SPIN ALIGNMENTS OF SPIRAL GALAXIES WITHIN THE LARGE-SCALE STRUCTURE FROM SDSS DR7

The galaxy sample we use here is the New York University Value-Added Galaxy Catalog⁷ (NYU-VAGC; Blanton et al. 2005; Padmanabhan et al. 2008; Adelman-McCarthy et al. 2008), which is based on the multi-band imaging and spectroscopic survey SDSS DR7 (Abazajian et al. 2009). As the completion of the survey phase SDSS-II, DR7 features an imaging survey in five bands over a continuous $7646\deg ^2$ region in the Northern Galactic Cap. The continuity over this large area is essential and critical to reconstruct the real-space tidal field from the SDSS survey. From the NYU-VAGC, we select 639, 555 galaxies in the Main Galaxy Sample with an extinction-corrected apparent magnitude brighter than r = 17.72, with redshifts in the range 0.01 ⩽ z ⩽ 0.20 and with redshift completeness ${\cal C}_z > 0.7$. A small subset of galaxies have redshifts adopted from the Korea Institute from Advanced Study Value-added Galaxy Catalog (Choi et al. 2007, 2010).

Using the adaptive halo-based group finder developed by Yang et al. (2005, 2007), we construct a catalog of 472, 532 galaxy groups (about 404, 300 groups with single galaxies). For each group, we assign a halo mass according to the ranking of its characteristic group luminosity, defined as the total luminosity of all group members with ^0.1M_r − 5log h ⩽ −19.5, where ^0.1M_r − 5log h is the r-band absolute magnitude, K corrected and evolution corrected to z = 0.1, using the method described by Blanton et al. (2003). In order to ensure sample completeness, in what follows, we focus our analysis on the volume covering the redshift range 0.01 ⩽ z ⩽ 0.12, and within the largest continuous region in the Northern Galactic Cap. Using the complete galaxy groups with masses of M_h ⩾ M_th = 10¹² h⁻¹ M_☉, Wang et al. (2012) constructed the mass, tidal, and velocity (SDSS-MTV) fields in this volume. We use the method developed by Wang et al. (2009) to correct for the peculiar motion of groups as well as the finger-of-god effect. We have tested that our alignment signals are insensitive to the boundary effect. The smoothing length scale we used to reconstruct the tidal field is R_s = 2.1 h⁻¹Mpc because this value provides the best agreement with the visual classification of the large-scale structure (Hahn et al. 2007a, 2007b). The smoothing scale R_s is related to the mass M_s contained in the Gaussian filter at mean density $\bar{\rho }$ via $M_s = (2\pi)^{3/2} \bar{\rho } R_s^3$, thus a smoothing scale R_s = 2.1 h⁻¹ Mpc corresponds to the mass M_s = 10¹³ h⁻¹ M_☉.

Following Wang et al. (2012), we calculate the tidal tensor using

$$\begin{equation} T_{ij}({\boldsymbol{x}}) = \frac{\partial ^2 \phi }{\partial x_i \partial x_j}, \end{equation} \tag{ 1 }$$

where i and j are indices with values of 1, 2, or 3, and ϕ is the peculiar gravitational potential, which can be calculated from the distribution of dark matter halos with masses M_h above some threshold value M_th through the Poisson equation. At the position of each group, the tidal tensor is diagonalized to obtain the eigenvalues λ₁, λ₂, and λ₃ (λ₁ ⩾ λ₂ ⩾ λ₃), as well as the corresponding eigenvectors t₁, t₂, and t₃.

According to the Zel'dovich theory (Zeldovich 1970), the sign of the eigenvalues of local tidal tensor can be used to classify the group's environment into one of four categories (Hahn et al. 2007a, 2007b):

• 1.  
  cluster: a point where all three eigenvalues are positive,
• 2.  
  filament: a point where T_ij has one negative and two positive eigenvalues,
• 3.  
  sheet: a point where T_ij has two negative and one positive eigenvalues,
• 4.  
  void: a point where all three eigenvalues are negative.

The direction of a filament at the location of a filament group can be identified with the eigenvector corresponding to the single negative eigenvalue. Likewise, the normal vector of a sheet for a sheet group is the eigenvector corresponding to the single positive eigenvalue of the tidal tensor.

We adopt the morphological classifications of galaxies from the GZ2 catalog (Willett et al. 2013),⁸ which provides robust classifications into elliptical and spiral for 304, 122 galaxies. In the following analysis, we only use spiral galaxies with $gz2\_class$ strings beginning with "S." This results in a sample of 120, 102 spirals in our survey volume. Based on the group catalog, the sample contains 90, 761 centrals and 29, 341 satellite galaxies, among which 74, 713 centrals and 28, 514 satellites have reliable halo masses. To compare with the results from N-body simulations, we focus on the 74, 713 central galaxies with reliable halo masses. Based on local tidal fields described in the last subsection, 12, 386 galaxies (16.6%) are located in cluster environments, 49, 898 (66.8%) in filament, 11, 909 (15.9%) in sheet, and 520 (0.7%) in void.

