A fractal cascading model for the large scale galaxy distribution

0083~5656/90 $0.00 + .50

Vistas in Astronomy, Vol. 33, pp. 323-335, 1990 Printed in Great Britain, All rights reserved.

© 1990 Pergamon Press plc.

A FRACTAL CASCADING MODEL FOR

T H E L A R G E SCALE G A L A X Y D I S T R I B U T I O N C. Castagnoli ~z A. Provenzale I s t i t u t o di C o s m o g e o f i s i c a del C.N.I't. and I s t i t u t o di Fisica G e n e r a l e d e l l ' U n i v e r s i t ~ di T o r i n o Corso F i u m e 4, 1 - 1 0 1 3 3 Torino, I t a l y

We discuss a family of fractal cascading models which provide a phenomenological description of the large scale distribution of galaxies. These models are based on a modification of the random ¢ model introduced in the study of turbulence. We consider both fractal cascades without an upper cutoff and models which provide an homogeneous distribution at the largest scales. In particular, we introduce a cascading process which naturally provides a smooth transition from small-scale fractality to large-scale homogeneity. This model may also account for the multifractal behavior of the LSS. In this context, the notion of dressed fractal dimension is introduced. Abstract:

1. I n t r o d u c t i o n A widely debated issue in Cosmology is the behavior of the matter distribution at sufficiently large scales (see for example Peebles, 1980). The classic point of view, expressed for example by the ideas of Einstein and Hubble, is that the galaxy distribution may be very irregular on relatively small scales, but it must approach homogeneity when sampled at sufficiently large scales. In this case, a meaningful average density of the visible matter content of the Universe may be defined. In recent years, however, several observations of the galaxy and cluster distributions renewed the idea of a possibly inhomogeneous large scale structure (LSS) of the Universe. The existence of large voids and superclusters and the detection of large peculiar velocities of galaxies at very large scales, together with the powerlaw form of the two-point correlation function of galaxies and clusters, have raised some doubts on the very existence of a length scale above which the matter distribution becomes homogeneous. In this context, 'hierarchical' models have been resurrected (Mandelbrot, 1982; Pietronero, 1987; Coleman et al., 1988). The main characteristic of these models is the continuous dependence of the matter density upon the sample depth, without an upper cutoff beyond which a mean density can be defined. These models are in general based upon the concept of a self-similar matter distribution. A simple way to generate a self-similar distribution of points is the use of a fractal cascading process called the ~ model, which has been introduced some years ago in the study of turbulence (Frisch et al., 1978). In Section II we discuss the basic concepts used in the fractal analysis of the galaxy distribution and in Section III we introduce the simplest version of the cascading B model. On the other hand, the impossibility of meaningfully defining an average matter density in the Universe is not easy to accept. Contrarily to a purely self-similar behavior, the fractal nature of the large scale matter distribution in the Universe may in fact be limited by an upper homogeneity cutoff. In particular, the uniformity of the cosmic microwave background points toward an homogeneous matter

323

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C. Castagnoli ~ A. Provenzale

distribution at sufficiently large scales, a fact which is also suggested by standard big bang theories. In contrast to the monofractal approach, Peebles (1988 &: 1990) has shown that the present data do not support a simple self-similar distribution of visible matter, and he saw 'no hope for a pure renormalizable fractal Universe'. Also, the analysis of the matter distribution at the largest scales reveals a strong tendency toward homogeneity, displaying a smooth transition from small and intermediate scale fractality to large scale homogeneity (see for example Martinez, 1990). In addition, Jones et al. (1988), Martinez & Jones (1990) and Martinez et al. (1990) have analyzed the multifractal properties of the galaxy distribution in the CfA sample, finding evidence of a non trivial multifractal behavior of the matter distribution, at least for scales up to about 5-10 Mpc (for an introduction to multifractals see for example Paladin ~z Vulpiani, 1987). For all these reasons, fractal models of the LSS with an upper homogeneity cutoff have been developed. Among these, Calzetti et al. (1988) proposed a family of models based on a random, homogeneous distribution of basic cells with approximate diameter do (corresponding approximately to the supercluster scale), each of which undergoes a self-similar breaking cascade which generates a mono-fractal matter distribution inside the cells. In this model the matter distribution is thus a simple self-similar fractal for scales smaller than do and it is homogeneous and space filling for scales larger than do. In a previous paper (Castagnoli &: Provenzale, 1990) we have proposed a different model based on a modification of the cascading ~ model of turbulence. These models, described in Section IV, generate a matter distribution whose fractal properties vary with the length scale, as observed in the analysis of real galaxy catalogs. Also, a smooth transition from small-scale fractality to large-scale homogeneity is naturally obtained. A further property of this family of models is that the scale dependence of the fractal behavior provides a 'dressing' of the asymptotic value of the fractal dimension to a larger effective value which is the one practically observed. We believe this family of models may be of some use in the description of the large scale distribution of galaxies. Clearly, for the time being these models are nothimg more than euristic descriptions whose physical basis and motivations have to be carefully assessed. 2. C o r r e l a t i o n F u n c t i o n a n d Fractal D i m e n s i o n

