id (pId), comoving position vector ( r ), spatial grid index, virialmass ( M ), coordinate vector of the largest shape ellipsoid axis ( e ) of each Rockstar halo.The integer value of pId is used to exclude the subhalos from our analysis. For the case ofa distinct halo that is not a subhalo hosted by any other larger halo, the parent id has thevalue of pId=-1. The coordinate vector, e , is a measure of the most elongated axis of anellipsoid to which the shape of a given Rockstar halo was fitted. From here on, the unitcoordinate vector of the largest shape ellipsoid axis, ˆ e ≡ e /e , will be called a shape vector .We make a sample of the distinct galactic halos by selecting only those from the Rock-star catalog which meet two conditions of pId= − . ≤ M/ (10 h − M ⊙ ) < . ≤ M/ (10 h − M ⊙ ) < ≤ M/ (10 h − M ⊙ ) < ≤ M/ (10 h − M ⊙ ) <
10 (medium-mass galactic halos) and10 ≤ M/ (10 h − M ⊙ ) <
50 (high-mass galactic halos), respectively. We exclude the subha-los from the analysis, since the effect of the large-scale tidal field on the subhalos are likelyto be negligible compared with that of the internal nonlinear tidal fields inside the host halosThose halos with
M < . × h − M ⊙ are excluded on the ground that the measurementsof the shape and spin vectors of those halos are likely to be contaminated by the shot noisedue to the small number of the component DM particles (Bett et al. 2007). The group andcluster size halos with M ≥ × h − M ⊙ are also excluded since the measurements of theirshapes and spins should be severely affected by their dynamical states, internal structuresand recent merging events. doi:10.17876/cosmosim/smdpl/ ρ ( r ), defined on the 512 grids at z = 0 viathe CosmoSim database. Then, we calculate the dimensionless density contrast field, δ ( r ) =[ ρ ( r ) − h ρ i ] / h ρ i , where the ensemble average, h ρ i , is taken over all the grids. With the help ofthe numerical recipe code that performs the Fast Fourier Transformation (FFT) (Press et al.1992), we compute the Fourier amplitude of the density contrast field, ˜ δ ( k ), where k = k (ˆ k i )is the wave vector in the Fourier space. The inverse FFT of ˜ T ij = ˆ k i ˆ k j ˜ δ ( k ) exp (cid:0) − k R f / (cid:1) for i, j ∈ { , , } leads us to have the tidal field, T ( r ), smoothed on the scale of R f .For each subsample, we take the following steps. First, at the grid where each halo isplaced, we perform a similarity transformation of T ( r ), to find its eigenvectors { ˆ u i } i =1 aswell as the eigenvalues { λ i } i =1 . Second, we calculate, {| ˆ u i · ˆ e |} i =1 , whose values lie in therange of [0 , ,
1] into seven bins with equal length of ∆ = 1 / n h,i , whose values of | ˆ u i · ˆ e | fall in each bin foreach i ∈ { , , } . Third, the probability densities of | ˆ u i · ˆ e | at each bin are determined as p ( | ˆ u i · ˆ e | ) = n h,i / ( N t ∆) where N t is the total number of the galactic halos contained in eachsubsample.Figure 1 plots p ( | ˆ u · ˆ e | ) (left panel), p ( | ˆ u · ˆ e | ) (middle panels) and p ( | ˆ u · ˆ e | ) (rightpanels) as filled circular dots for the cases of the lowest-mass (top panels), low-mass (sec-ond from the top panels), medium mass (second from the bottom panels), and high-mass(bottom panels) galactic halos. To obtain these results, we smooth ˆ T ( r ) on the scale of R f = 5 h − Mpc. As can be seen, for all four subsamples, the shape vector, ˆ e , shows astrong inclination (anti-inclination) toward the minor (major) eigenvector, ˆ u (ˆ u ), whileit shows no alignment with the intermediate eigenvector, ˆ u . Note also that the higher-mass galactic halos exhibit stronger alignment (anti-alignment) tendency between ˆ e andˆ u (ˆ e and ˆ u ), which are consistent with the previously reported numerical and observa-tional results (e.g., Hahn et al. 2007b; Zhang, Yang & Faltenbacher 2009; Joachimi et al.2013; Zhang et al. 2013; Chen et al. 2016; Hilbert et al. 2017; Xia et al. 2017; Piras et al.2018, and references therein).To compare the analytic model presented in Section 2.1 against these numerical results,we first calculate the mean values of ˆ λ i and d t averaged over the galactic halos contained ineach subsample as h ˆ λ i i = 1 N t N t X α =1 ˆ λ α,i = 1 N t N t X α =1 , ˜ λ α,i qP j =1 ˜ λ α,j , i ∈ { , , } , (9)˜ λ α,i = λ α,i − X j =1 λ α,j , i ∈ { , , } , (10) 9 – h d t i = 1 N t N t X α =1 " − X i =1 ˆ λ α,i | ˆ u α,i · ˆ e α | , (11)where { λ α,i } i =1 denotes a set of the three eigenvalues of T at the grid where the α th DM haloof a given subsample is located, and ˆ e α is the shape vector of the α th DM halo. Note that h ˆ e i | ˆ T i in Equation (8) is approximated by | ˆ u α,i · ˆ e α | in Equation (11) since the measuredvalues in numerical realizations are believed to be close to the expectation values in theory.Substituting these mean values of h ˆ λ i i and h d t i for ˆ λ i and d t respectively in Equations(4)-(5), we evaluate the analytical model and plot them as red solid lines in Figure 1. Ascan be seen, for all of the four cases of the halo mass ranges, the analytic model with theempirically determined parameter d t describes very well not only the alignments of ˆ e withˆ u but also simultaneously its anti-alignment with ˆ u and no correlation with ˆ u as well,even though no fitting process is involved. Figure 2 plots h d t i for the four different cases ofthe mass ranges, showing quantitatively how the strength of the shape alignments increaseswith the increment of M .Smoothing ˆ T on three larger scales, R f = 10 ,
20 and 30 h − Mpc, we repeat the wholecalculations, the results of which are shown in Figure 3 for the case of the high-mass galactichalos. The analytic model with the empirically determined parameter d t agrees quite wellwith the numerical results for all of the three cases of R f . Figure 4 shows quantitativelyhow the increment of R f weakens the shape alignments. Although the alignment tendencybecomes weaker as R f increases, the shape vector, ˆ e , still shows significant alignment (anti-alignment) with ˆ u (ˆ u ) even for the case of R f = 30 h − Mpc, which is consistent with thefindings of the previous works (e.g., Xia et al. 2017).True as it is that our analytic model shows good quantitive agreements with the numer-ical results for the case of the high-mass galactic halos, it is not perfect. Some discrepanciesare found in the behaviors of p ( | ˆ u · ˆ e | ) and p ( | ˆ u · ˆ e | ) between the analytic model and thenumerical results, as can be seen in the bottom panel of Figure 1. The former describes aslightly milder increase of p ( | ˆ u · ˆ e | ) with | ˆ u · ˆ e | and a slightly milder decrease of | ˆ u · ˆ e | with | ˆ u i · ˆ e | than the latter especially for the case of the high-mass galactic halos. However, Figure3 shows that the increment of R f improves the agreements between the analytic model andthe numerical results, which in turn implies that the discrepancies may be caused by theuncertainties associated with the approximations of p ( e | T ) as a multivariate Gaussian dis-tribution and T as a Gaussian random field made to derive the analytic model. The largerthe scales are, the more valid these assumptions become. It explains why the analytic modelworks better at R f > h − Mpc. 10 –
