Spin Parity of Spiral Galaxies III – Dipole Analysis of the Distribution of SDSS Spirals with 3D Random Walk Simulations

Number statistics can be used to study the S/Z number asymmetry from equipartition. The significance level of the number asymmetry is given by

Another option is to look for any statistically significant dipole in the spatial distribution of S/Z-spirals.

Let $\mathbf{\Omega}^{i}(l^{i},b^{i},d^{i})=(\Omega_{l}^{i},\Omega_{b}^{i},\Omega_{d}^{i})$ be the spin vector of the $i$-th galaxy with galactic longitude $l^{i}$, galactic latitude $b^{i}$, and distance $d^{i}$. The distance $d^{i}$ to the galaxy can be approximated for the nearby universe by $d^{i}=cz^{i}/H_{0}$, where $c$ is the speed of light, $z^{i}$ is the spectroscopic/photometric redshift of the $i$-th galaxy and $H_{0}$ is the Hubble constant.

Measuring the 3D vector $\mathbf{\Omega}^{i}$ is nontrivial, but one can easily judge the sign $h^{i}$ of $\Omega_{d}^{i}$. This can be done by examining the spiral winding sense to see if it is S-wise ($h^{i}=-1$) or Z-wise ($h^{i}=+1$), since all spiral galaxies can be assumed to be trailing (Iye et al. , 2019, : Paper I). The net effect of misidentifying S-spirals vs Z-spirals and improbable presence of leading spiral arms would be reduction of the dipole strength, should it exist. For simplicity, hereafter we assume that all the $\mathbf{\Omega}^{i}$ have an unit length, namely $\Omega_{d}^{i}=h^{i}$.

By compiling the spin parity, $h^{i}=\pm 1$, together with the coordinates of spiral galaxies from image archives such as SDSS, Pan-Starrs1, ESO-DR2, DES, Subaru HSC, and others, we estimate that one can produce a spin parity catalog of up to a million spiral galaxies in the 3D volume within 1 Gpc of the Earth. We are developing an analysis scheme to probe any partial volume within this volume, not necessarily centered on the Earth. This will be discussed in a forthcoming paper. For the moment, however, we consider a local volume centered on the Earth.

To analyze the spin vector distribution in terms of spherical harmonics expansion, one can first examine on its dipole component $Y_{1}^{0}$, quadrupole component $Y_{2}^{0}$ and higher components.

In fact, $D_{max}$ can be obtained simply by calculating a vector $\mathbf{G}$, which is the vector sum of the unit radial vectors pointing to the direction of the $i$-th galaxy multiplied by the helicity $h^{i}$. The inner product of $\mathbf{G}$ with $\mathbf{P}$ takes the maximum value when $\mathbf{P}$ is pointing in parallel to $\mathbf{G}$, and then $D_{max}$ is $\|\mathbf{G}/N\|$.

We assume an extreme case of perfect dipole segregation, where all S-spirals are in the northern hemisphere and all Z-spirals are in the southern hemisphere as shown in Figure 2(a). The weight factor $\cos\theta^{i}$ of Equation 2 shows that the galaxies toward the dipole axis with small $\theta^{i}$ add to $D_{max}$ while those near the equator with $\theta^{i}\sim\pi/2$ reduce $D_{max}$. It is easy to see by integration that the expected mean amplitude $D_{max}^{perfect}$ of all sky sampling is 0.5. Perfect but random dipole distribution produces fluctuation around this expected mean amplitude of 0.5. Monte Carlo simulation shows that the associated standard deviation from this mean amplitude for perfect random dipole distributions is $\sim 0.3/\sqrt{N}$.

Although the currently available data from imaging surveys are growing rapidly, the data do not yet fill the entire sky uniformly. There may be observational biases that affect evaluation of S/Z number asymmetry and/or the evaluation of the observed dipole vector in the S/Z distribution. Let us examine the effect of non-uniform sky coverage for the perfect dipole distribution case mentioned in the previous subsection.

When the sky sampling is limited to a certain narrow cone direction, the dipole amplitude can take any value in the range $0\leq D_{max}\leq 1$ depending on the direction to the sample. The largest amplitude $D_{max}=1$ is obtained when the sampling is toward the pole, $\theta=0$ or $\pi$, and the lowest amplitude $D_{max}=0$ is observed toward the equator $\theta=\pi/2$. Therefore depending on the direction of the biased small sky sampling, the resulting $D_{max}$ can be larger or smaller than the whole sky sampling.

Consider a non-uniform sampling of the complete dipole distribution covering only the northern hemisphere as shown in Figure 2(b). There would be a conspicuous number count asymmetry. However, the dipole strength observed would be equal to that of the whole sky sampling. If the sampling were limited to the eastern hemisphere as shown in Figure 2(c), there would be no number asymmetry. The dipole amplitude, again, would be equal to that of full sky sampling.

The number asymmetry and the dipole amplitude thus provide complementary information to analyze large scale symmetry-breaking in the spin distribution of galaxy ensembles.

For a uniformly randomly distributed set of $N$ vectors with $\mathbf{\Omega^{i}}$, the resultant vector sum $\mathbf{D_{max}}$ will have an isotropic distribution with non-zero amplitude, unless the summation has incidentally completely canceled $\mathbf{D_{max}}$.

As $D_{max}$ can be calculated by $\|\mathbf{G}/N\|$, our problem is equivalent to a well-studied mathematical problem, 3D random walk (random flight). Namely, it is equivalent to find the distribution of distance to the final point vector $\bf{R}$ of a particle that starts from the origin and takes $N$ steps of a 3D random walk. One can evaluate the mean amplitude of $D_{max}$ and the standard deviation around the mean amplitude as a function of $N$.

The distribution of $D_{max}^{2}$, therefore, follows the chi-squared distribution for three degrees of freedom. The distribution of $D_{max}$, hence, follows the chi distribution¹¹1Weisstein, Eric W. ”Chi Distribution.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ChiDistribution.html, a square root of chi- squared distribution. The expected mean amplitude $\overline{D}_{max}$ is

The associated standard deviation from this expected mean distance is given by

We can use these formulae to evaluate the statistical significance $\sigma_{D}$ of the observed spin dipole strength $D_{max}$ in any ensemble of spiral galaxies using

Consider an ensemble of galaxies where a perfect dipole system and a uniform random system are mixed with fractions $p$ and $1-p$, respectively. The observed dipole vector $\mathbf{D_{max}}$ would be, on average, a vector sum of $\mathbf{D_{max}^{perfect}}$ with an amplitude $0.5p$ and a random vector with an amplitude $(1-p)\times 0.921/\sqrt{N}$ in an arbitrary direction. To discern the intrinsic dipole from the random dipole with a statistical significance of $s$-sigma, the following relation is required

This implies

The Equation 8 indicates that between one hundred thousand or one million spirals are necessary to detect a 3%, or 1% residual inherent dipole system at $5\sigma$ confidence level, respectively.

Real data are not always obtained uniformly. The quantitative discussion of the non-uniform sampling of a random distribution in the general case is not straightforward. We examine, however, the actual S/Z data of SDSS spirals in the next section and compare the observed dipole amplitude with those expected from simulated random distributions.

To apply our formulation of dipole anisotropy to real data, we studied two samples using Shamir’s catalog based on SDSS photometric data (Shamir, 2017a)²²2https://data-portal.hpc.swin.edu.au/dataset/data-for-galaxy-assymetry-experiment. The first sample retains all 162,516 spirals from the original catalog. The original sample shows an S/Z dipole signal of $D_{max}=0.00489$, with its axis pointing toward $(l,b)=(189^{\circ},+15^{\circ})$. This axis coincides with that reported in Shamir (2020a), ($\alpha=88^{\circ},\delta=+36^{\circ}$), which is ($l=175^{\circ},b=+5^{\circ}$) for an SDSS sample, considering the $1\sigma$ estimation error of about $30^{\circ}$ in both coordinates.

For calibration, we made 50,000 independent Monte Carlo simulations by assigning $h^{i}=\pm 1$ randomly to the 162,516 spirals and measured the simulation’s $\mathbf{D_{max}}$. The resulting $\mathbf{D_{max}}$ shows an isotropic distribution with an ensemble mean amplitude of $D_{max}=0.00225$ and an associated amplitude standard deviation of $\sigma=0.00105$, as shown in Table 1. The observed $D_{max}$ from the original catalog, therefore, has an amplitude, that is larger than the mean amplitude by $\sigma_{D}=2.52$. The number dominance of Z-spirals over S-spirals here is $\sigma_{N}=4.89$.

The second sample we studied is a volume-limited sample retaining 111,867 spirals with measured redshift in the range $0.01\leq z\leq 0.1$. To avoid any possible effect of local peculiar motions, 162 nearby spirals at $z\leq 0.01$ were removed from this volume-limited sample. This sample shows a stronger S/Z dipole signal $D_{max}=0.00773$ with its axis pointing toward $(l,b)=(138^{\circ},-38^{\circ})$.

The ${D_{max}}$ value observed for this sample, together with that of 50,000 simulated mock samples is shown in Figure 3. The observed $D_{max}$ from the original catalog limited to a volume defined by redshift, therefore, has an amplitude, that is larger than the mean amplitude by $\sigma_{D}=4.00$.This amplitude happens to be close to the value $4.34\sigma$ reported in Shamir (2020a). The number dominance for this sample is $6.36\sigma$.

Upon re-examining Shamir’s original catalog, we found significant duplication of entries in its tables. Apparently, Shamir (2017a) used PhotoObjAll to search the SDSS catalog. According to the Table Description of DR8³³3http://skyserver.sdss.org/dr8/en/help/docs/tabledesc.asp, the view of PhotoPrimary, instead of PhotoObjAll, should be used to avoid duplicate detections. PhotoObjAll returns every entry from the searched images without checking for duplication of the same object.

In fact, 34,198 spiral galaxies were found to have multiple entries with their coordinates within 3 arcsec distance of each another. For instance, one of the galaxies in his catalog, SDSS J031945.63-000437.9 (objid=1237666300555427954) at $z=0.037$, has 89 separate entries. We confirmed by visual inspection that all of them are the same galaxy.

A total of 106 spirals had contradicting S/Z classification among the duplicated entries. We rectified these contradictions by assigning a spin parity via visual inspection in some cases. In other cases, where the voting was significantly split, we adhered to the outcome of majority voting.

After removing the duplicated entries, the number of spiral galaxies is reduced to 72,888, that is $45\%$ of the original number. The number of spirals in the volume-limited sample range $0.01\leq z\leq 0.1$ is 48,089 (23,819 S-spirals and 24,270 Z-spirals), 43% of the 111,867 counted in the original catalog. The measured amplitudes for these ’cleaned’ samples with and without the volume limitation are summarized in Table 1.

The observed dipole for the entire cleaned sample set has $D_{max}=0.00535$ with its axis pointing toward $(l,b)=(192^{\circ},+79^{\circ})$. 50,000 random mock simulations for the cleaned sample shows a mean amplitude $D_{max}=0.00336$ and a standard deviation $\sigma=0.00154$. Therefore, the observed dipole deviates from the expected mean strength only at a level of $\sigma_{D}=1.29$. Number dominance of Z-spirals over S-spirals is also only at the $\sigma_{N}=1.87$ level.

The observed dipole for the cleaned sample limited to the redshift range $0.01\leq z\leq 0.1$, is $D_{max}=0.00468$ with its axis pointing toward $(l,b)=(106^{\circ},+57^{\circ})$. Results of 50,000 random mock simulations for the cleaned sample shows a mean amplitude $D_{max}=0.00414$ and a standard deviation $\sigma=0.00188$, as shown in Figure 4.

Therefore, the observed dipole deviates from the expected mean strength only at the $\sigma_{D}=0.29$ level. Column (13) of Table 1, obtained from Equation 6 for uniform sampling is not necessarily correct for non-uniform sampling. However, the fact that Column (13) shows values close to the values in column (12), obtained from actual sampling, suggests their relevance. Finally, the number dominance of Z-spirals over S-spirals is also only at the $\sigma_{N}=2.06$ level.

Comparison of Figures 3 and  4 clearly shows that the apparent pseudo dipole signal observed in Figure 3 came from massively duplicated data in the original catalog.

As a final remark, Longo (2011) made a similar analysis of his sample of 15,158 spirals with $z\leq 0.085$ from SDSS DR6. He used the terms left-handed (S-spiral in the present paper) and right-handed (Z-spiral). He found a dipole asymmetry by plotting the distribution of $A=(Z-S)/(Z+S)$. The dipole strength under his definition was $-0.0408\pm 0.011$, found with an axis pointing at $(l,b)=(52^{\circ},+68.5^{\circ})$ at $5.6\sigma$. The relation of his study to the current work has yet to be investigated.