In this paper we provide a very simple introduction to the ~ model and to its fractal properties. A more complete discussion may be found in Castagnoli ~ Provenzale (1990). The classic definition of a fractal set is 'a mathematical object whose fractal (Hausdorff) dimension D is strictly larger than its topological dimension D r . ' The rigorous definition of Hansdorff dimension may be found for example in Mandelbrot (1982), for simplicity here we use an operational definition based on the calculation of the correlation integral c ( r ) introduced by Grassberger Procaccia (1983) in the study of strange attractors. This method is particularly appropriate to study the fractal dimension of a point distribution; in this case c ( r ) is given by N

C(r) = lim E

O (t-'X,-

Xj')

(1)

where O is the Heaviside step function and N is the total number of points; c ( r ) measures the probability that two randomly selected points with positions .f~ and )~j are closer than the distance r. For a fracta/distribution one has: lim ccr) = #o2

r~o

(2)

Large Scale Galaxy Distribution

325

where D2 is the correlation dimension of the point set; D2 is a lower bound to the Hausdorff dimension D of the set. For a point distribution ('dust') the topological dimension is D r =0. For a fractal dust the fractal dimension D must be larger than 0, homogeneity corresponds to D=3. An important property of a fractal set is its scaling behavior. Scaling means that the structures present at a certain scale are related to the structures at the other scales by a general recursive relation. A fractal set may be deterministic, i.e. it may be generated by a determinitic algorithm (like a Koch curve or a Cantor set), or may be a stochastic fractal associated for example with the output of a stochastic process. In the latter case the scaling properties refer to the probability distribution describing the fractal set. In general, the fractal sets which are most useful in physics and cosmology are stochastic fractals. An important class of fractals are self-similar mono-fractals, which are characterized by a single fractal dimension. These fractal sets have the property that every part of the set is an exact (in a statistical sense) replica of the whole set. In this case all the moments of the probability distribution scale with the same scaling exponent, and the scaling properties are scale-independent. More complex fractals are represented by the so-called multifractal sets. The different moments of the probability distribution of a multifractal set scale with a different scaling exponent. For these sets one may also say that the scaling properties depend on the scale range which is considered. If the large scale matter distribution is a pure fractal, then a meaningful average density cannot be defined, since the average density is vanishing in the limit of an infinite sample volume. Practically, by considering larger and larger sample volumes one would find an ever decreasing average density of the Universe (Pietronero, 1987, Calzetti et al., 1988). This problem does not exist for a scale-dependent fractal set, when the fractal properties depend on the length scale considered and the distribution approaches homogeneity at large scales. An average matter density may thus be defined at scales larger than the homogeneity scale. An important quantity used in the study of the large scale galaxy distribution is the two point correlation function ~. This is defined as 6N = n(1 + ~(r)) 6V

(3)

where 6N is the average number of galaxies in a small volume ~V placed at a distance r from a generic galaxy of the sample; n is the average number density of galaxies. For a random, homogeneous distribution of points ~(r)=0. The correlation function measures the probability, in excess with respect to a random homogeneous distribution, of finding a galaxy at a distance r from another galaxy. If the distribution of galaxies is correlated, then ~(r) > 0, while if the galaxy positions are anticorrelated one observes ~(r) < 0. It is easy to see that the correlation function ~(r) may be easily obtained from the correlation integral c ( r ) by the formula N

