Observational evidence of fractality in the large-scale distribution of galaxies
BBAAA, Vol. 62, 2020 Asociaci´on Argentina de Astronom´ıaA.M. V´asquez, F.A. Iglesias, M.A. Sgr´o & E.M. Reynoso, eds. Bolet´ın de art´ıculos cient´ıficos
Observational evidence of fractality in the large-scaledistribution of galaxies
T. Canavesi & T.E. Tapia Insituto de F´ısica de la Plata, CONICET-UNLP Facultad de Ciencias Astron´omicas y Geof´ısicas de la Plata - UNLP Wolfram Research, EE.UU.Contact / tcanavesi@fisica.unlp.edu.ar
Resumen / Usando una muestra de 133 991 galaxias distribuidas en la region del cielo 100 ◦ < α < ◦ y7 ◦ < δ < ◦ , extra´ıda del cat´alogo SDSS NASA/AMES Value Added Galaxy Catalog (AMES-VAGC), estimamosla dimensi´on fractal usando dos m´etodos. El primero usa un algoritmo para estimar la dimensi´on de correlaci´on.El segundo, una nueva aprocimaci´on, crea un grafo usando los datos y estima la dimensi´on del grafo basadounicamente en la informaci´on de conectividad. En ambos m´etodos encontramos una dimensi´on D ≈ Abstract / Using a sample of 133 991 galaxies distributed in the sky region 100 ◦ < α < ◦ and 7 ◦ < δ < ◦ ,extracted from the SDSS NASA/AMES Value Added Galaxy Catalog (AMES-VAGC), we estimate the fractaldimension using two different methods. First, using an algorithm to estimate the correlation dimension. Thesecond method, in a novel approach, creates a graph from the data and estimates the graph dimension purelyfrom connectivity information. In both methods we found a dimension D ≈ Keywords / large-scale structure of universe — cosmology: observations — cosmology: miscellaneous
1. Context
The cornerstone of modern Cosmology is the cosmo-logical principle, which assumes that the Universe ishomogeneous and isotropic at large scales. From thisstatement one could ask: • At what scale is the universe homogeneous? • Does the universe show a fractal structure?Several approaches to measure fractality have been pro-posed, Chac´on-Cardona et al. (2016), Kamer et al.(2013), Chac´on & Casas (2009). As a classical method,if we count the number of points N inside a growingsphere of radius r , we expect for an homogeneous dis-tribution a power law relation of the form N ( r ) ∼ r D , (1)with the dimension D = 3. Observations in the lastdecades have allowed us making more accurate calcula-tions of the dimension of galaxy distribution, as wellas any other cosmological parameter. This work es-timates the dimension of the spatial distribution of asample of 133 991 galaxies extracted from the SDSSNASA/AMES Value Added Galaxy Catalog (AMES-VAGC) (cid:63) using two approaches. First, using the defi-nition of correlation dimension. Second, in a novel way,by creating a graph from the galaxy spatial distributionand estimating its dimension purely from connectivityinformation. (cid:63) https://cdsarc.unistra.fr/viz-bin/cat/J/ApJ/799/95
2. Methodology
In the first approach we use the correlation integral.In agreement with Bagla et al. (2008), the correlationcorrelation integral C is defined as C ( r ) = 1 N M M (cid:88) i =1 n i ( r ) , (2)where N is the number of galaxies in the sample, M isthe number of galaxies chosen as centers of the growingspheres, n is the number of galaxies reached by the grow-ing spheres of radius r with center in the ith-galaxy, andthe summation is carried over the set of spheres. Thecorrelation integral is defined similarly to (1), then thecorrelation dimension is D = d log C ( r ) d log r . (3)In our results the growing spheres moves in steps of2 Mpc. Then, the numeric estimation of the dimensionis calculated as: D ( r ) = log C ( r + 1) − log C ( r )log( r + 1) − log( r ) , (4)where r now moves in discrete steps.In the second method a graph is constructed fromthe position of the galaxies. Each galaxy represents anode, and an undirected edge will join two nodes if thedistance between the nodes is less than 10 Mpc. Then,to estimate the dimension of the resulting graph we usegrowing spheres whose center is a node X in the graph, Poster contribution 1 a r X i v : . [ a s t r o - ph . C O ] F e b bservational evidence of fractality in the large-scale distribution of galaxiesand we count the number of nodes we have inside eachsphere until we reach the boundary of the graph. Ifwe make an adjustment taking into account the numberof nodes reached by each sphere as a function of theradius of the sphere, we can adjust a dimension D . Thisapproach is similar to our first method, but in this casewe just use connectivity information. For more detailssee chapter 4.5 of Wolfram (2020).The following computations were done using Math-ematica 12.1 Inc., and the software developed in theWolfram Physics Project (cid:63)(cid:63) .