Now, we would like to investigate whether or not the strength of the alignments betweenthe shapes of the galactic halos and the tidal eigenvectors depend on the types of the cosmicweb. Following the conventional scheme (Hahn et al. 2007a), we classify the galactic halosof each subsample into the knot, filament, sheet and void halos according to the signs of theeigenvalues of T at the grids where the halos are located: λ > → knot , (12) λ > , λ < → filament , (13) λ > , λ < → sheet , (14) λ < → void . (15)Using only those galactic halos embedded in the same type of the cosmic web, we redothe whole analysis described in Section 2.2. Figures 5-8 show the same as Figures 1 butonly with the knot, filament, sheet and void halos, respectively, showing how the shapealignment depends on the web environment. Figure 9 plots h d t i versus M for the fourdifferent cases of the web type. As can be seen, the value of h d t i increases more sharplywith the increment of M for the cases of the sheet and void halos than for the cases of theknot and filament counterparts, which indicates that the shapes of the galactic halos in therelatively low-density regions tend to be more strongly aligned with those in the relativelyhigh-density regions. Given that the galactic halos located in the knot and filament regionsare expected to have formed earlier and undergone more severe nonlinear evolutions thanthose in the sheet and void regions (Gao & White 2007), the results shown in Figure 2 implythat the nonlinear evolution in denser environments will play a decisive role in diminishingthe strength of the tidally induced shape alignments of the galactic halos.It is interesting to note that the results shown in Figures 5-9 are in direct contradictionwith that of Xia et al. (2017) who found the strongest shape alignments of the halos in theknot environments. We think that this apparent inconsistency between our and their resultsmay be related the difference in the web classification scheme. In their analysis, the types ofthe cosmic web are classified according to the signs of the eigenvalues of the Hessian matrixof the density field. Whereas in our analysis the eigenvalues of the Hessian matrix of thegravitational potential field (i.e, tidal field) are used for the web classification.Figures 10-13 show the same as Figures 3 but with only those high-mass galactic haloslocated in the knot, filament, sheet and void environments, respectively. Figure 14 plots h d t i versus R f for the four different cases of the web type. The decrement of the alignmentstrength with the increment of R f is found for all of the four types of the cosmic web. The void 11 –(knot) galactic halos show the most (least) rapid change of h d t i with R f . The web-dependenceof the rate of the change of h d t i with R f shown in Figure 14 implies that the strength of thetidally induced shape alignments of the galactic halos is determined not only by the differencebetween R g and R f but also by the strength of the cross correlations between the tidal fieldssmoothed on different scales. In the denser knot and filament environments, although thenonlinearity diminishes the strength of the initially induced shape alignments with the large-scale tidal fields, the stronger cross correlations between the tidal fields smoothed on differentscales slow down the rate of the decrement of the strength of the shape alignments with theincrement of R f . Whereas, in the less dense sheet and void regions where the strongestsignals of the shape alignments are found on the scale of R f = 5 h − Mpc, the weaker cross-correlations between the tidal fields on different scales cause the strengths of the shapealignments to decrease quite rapidly as R f increases.Figures 5-14 clearly demonstrate that our analytic model with the empirically deter-mined parameter, Equations (5)-(8), makes a quantitative success in describing simultane-ously and consistently the amplitudes and behaviors of the three probability density func-tions, { p ( | ˆ u i · ˆ e | ) } i =1 , for all of the cases of the galactic mass ranges M , the smoothing scales R f and the types of the cosmic web. This notable success of our analytic model confirmsthe validity of the key assumption made for Equation (2) that the covariances of the shapesof the galactic halos have a linear dependence on the large-scale tidal fields (Catelan et al.2001; Lee & Pen 2008; Hui & Zhang 2008).
3. Tidally Induced Spin Alignments3.1. Analytic Models
Employing the analytic model based on the LTT theory developed by Lee & Pen (2000,2001), Lee (2004) derived the probability density functions of the coordinates of the unitspin vectors, ˆ s , given ˆ T (see also Lee et al. 2018): p ( | ˆ u i · ˆ s | ) = 12 π Z π " Y n =1 (cid:16) c t − c t ˆ λ n (cid:17) − X l =1 | ˆ u l · ˆ s | c t − c t ˆ λ l ! − dφ jk , (16)where c t is the spin correlation parameter in the range of [0 ,
1] (Lee & Pen 2001). The largervalue of c t is translated into the stronger ˆ u -ˆ s alignment. Although Equation (16) is quitesimilar to Equation (5), there is an obvious difference: the former is expressed in terms of ˆ λ i ,while the latter in terms of ˆ λ i . This difference originates from the fact that the covariancesof the spin vectors of the galactic halos have a quadratic dependence on ˆ T according to theLTT theory (Doroshkevich 1970; White 1984). 12 –The core assumption that underlies Equation (16) is that the rescaled covariance, h s i s j | ˆ T i , can be written as (Lee & Pen 2000) h s i s j | ˆ T i = 1 + c t δ ij − c t X k =1 ˆ T ik ˆ T kj . (17)Solving Equation (17) for c t in the principal frame of ˆ T gives (Lee & Pen 2001) c t = 103 − X i =1 ˆ λ i h ˆ s i | ˆ T i . (18)Equation (18) enables us to evaluate the value of c t directly from the values of ˆ s , { ˆ λ i } i =1 and { ˆ u i } i =1 . Several observational and numerical studies showed that this analytic model,Equations (16)-(18), was indeed useful and adequate in describing the tidally induced spinalignments especially in the sheet environments (e.g., Navarro et al. 2004; Trujillo et al. 2006;Lee & Erdogdu 2007; Lee et al. 2018). As mentioned in Section 1, however, the LTT theorybreaks down on the mass scale below M t ∼ h − M ⊙ . The numerical analyses based onrecent large high-resolution N -body simulations found that the spin flip, a transition of thetendency from the ˆ u -ˆ s ( M > M t ) alignments to the ˆ u -ˆ s alignments ( M ≤ M t ) occurs(Arag´on-Calvo et al. 2007; Codis et al. 2012; Veena et al. 2018) and that the value of thetransition mass scale, M t , depends on the type of the cosmic web (Libeskind et al. 2013).Now, we would like to construct a new model that might describe quantitatively thetransition of the spin alignment tendency at M t and its dependence on the type of the cosmicweb. In the light of the previous studies which claimed that the nset of the non-Gaussianityof the tidal fields even on large scales would cause the covariance, h s i s j | ˆ T i , to scale linearlywith ˆ T (Hui & Zhang 2008; Lee & Pen 2008), we first modify Equation (17) into h s i s j | ˆ T i = (1 + c t + d t )3 δ ij − c t X k =1 ˆ T ik ˆ T kj − d t ˆ T ij , (19)where two spin correlation parameters, c t and d t , both lying in the range of [0 , s to ˆ u and to ˆ u , respectively. If the first spin correlation parameter, c t ,is close to zero and the second spin correlation parameter, d t , is close to unity, then the spinvectors ˆ s will show strong alignments with ˆ u just like the shape vectors, ˆ e . If c t is close tounity and d t is close to zero, then it will be reduced to the original model, Equation (16),which describes the ˆ u -ˆ s alignments. If both of the parameters are close to zero, then the In Lee et al. (2018), there was a typo in the formula. It is corrected here.