1 q-~.(r) = 4mr

dC(r)

2

dr

For a fractal distribution of points with correlation dimension D~, one has c ( r ) r D2 and consequently l+~(r) = r D~-s. Obviously, when ~(r) :~ 1 one has that ~(r) ~. r D2-3. It is a well-known experimental result that the real galaxy distribution has a correlation function with power-law form, ~(r) = (r/to) ~, with "/ approximately 1.7 or 1.8 and r0 of the order of 5 Mpc, for distances up to about ten Mpc. This result implies a correlation dimension D2 of about 1.2 or 1.3 for the galaxy distribution

(4)

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Fig. la. Two-dimensional projection of a self-similar point distribution generated by the simplest version of the ~ model• At each iteration step each object breaks down into 8 smaller 'son' objects. Only 3 sons over eight survive until the next iteration and are able to break again.

at these scales. At larger scales there is evidence that the galaxy distribution becomes more and more homogeneous (see for example Martinez, 1990). 3. T h e S e l f - S i m i l a r Z M o d e l The simplest version of the Z model is a three-dimensional breaking cascade from a parent object (e.g. a cube) with linear size lo into M smaller objects with linear size 11= l o / M 1/3. O f these, only m cubes remain 'active' and are able to break again at the second iteration. By repeating several times this procedure one obtains a fra~tal distribution of points. The fractal dimension of the distribution may be immediately found by using a box-counting algorithm. The fractal dimension of a point set may in fact be computed also as Do = lira

1on n(r)

, 4 0 log 1/r

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dimension. Apart from statistical fluctuations, for simple fractals one has D2 ---- Do, while for multifractal sets D2 < Do. In general, while D2 provides a lower limit to the Hausdorff dimension, the box-counting dimension Do gives an upper limit. In the case of the fractal cascade described above it is clear that log m k Do =

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since r = ( 1 / M 1 / S ) k and n(r) = rn k, where k denotes the k-th iteration in the breaking cascade. By choosing for example M = 8 (SO that /k+~ = lk/2) and m ---- 3 (i.e. only three cubes over eight 'survive' at each iteration) one gets Do = log 3/log 2 ~ 1.58. Fig. l a reports a projection of this cascading process with M --'-- 8 , r n = 3 . A small random perturbation has been added to the positions of the points at the smallest scales, in order to simulate the effects of small-scale, non-hierarchical processes. Fig. lb reports the correlation integral c ( r ) for this point distribution; the linear fit of log c(r) versus log r gives a dimension D2 ~ 1.6. The 'knee' in tog c(~) at small values of ~ is generated by the small scale random component, as observed in the analysis of real data. Fig. lc reports the correlation function ~(~) as obtained from Eq. (4). A better version of the cascading ~ model is obtained by fixing the probability of survival of each object in the breaking cascade, rather than fixing the number of

(6)

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Fig. lc. Two-point correlation function of the same distribution. the surviving objects. Each parent object may thus generate a different number of 'surviving sons'. The average number of surviving sons for each object is thus ma~ = p M . The dimension of the point distribution obtained by iterating this cascading process is then log ( p M ) k Do =

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as it is obvious from the definition of ma~. Fig. 2a reports the projection of a point distribution obtained with M --- 8 and p = 0.3. From Eq. (7) the dimension is Do ~ 1.26. Fig. 2b reports the correlation integral C(r) and Fig. 2c reports the correlation function ~(r). Again scaling is evident, giving D2 ~ 1.25 and -~ ~ 1.75, as observed in the analysis of real data. The distribution in Fig. 2a is self-similar, i.e. no upper homogeneity cutoff is present. For this kind of distributions the dimension of the voids increases with the sample size, and the matter density vanishes in the limit of an infinite sample. While of some interest for describing some of the properties of the galaxy distribution at intermediate scales, these selfsimilar models are probably inadequate for modelling the real behavior of the LSS at all scales. In addition, no multifractal behavior is present in this type of model. In the next Section we introduce a modification of the E-model which may easily describe both the transition to large-scale homogeneity and the scale-dependence of the fractal properties of the galaxy distribution.

