Theology Project

Fractals and Cosmological Large-Scale Structure Author(s): Xiaochun Luo and David N. Schramm Source: Science, New Series, Vol. 256, No. 5056 (Apr. 24, 1992), pp. 513-515 Published by: American Association for the Advancement of Science Stable URL: https://www.jstor.org/stable/2877211 Accessed: 12-04-2019 01:28 UTC

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* REPORTS

Fractals and Cosmological Large-Scale Structure

Xiaochun Luo and David N. Schramm

Observations of galaxy-galaxy and cluster-cluster correlations as well as other large-scale structure can be fit with a "limited" fractal with dimension D = 1.2. This is not a "pure" fractal out to the horizon: the distribution shifts from power law to random behavior at some large scale. If the observed patterns and structures are formed through an aggregation growth process, the fractal dimension D can serve as an interesting constraint on the properties of the stochastic motion responsible for limiting the fractal structure. In particular, it is found that the observed fractal should have grown from two-dimensional sheetlike objects such as pancakes, domain walls, or string wakes. This result is generic and does not depend on the details of the growth process.

The origin of cosmological large-scale structure is probably the most pressing prob- lem in physical cosmology today. Under- standing the observed structure requires the use of quantitative methods to describe it. The measurement of the two-point galaxy- galaxy correlation function, that is, gg : (r)18, for separations r up to -10 Mpc, marked the beginning of quantitative at- tempts to understand the large-scale struc- ture of the universe (1). The two-point correlation function for clusters of galaxies appears to have similar behavior, but with higher amplitude (2, 3). Initial worries about projection effects biasing the results have been minimized somewhat by the result of West and Van den Bergh (4) for cD galaxies (cD's are associated with the core of rich clusters) and that of Lahav et al. (5) for x-ray clusters; these show the same behavior as the clusters. Several years ago, Szalay and Schramm (6) showed that the correlation functions could be written in a unified way by using a dimensionless variable r/L, where L is the average separation of objects in the catalog being examined: ((r) = 1 (L) (r/L) -1.8, They found the correlation amplitude 3 (P 0.35) is a constant for all clusters of galaxies and is unity for galaxies. The slightly larger correlation for the galax- ies in this scale-free approach is probably an indication of gravitational clustering.

The near constant behavior of 1 for clusters indicates that the clustering process may be roughly scale invariant or, in other words, that the structure is a fractal. But it is not a true fractal because it does not show power law behavior to infinite scale. State- ments about a so-called "fractal universe" are therefore excessive. At scales > 100 Mpc, the data are sufficiently poor that the power law correlation is not evident, and at very large scales we know that the universe is isotropic and not fractal from microwave observations and the relatively smooth dis-

University of Chicago, 5640 South Ellis Avenue, Chi- cago, IL 60637, and National Aeronautics and Space Administration-Fermilab Astrophysics Center, Box 500, Batavia, IL 60510.

tribution of objects on large scales (7). Thus, at best, the fractal is a limited fractal.

It is interesting that as the sampling of the universe gets larger and deeper, more observations appear to continue to support this limited-fractal hypothesis. As was shown in Bahcall and Chokshi [in (9) ], Fig. 1 shows a summary of the current situation and our error bar estimates, with data points for correlation of superclusters (8), quasars (9), x-ray clusters (5), and the cD's at the center of superclusters (4), as well as recent work by Efstathiou [in (10)] with the Automated Plate Measuring (APM) survey that has supported this basic clustering be- havior. For 10 < L : 100 Mpc/h, 3(L) is nearly constant; the current best fit value is 1 = 0.26. Note that a power law correla- tion function with index 1.8 corresponds in three-dimensional (3-D) space to a fractal with D = 1.2.

The following questions arise when we

1.5

1.0

0.5

,, ;H;, , ;. .. .. .. .. .. . ......

o

0 20 40 60 80 100 120

L (Mpc/h)

Fig. 1. The two-point correlation function can

be expressed in the scale-invariant form: t(r) = (rIL)1 8, where L = n113 is the mean distance

between objects in a catalog, n is the mean

density, and p is the dimensionless correlation amplitude. The best fit (broken line) to the

updated observational data gives p = 0.26. The error bars represent +50% uncertainty in the density, +50% uncertainty in the correlation amplitude, and +20% uncertainty in determin- ing the power law index. (A), Galaxies; (0), quasi-stellar objects; (O), Schectman; (O), cDs; (x), APM; (A), x-ray; (O), Bahcall-Soniera (R ? 1); and (-), supercluster.

discuss the possible fractal structures in the universe: How far out does the fractal corre- lation extend? What can we learn from the fractal dimension D = 1.2? What physical process can give rise to a fractal structure in the distribution of observable objects? At present these questions have ambiguous an- swers. Most researchers of fractal large-scale cosmological structure have either tried to assume a pure fractal structure (11, 12) or to emphasize how a pure fractal cannot explain the structure of the universe because of the isotropy of microwave radiation and the relatively uniform distribution of objects at large distances (7). A point that can be lost in such arguments is that if the universe is fractal-like for some range of scale, then some insight might be gained by looking at how such fractals can develop, even though the fractal is eventually truncated.

