A mechanism to explain galaxy alignment over a range of scales

There currently exists considerable evidence that the galaxies show alignment with one another over Mpc to Gpc distance scales, although the observations still lack firm confirmation. The first evidence was presented in Hutsemékers (1998) who found that the polarization of visible radiation from distant quasars is aligned over very large distance scales of order Gpc. The signal was confirmed by further investigations (Hutsemékers & Lamy, 2001; Jain et al., 2004; Hutsemékers et al., 2005). A signal of alignment of radio polarizations has also been observed on smaller scales of order 100 Mpc (Tiwari & Jain, 2013; Pelgrims & Hutsemékers, 2015). Mandarakas et al. (2021) claim to find four regions of radio galaxy axis alignment on distance scale of order $\approx 400$ Mpc at a significance level of more than $5\sigma$. Furthermore several studies have claimed alignment of radio galaxy axis on distance scales of order 20 to 60 Mpc (Taylor & Jagannathan, 2016; Contigiani et al., 2017; Panwar et al., 2020). However, the signal of alignment appears to be absent in Astrogeo database radio jets (Blinov et al., 2020) and in polarization of radio sources observed with IRAM (Institute for Radio Astronomy in the Millimeter Range) 30m telescope (Tiwari & Jain, 2019).

There exist many theoretical attempts to explain these observations. It has been proposed that the effect is related to the correlation of magnetic field on cosmological (Agarwal et al., 2011; Agarwal et al., 2012) or spercluster distance scales (Tiwari & Jain, 2016). A hypothetical light pseudoscalar field has also been invoked to explain the large scale optical alignment (Jain et al., 2002; Agarwal et al., 2011; Piotrovich et al., 2009; Agarwal et al., 2012). One potential problem with this explanation is the relatively large circular polarization predicted by this phenomenon (Hutsemékers et al., 2010; Payez et al., 2011). However, there could be sources of decoherence, for example a fluctuating background magnetic field can significantly reduced the circular polarization in comparison to linear polarization (Agarwal et al., 2012). Other explanations include, vector perturbations (Morales & Saez, 2007), dark energy (Urban & Zhitnitsky, 2010), cosmic strings (Poltis & Stojkovic, 2010; Hackmann et al., 2010) violation of isotropy (Ciarcelluti, 2012) and a superhorizon perturbation (Chakrabarty, 2016).

Recently, Tovmassian & Torres-Papaqui (2020) explored the distribution of position angles (PA) of galaxies inside clusters and reported alignment, i.e. the galaxies’ PAs are not oriented randomly. If the Universe is homogeneous and isotropic on large scales, we expect random orientations of galaxies if a large enough volume is considered. However, for cluster scales the local physical processes will dominate, and the orientation of galaxies can be anisotropic. The alignment of galaxies within cluster is presumably related to galaxy formation, the tidal gravitational fields in the large-scale structure can align nearby galaxies (Tempel & Libeskind 2013; Zhang et al. 2013; Kiessling et al. 2015 and references therein). The orbital angular momentum and hence the background magnetic field must also play a role in such an alignment, which is so far not completely understood. Furthermore, the visible galaxy shapes are distorted from weak-lensing and contribute to galaxy alignment (Miralda-Escude, 1991; Van Waerbeke et al., 2000; Bartelmann & Schneider, 2001; Catelan et al., 2001). To explore galaxy formation there have been many efforts over several decades for the study of the distribution of the PA of galaxies in clusters (Sastry, 1968; Adams et al., 1980; Fong et al., 1990; Fuller et al., 1999; Chen et al., 2019). Tovmassian & Torres-Papaqui (2020) further explore the galaxy alignment within clusters and assume alignment if, in a cluster, the galaxy number at one $90^{\circ}$ position angle interval is more than twice higher than the other $90^{\circ}$ interval and report relatively large alignment for galaxies in isolated clusters. Tovmassian & Torres-Papaqui (2020) propose that the galaxies in clusters could have been aligned at the beginning but with time the accretion and mutual interactions of galaxies could have randomized galaxy orientations, and thus in the rich clusters the alignment signal may not be significant.

The signal of alignment in radio galaxy PAs has also been probed (Panwar et al., 2020) in the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST; Becker et al. 1995; Helfand et al. 2015) catalog. The authors report a significant signal of alignment for scales less than $1^{\circ}$ which corresponds to a distance scale of about 28 Mpc, consistent with what has been observed earlier (Taylor & Jagannathan, 2016). With their 3D analysis, Panwar et al. (2020) report mild alignment at larger distances.