In order to obtain a more reliable detection of the spin vectors s of galaxies, in this paper, we only use a sample of spiral galaxies. If a spiral galaxy were a thin circular disk, the inclination angle ζ, between the plane of the disk and the line of sight, can be determined from the projected minor-to-major axis ratio b/a. Many studies used a simple model for the finite disk thickness to relate the apparent axis ratio b/a and an intrinsic flatness parameter f to obtain the inclination angles,⁹ ζ, of galaxies (Haynes & Giovanelli 1984; Lee & Erdogdu 2007; Varela et al. 2012). Here we follow the same practice and obtain ζ through

$$\begin{equation} \sin ^2\zeta = \frac{(b/a)^2 - f^2}{1-f^2} \end{equation} \tag{ 2 }$$

for b/a > f and set ζ = 0 for b/a < f. According to Haynes & Giovanelli (1984), the intrinsic flatness f depends on the morphological type of galaxies. However, we will use an average value of 0.14 (Varela et al. 2012), since the variation of the flatness f does not significantly affect the calculation of the inclination angle ζ. We have repeated our entire calculation of the alignment assuming f = 0, and found that the results are very similar to those using f = 0.14.

Figure 1 shows the probability distribution of the inclination angles of galaxies. We find that the distribution differs significantly from a uniform distribution. This is most likely due to the fact that spiral galaxies have bulges that distort the projected axis ratio. If the b/a axis ratio is measured around an isophote where the bulge contributes significantly, then the result can be affected. This effect becomes more and more significant for systems that are close to edge-on. Because of this, our following analysis will focus on galaxies with sin ζ > 0.5, which results in a sample of 54, 457 centrals and 18, 976 satellites.

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From a two-dimensional image of a galaxy, projected on the sky, we cannot decide which side of the galaxy is closer to us, which means that in the determination of ζ using Equation (2), a single value of b/a corresponds to two values of the inclination angle: ±|ζ|. To avoid this problem, many studies constrain their analyses to edge-on (ζ ∼ 0) and face-on (ζ ∼ ±π/2) galaxies, which usually results in small samples of about a few hundred galaxies (e.g., Trujillo et al. 2006; Slosar & White 2009; Jones et al. 2010). In this paper, we use all the spirals selected from the GZ2 catalog. Throughout, we apply the positive sign of ζ for all the spirals. We have tested and found no difference in the alignment signal when using the negative sign for ζ. However, the sign ambiguity is expected to reduce the strength of the alignment signal, as we quantify in the following section.

Following Trujillo et al. (2006), Lee & Erdogdu (2007), and Varela et al. (2012), we compute the components of the spin vector s as

$$\begin{eqnarray} s_x & =& \cos \alpha \cos \delta \sin \zeta \nonumber \\ && + \cos \zeta (\sin \phi \cos \alpha \sin \delta - \cos \phi \sin \alpha), \nonumber \\ s_y & =& \sin \alpha \cos \delta \sin \zeta \nonumber \\ && + \cos \zeta (\sin \phi \sin \alpha \sin \delta + \cos \phi \cos \alpha), \nonumber \\ s_z & =& \sin \delta \sin \zeta - \cos \zeta \sin \phi \cos \delta, \end{eqnarray} \tag{ 3 }$$

where α and δ are the right ascension and declination of a galaxy, respectively. ζ is the inclination angle calculated from Equation (2). The position angle ϕ increases counterclockwise from north to east in the plane of the sky, and is specified by the 25 mag per square arcsecond isophote in the r band.

To estimate the correlation between spin vectors of galaxies and the large-scale structure, we compute the cosine of the angle, θ_ST, defined as the angle between the spin axis of the galaxy, s, and the three principal axes t₁, t₂, and t₃ of the local tidal tensor. To quantify the alignments and their significance, we generate 100 random samples in which the orientations of the principal axes of the tidal tensors are kept fixed, but the spin axes of the galaxies are randomized. The alignment strength is then expressed by the normalized pair count (e.g., Yang et al. 2006):

$$\begin{equation} P(\cos \theta _{\rm ST}) = \frac{N(\cos \theta _{\rm ST})}{\langle N_{\rm R}(\cos \theta _{\rm ST}) \rangle }, \end{equation} \tag{ 4 }$$

where N(cos θ_ST) is the number of pairs for each bin in cos θ_ST, and 〈N_R(cos θ_ST)〉 is the average number of such pairs obtained from the 100 random samples. We use σ_R(cos θ_ST)/〈N_R(cos θ_ST)〉, where σ_R(cos θ_ST) is the standard deviation of N_R(cos θ_ST) obtained from the 100 random samples, to assess the significance of alignment in each bin. Since the significance is quantified with respect to the null hypothesis, we plot the error bars on top of the P(cos θ_ST) = 1 line. In addition to the normalized pair count, we also calculate the mean angle 〈cos θ_ST〉 and $\sigma _{\cos \theta _{\rm ST}}$, which is the standard deviation of 〈cos θ_ST〉_R for the 100 random samples. In this paper, cos θ_ST is restricted to the range [0, 1]. In the absence of any alignment, P(cos θ_ST) = 1 and 〈cos θ_ST〉 = 0.5. cos θ_ST = 1 implies that the galaxy spin is parallel to the vector in question, while cos θ_ST = 0 indicates that it is perpendicular to it.

