Continuous Filament Network of the Local Universe

The notion of a cellular large-scale structure (LSS; de Lapparent et al. 1986) of the universe has long been recognized as an important component of the astronomical picture of the world (Bernardeau et al. 2002). Optical radiation, coming from the stellar content of a galaxy, is the most accessible for observations, but a much larger mass is concentrated in dark matter (DM), whose distribution can be detected by galactic flows (Courtois et al. 2013), gravitational lensing (Kaiser & Squires 1993), X-ray radiation of clusters (Reiprich & Böhringer 2002), via the Sunyaev-Zeldovich effect (Sunyaev & Zeldovich 1980), and other methods (Shulga et al. 2014). The spatial structure of the cosmic web is reflected primarily in needle-like filaments. Clusters of galaxies occupy a small volume, voids contain a small number of galaxies, and walls contain a number of filaments (van de Weygaert & Schaap 2009; Cautun et al. 2013). Through the structure of the cosmic web one can determine the properties of DM (Alexander et al. 2022) and the expansion laws of the universe, or validate the theory of inflation (Springel et al. 2006). The localization of voids (Mao et al. 2017; Peper et al. 2023) can aid cosmological tests (Paz et al. 2023) and the study of the few galaxies that occupy them (Porter et al. 2023).

Two principles lie at the heart of the LSS theory: (i) the hypothesis of the Gaussian distribution of initial density perturbations, and (ii) Zeldovich's approximation within which one can describe their evolution (van de Weygaert 2002). Most of the methods for describing the LSS have been tested via numerical simulations (Libeskind et al. 2018; Ganeshaiah Veena et al. 2019; Bonnaire et al. 2020; Banfi et al. 2021; Zhang et al. 2022), and only a few have been applied to actual observational surveys (Sousbie 2011; Sousbie et al. 2011; Tempel et al. 2014; Chen et al. 2016; Carrón Duque et al. 2022). A detailed comparison of several modern methods of filament detection is given by Libeskind et al. (2018).

A common method for detecting cosmic structures in surveys like the Sloan Digital Sky Survey (SDSS; York et al. 2000; Abazajian et al. 2009) is a topological method called Discrete Persistent Source Extractor (DisPerSE, Sousbie 2011; Sousbie et al. 2011), based on the Delaunay tesselation of the discrete sample of points symbolizing galaxies, which allows one to extract filaments, walls, and voids. DisPerSE was applied to illustrate the effect of modified theories of gravity on the length, mass, and thickness of filaments (Ho et al. 2018). A method of approximating the distribution of galaxies with cylinders (the Bisous model; Stoica et al. 2010) was used by Tempel et al. (2014) and Muru & Tempel (2021). As a result, the galaxies were grouped into small cylinders (with at least three galaxies inside each cylinder) and were combined into a larger filament net. It was confirmed that about 40 per cent of the mass is gathered in the filaments (Forero-Romero et al. 2009; Jasche et al. 2010), while occupying only 8 per cent of the volume. The detection of filaments was also performed by Chen et al. (2016) and Carrón Duque et al. (2022) in redshift layers with radial thicknesses of about 20 Mpc. This approach allows to compensate for the effect of peculiar velocities of galaxies in local gravitational fields by solving a simplified two-dimensional problem, mitigate redshift-space distortions, correlate the projections with maps of the cosmic microwave background, trace the evolution of the LSS throughout cosmic time, etc. One particular disadvantage is the insensitivity to filaments oriented close to the line of sight and inaccurate detections regarding filaments spanning a few neighboring redshift layers, but the choice of the thickness of the redshift layers is expected to balance the biases and errors. Chen et al. (2016) used a variant of a ridge filter applied directly to data, and obtained a filament net within 0.05 < z < 0.7, whereas Carrón Duque et al. (2022) extended the sample to redshifts up to z = 2.2 and employed spherical coordinates to bypass projection effects. The resulting nets are consistent with the local overdensities of the cosmic web, but both works also contain some lone, very short filaments that are not connected to the rest of the network and are not obviously associated to any bigger structure, as well as gaps in the overall network.