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Fig. 2a. Two-dimensional projection of a self-similar point distribution generated by a cascading random ~ model. At each iteration step each object breaks down into 8 smaller 'son' objects. Each son has a constant probability p of surviving until the next iteration and of breaking again. 4. A S c a l e - D e p e n d e n t ~ M o d e l The simplest way of generating a point distribution with an upper homogeneity cutoff is to consider an abrupt jump from self-similarity to homogeneity, as done for example in the model introduced by Calzetti et al. (1988)• In the framework of the ~ model this would correspond to a jump of the survival probability from a value p = : for scales larger than the homogeneity scale do to a value p < 1 for scales smaller than do. A simple way of generating both a smooth transition from fractality to homogeneity and a scale-dependent fractal behavior is to allow the survival probability p to be a function of the size of the breaking object, i.e. p = p(l~). For a possible physical interpretation of a scale-dependent survival probability see the discussion in the next Section. Clearly, the form of the function p(Ik) would have to be given by a detailed physical mechanism. Following for now a purely euristic approach, one of the simplest choices of function p(lk) may be p(lk) --'-- 1

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that is, the distribution is homogeneous and space-filling for lk > do, it is a pure fractal for lk < all, and it has scale-dependent fractal properties at intermediate scales. In Fig. 3a we show a projection of a point distribution generated by this model with do = 1 (in arbitrary units, obviously), d l = 1/4 and po = 1/4. This value of po is somewhat 'noble' since it would give an asymptotic dimension D,, = 1 (from Eq. (7) with p = po), a value which has already been considered by several authors (see the historical discussion given by Mandelbrot, 1982) and which, more importantly, is obtained in cosmic string scenarios (see e.g. Turok, 1986 ~: Schramm, 1988). As one can see, the distribution depicted in Fig. 3a bears several resemblances with the observed distribution of galaxies, with clusters, voids and filaments. Also, the dimension of the voids does not increase without bound, and a meaningful average density can be defined at sufficiently large scales. Fig. 3b reports the correlation integral c(~) for the distribution shown in Fig. 3a. Again, scaling is evident at intermediate scales, but now three regions can be detected in the correlation integral c(r). At very small scales, log c(,.) is very steep, owing to the small-scale random noise added to the galaxy positions. At intermediate scales, a linear dependence of log c(r) versus tog r is present, but now at larger scales the slope of log c(r) grows again, due to the transition to large scale homogeneity. In addition, the linear fit of log c ( r ) versus log ,. at

Large Scale Galaxy Distribution 105

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Fig. 2c. Two-point correlation function of the same distribution. intermediate scales gives D2 ~ 1.2, a value which is larger than the asymptotic value Do,. This behavior has been observed in a variety of situations (see Castagnoli ~z Provenzale, 1990)i always the smooth transition from small scale fractality to large scale homogeneity is responsible for a dressing of the fractal dimension from the asymptotic value Da, (fixed by the value of po) to a larger, 'effective' value D~j which is practically observed. A plausible speculation is that a mechanism of this kind may be responsible for the dressing of the fractal dimension of the real galaxy distribution from an asymptotic value D,, = 1 (related for example to the presence of cosmic strings) to the observed value D,j ~. 1.2. Finally, Fig. 3c reports the two-point correlation function for the distribution shown in Fig. 3a. Scaling is evident, but now the correlation function becomes flatter at large scales, due to the presence of homogeneity. In a previous paper (Castagnoli ~: Provenzale, 1990) we have considered a different dependence of the survival probability on the size of the objects, namely p ( l k ) ---- 1 v ( l , ) = po + ( 1 -

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Da° ----- 1. The effective dimension D~/ was found to be a function of L, growing from D ~ / ~ 1 for L ~ do to D~! ~ 2 for L ~ 0. In particular, a point distribution closely resembling the real galaxy distribution was shown to be consistent with a dressing of the asymptotic dimension Do. = 1 to an effective value D ~ ~ 1.2, in agreement with the observed dimension.