Our basic idea is that there exists a background magnetic field which has spatial correlations over wide range of length scales. This field interacts with the charged particles in the radio galaxies and tends to orient their jet axis along the background field direction. Hence the galaxy axes acquire the spatial correlations of the background field. Detailed numerical simulations are required in order to determine the precise nature of the induced correlations. Such simulations are so far not available in literature. However, we expect that a sufficiently strong background magnetic field acting for large enough time will have a tendency to align the individual galaxies within a cluster. Hence, as a simple approximation, we assume that the jet axes are aligned in exactly the same direction as the background magnetic field at the position of the galaxy. A more detailed study may relax this assumption by proposing a suitable distribution function of jet axes, which may lead to only partial correlations to the background field direction. We postpone this to future work. We point out that this mechanism would also lead to correlations in the integrated radio polarizations. This is because these polarization angles are known to be correlated with the jet axis (Gabuzda et al., 1994; Lister & Smith, 2000; Pollack et al., 2003; Helmboldt et al., 2007; Joshi et al., 2007). The model is described in more detail in Tiwari & Jain (2016).

The origin of the background magnetic is assumed to be primordial and has correlations over wide range of length scales, including very large distances of order Gpc. Such correlations may explain the alignment signal of large distance scale quasar optical polarizations Hutsemékers (1998) by inducing intrinsic alignments in quasars on such large distance scale. We assume that the magnetic field can be defined by a single power law model over these wide range of scales from Mpc to Gpc. Here we shall be interested in the application of the model on alignment at relatively small distance scales of order less than 30 Mpc.

Let $\vec{B}_{i}(\vec{r})$ represents the $i^{\rm th}$ component of cluster magnetic field at position $\vec{r}$. We decompose it in terms of its Fourier components $b_{i}(\vec{k})$, such that,

where $V$ is the volume. It is reasonable to assume that the magnetic field is uncorrelated in Fourier space. Hence we can write the two point correlations as (Tiwari & Jain, 2016),

where $M(k)$ is the spectral function,

and $k=|\vec{k}|$. The two point correlations in real space can be expressed as,

where $W(kr_{c})$ is a window function,

and $r_{c}=k_{c}^{-1}$ is a suitably chosen cutoff scale.

It is reasonable to assume a power law form of the spectral function, that is,

On the cluster scale, simulations suggest that the spectral index may be roughly equal to $-2.7$ (Dolag et al., 2002). The parameter $A=V\pi^{2}B_{0}^{2}\frac{3+n_{{}_{b}}}{k_{c}^{3+n_{{}_{b}}}}$, is fixed by demanding $\sum_{i}\langle B_{i}(\vec{r})B_{i}(\vec{r})\rangle=B_{0}^{2}$, where $B_{0}$ is the magnetic field averaged over the scale $r_{c}$. We add that in our model we assume one-to-one mapping between radio galaxy orientation and the magnetic field and thus the magnitude $B_{0}$ remains insignificant in our analysis. Relaxing this assumption or considering $B_{0}$ dependent mapping between radio galaxy orientation and the magnetic field or both will result in a weaker alignment. Choosing a coordinate system such as $\vec{k}$ points in $\hat{z}$ direction, $P_{xx}=P_{yy}=1$ and all other components $P_{ij}=0$. The only non-zero terms, $xx$ and $yy$ are therefore,

We substitute $A=V\pi^{2}B_{0}^{2}\frac{3+n_{{}_{b}}}{k_{c}^{3+n_{{}_{b}}}}$ in equation 7, and plot magnetic field correlations in figure 1.

We generate the correlated magnetic field and assume the magnetic field direction as a proxy to galaxy PA. For generating the magnetic field, we follow the procedure described in Agarwal et al. (2012) and in Tiwari & Jain (2016). We consider 3D discrete space consisting of a large number of grids of equal size. The magnetic field is assumed to be uniform in each grid. The magnetic field in the k-space is uncorrelated, with its component $b_{k}$ along $\vec{k}$, equal to zero. For simplicity, we use polar coordinate ($\vec{k}$, $\theta$, $\phi$), then the remaining two orthogonal component of the magnetic field in k-space, $b_{\theta}$ and $b_{\phi}$ can be considered independent. Next, the k-space magnetic field is obtained by setting $b_{k}=0$, and generating $b_{\theta}$, $b_{\phi}$ by assuming normal distribution:

where, N is the normalization factor. We next obtain the real space magnetic field by Fourier transforming the k-space field. We consider a large volume in our simulations, and thus $r_{\rm max}\approx\infty$, i.e., the lower limit of the integration in equation 4, $k_{\rm min}=r^{-1}_{\rm max}$ is set to zero. The real space magnetic field thus obtained follows the correlations governed by field spectrum in equation 6.