3. Results
Fig. 1 shows the mean number of galaxies reached bythe growing spheres vs the radius of the spheres. Errorbars considering 1 σ ranges in the distribution of valueswere calculated. Even error bars are not big enough tobe noticed in Fig. 1, errors propagate and are visible inFig. 2. Also, Fig. 1 shows a fit of the form ar D (thered line), and the parameters of the fit are shown inTable. 1. A dimension D = 2 . [ Mpc ] N u m be r o f ga l a x i e s D Figure 1: Mean number of galaxies reached by the growingspheres vs the radius of the spheres. One hundred galaxieswere used as centers of the growing spheres. Dots are ob-tained from the computation, and solid red line indicates afit of the form ar D . From the fit, the obtained dimension isshown in the red box.Table 1: Parameters of the fit plotted in Fig. 1, includingthe standard error and the t-statistic.Parameter Estimate Standard Error t-Statistic a D Fig. 2 shows the dimension D as a function of theradius r . The decay of the dimension at large scalesis due to boundary effects of the dataset. A dimension (cid:63)(cid:63) of D = 2 in the scales of [20 ,
30] Mpc transitions andreaches D = 3 at the scales [60 ,
70] Mpc. [ Mpc ] D i m en s i on Figure 2: Dimension vs radius. Dimension is plotted alongwith gray error bars. Dashed lines correspond with a dimen-sion of 2 and 3. .
After building a graph using the galaxies as ex-plained in Sec. 2., we can estimate the dimension of thegraph counting the number of nodes that fall within asphere of increasing radius and whose limit is the bound-ary of the graph. We can see this process in a qualitativeway in in Fig. 3. In Fig. 4 we can see the estimation ofthe dimension starting for a single point in the graph,in this case the center and using a fit of the form ar D whose adjustment parameters can be found in the inTable 2. In Fig. 5 we show the dimension estimationusing a graph of 51 845 nodes and considering differentcentral nodes. We do this because the result may de-pend on certain central node X , then we average thecounting using several central nodes. The error bars in-dicate 1 σ ranges in the distribution of values obtainedfrom different central nodes X . Figure 3: A representative graph using 1853 nodes. Startingfrom a central node X in the graph, we count the numberof nodes in the graph that can be reached by going out atmost a distance r . In red color we can observe this processstarting from the center of the graph. The number at theright of each graph is the total number of nodes counted upto a distance r . BAAA, 62, 2020 anavesi & TapiaFigure 4: A estimation of the dimension starting for a uniqueposition in the graph.Table 2: Parameters of the fit plotted in Fig. 4, includingthe standard error and the t-statistic.Parameter Estimate Standard Error t-Statistic a D
4. Conclusions • Using the two methods, either the integral correla-tion or the graph approximation, a fractal dimensionis found before the transition to homogeneity. • Using the first approach, a transition from D = 2 to D = 3 starts in the interval [20 ,
30] Mpc and endsin the interval [60 ,
70] Mpc. Other works report atransition to homogeneity at larger scales, as Nteliset al. (2017), Scrimgeour et al. (2012). • Using the graph approach we found a region wherethe dimension of the graph is around D = 3, and aregion where the graph dimension is less than three,as seen with the first method. • We add evidence about the non-homogeneity of thedistribution of galaxies at certain scales. Findinga transition from D = 2, corresponding to a dis-tribution of galaxies where the matter is uniformlydistributed on spherical surfaces surrounding the ob-servation point, to D = 3 corresponding to a homo-geneous and isotropic distribution of galaxies. • Calculate the dimension of galaxy distribution usingdifferent methods. • Use recent astronomical catalogs to consider moregalaxies in the computations. • Study the implications of fractality in the dynamicsof galaxies using general relativity. • Apply the methods used in this work to star forma-tion regions, and compare with previous results asin Canavesi & Hurtado (2020). • Analyze the implications of assuming a fractal grav-itational model in astronomical and cosmologicalscales, as in Canavesi (2020).
Figure 5: Dimension estimation as a function of the distance r from the central nodes X . The graph has 51 845 nodes andseveral central nodes X are used. The error bars indicate1 sigma ranges in the distribution of values obtained fromdifferent central nodes X . Acknowledgements:
Thanks to the Wolfram Physics Project teamfor providing us with all the necessary software tools.
References