5.

Discussion

and

Conclusions

In this paper we have introduced a modification of the ~ model of turbulence which m a y be used to describe the large scale galaxy distribution• This model provides a s m o o t h transition from small-scale fractality to large-scale homogeneity and generates a scale-dependent fractal behavior. In addition, the fractal dimension of the galaxy distribution is 'dressed' from an asymptotic value Dao to an observed, 'effective' value D,]. Point distributions which closely resemble the real galaxy catalogs are associated with a dressing of the fractal dimension from a value D~o = 1 to a value D , I ~ 1 . 2 . We believe these results m a y be of some interest since the "

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Fig. 3b. Correlation integral C(r) of the distribution shown in Fig. 3a. value D~0 = 1 is obtained for example in the cosmic string scenario, while the value D,j ~. ].2 is the one which is practically observed in the analysis of galaxy catalogs. A dressing effect such as that depicted here may thus be present. Along this line of reasoning, we recall that Luckas & Novikov (1988), stimulated by the results of Jones et al. (1988) on the multifractal nature of the galaxy distribution, have recently suggested that the fractal dimension of the LSS may have a physically motivated scale-dependence. In this picture, the fractal dimension D would grow from D ~ 1 at small scales, where a filamentary structure dominates, t o D ~ 2 at large scales, where the presence of pancakes (see e.g. Shandaxin 8z Zeldovich, 1989) dominates the LSS, to D ~ a at very large scales, where homogeneity and space-filling distributions are finally found. Without entering the details of the appropriateness of such a picture, we note that the cascading Z model introduced here would be perfectly consistent with this point of view, and it could also be used to provide an operational model for such a scenario. A final point concerns the physical meaning of the term active objects, a basic feature of the # model introduced here. In the study of turbulence, the set of active objects corresponds to the set of active regions of dissipation, and the fact that not all objects in the breaking cascade remain active (i.e. not all objects 'survive') implies that the support of dissipation is not space-filling. Alternatively, one may think of the active objects as representing the turbulent eddies in an energy cascade

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Fig. 3c. Two-point correlation function of the same distribution. process (either a three-dimensional direct cascade or a two-dimensional inverse cascade, see e.g. Frisch et al., 1978). At the smallest scales, the turbulent eddies do not break any more and are directly dominated by dissipation. In a cosmological context, the interpretation of the/~ model in terms of a turbulent cascade is somewhat delicate. There are however at least two possible scenarios in which the introduction of a 'cosmological' fractal cascading process may be quite natural. In a Universe dominated by hot dark matter, large scale strucures form first (top-down scenario), followed by a fragmentation process in which both hydrodynamical processes and gravitational instability may play an important role. In this framework the fragmentation process may be well modelled by the cascading/~ model. The varying survival probability in the breaking cascade would naturally be generated by varying local conditions. In this case the cascade is direct, in analogy with three-dimensional turbulence. The second scenario may be a cold dark matter Universe. In this case the small scale structures would form first (bottom-up scenario), and the largest structures form by subsequent gravitation instability. The /~ model may be used to simulate the hierarchical gravitational clustering; in this case the varying survival probability might be for example associated with a varying value of the Jeans length. This case would correspond to the inverse cascade of two-dimensional turbulence. Apart from the above general speculations, it is clear that the cascading/~ model

Large Scale Galaxy Distribution

335

introduced in this paper is, for now, only a possible phenomenological description of the large scale galaxy distribution. The physical mechanisms associated with the cascading process must be carefully assessed, and future explorations must consider this point. In our opinion, an interesting approach would also be to compare the statistical properties of the real galaxy distributions with those generated by the cascading ~ model, in order to derive the form of the dependence of the survivival probability p(lk) on the size tk of the breaking objects. Solving this 'inverse problem' could perhaps provide a useful framework for the description of the large scale galaxy distribution. Acknowledgements We acknowledge useful discussions with E. Bertschinger, A. Blanchard, B. Dubrulle, U. Frisch, P. Galeotti, F. Lucchin, S. Matarrese and A. Vulpiani. References

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