If the fractal behavior is real, gravity alone cannot be used to explain it because the clustering amplitude of clusters would then not be higher than that of galaxies. Although some form of biasing (13) may be useful here, we instead see if accepting the fractal interpretation offers any useful in- sights. In particular, let us assume that some sort of fractal seed or growth process provides the fractal correlation while gravity enhanc- es correlation amplitude on small scales. We find that applying fractal analysis techniques to large-scale matter distribution in the uni- verse yields some interesting results.

There are two basic requirements to form large-scale structure: (i) primordial seeds or fluctuations (density perturbations) and (ii) the aggregation of matter to the seed (growth process). The correlation of seeds or density perturbations and the scaling behav- ior of growth processes are all responsible for any fractal structure we observe today, and it is interesting to find that most structure formation theories can be fitted in the cate- gory of emphasizing one or the other.

In a continuous clustering model (14)- for example, the variant of Mandelbrot ( 1), in which galaxies are placed on each step of a Levy flight (11)-the correlation between seeds is fully responsible for the fractal distribution of observed objects. The model is simple and successful in reproduc- ing the observed correlation functions. For the Mandelbrot model the fractal dimen- sion D enters the program through the ansatz of the probability distribution of Levy flight: for a random walk with step size l, P(l) = 0 for l < l1o; and P(l) = Dl/D+1 for 1 > lo0 Thus the model is more an empirical computational device than a true physically motivated growth process. It also has the problem of no natural truncation of the fractal at large scales. On the other hand, in the random Gaussian fluctuation model (15), the seeds are randomly distrib- uted in a Gaussian manner. If the ampli-

SCIENCE * VOL. 256 * 24 APRIL 1992 513

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tude of the fluctuations is scale invariant, the model is able to reproduce the two- point correlation on a small scale (, 10 Mpc). When the scale gets larger, some problems appear, as illustrated by the excess power observed on larger scales relative to the falloff in the model (10). (If biasing is invoked to fit the cluster correlations from the galaxy correlations, then the cluster correlation function is directly proportional to the galaxy correlation function. If the galaxy function should unambiguously be negative, then so should the cluster func- tion on that scale. As of this time, the data are too ambiguous for this test to be made.)

Numerical modeling with N-body simu- lations has become the prime tool used in cosmology for exploring the aggregation of matter to seeds beyond linear gravitational perturbation theory [see (16)]. In general, matter undergoes a stochastic motion in space until it is gravitationally bound by seeds to form clumps, and the growth rate of the clump is controlled by the diffusing flux of matter onto the seed. The underly- ing physics of this kind of growth process can be modeled by DLA (diffusion-limited aggregation), and studies with the model show that the resulting aggregate has a well-defined scaling behavior (17).

In particular, the two assumptions used by Ball and Witten (18) in deriving their causality limits are that the aggregate grows by absorbing particles doing a random walk and that the aggregate is limited by diffu- sion. Both of these assumptions seem well justified in the cosmological case. In the growth of a traditional DLA fractal, the interaction of the diffusing particles with the aggregate is short ranged, and the ag- gregate does not grow until the diffusing particles are attached, so the aggregate is connected. In the cosmological case, be- cause gravity is long ranged, the star cluster is more loosely bound. In this report we do not go into the details of a particular growth model but rather show that based on the growth process, the fractal dimension D can serve as an interesting constraint on the growth space, which is defined as the possi- ble trajectories of the stochastic motion of matter clumps. The overall space is 3-D, but the stochastic motion is not necessarily 3-D.