The ambiguities of different algorithms and observational restrictions urge the development of more careful methods for detecting and connecting filaments. A way to perform general statistical analyses of the LSS is to generate the apparent distribution of galaxies assuming a certain matter density field (Voitsekhovski & Tugay 2018). This will determine the overall statistical characteristics of the filaments and therefore describe the LSS of the universe as a whole. It also allows to produce mock data sets with statistical properties matching those of actual observational surveys. Nowadays, only the closest affiliates of the Local Supercluster (Kim et al. 2016; Castignani et al. 2022), which are associated with the accumulation of Virgo galaxies, are most prominently identified.

Early galaxy color studies were based on the development and implementation of different photometric systems with appropriate photometric stellar standards ³ (de Vaucouleurs 1961; Sandage & Visvanathan 1978; Fukugita et al. 1995). Detailed color studies were conducted for early-type (elliptical) galaxies (Bernardi et al. 2003, 2005), as well as for late-type (spiral) galaxies (Tully et al. 1982; Peletier & de Grijs 1998; Fraser-McKelvie et al. 2016).

Galaxy color distributions are commonly explained by a halo model that links the galaxy color to the mass of the DM halo (Skibba & Sheth 2009; Hearin & Watson 2013). Colors are also correlated with luminosities and morphological types of the galaxies (Kodama & Arimoto 1997; Kodama et al. 1998). Such relations were analyzed by Mobasher et al. (1986) and Bower et al. (1992a, 1992b) for nearby galaxies in the Virgo and Coma clusters, and by Lange et al. (2015) for more distant galaxies at z < 0.1. An important feature of galaxy colors is a bimodal distribution that relates to spiral and elliptical galaxies (Strateva et al. 2001). This is clearly seen in the SDSS (Baldry et al. 2004) and Millenium Galaxy Catalog (Driver et al. 2006), as well as for distant galaxies at z < 2.5 (Brammer et al. 2009). Bimodality was also found in the two-dimensional color–color distribution (Park & Choi 2005). The bimodal character appears in cosmological computer simulations as well, e.g., Illustris (Nelson et al. 2018), or Evolution and Assembly of GaLaxies and their Environment (Trayford et al. 2015). Colors are connected with environment, in dependence on the surface brightness and luminosity (Blanton et al. 2005). The existence of the green valley, though, has been challenged (Eales et al. 2018), and explained to arise as a consequence of the Malmquist bias in the submillimeter wave band.

Redder and more massive galaxies are located closer to filaments (Luber et al. 2019). The surplus of massive red elliptical galaxies in the SDSS filaments was analyzed by Kuutma et al. (2017, 2020), and Welker et al. (2020) presented similar results for the Sydney Australian Astronomical Observatory Multi-object Integral field spectrograph galaxy survey and determined the angles between the galaxies' spins and filaments. A corresponding study of the orientation of X-ray clusters in relation to the filaments was performed by Shevchenko & Tugay (2017).

We present the results of applying a ridge filter within the framework of image analysis to build a continuous net of filaments for radial layers of the galaxy distributions up to redshift z = 0.1 within a 120° × 60° solid angle. This paper is organized as follows. Section 2 gives a description of the utilized galaxy sample and discusses its completeness. In Section 3 the algorithm for filament detection is outlined. In Section 4 we test our approach on a simulated data set. In Section 5 we provide a comparison of our filaments with similar outcomes by Chen et al. (2016). Section 6 is devoted to validating the extracted filament net through applications: a statistical analysis of distances of various astronomical sources to the nearest filament is performed in Section 6.1, and in Section 6.2 we examine how the g − r color of galaxies changes with the distance from the nearest filament. The discussion and concluding remarks are gathered in Section 7. We utilize the cosmological parameters H₀ = 70 km s⁻¹ Mpc⁻¹ and Ω = 0.32 throughout.

We selected some astrophysical sources to test if different types of objects prefer to reside in different distances ${ \mathcal D }$ from the backbone of the filament net. Upon the constraints 120° ≤ R.A. ≤ 240°, 0° ≤ decl. ≤ 60°, 0.02 ≤ z ≤ 0.1, we gathered the following samples, with respective number of sources fulfilling the above criteria:

• 1.  
  Seyfert 1 and 2 galaxies (Véron-Cetty & Véron 2010)—1755 sources. Objects with spectrum_classification = S1 or S2 are chosen as secure Seyfert galaxies.
• 2.  
  All Sky Wide-field Infrared Survey Explorer (AllWISE) active galactic nuclei (AGN; Secrest et al. 2015)—676 sources from the AllWISE catalog of mid-infrared AGN ¹² . Only objects with spectroscopically obtained redshifts were used (redshift_flag = s), which effectively narrowed the range to 0.02 ≤ z < 0.1.
• 3.  
  Radio galaxies (van Velzen et al. 2012)—67 sources (Fanaroff–Riley class II (FR II) radio galaxies) in the local universe.
• 4.  
  FR II—148 sources obtained by crossmatching the van Velzen et al. (2015) catalog of FR II radio quasars with the SDSS DR7 Main Galaxy (Strauss et al. 2002) and Luminous Red Galaxy (Eisenstein et al. 2001) samples, with a maximum distance between positions of the radio source and optical counterpart of 20''. The redshifts and positions were taken from the SDSS.
• 5.  
  Low-surface-brightness galaxies (LSBGs; Honey et al. 2018)—146 sources. LSBGs have typically 5–10 times smaller surface brightness than regular galaxies and can be of various morphologies (spirals, irregulars, dwarfs). The redshifts were computed from the reported distances ¹³ d by solving

  $$\begin{eqnarray}&&d={d}_{p}(z)\end{eqnarray} \tag{ 3 }$$

  to place the galaxies in the appropriate redshift bins in our filament catalog.
• 6.  
  Million Quasars (MILLIQUAS) AGN (Flesch 2015)—9059 sources from the MILLIQUAS catalog. ¹⁴ Here, like in the case of AllWISE AGN, the photometric redshifts (i.e., those rounded to 0.1 according to the catalog description) were discarded, resulting in the effective range 0.02 ≤ z < 0.1. The types of sources are: unclassified AGNs, type-1 quasars, BL Lac objects, and narrow-emission-line galaxies.
• 7.  
  Dwarf galaxies (Reines et al. 2013)—88 sources; the stellar mass is within $8.6\leqslant \mathrm{log}\left({M}_{\star }/{M}_{\odot }\right)\leqslant 9.48$. These are all nearby galaxies, with z ≤ 0.054.

For comparison, samples of randomly located points (8000 in total, i.e., 500 per redshift bin), drawn from a uniform distribution on the sphere, were generated according to

$$\begin{eqnarray}&&x=360^\circ v,\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&y=\displaystyle \frac{180^\circ }{\pi }\arcsin (2u-1),\end{eqnarray} \tag{ 4b }$$

where $v\sim { \mathcal U }\left(\tfrac{1}{3},\tfrac{2}{3}\right)$ and $u\sim { \mathcal U }\left(\tfrac{1}{2},\tfrac{1}{2}\left(\sin \tfrac{\pi }{3}+1\right)\right)$ ensure 120° ≤ x ≤ 240°, 0° ≤ y ≤ 60°. The centroids of the voids in our filament net (978 locations) were calculated, too, as one cannot be further from a filament than in the center of a void. They are available in our catalog as well (Appendix A).

The results are displayed in Figure 6. All PDFs of the distances ${ \mathcal D }$ from the nearest filament exhibit a narrow peak at about 1−2 Mpc. This is in agreement with the typical widths of simulated filaments (Colberg et al. 2005; Galárraga-Espinosa et al. 2020). The PDF of AllWISE AGN seems quite close to the random sample. The medians m and skewness s are calculated from the samples, and their sample standard errors (SEs) are

$$\begin{eqnarray}&&\mathrm{SE}(m)=\displaystyle \frac{1}{2\sqrt{N}f(m)},\end{eqnarray} \tag{ 5 }$$

where f(m) is the PDF evaluated at m (Rider 1960), and

$$\begin{eqnarray}&&\mathrm{SE}(s)=\sqrt{\displaystyle \frac{6N(N-1)}{(N-2)(N+1)(N+3)}}\end{eqnarray} \tag{ 6 }$$

for a sample size N (Matalas & Benson 1968). The skewness measure is important, as for skewed distributions the mean, median, and mode do not align. The mean of a skewed distribution is uninformative in most cases; hence we report the medians instead. For all source types, the medians are significantly smaller than for a random sample (including AllWISE AGN). The PDF for voids is the most symmetric one, with a median of about 14 Mpc.