13 –spin vectors of the galactic halos will be random having no correlations with the large-scaletidal fields.Replacing Equation (17) by Equations (19) in the original derivation of Equation (16), itis straightforward to show that the probability density functions, p ( | ˆ u i · ˆ s | ), can be expressedas p ( | ˆ u i · ˆ s | ) = 12 π Z π " Y n =1 (cid:16) c t − c t ˆ λ n + d t − d t ˆ λ n (cid:17) − × " X l =1 | ˆ u l · s | c t − c t ˆ λ l + d t − d t ˆ λ l ! − dφ jk . (20)Equation (18), which was originally derived in the LTT theory, holds true even when thecovariance, h ˆ s i ˆ s j | ˆ T i , has an additional term, since the second and third terms in Equation(19) are uncorrelated due to h ˆ T ik ˆ T kl ˆ T lj i = 0 (see Appdendix E in Lee & Pen 2001). Thus,the same formula as Equation (18) can be used to obtain the value of c t for this new model.Likewise, the same formula as Equation (8) but with ˆ e replaced by ˆ s can be used to obtainthe value d t as d t = − X i =1 ˆ λ i h ˆ s i | ˆ T i . (21)In Section 3.2, we will numerically test three models for the galaxy spin alignments, model I , model II and model III . The model III is Equation (20) with two non-zeroparameters, c t and d t . The model I is Equation (20) with d t = 0. It is identical to theoriginal model based on the LTT theory, Equation (16). The model II is Equation (20)with c t = 0. It has the same functional form as Equation (5) for the tidally induced shapealignments. To numerically obtain three probability density functions, { p ( | ˆ u i · ˆ s | ) } i =1 , we performthe exactly same calculations as presented in Section 2.2, but with ˆ e replaced by ˆ s . For theevaluation of the three analytic models, we first determine the ensemble values of h c t i and h d t i for each subsample as, h c t i = 1 N t N t X α =1 " − X i =1 ˆ λ α,i | ˆ u α,i · ˆ s α | , (22) 14 – h d t i = 1 N t N t X α =1 " − X i =1 ˆ λ α,i | ˆ u α,i · ˆ s α | , (23)and put these ensemble average values into Equation (20) to evaluate the model III . Putting h c t i ( h d t i ) into Equation (20) and setting h d t i ( h c t i ) at zero, we evaluate the mode I ( modelII ). Figure 15 plots the numerically obtained probability density functions, { p ( | ˆ u i · ˆ s | ) } i =1 (filled dots), and compares them with the model I (blue lines), model II (green lines) and model III (red lines). As can be seen, the three functions, { p ( | ˆ u i · ˆ s | ) } i =1 , have much loweramplitudes than { p ( | ˆ u i · ˆ e | ) } i =1 displayed in Figure 1. It indicates that the spin vectors of thegalactic halos are much less strongly aligned with the large-scale tidal fields than the shapevectors, which is consistent with the results of the previous numerical and observationalstudies (e.g., Hahn et al. 2007b; Forero-Romero et al. 2014; Zhang et al. 2015).The occurrence of the spin-flip phenomenon is indeed witnessed: For the case of thelower mass galactic halos with M < h − M ⊙ , the unit spin vectors, ˆ s , tend to bealigned not with the intermediate eigenvectors, ˆ u , but with the minor eigenvectors, ˆ u ,while the high-mass galactic halos with M ≥ h − M ⊙ , exhibit the stronger alignments ofˆ s with ˆ u rather than with ˆ u , which is quite consistent with the previous numerical results(e.g., Arag´on-Calvo et al. 2007; Hahn et al. 2007b; Codis et al. 2012; Libeskind et al. 2013;Dubois et al. 2014; Chen et al. 2016; Veena et al. 2018).The strength of the ˆ u -ˆ s alignments tends to decrease with M , while the strengths ofthe ˆ u -ˆ s alignments increases with M . These opposite trends can be quantitatively describedby the variation of the first and second spin correlation parameters with M as shown in thetop and bottom panels of Figure 16, respectively. As can be seen, h d t i is larger than h c t i in the lower mass range of M < h − M ⊙ but drops below h c t i in the higher mass rangeof M ≥ h − M ⊙ . The transition mass scale of the spin-flip corresponds to the momentwhen h d t i becomes lower than h c t i .For the case of the lowest and low-mass galactic halos with M < × h − M ⊙ , bothof the models II and III succeed in matching simultaneously the amplitudes and behaviorsof the three numerically obtained probability density functions. The model II is almostidentical to the model III in these low-mass ranges, since the values of h c t i obtained viaEquation (22) are low for these cases. It is also worth noting that the signal of the strongˆ u -ˆ s anti-alignments is found to increase with M whose behavior is well described by bothof the model II and III . The success of the model II and model III and the failure ofthe model I in describing the amplitudes and behaviors of p ( | ˆ u · ˆ s | ) and p ( | ˆ u · ˆ s | ) are alsofound for the case of the medium-mass halos (second from the bottom panels in Figure 15). 15 –It is, however, interesting to note that in this medium-mass range the spin vectors, ˆ s ,exhibit a weak but non-negligible alignment with the intermediate eigenvectors, ˆ u , whichtendency is properly described by both of model I and model III but not by the modelII . For the case of the high-mass galactic halos, the unit spin vectors, ˆ s , turn out to be morestrongly aligned with ˆ u than with ˆ u , which cannot be described by the model II . But, thealignments of ˆ s with ˆ u and its anti-alignments with ˆ u are still well described by the modelII and III but not by the model I . Thus, it is only the model III that agrees concurrentlyand consistently with the numerically obtained three probability density functions, p ( | ˆ u · ˆ s | ), p ( | ˆ u · ˆ s | ), and p ( | ˆ u · ˆ s | ), in all of the four mass ranges.Figure 17 shows the same as the bottom panels of Figure 15 but for the cases thatthe tidal fields are smoothed on three larger scales of R f = 10 ,
20 and 30 h − Mpc, in thetop, middle and bottom panels, respectively. As can be seen, the increment of R f decreasesthe alignment strengths even more rapidly than for the case of the shape alignments. Notethat it is only the model III that succeeds in making good simultaneous descriptions of theamplitudes and behaviors of { p ( | ˆ u i · ˆ s | ) } i =1 for all of the three cases of R f .Although the model III achieves overall good agreements with the numerical results,some discrepancies between its description and the numerical results are found. As can beseen in Figures 15-17, the numerically obtained three probability functions display substan-tially fluctuating behaviors especially for the case of the high-mass galactic halos. However,the increment of R f reduces these discrepancies as shown in Figure 17, which implies thatthe inaccuracies associated with the approximations of T as a Gaussian random field and p ( s | T ) as a multivariate Gaussian distribution in the derivation of Equation (20) should belargely responsible for these discrepancies.The uncertainties involved in the measurements of the spin vectors of the galactic halosmay be another source of the discrepancies. Since the spin direction of a galactic halo isdominantly determined by the positions and velocities of the outmost DM particles from