Figure 2 shows the normalized pair count P(cos θ_ST) of the cosine of the angle between the galaxy spin vector and the eigenvectors t₁, t₂, and t₃ of the local tidal tensor. In the left panel, the calculation of the alignment is based on all 74, 713 centrals and 28, 511 satellite spiral galaxies selected from GZ2. In the right panel, we repeat the measurements for galaxies with sin ζ > 0.5. As expected, galaxies with sin ζ > 0.5 have stronger alignment signals because of the reduced uncertainty in the values of ζ. The average cosines and their errors are displayed for reference.

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Figure 2 shows that the galaxy spin vector has a weak tendency to be aligned with the intermediate axis t₂, of the local tidal tensor, which is consistent with the theoretical prediction using the linear TTT (Lee & Erdogdu 2007). Similar trends have also been found in simulation data. Based on the tidal field traced by halos from the N-body simulation, Wang et al. (2011) found that the spin vector of halos with masses larger than 10¹² h⁻¹ M_☉ shows significant alignment with the intermediate axis of the tidal field. Using a cosmological hydrodynamic simulation of galaxy formation in a cosmic filament, Hahn et al. (2010) found that the spin vector has a weak alignment with the intermediate axis of the tidal field for their medium mass sample at z = 0 (see their Figure 11). We can also see that the galaxy spin tends to be perpendicular to the minor axis t₃, which is consistent with the result shown in the right panel of Figure 2 in Lee & Erdogdu (2007). The alignment with the major axis t₁ is almost random.

It is interesting to see that the alignment signal for satellites is as strong as that for centrals. This may suggest that satellites can preserve their alignment with the large-scale tidal field after they have fallen into their host halos.

Since we have the halo mass for each central galaxy, we can investigate how the spin–tidal field alignment signal depends on the halo mass of groups. To that extent, we split the central galaxies into four sub-samples of halo masses. Figure 3 shows the alignment signals as a function of halo mass. As one can see, the alignment signals in the middle panels appear to be slightly stronger for centrals in more massive halos. It is also interesting to see that there is a transition of the alignment signals in the lower left panel, though the error bars are very large due to the small number of galaxies with halo masses larger than 10¹³ h⁻¹ M_☉.

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Figure 4 shows the alignment between the three eigenvectors of the local tidal tensor and the spin vector of central galaxies that are located in cluster (red), filament (orange), sheet (green), and void (blue) environments. Here we see that, for spiral galaxies in cluster environments (first row), the spin vectors have a significant alignment with the intermediate axis of the local tidal tensor. The upper right panel indicates that the spin vectors of galaxies in cluster environments have a strong anti-alignment with the minor axis of the local tidal tensor. For galaxies in sheet and filament environments, the alignments between spin vectors and the intermediate eigenvectors are rather weak. Due to the small number of galaxies in void environments, we cannot obtain any useful information for these galaxies.

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According to the definition of filament, the right panel in the second row in Figure 4 can be considered to be the normalized pair count for the cosine of the angle between the spin vector and the direction of the filament in which the galaxy resides. There is a weak tendency for filament galaxies to have their spins preferentially perpendicular to their filaments, consistent with the prediction of the TTT. However, some previous investigations found opposite alignments. Using 69 edge-on galaxies (b/a < 0.2) in filaments detected from SDSS DR5, Jones et al. (2010) found that only 14 galaxies have their spin axes aligned perpendicular to their host filaments. This indicates that the spin of the edge-on galaxies tends to be parallel to their hosted filaments. However, as we have tested, if one were to use only edge-on or face-on galaxies, one needs to have a fairly large sample of filament directions to get statistically meaningful results. Otherwise, the signals may not be reliable. For example, if we have only one filament that lies along the line-of-sight direction, the edge-on or face-on galaxies will all have perfect perpendicular or alignment signals by selection.

Based on a galaxy sample constructed from SDSS DR8, Tempel & Libeskind (2013) found that the spin axes of spiral galaxies tend to align with the host filaments that are opposite to our alignment signals. This discrepancy may be due to the different methods used to identify filamentary structures. In their investigation, filaments are traced by thin cylinders placed on the galaxy number density field, while the filaments in our samples are obtained from the continuous potential field constructed using the method of Wang et al. (2009).

For spiral galaxies in sheet environments, the left panel in the third row in Figure 4 may be considered the normalized pair count for the cosine of the angle between the spin and the normal of the sheet in which the galaxy resides. The results indicate that the spin vectors of sheet galaxies have a very weak tendency to align along the normal of the sheet. Using a sample of disk galaxies lying in the surfaces of voids obtained from SDSS DR7, Varela et al. (2012) found a significant tendency for galaxies around very large voids to have their spins aligned with the radial direction of the voids. This trend is consistent with our results. Using 178 edge-on and 23 face-on galaxies from the 2dFGRS and SDSS DR3, Trujillo et al. (2006) found that spiral galaxies located on the walls of the largest voids have spins lying preferentially on the void surface. The disagreement may again be due to the small number of galaxies Trujillo et al. (2006) used.