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The Seyfert, AllWISE, and MILLIQUAS samples (i.e., AGNs) have their medians at about 4 Mpc. For the remaining samples, the medians are at 2–3 Mpc, with LSBGs and dwarfs slightly closer to filaments than FR II radio galaxies. The skewness of the AllWISE and random samples are comparable, but the difference in their medians indicate different weight in the tails of the distributions. Overall, one can say that all types of objects examined herein tend to gather near the filaments, and no striking differences between them are visible.

All above sources are plotted on a celestial map in consecutive redshift bins in Figure 7. We observe that the objects trace our filament net rather tightly. Finally, the catalog of giant radio sources (GRSs; Kuźmicz et al. 2018), while containing 349 sources in total, has only 13 sources fulfilling our R.A., decl., and z constraints. This sample is too small to make any statistical inferences, but we place the relevant GRSs on the celestial maps for illustration. Likewise, we display the three new GRSs and three GRS candidates from the Radio sources associated with Optical Galaxies and having Unresolved or Extended morphologies I (ROGUE I) catalog ¹⁵ (Kozieł-Wierzbowska et al. 2020), and three (two) GRSs (GRS candidates) from the upcoming ROGUE II catalog (N. Żywucka, priv. comm.).

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We ask the question whether the g − r color of the galaxies changes with the distance ${ \mathcal D }$ from the filaments. For that purpose, we examine the color distributions in five consecutive distance bins: for the first redshift bin, i.e., z ∈ [0.02, 0.025], we choose galaxies falling in the ranges of ${ \mathcal D }=0\mbox{--}1$ Mpc, 1–2 Mpc, 2–3 Mpc, 3–4 Mpc, and 4–5 Mpc. In subsequent redshift bins, the distance bin sizes are increased by a factor of 16/15, so that for the last redshift bin, z ∈ [0.095, 0.1], the distance bins are ${ \mathcal D }=0\mbox{--}2$ Mpc, 2–4 Mpc, 4–6 Mpc, 6–8 Mpc, and 8–10 Mpc. This adaptive binning is justified by the observation that the number of galaxies in each redshift bin is quite similar (Figure 1(a)), we have a constant field of view (120° × 60°), so that if a size of ${ \mathcal D }=1$ Mpc at redshift z = 0.02 has an angular size θ, a size of ${ \mathcal D }=1$ Mpc at z = 0.1 becomes θ/5. The distance bins are chosen so that a similar number of galaxies is in each, for corresponding redshift bins.

Figure 8 shows the resulting color distributions. The left peak in each panel corresponds to the blue cloud, while the right peak is the red sequence. We observe the well-known reddening with increasing redshift (Sandage 1973). It appears that the ratio of blue to red galaxies increases with the distance ${ \mathcal D }$ to the filament. This effect has the following physical explanation (Hogg et al. 2004; Berlind et al. 2005): galaxies in filaments have more interactions with neighbors, especially at filament intersections and in clusters. The interaction of galaxies with each other (Schweizer & Seitzer 1992; Kannappan et al. 2009), or even with dense intergalactic medium (McNamara & O'Connell 1992), accelerates star formation (Bower et al. 1998) and eventually leads to the exhaustion of interstellar gas. As a result, not much of the interstellar gas remains (Kauffmann & Charlot 1998). A blue color indicates the presence of several blue stars within: the brightest, most massive, and short-lived stars. Blue color is typically an indicator of current star formation rate (Searle et al. 1973). Isolated galaxies, e.g., in voids, did not experience much interaction in their past, so they spend their gas on star formation continuously and at a low rate. Otherwise, in dense regions of LSS, many galaxies have lost most of their gas in interactions, so they have a small number of young blue stars, and these galaxies are seen as red. Similar environment dependencies were studied for different samples of SDSS galaxies (Ball et al. 2008), and in particular in the Galaxy Zoo project (Bamford et al. 2009; Skibba et al. 2009; Darg et al. 2010).