thehalo center, its measurement would depend sensitively on the dynamical state of the galactichalo, halo-finding algorithm and definition of the virial radius. If a high-mass galactic halohas yet to be fully relaxed and/or in the middle of merging, containing multiple substructures,the measurement of its spin direction is likely to suffer from substantial uncertainties, whichin turn would cause mismatches between the analytical and the numerical results on the spinalignments with the large-scale tidal field.Figure 18 shows how the first and second spin parameters vary with R f for the high-mass galactic halos in the top and bottom panels, respectively. As can be seen, both ofthe parameters decrease with the increment of R f . The two parameters, however, showdifferent variations with M . The first spin parameter, h c t i , decreases more rapidly with 16 –the increment of R f than the second spin parameter, h d t i . It is found that h c t i > h d t i at R f ≤ h − Mpc, while c t < d t at R f = 30 h − Mpc. This result implies that the occurrenceof the spin flip phenomenon is contingent on the sizes of the large-scale structures. Supposethat the galaxies with masses in the range of 0 . ≤ M/ (10 h − M ⊙ ) ≤
50 embedded in acoherent large-scale structure like a filament with size R f ≥ h − Mpc. According to ourresults, the spin vectors of those galaxies would not flip, with their spins always aligned withthe elongated axes of the host filament since h d t i is always higher than h c t i in the given massrange (see Section 3.3). Following the same procedure as presented in Section 2.3, we investigate how the proba-bility density functions, { p ( | ˆ u i · ˆ s | ) } i =1 , depend on the type of the cosmic web. Figures 19-22show the same as Figure 15 but only with the galactic halos located in the knot, filament,sheet and void environments, respectively. In the knot environments (Figure 19), the unitspin vectors, ˆ s , of the galactic halos are found strongly aligned with the minor eigenvectorˆ u in all of the four mass ranges (i.e., no spin-flip). For the cases of the lowest-mass, low-mass and medium-mass knot galactic halos, we find ˆ s to be slightly anti-aligned rather thanaligned with ˆ u , while the high-mass knot galactic halos show weak ˆ u -ˆ s alignments. Both ofthe model II and model III describe well the ˆ u -ˆ s alignment and the ˆ u -ˆ s anti-alignment.However, model II cannot describe the observed tendency of the ˆ u -ˆ s anti-alignment in themass scale of 5 ≤ M/ (10 h − M ⊙ ) <
10 while the model III can. It is interesting to seethat the model I describes better the observed ˆ u -ˆ s anti-alignments in the medium-massrange better than the model II although it still notoriously fails in describing the observedstrong ˆ u -ˆ s alignments and ˆ u -ˆ s anti-alignments.The filament galactic halos yield much stronger ˆ u -ˆ s alignment and ˆ u -ˆ s anti-alignmentin all of the four mass ranges than the knot counterpart, although the behaviors of { p ( | ˆ u i · ˆ s | ) } i =1 between the two cases are quite similar to each other (Figure 20). The high-massfilament galactic halos show a substantial ˆ u -ˆ s alignment whose strength is comparable tothat of the ˆ u -ˆ s (i.e., the occurrence of the spin flip). Although the model III works quitewell in matching the numerically obtained probability density functions, it is interesting tonote that the model II gives a better description of p ( | ˆ u · ˆ s | ) than the model III in themass range of M < h − M ⊙ .The sheet galactic halos exhibit a different trend (Figure 21). Their spin vectors tendto lie in the plane spanned by ˆ u and ˆ u , being orthogonal to ˆ u . The increment of M leadsto the stronger ˆ u -ˆ s alignment and ˆ u -ˆ s anti-alignment but weaker ˆ u -ˆ s alignment. For the 17 –case of the lowest-mass and low-mass sheet galactic halos, the ˆ u -ˆ s alignment tendency isweaker than the ˆ u -ˆ s alignment. For the case of the medium-mass sheet galactic halos, theˆ u -ˆ s alignment begins to exceed in strength the ˆ u -ˆ s alignment (i.e., occurrence of the spinflip). The strongest signal of the ˆ u -ˆ s alignments is found from the high-mass sheet galactichalos, which result is consistent with the previous numerical finding of Hahn et al. (2007a).As can be seen, only the Model III succeeds in describing simultaneously and consistentlythe behaviors of { p ( | ˆ u i · ˆ s | ) } i =1 , fairly well for the case of the sheet galactic halos in all ofthe four mass ranges.This result is inconsistent with the observational finding of Zhang et al. (2015) thatthe galaxies in the knot environments exhibited the strongest spin alignments with the tidalfields. We suspect that two factors may have caused this inconsistency between the numericaland observational results on the web dependence of the spin alignments. First, the differencein the way in which the tidal fields were constructed. In the work of Zhang et al. (2015), thetidal fields, ˆ T , were constructed from the spatial distributions of the galaxy groups, while inthe SMDPL the spatial distribution of the DM particles were used. Second, the differencein the measurements of ˆ s : In the observational analysis of Zhang et al. (2015), the unitspin vectors ˆ s were determined from the luminous parts of the galaxies while in the currentnumerical analyses, all of the constituent DM particles determine ˆ s .The weakest spin alignments with the large-scale tidal fields are found in the voidenvironments (Figure 22). Although the signals are quite lower than those yielded by thesheet galactic halos, the behaviors of { p ( | ˆ u i · ˆ s | ) } i =1 obtained from the void galactic halos arequite similar to those from the sheet galactic halos: the alignments of ˆ s with ˆ u and ˆ u . Theformer (the latter) alignment become stronger (weaker) with the increment of M . For thelowest-mass and low-mass void galactic halos (top two panels), the ˆ u -ˆ s alignment is slightlystronger than the ˆ u -ˆ s alignment. Only the Model III pulls it off to describe simultaneouslythe behaviors of { p ( | ˆ u i · ˆ s | ) } i =1 . For the case of the medium-mass and high-mass void galactichalos, however, the large errors make it difficult to interpret the numerical results and tomake a fair comparison of them with the three models.Figures 23 and 24 plot h c t i and h d t i versus M for the four different web types, re-spectively. Although the increment of the first spin correlation parameter, h c t i , with M isuniversally shown, the increment rate sensitively depends on the web type. The most (least)rapid change of h c t i with M is found from the sheet (knot) galactic halos. Meanwhile, thesecond spin correlation parameter, h d t i , does not show strong variations with M . For the caseof the high-mass filament and void galactic halos, however, it shows an abrupt decrementwith M .Defining the transition mass, M t , as the one beyond which h c t i exceeds h d t i , we expect 18 –the galactic halos with M > M t ( M ≤ M t ) to exhibit the preferential ˆ u -ˆ s (ˆ u -ˆ s ) alignment.The results shown in Figures 23 and 24 imply that the value of M t depends on the webtype. as shown in Libeskind et al. (2013). For the case of the knot galactic halos, no spinflip occurs in the given whole mass range since h d t i is always larger than h c t i . The spinflip of the filament (sheet) galactic halos is expected to occur around M t ∼ × h − M ⊙ ( M t ∼ h − M ⊙ ), while the void galactic halos show the lowest transition mass scale, M t ∼ × h − M ⊙ ).As done in Section 2.3, smoothing the tidal fields on three larger scales R f and repeatingthe whole calculation for each case of R f , we investigate the dependence of the tendency andstrength of the spin alignments on R f for the case of the high-mass galactic halos. Figures25-28 plot the same as the bottom panels of Figures 19-22, respectively, but for the casesof R f = 10 , , h − Mpc. As can be seen, whatever type of the cosmic web the galactichalos are embedded in, the increment of R f always decreases the alignment strength, whichis well described by the model III .For the cases of the high-mass galactic halos in the knot and filament regions, theincrement of R f just decreases the strength of the spin alignments but does not changeits tendency (Figures 25-26). However, for the case of the high-mass sheet galactic halos(Figure 27), it changes both of the strength and the tendency of the spin alignments. Onthe scales of R f = 10 and 20 h − Mpc, the high-mass sheet galactic halos show the strongerˆ u -ˆ s alignments than the ˆ u -ˆ s alignments. But, on the larger scale of R f = 30 h − Mpc,we witness a different tendency, the ˆ u -ˆ s alignments seem slightly stronger than the ˆ u -ˆ s alignments. In other words, if the sheet environment is defined on the scale equal to orlarger than 30 h − Mpc, no spin-flip will occur in the given mass range. Since both of ˆ u andˆ u span the plane of a sheet (Zel’dovich 1970), our result shown in Figure 27 supports theclaim of Hahn et al. (2007b) that the spin vectors of the DM halos have a universal tendencyof lying in the plane of the sheet, regardless of the halo mass.
4. Summary and Discussion
To study the large-scale tidal effect on the spin and shape orientations of the galaxiesand the spin-flip phenomenon, we have considered three different analytic models, the modelI , model II and model III . The model I , Equation (16), which was originally developedby Lee & Pen (2000) based on the LTT theory, describes the alignment tendency betweenthe galaxy spin vectors, ˆ s , and the intermediate eigenvectors, ˆ u , of the large-scale tidalfield, T . The model II , Equation (5), has been constructed here to describe the alignments(anti-alignments) of the galaxy shapes, ˆ e , with the minor (major) eigenvectors, ˆ u (ˆ u ) of T . 19 –This model is based on the first order Lagrangian perturbation theory according to whichthe major principal axes of the inertia momentum tensors of the galactic halos are perfectlyaligned with the minor principal axes of the local tidal tensors in the Lagrangian regime.The model III , Equation (20), is a practical formula constructed by combining the modelI and model II to describe simultaneously the tidally induced shape and spin alignments.The model I ( model II ) carries a single parameter, c t ( d t ), which measures the strengthof the alignment with ˆ u (ˆ u ). The model III carries two parameters, c t and d t , whoserelative ratio determines the transition mass scale for the occurrence of the spin-flip. The firstparameter, c t , would reach the maximum value of unity, if the inertia momentum tensors ofthe galaxies are uncorrelated with the surrounding tidal tensors, while the second parameter, d t , will attain the value of unity if the two tensors are perfectly correlated. These parameterscan be empirically determined by Equation (22) directly from the measured values of ˆ e andˆ s in the principal frame of ˆ T without resorting to any fitting procedure.To numerically test the three analytic models, we have utilized the density fields andthe Rockstar halo catalogs extracted from the SMDPL simulations (Klypin et al. 2016).Constructing the unit traceless tidal tensor, ˆ T , smoothed on the scale of R f = 5 h − Mpcfrom the density fields given on the 512 grids that constitute the simulation box of volume400 h − Mpc and selecting the galactic halos in the mass range of 0 . ≤ M/ (10 h − M ⊙ ) ≤
50 from the Rockstar catalog, we have first numerically obtained the probability density func-tions of the tidally induced shape alignments, { p ( | ˆ u i · ˆ e | ) } i =1 (see Figures 1-3). The numericalresults have clearly shown that ˆ e has a tendency to be strongly aligned (anti-aligned) with ˆ u (ˆ u ) but no correlation with ˆ u . Investigating the dependence of the strength of the tidallyinduced shape alignments on M , R f , and the type of the cosmic web, it has been found thatthe more massive galactic halos yield stronger ˆ u -ˆ e alignments (ˆ u -ˆ e anti-alignments) andthat the increment of R f weakens the alignment tendency (see Figures 4). These numeri-cal results are consistent with what the previous works already found (Joachimi et al. 2013;Zhang et al. 2013; Chen et al. 2016; Hilbert et al. 2017; Xia et al. 2017; Piras et al. 2018).The strongest (weakest) ˆ u -ˆ e alignments are found from the void (knot) galactic halos(see Figures 5-8), which seem inconsistent with the previous numerical result that the DMhalos showed the strongest shape alignments in the knot environments (Xia et al. 2017).This inconsistency has been ascribed to the different classification schemes used in the twoanalyses. The sheet galactic halos yield much stronger shape alignment tendency than theknot and filament galactic halos in the whole mass range, which result is consistent with whatHahn et al. (2007a) found. In the lowest and low mass range (0 . ≤ M/ [10 h − M ⊙ ] < ≤ M/ [10 h − M ⊙ ] <
10) and high-mass (10 ≤ M/ [10 h − M ⊙ ] <
50) 20 –ranges, the shape alignments of the filament galactic halos become stronger than the knotcounterparts (Figure 9). These numerical results imply that the void and sheet galactic halosretain best the tidally induced shape alignments, while the evolution of the galactic halos inthe dense environments like the knots and filaments has an effect of deviating the directionsof their shapes from the tidally induced inclinations.The comparison with the numerical results revealed the success of the model II indescribing the amplitudes and behaviors of { p ( | ˆ u i · ˆ e | ) } i =1 , for all of the cases of M , R f and the type of the cosmic web. For the shape alignments, the model III turns out to beidentical to the model II . For all of the four cases of the web type, the increment of R f hasbeen found to decrease the strength of the tidally induced shape alignments but improve theagreements between the model III and the numerical results (Figure 10-14). We interpretthis result as an evidence supporting the scenario that the nonlinear evolution has an effectof diminishing the strength of the tidally induced shape alignments.In a similar manner, we have numerically determined the probability density functionsof the tidally