The aggregate grows by absorbing parti- cles that are randomly moving in d-dimen- sional growth space and the outer radius R of the aggregate grows with time, but dR/dt is limited by some value v, which is propor- tional to the density u of moving particles, because of the "shadow" effect, in which parts of a cluster begin to block the interior sites. In our case, the "shadow" effect also occurs for a different reason-when the material is used up, the sites adjacent to the "void" cannot grow. So, dR/dt < v u. The quantity dR/dt is related to the change

of mass M (= RD) of the aggregate by dR/dt = (dM/dt)/(dM/dR). The quantity dM/dt is also the rate at which the diffusing particles are first bounded by the aggregate:

dM - UR d-2 (1) dt

So

dR URd-l-D < V = U (2) dt

thus

d-1-D < 0 (3)

or

D 2 d-1 (4)

This is Ball and Witten's causality bound (18) on the fractal grown from a diffusion- limited process. The observed fractal di- mension D = 1.2 implies that the dimen- sion d of the growth space is less than 2.2. In other words, the growth space should involve a two-dimensional sheetlike object. This fact can constrain the properties of topological defects that might serve as seeds for large-scale structure. This result favors light domain walls (19), wakes of string (20), superconducting strings (the explo- sive model) (21), the pancake model (22), or collapsing textures (23). (Of course it says nothing about other problems these models may have, such as the microwave background y-parameter constraint on the explosive models, and so on.)

One consequence of embedding a fractal structure generation mechanism into the well- established big bang framework (7) is the prediction that the fractal correlation should break down at some scale. As pointed out by Peebles (1), a pure fractal contradicts the observed large-scale angular correlation func- tion. It also has problems with microwave background isotropy. Because the growth pro- cess is limited by the diffusion of particles onto the aggregate, it can drop below the expan- sion rate of the universe. Furthermore, when the random motion of the matter is not constrained, the growth will be 3-D. From Eq. 4 we know that it is impossible to grow a D = 1.2 pure fractal in three dimensions with any kinematic growth process.

The breakdown scale of the fractal corre- lation can be estimated from the constraint of microwave background anistropy 8T/T s i0-5 in the extreme case of a sheetlike seed model, for example, light domain walls from a late-time phase transition. In the late time phase transition scenario, the cosmological seed and density perturbation is generated after the decoupling of the microwave back- ground, which minimizes the cosmic black- body radiation (CBR) anisotropy (17, 24, 25). However, significant non-Gaussian fluctuations can be produced, which may have large amplitudes (24). To grow a fractal

extending to scale L, the aggregation of matter onto the seeds will perturb the cosmic microwave background (24),

8T/T = (9/32)(31Tr)112(HOL)3flwall (5)

where Ho is the Hubble constant and fQwall is the ratio of present density of walls to the critical density, and the density perturbation 8p/p induced by a wall is estimated to be

8p/p - (3T, /20)Awall (6) The fractal growth process can only proceed when ip/p > 1 or 8T/T ; (HOL)3. So &T/T < i0-5 implies L > lOOh-1 Mpc, where the normalized Hubble constant is h _ Ho/(100 km s-1 Mpc-1). This is a natural result because the horizon size at the time of struc- ture formation serves as a cutoff for the fractal correlation. The horizon size R at the time of a late-time phase transition (z = 1000), R 3000 Mpc (h'\1 + z)-1 lOOh-1 Mpc = 200 Mpc, with h = 0.5. This agrees reason- ably well with the previous argument.

The fractal argument not only casts fur- ther doubt on the 3-D-filling Gaussian fluctuation model with cold dark matter, but it also helps point the way toward plausible solutions.

REFERENCES AND NOTES

1. P. J. E. Peebles, The Large Scale Structure of the Universe (Princeton Univ. Press, Princeton, NJ, 1980).

2. N. Bahcall and R. Soniera, Astrophys. J. 270, 20

(1983). 3. S. A. Shectman, Astrophys. J. Suppl. Ser. 57, 77

(1985). 4. M. West and S. Van den Bergh, Astrophys. J. 373,

1 (1991). 5. 0. Lahav, A. Edge, A. C. Fabian, A. Putney, Mon.

Not. R. Astron. Soc. 238, 881 (1989). 6. A. S. Szalay and D. N. Schramm, Nature 314, 718

(1985). 7. P. J. E. Peebles, D..N. Schramm, E. L. Turner, R.

G. Kron, Fermilab-Pub-91/35-A (Fermilab Astro- physics Center, Batavia, IL, 1991).

8. N. Bahcall and W. Burgett, Astrophys. J. 300, L35

(1984). 9. N. Bahcall and A. Chokshi, in preparation; L. Z.

Fang, Y. 0. Chu, X. F. Zhu, Astrophys. Space Sci. 115, 99 (1985); Y. 0. Chu and X. F. Zhu, Astron. Astrophys. 205, 1 (1988).

10. For a review of the APM survey, see G. Efstathiou, Phys. Scr. T36, 87 (1991). See also A. Picard, Astrophys. J. 386, L7 (1991); Astron. J. 102, 455

(1991). 11. B. B. Mandelbrot, The Fractal Geometry of Nature

(Freeman, New York, 1983). 12. P. H. Coleman, L. