Do galactic bars depend on environment?: An information theoretic analysis of Galaxy Zoo 2

Observations suggest that a significant fraction of spiral galaxies in the present universe are barred (Eskridge et al., 2000; Marinova & Jogee, 2007; Barazza, Jogee, & Marinova, 2008). Even our Milky Way is known to host a bar-like structure (Binney et al., 1991; Wegg, Gerhard, & Portail, 2015). Bars are extended linear structures that results from disc instabilities (Toomre, 1964). The stellar bars transfer angular momentum to the outer disc (Lynden-Bell, 1979; Athanassoula, 2002; Berentzen et al., 2007) and also help to redistribute angular momentum between the disk and the surrounding dark matter halo (Weinberg, 1985; Debattista & Sellwood, 2000; Athanassoula, 2003; Berentzen, Shlosman, & Jogee, 2006). They are efficient in driving gas into the central regions of galaxies which can trigger starbursts and AGN activity and also help to grow a central bulge component (Schwarz, 1981; Kormendy, 1982; Shlosman, Frank, & Begelman, 1989; Hunt & Malkan, 1999; Knapen et al., 1995; Knapen, Shlosman, & Peletier, 2000; Kormendy & Kennicutt, 2004; Jogee, Scoville, & Kenney, 2005; Laurikainen, Salo, & Buta, 2004; Sheth et al., 2005; Laurikainen et al., 2007). The bars can thus play a driving role in the evolution of disk galaxies.

It is still not clear if the formation and evolution of bars are purely governed by internal secular processes. Galaxies form and evolve in diverse environments in the cosmic web. They form at the centre of the dark matter halos (White & Rees, 1978) which are embedded in different environments of the cosmic web. The mass, shape and angular momentum of the dark matter halos are known to be influenced by their large-scale environments (Hahn et al., 2007, 2007b). So the environment may impart an indirect influence on the formation and evolution of galactic bars. The different assembly history of the dark matter halos causes the early-forming low mass halos to cluster more strongly as compared to the late-forming halos of similar mass (Croton, Gao & White, 2007; Gao & White, 2007). Such clustering bias of dark matter halos may indirectly affect the formation of galactic bars. The direct external influence of environments may also play a role in the formation and evolution of bars in spiral galaxies. Studies with N-body simulations suggest that tidal interactions and the passage of a companion galaxy can trigger the formation of bars in disk galaxies (Byrd & Valtonen, 1990; Gerin, Combes, & Athanassoula, 1990; Berentzen et al., 2004; Martinez-Valpuesta et al., 2017; Łokas, 2018; Ghosh et al., 2020). These trends have been also supported by observations (Elmegreen, Elmegreen, & Bellin, 1990; Giuricin et al., 1993; Méndez-Abreu et al., 2012).

It would be interesting to know whether the formation and evolution of galactic bars are influenced by their environment. A significant number of studies have been carried out to test the correlation between the environment and the presence of bars in spiral galaxies. Thompson (1981) studies the radial distribution of barred galaxies in the Coma cluster and find that a significantly larger fraction of barred galaxies is found at the cluster core. Giuricin et al. (1993) use NGC catalogue to study the effect of local galaxy density on the presence of bars and find that the early type and low luminosity spiral galaxies in high density environments tend to be barred. Eskridge et al. (2000) find a slightly higher fraction of barred galaxies in the Fornax and Virgo clusters compared to the fields. Barway, Wadadekar, & Kembhavi (2011) study bar fraction in lenticular galaxies using SDSS DR7 and find a higher fraction of barred galaxies in clusters than in the fields. Skibba et al. (2012) study the environmental dependence of bars in spiral galaxies using the Galaxy Zoo 2 project and find that the redder galaxies with higher stellar mass are more likely to have bars. They reported a significant bar-environment correlation which shows that the barred galaxies more frequently occur in denser environments than their unbarred counterparts.

Contrary to these findings, several studies reported no dependence of bars on the environment. van den Bergh (2002) investigate the dependence of bar frequency in fields, groups and cluster environments and find no evidence for any role of environment on the formation of bars. Li et al. (2009) studied the projected redshift-space two-point cross-correlation functions of barred and unbarred galaxies in the SDSS and find that at a fixed stellar mass, the clustering of barred and unbarred galaxies are indistinguishable over the scales 20 kpc - 30 Mpc. Barazza et al. (2009) find that the fraction and properties of bars in clusters and fields are quite similar. Aguerri, Méndez-Abreu, & Corsini (2009) study bar fraction as a function of local galaxy density using SDSS DR5 and find that there is no difference in the local galaxy density of barred and unbarred galaxies. Cameron et al. (2010) investigate the evolution of bar fraction in the COSMOS field and reported that the evolution of the barred galaxy populations does not depend on the large-scale environmental density. Méndez-Abreu, Sánchez-Janssen, & Aguerri (2010) use HST ACS data to study the bar fraction in the Coma cluster and find that the bar fraction does not vary significantly in the centre and outskirts of the cluster. Martínez & Muriel (2011) study the relationship between the fraction of barred spirals and a number of environmental parameters and find that the fraction of barred spirals is insensitive to their environment. Lee et al. (2012) use SDSS DR7 to study the dependence of bars on environment and find that the fraction of barred galaxies are independent of their large-scale environment when the other galaxy properties are fixed. Marinova et al. (2012) study bars in massive disk galaxies using data from the HST ACS Treasury survey of the Coma cluster and find that the bar fraction does not show a statistically significant variation across environments.

Clearly, the correlation between the occurrence of bars in spirals and their environment still remains a debated issue and there are no clear consensus on the bar-environment correlation.

Most of the studies in this field are plagued by smaller size of data samples which made it difficult to derive statistically meaningful conclusions. The SDSS (York, et al., 2000) is the largest and most successful redshift survey to date which has provided the most detailed three dimensional map of the nearby universe and a wealth of information about the individual galaxies. The Galaxy Zoo (Lintott et al., 2008, 2011) is a citizen science project based on the SDSS and HST data which invites volunteers to help in the morphological classification of a large number of galaxies by visual inspection of their images. We plan to use data from the Galaxy Zoo 2 project (Willett et al., 2013) for the present work.

Pandey & Sarkar (2017) proposed an information theoretic framework to study the correlation between the morphology of a galaxy and its large-scale environment. They considered spirals and ellipticals as two distinct morphological classes and find a synergic interaction between morphology and environment up to a length scale of $30{\,h^{-1}\,{\rm Mpc}}$. Recently Sarkar & Pandey (2020) show that the observed excess mutual information between morphology and environment are statistically significant at $99.9\%$ confidence level. Another study by Bhattacharjee, Pandey, & Sarkar (2020) show that a conditioning on stellar mass does not explain the statistically significant mutual information between morphology and environment on larger length scales. Galactic bar is an important morphological feature based on which a spiral galaxy is further classified as barred or unbarred. It would be natural to ask if there exists a bar-environment correlation similar to the correlation observed between morphology and environment.

We would like to measure the mutual information between barredness of a galaxy and its environment on different length scales. One can randomize the information about barredness and also shuffle the spatial distribution after dividing it into smaller sub-cubes. Comparing the mutual information in these distributions with that from the original distribution would allow us to test the statistical significance of any observed correlation between barredness and environment (Sarkar & Pandey, 2020). In this work, we use this information theoretic framework to test the large-scale environmental dependence of galactic bars if any.

The local density of galaxies governs the various external influences such as tidal interactions which may act as an external trigger for bar formation. Comparing the number density of galaxies at the locations of the barred and unbarred spirals can elucidate this issue. Keeping this in mind, we also separately test any effects of local density on the presence of galactic bars in spiral galaxies.

We now have a magnitude limited sample of the spiral galaxies defined within a cubic region of size $L{\,h^{-1}\,{\rm Mpc}}$. We divide this cube in $N_{d}=n_{g}^{3}$ voxels with size $d=(\frac{L}{n_{g}}){\,h^{-1}\,{\rm Mpc}}$. One can have different sets of voxels by changing the number of grids $n_{g}$. Different choices of $n_{g}$ allow us to study the environment on different length scales.

Let us now define a discrete random variable $X$ that represents the environment at a certain length scale. The probability of finding a randomly selected galaxy in the $i^{th}$ voxel will be $p(X_{i})=\frac{N_{i}}{N}$, where $N_{i}$ is the number of galaxies in the $i^{th}$ voxel and $N=\sum_{i=1}^{N_{d}}{N_{i}}$ is the total number of galaxies in the entire cube. The random variable $X$ have $N_{d}$ outcomes ($\{X_{i}:i=1,2,....N_{d}\}$).

The information entropy (Shannon, 1948) associated with the random variable $X$ on length scale of $d{\,h^{-1}\,{\rm Mpc}}$ is given by,

Let us now define another variable $Y$ that represents the presence or absence of bar in spiral galaxies. Here, $Y$ only takes two values $Y_{1}$ for barred & $Y_{2}$ for unbarred. If there are $N_{1}$ barred and $N_{2}$ unbarred galaxies in the cube then the Shannon entropy associated with the variable $Y$ will be

Here, $N=N_{1}+N_{2}$ is the total number of galaxies in the sample. Any variation in the voxel size will not change the number of barred and unbarred galaxies and the value of $H(Y)$ is independent of the grid size.

We also calculate the joint entropy for the variables $X$ and $Y$. Which is given by,

Here $N_{ij}$ is the number of galaxies that resides in the $i^{th}$ voxel and belongs to the $j^{th}$ morphology. So we have,

If the two variables $X$ and $Y$ are uncorrelated then $H(X)+H(Y)=H(X,Y)$. Otherwise the joint entropy would be smaller than the some of the individual entropies, i.e. $H(X,Y)<H(X)+H(Y)$.

We calculate the mutual information between the two variables $X$ and $Y$ as,

The mutual information measures the amount of information shared between two random variables. In other words, it is the reduction in uncertainty in the outcome of one random variable due to the pre-existing knowledge of the other. Higher the mutual information, greater is the association between the two variables. The mutual information measures the association between two random variables irrespective of the nature of the random variables and their relationship.

Understanding the role of bars in spiral galaxies are central to understanding their formation. Bars are known to be the most efficient means to redistribute materials inside a galaxy. It is yet not clearly known why some spiral galaxies host a bar while others do not. The detail process of bar formation may be governed by several factors. Both the internal secular processes and external triggers may induce the bar formation in a galaxy. Besides, the large-scale environment and the assembly history of dark matter halos may also have an influence on the formation of galactic bars. All these possibilities must be tested against observations to identify the most influential factors governing the formation of galactic bars.

We have calculated the mutual information between the barredness of a galaxy and its environment on different length scales using the SDSS and Galaxy Zoo 2 data. We randomize the bar/unbar classification of galaxies and measure the mutual information between the barredness and environment. There are no statistically significant change in the mutual information between barredness and environment after the classifications are randomized. We also shuffle the spatial distribution of SDSS galaxies after dividing it in smaller sub-cubes and randomly interchanging their spatial locations along with random rotations. We also do not observe any statistically significant difference in the mutual information between the barredness and the environment after the data is shuffled on different length scales. The analysis do not provide any strong evidence against the null hypothesis which suggests that the large-scale environment of barred and unbarred galaxies are similar and there are no correlations between the barredness of a galaxy and its large-scale environment.

We also separately test any possible influence of local density on the presence of galactic bars. We measure the local density at the locations of barred and unbarred galaxies and then compare their cumulative distribution functions using a Kolmogorov-Smirnov test. The test favours the null hypothesis which indicates that the local density of barred and unbarred galaxies are quite similar. A study of the bar fraction in nearby galaxy clusters suggests that the bar formation in low-mass galaxies are expected to be more susceptible to their environment than the bright or massive galaxies (Méndez-Abreu et al., 2012). The volume limited sample analyzed in this work consists of brighter galaxies for which the bar formation may be unaffected by their environment.

In the present work, we explore any possible role of the small-scale and large-scale environments of galaxies on the formation of galactic bars. Our analysis clearly indicates that the presence or absence of bars in spiral galaxies do not depend on either their small-scale or large-scale environments. This suggests that the formation of galactic bar is largely decided by the internal processes of the host galaxy.

The data underlying this article is available in https://skyserver.sdss.org/casjobs/ . The datasets were derived from sources in the public domain: https://www.sdss.org/ and http://zoo1.galaxyzoo.org .

We sincerely thank an anonymous reviewer for insightful comments and suggestions. The authors would like to thank the SDSS and Galaxy Zoo team for making the data public. The authors also acknowledge the efforts of the Galaxy Zoo 2 volunteers for the detailed visual morphological classifications of the SDSS galaxies, without which this work would not be possible.

BP would like to acknowledge financial support from the SERB, DST, Government of India through the project CRG/2019/001110. BP would also like to acknowledge IUCAA, Pune for providing support through associateship programme.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS website is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.