induced spin alignments, { p ( | ˆ u i · ˆ s | ) } i =1 , explored their dependences on M , R f and the web type, and compared the results with the three analytic models. The tidallyinduced spin alignments have been found significant but quite weak compared with theshape alignments (Figures 15-16), consistent with the results from the previous works (e.g.,Hahn et al. 2007b; Forero-Romero et al. 2014; Zhang et al. 2015). The occurrence of thespin-flip phenomenon has been witnessed. For the case of R f = 5 h − Mpc, the lowest-mass,low-mass and medium-mass galactic halos show strong ˆ u -ˆ s alignments and negligible ˆ u -ˆ s alignments, while the high-mass galactic halos exhibit strong ˆ u -ˆ s alignments, which resultshave confirmed the claims of the previous works (Arag´on-Calvo et al. 2007; Paz et al. 2008;Zhang, Yang & Faltenbacher 2009; Codis et al. 2012; Libeskind et al. 2013; Trowland et al.2013; Dubois et al. 2014; Chen et al. 2016; Veena et al. 2018).However, we have noted that the spin-flip does not occur abruptly at a certain fixedtransition mass scale. Rather it is a gradual transition of the spin alignment tendency thatproceeds over a broader mass range, depending on R f (Figures 17-18). For the case of R f = 5 h − Mpc, the high-mass galactic halos have been found to yield stronger ˆ u -ˆ s andweaker but significant ˆ u -ˆ s alignments, while the medium-mass galactic halos exhibit strongˆ u -ˆ s and much weaker ˆ u -ˆ s alignments. For the case of R f ≥ h − Mpc, however, thehigh-mass galactic halos exhibit stronger ˆ u -ˆ s and weaker but significant ˆ u -ˆ s alignments.The strengths of the tidally induced spin alignments have been also found to sensitivelyvary with the types of the cosmic web (see Figures 19-28), which supports the claim ofLibeskind et al. (2013). The strongest (weakest) signals of the tidally induced spin align-ments have been found from the sheet (void) galactic halos, while the filament galactic halos 21 –have been found to have stronger spin alignments than the knot counterparts in the wholemass range ( Figures 23-24). These results are inconsistent with the observational finding ofZhang et al. (2015) that the knot galaxies exhibited the strongest signals of the spin align-ments. We have suspected that this inconsistency might be related to the construction ofthe tidal field from the galaxy groups and the determination of the spin axes of the galaxiesfrom their stellar components in the observational analysis.Determining empirically h c t i and h d t i from the numerical data (Figures 23-24) anddefining the condition for the occurrence of the spin flip as h c t i > h d t i , we have quantita-tively investigated how the occurrence and the transition mass scale, M t , of the spin-flipphenomenon depend on the size and type of the cosmic web and found the following:1. Regardless of the web type, the transition mass scale, M t , of the spin-flip increaseswith the increment of R f .2. The knot galactic halos do no show any spin-flip phenomenon. That is, the unit spinvectors, ˆ s , of the knot galactic halos are always preferentially aligned with ˆ u ratherthan with ˆ u in the whole mass range, regardless of the value of R f (Figure 25).3. For the case of the filament galactic halos, the spin flip occurs around M t ∼ × h − M ⊙ when R f = 5 h − Mpc. At the larger scale of R f > h − Mpc, the value of M t exceeds the galactic mass scales, i.e, M t > × h − M ⊙ (Figure 26).4. In the sheet environment, the transition mass scale has a lower value than in thefilaments: M t ∼ h − M ⊙ when R f = 5 h − Mpc. Only when R f reaches 30 h − Mpc,the value of M t becomes larger than the galactic mass scale (Figure 27).5. The void galactic halos yield the lowest transition mass scale, M t ∼ × h − M ⊙ when R f = 5 h − Mpc. At the larger scales, the number of the void galactic halos istoo low to produce any significant signals (Figure 28).It is interesting to note that our results on the web and mass dependence of the spin-flipphenomenon are consistent with the theoretical explanation of Codis et al. (2015), accord-ing to which the misalignments between the inertia momentum and tidal tensors in theanisotropic environments like the filaments and sheets are largely responsible for the occur-rence of the spin flip. In line with their theoretical explanation, we interpret no occurrence ofthe spin flip in the knot environments as an evidence for the stronger alignments between thetwo tensors in the dense environments. In other words, in the knot regions where the tidaltensors are more isotropic, the inertia momentum and tidal tensors may be more strongly 22 –aligned with each other, which plays a role in suppressing the occurrence of the spin-flip ofthe knot galaxies.It has also been clearly demonstrated in the current work that the model III succeeds indescribing consistently and simultaneously the numerical results of the tidally induced shapeand spin alignments for all of the cases of M , R f and type of the cosmic web, while the modelI and model II fail. Showing that the model III works better as R f increases, we haveascribed the slight mismatches between the numerical results and the model III to theinaccuracies caused by the approximations of p ( s | T ) as a multivariate Gaussian distributionand ˆ T as a Gaussian random field made in the construction of the model III . We also suspectthat the uncertainties in the measurements of ˆ s and ˆ e caused by the simple assumptions ofeach galactic halo having a perfect ellipsoidal shape and no substructure in a completelyrelaxed dynamical state must contribute to the mismatches.We conclude that the model III is an effective practical model for the spin and theshape alignments of the galactic halos with the large-scale tidal fields, providing an analytictool with which the condition of the spin flip occurrence as well as its dependence on theproperties of the large-scale structures can be quantitatively described. Its good accord withthe numerical results supports the scenario that the occurrence of the spin flip phenomenonis associated more with the geometrical properties of the large-scale tidal field as well as theinteractions of the galactic halos with the cosmic web rather than with the physical processesduring the nonlinear evolution (see Libeskind et al. 2013; Codis et al. 2015; Wang & Kang2017; Veena et al. 2018).Given that the model III is expressed in terms of the linear quantities, it may provideanother independent probe of the background cosmology. For this purpose, however, a coupleof back-up works will have to be done. First, as suspected in our analysis, differences in theschemes used to to construct the tidal fields, to measure the shape and spin axes of thegalaxies, and to classify the cosmic web would yield different patterns in the dependenceof the tidally induced shape and spin alignments on the sizes and types of the cosmic web.Thus, it will be necessary to test the robustness of the model III REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