Our results show that we are able to explain the significant signal observed in the FIRST catalog for angles less than 1 degrees using with $n_{{}_{b}}=-2.62$. With this value we are also able to explain the alignment seen inside clusters over distance scales less than 2 Mpc to some extent. However the alignment signal in this case shows a relatively mild dependence on the spectral index and does not necessarily require the value $n_{{}_{b}}=-2.62$. Furthermore, the same model explains the alignment signal observed in JVAS/CLASS data (Tiwari & Jain, 2013, 2016) with slightly higher spectral index equal to $-2.74\pm 0.04$. Apparently it seems that the galaxy alignments at all scales and at all distances (redshifts) can be modeled as a power law in k-space. The best fit spectral index slightly differs for different data sets. However, the data is likely to have considerable errors and hence the difference is not too significant. Also, relaxing one-to-one mapping between radio galaxy orientation and magnetic field, and counting on evolution due to structure formation may result a consistent background primordial spectral index $n_{b}$. Alternatively, it is possible that the power law model is too simple. A simple generalization would be to introduce a scale and position dependent spectral index. One can further generalize the model and consider background magnetic field magnitude, $B_{0}$, dependent mapping between radio galaxy orientation and magnetic field. With these generalizations the effective value of $B_{0}$ and $n_{{}_{b}}$ could differ for different data sets.

For the FIRST data, the theoretical estimate of the statistic $S_{D}$ starts to deviate from data for angles larger than 1 degree. This may arise if the spectral index of the magnetic field changes significantly beyond this angle. Another possible explanation is that here we are using only two dimensional analysis. Hence beyond a certain angular distance it may no longer be reasonable to assume that most the sources within this angular separation are spatially close to one another. A proper analysis is only possible once we have redshift information for these sources. The data set for which this parameter is available is rather limited and hence a detailed analysis is best postponed to future work.

In this paper we have argued that the alignment seen in the radio galaxy axis over distance scales of few Mpc to several tens of Mpc may be explained in terms of correlations in magnetic field in real space. We emphasize that the magnetic field correlations of similar nature (i.e. approximately same spectral index) can explain the radio polarization alignment observed with JVAS/CLASS data over 100 Mpc scale (Tiwari & Jain, 2016) and potentially also explain the quasar alignment observed over Gpc (Agarwal et al., 2012). Hence we find that, remarkably, the magnetic field correlations seem to explain observed alignment signals over all scales. The magnetic field is assumed to be uncorrelated in Fourier space, i.e. it is proportional to $\delta_{\vec{k},\vec{q}}$, with its two point correlations described by a spectral function $M(k)=Ak^{n_{{}_{b}}}$ with the spectral index $n_{{}_{b}}<0$. Theoretical simulations suggest $n_{{}_{b}}\approx-2.7$. The magnetic field acquires correlations over large distances in real space, which increase with increase in the magnitude of $n_{{}_{b}}$, i.e. as the spectral index becomes more negative. We assume a simple model in which we set the galaxy axis along the direction of the background magnetic field. A future refinement of our work would be to relate the alignment in galaxy axis to those in the magnetic field through simulations. This will allow us to go beyond our assumption that the galaxy axis points along the background magnetic field direction. Using the model described above, we determine the spectral index that fits the data for the alignment signal seen in galaxies inside clusters on distance scale of 2 Mpc Tovmassian & Torres-Papaqui (2020) as well as on large scale of about 28 Mpc, seen in the FIRST catalog Panwar et al. (2020). We find that FIRST alignment signal can be explained if we choose the spectral index $n_{{}_{b}}=-2.62$. In our model the alignment within clusters at distance less than 2 Mpc does not show a strong dependence on $n_{{}_{b}}$ nonetheless, considering the best fit spectral index using the FIRST or JVAS/CLASS data the model also provides a reasonable explanation of the galaxy alignment at such short distance scales. The value $n_{{}_{b}}=-2.62$ is dominantly selected by analysis of the FIRST data. It is encouraging that the value obtained by this analysis is close to that expected from simulations.

We thank Mohit Panwar for making FIRST catalog used in their work available to us. This work is supported by the RFIS grant (No. 12150410322) by the National Natural Science Foundation of China (NSFC) and the support by the National Key Basic Research and Development Program of China (No. 2018YFA0404503) and NSFC Grants 11925303 and 11720101004 and through the DST-SERB grant number EMR/2016/004070.

The data that support the analysis and plots within this paper are available from the corresponding author upon reasonable request.