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To verify whether this effect is statistically significant in our sample, we performed Kolmogorov–Smirnov (KS) and Anderson–Darling (AD) tests in the following way: we perform pairwise tests between all five color distributions in each redshift bin. For instance, in the first bin, z ∈ [0.02, 0.025], we first compare the colors of galaxies closest to the filaments, i.e., those within a distance range of ${ \mathcal D }=0\mbox{--}1$ Mpc, with the color distribution of those falling in the second closest range, ${ \mathcal D }=1\mbox{--}2$ Mpc, and record the output of the tests (i.e., reject or not reject the null hypothesis). This is then repeated for the first distance range and the third one, second and third, and so on, eventually testing all pairs, and similarly for the subsequent redshift bins. The results are gathered in Figure 9. The labels numbered 1 to 5 denote the increasing ranges of distance ${ \mathcal D }$, color coded in the same way as in Figure 8. The check marks indicate instances when the tests imply that the respective distributions are indeed different, and cross marks denote cases in which the null hypothesis that the distributions are the same cannot be rejected. While the effect of the increasing ratio of blue to red galaxies with the distance ${ \mathcal D }$ to the filament is rather clear in the closest redshift bins, it becomes much less prominent for greater redshifts. This is likely associated with the insufficient sample size for greater cosmological distances, other selection biases, and observational uncertainties. On the other hand, these outcomes are generally consistent with those of Lee et al. (2021) who found that the g − r color correlates with the distance ${ \mathcal D }$ from the filaments around the Virgo cluster, and Castignani et al. (2022) who demonstrated that early-type galaxies are indeed located closer to filaments than late-type galaxies.

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Different methods applied to large observational surveys yield different decomposition of the LSS into filaments (Rost et al. 2020). One can see that segmentation of the same sample of galaxies results in a significantly different cosmic web; in particular, with the filament network composed of many disjoint parts, the physical reality of a particular map of the LSS is uncertain. To reliably detect real filaments, new robust methods for their detection should be developed, possibly including other observational strategies. One such approach might be a direct observation of filaments via the 21 cm H I line, possible with future giant radio telescope arrays like the Square Kilometre Array (Kooistra et al. 2017). A detailed filament network is an essential tool for modern cosmography and can be used for a topological description of LSS.

The ridge detection algorithm from Section 3 readily generalizes to three-dimensional spaces (Lindeberg 1998). The framework of image analysis that we built around it can, in principle, be applied to three-dimensional images (i.e., stacks of two-dimensional images), composed of voxels instead of pixels, as well. We leave the implementation and exploration of such a generalization to a future work. It should be noted that the uncertainties in redshift are larger than those of celestial positions, leading to very asymmetric uncertainty distributions in the physical space and posing difficulties in constructing a truly three-dimensional LSS. The benefits of two- or three-dimensional catalogs depend on the particular application and scientific objectives.

We compiled a catalog of filaments within a solid angle 120° ≤ R.A. ≤ 240°, 0° ≤ decl. ≤ 60°, covering the redshift range 0.02 ≤ z ≤ 0.1 with 16 equally sized nonoverlapping bins in celestial projections. These filaments form a continuous net, with no stray filaments, and very few tails. This continuity is an advantage of our LSS model compared to previous works. Our LSS model is simpler in its principle than that of, e.g., Sousbie (2011); instead of a system of doubtful voids with too complicated shapes and filamentary boundaries, we obtained a cleaner network of smooth filaments. As an illustration of potential applications and validation of such a filament network, we compared the distributions of the distance ${ \mathcal D }$ to the nearest filament of various astrophysical sources (Seyfert galaxies and other AGN, radio galaxies, LSBGs, and dwarf galaxies; Section 6.1), and the dependence of the g − r color distribution on the distance ${ \mathcal D }$ to the nearest filament (Section 6.2), obtaining outcomes in tune with other studies. The full catalog (Appendix A) is available online in machine-readable format.