28 –Fig. 1.— Probability density distributions of three coordinates of the unit shape vectors, ˆ e ,of the lowest-mass (top panel), low-mass (first middle panel), medium-mass (second middlepanel) and high-mass (bottom panel) galactic halos in the principal frame spanned by theminor (ˆ u ), intermediate (ˆ u ), and major (ˆ u ) eigenvectors of the tidal fields smoothed onthe scale of R f = 5 h − Mpc. In each panel, the numerical results are plotted as blackfilled circular dots with Poisson errors, while the analytic model, Equations (1)-(4), withthe empirically determined value of d t is shown as red solid line. The uniform constantprobability density is depicted as black dotted line. 29 –Fig. 2.— Mean values of the shape correlation parameter d t averaged over the galactic halosbelonging to four different mass ranges when the local tidal fields are smoothed on the scaleof R f = 5 h − Mpc. 30 –Fig. 3.— Same as the bottom panels of Figure 1 but for the cases of three larger smoothingscales of R f = 10 , , h − Mpc. 31 –Fig. 4.— Mean values of the shape correlation parameter d t averaged over the high-massgalactic halos as a function of the smoothing scale, R f . 32 –Fig. 5.— Same as Figure 1 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the tidal fields, λ , λ , λ , are positive. 33 –Fig. 6.— Same as Figure 1 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 34 –Fig. 7.— Same as Figure 1 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3
28 –Fig. 1.— Probability density distributions of three coordinates of the unit shape vectors, ˆ e ,of the lowest-mass (top panel), low-mass (first middle panel), medium-mass (second middlepanel) and high-mass (bottom panel) galactic halos in the principal frame spanned by theminor (ˆ u ), intermediate (ˆ u ), and major (ˆ u ) eigenvectors of the tidal fields smoothed onthe scale of R f = 5 h − Mpc. In each panel, the numerical results are plotted as blackfilled circular dots with Poisson errors, while the analytic model, Equations (1)-(4), withthe empirically determined value of d t is shown as red solid line. The uniform constantprobability density is depicted as black dotted line. 29 –Fig. 2.— Mean values of the shape correlation parameter d t averaged over the galactic halosbelonging to four different mass ranges when the local tidal fields are smoothed on the scaleof R f = 5 h − Mpc. 30 –Fig. 3.— Same as the bottom panels of Figure 1 but for the cases of three larger smoothingscales of R f = 10 , , h − Mpc. 31 –Fig. 4.— Mean values of the shape correlation parameter d t averaged over the high-massgalactic halos as a function of the smoothing scale, R f . 32 –Fig. 5.— Same as Figure 1 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the tidal fields, λ , λ , λ , are positive. 33 –Fig. 6.— Same as Figure 1 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 34 –Fig. 7.— Same as Figure 1 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 35 –Fig. 8.— Same as Figure 1 but with only those galactic halos located in the void environmentswhere 0 > λ > λ > λ . 36 –Fig. 9.— Same as Figure 2 but with those galactic halos embedded in four different typesof the cosmic web. 37 –Fig. 10.— Same as Figure 3 but with only those galactic halos located in the knot environ-ments. 38 –Fig. 11.— Same as Figure 3 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 39 –Fig. 12.— Same as Figure 3 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3
28 –Fig. 1.— Probability density distributions of three coordinates of the unit shape vectors, ˆ e ,of the lowest-mass (top panel), low-mass (first middle panel), medium-mass (second middlepanel) and high-mass (bottom panel) galactic halos in the principal frame spanned by theminor (ˆ u ), intermediate (ˆ u ), and major (ˆ u ) eigenvectors of the tidal fields smoothed onthe scale of R f = 5 h − Mpc. In each panel, the numerical results are plotted as blackfilled circular dots with Poisson errors, while the analytic model, Equations (1)-(4), withthe empirically determined value of d t is shown as red solid line. The uniform constantprobability density is depicted as black dotted line. 29 –Fig. 2.— Mean values of the shape correlation parameter d t averaged over the galactic halosbelonging to four different mass ranges when the local tidal fields are smoothed on the scaleof R f = 5 h − Mpc. 30 –Fig. 3.— Same as the bottom panels of Figure 1 but for the cases of three larger smoothingscales of R f = 10 , , h − Mpc. 31 –Fig. 4.— Mean values of the shape correlation parameter d t averaged over the high-massgalactic halos as a function of the smoothing scale, R f . 32 –Fig. 5.— Same as Figure 1 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the tidal fields, λ , λ , λ , are positive. 33 –Fig. 6.— Same as Figure 1 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 34 –Fig. 7.— Same as Figure 1 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 35 –Fig. 8.— Same as Figure 1 but with only those galactic halos located in the void environmentswhere 0 > λ > λ > λ . 36 –Fig. 9.— Same as Figure 2 but with those galactic halos embedded in four different typesof the cosmic web. 37 –Fig. 10.— Same as Figure 3 but with only those galactic halos located in the knot environ-ments. 38 –Fig. 11.— Same as Figure 3 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 39 –Fig. 12.— Same as Figure 3 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 40 –Fig. 13.— Same as Figure 3 but with only those galactic halos located in the void environ-ments where 0 > λ > λ > λ3
28 –Fig. 1.— Probability density distributions of three coordinates of the unit shape vectors, ˆ e ,of the lowest-mass (top panel), low-mass (first middle panel), medium-mass (second middlepanel) and high-mass (bottom panel) galactic halos in the principal frame spanned by theminor (ˆ u ), intermediate (ˆ u ), and major (ˆ u ) eigenvectors of the tidal fields smoothed onthe scale of R f = 5 h − Mpc. In each panel, the numerical results are plotted as blackfilled circular dots with Poisson errors, while the analytic model, Equations (1)-(4), withthe empirically determined value of d t is shown as red solid line. The uniform constantprobability density is depicted as black dotted line. 29 –Fig. 2.— Mean values of the shape correlation parameter d t averaged over the galactic halosbelonging to four different mass ranges when the local tidal fields are smoothed on the scaleof R f = 5 h − Mpc. 30 –Fig. 3.— Same as the bottom panels of Figure 1 but for the cases of three larger smoothingscales of R f = 10 , , h − Mpc. 31 –Fig. 4.— Mean values of the shape correlation parameter d t averaged over the high-massgalactic halos as a function of the smoothing scale, R f . 32 –Fig. 5.— Same as Figure 1 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the tidal fields, λ , λ , λ , are positive. 33 –Fig. 6.— Same as Figure 1 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 34 –Fig. 7.— Same as Figure 1 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 35 –Fig. 8.— Same as Figure 1 but with only those galactic halos located in the void environmentswhere 0 > λ > λ > λ . 36 –Fig. 9.— Same as Figure 2 but with those galactic halos embedded in four different typesof the cosmic web. 37 –Fig. 10.— Same as Figure 3 but with only those galactic halos located in the knot environ-ments. 38 –Fig. 11.— Same as Figure 3 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 39 –Fig. 12.— Same as Figure 3 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 40 –Fig. 13.— Same as Figure 3 but with only those galactic halos located in the void environ-ments where 0 > λ > λ > λ3 . 41 –Fig. 14.— Mean values of the shape correlation parameter averaged over the high-massgalactic halos belonging to the four different types of the cosmic web versus the smoothingscale R f . 42 –Fig. 15.— Probability density distributions of three coordinates of the unit spin vectors, ˆ s ,of the galactic halos in the principal frame spanned by three eigenvectors, { ˆ u , ˆ u , ˆ u } , ofthe local tidal fields smoothed on the scale of R f = 5 h − Mpc, for four different ranges of thehalo mass M . In each panel, the numerical results are plotted as black filled circular dotswhile the analytic model with the empirically determined parameters is shown as red solidline. The uniform probability density is depicted as black dotted line. 43 –Fig. 16.— Mean values of the first and second spin correlation parameters, c t and d t , averagedover the galactic halos belonging to four different mass ranges for the case of R f = 5 h − Mpcin the top and bottom panels, respectively. 44 –Fig. 17.— Probability density distributions, p ( | ˆ u · ˆ e | ) , p ( | ˆ u · ˆ e | ) , p ( | ˆ u · ˆ e | ) of the high-massgalactic halos for three different cases of the smoothing scale R f . 45 –Fig. 18.— Mean values of the first and second spin correlation parameters, c t and d t ,averaged over the high-mass galactic halos as a function of R f in the top and bottom panels,respectively. 46 –Fig. 19.— Same as Figure 15 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the local tidal fields, λ , λ , λ , are positive. 47 –Fig. 20.— Same as Figure 15 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 48 –Fig. 21.— Same as Figure 15 but with only those galactic halos located in the sheet envi-ronments where λ > > λ > λ . 49 –Fig. 22.— Same as Figure 15 but with only those galactic halos located in the void environ-ments where 0 > λ > λ > λ3
28 –Fig. 1.— Probability density distributions of three coordinates of the unit shape vectors, ˆ e ,of the lowest-mass (top panel), low-mass (first middle panel), medium-mass (second middlepanel) and high-mass (bottom panel) galactic halos in the principal frame spanned by theminor (ˆ u ), intermediate (ˆ u ), and major (ˆ u ) eigenvectors of the tidal fields smoothed onthe scale of R f = 5 h − Mpc. In each panel, the numerical results are plotted as blackfilled circular dots with Poisson errors, while the analytic model, Equations (1)-(4), withthe empirically determined value of d t is shown as red solid line. The uniform constantprobability density is depicted as black dotted line. 29 –Fig. 2.— Mean values of the shape correlation parameter d t averaged over the galactic halosbelonging to four different mass ranges when the local tidal fields are smoothed on the scaleof R f = 5 h − Mpc. 30 –Fig. 3.— Same as the bottom panels of Figure 1 but for the cases of three larger smoothingscales of R f = 10 , , h − Mpc. 31 –Fig. 4.— Mean values of the shape correlation parameter d t averaged over the high-massgalactic halos as a function of the smoothing scale, R f . 32 –Fig. 5.— Same as Figure 1 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the tidal fields, λ , λ , λ , are positive. 33 –Fig. 6.— Same as Figure 1 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 34 –Fig. 7.— Same as Figure 1 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 35 –Fig. 8.— Same as Figure 1 but with only those galactic halos located in the void environmentswhere 0 > λ > λ > λ . 36 –Fig. 9.— Same as Figure 2 but with those galactic halos embedded in four different typesof the cosmic web. 37 –Fig. 10.— Same as Figure 3 but with only those galactic halos located in the knot environ-ments. 38 –Fig. 11.— Same as Figure 3 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 39 –Fig. 12.— Same as Figure 3 but with only those galactic halos located in the sheet environ-ments where λ > > λ > λ3 . 40 –Fig. 13.— Same as Figure 3 but with only those galactic halos located in the void environ-ments where 0 > λ > λ > λ3 . 41 –Fig. 14.— Mean values of the shape correlation parameter averaged over the high-massgalactic halos belonging to the four different types of the cosmic web versus the smoothingscale R f . 42 –Fig. 15.— Probability density distributions of three coordinates of the unit spin vectors, ˆ s ,of the galactic halos in the principal frame spanned by three eigenvectors, { ˆ u , ˆ u , ˆ u } , ofthe local tidal fields smoothed on the scale of R f = 5 h − Mpc, for four different ranges of thehalo mass M . In each panel, the numerical results are plotted as black filled circular dotswhile the analytic model with the empirically determined parameters is shown as red solidline. The uniform probability density is depicted as black dotted line. 43 –Fig. 16.— Mean values of the first and second spin correlation parameters, c t and d t , averagedover the galactic halos belonging to four different mass ranges for the case of R f = 5 h − Mpcin the top and bottom panels, respectively. 44 –Fig. 17.— Probability density distributions, p ( | ˆ u · ˆ e | ) , p ( | ˆ u · ˆ e | ) , p ( | ˆ u · ˆ e | ) of the high-massgalactic halos for three different cases of the smoothing scale R f . 45 –Fig. 18.— Mean values of the first and second spin correlation parameters, c t and d t ,averaged over the high-mass galactic halos as a function of R f in the top and bottom panels,respectively. 46 –Fig. 19.— Same as Figure 15 but with only those galactic halos located in the knot environ-ments where all three eigenvalues of the local tidal fields, λ , λ , λ , are positive. 47 –Fig. 20.— Same as Figure 15 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 48 –Fig. 21.— Same as Figure 15 but with only those galactic halos located in the sheet envi-ronments where λ > > λ > λ . 49 –Fig. 22.— Same as Figure 15 but with only those galactic halos located in the void environ-ments where 0 > λ > λ > λ3 . 50 –Fig. 23.— Mean values of the first spin correlation parameter c t averaged over the knot(top-left panel), filament (top-right panel), sheet (bottom-left panel), and void (bottom-right panel) galactic halos in four different mass ranges. The smoothing scale R f is set at5 h − Mpc. 51 –Fig. 24.— Same as Figure 23 but for the second spin correlation parameter d t . 52 –Fig. 25.— Same as Figure 17 but with only those galactic halos located in the knot environ-ments. 53 –Fig. 26.— Same as Figure 17 but with only those galactic halos located in the filamentenvironments where λ > λ > > λ . 54 –Fig. 27.— Same as Figure 17 but with only those galactic halos located in the sheet envi-ronments where λ > > λ > λ . 55 –Fig. 28.— Same as Figure 17 but with only those galactic halos located in the void environ-ments where 0 > λ > λ > λ3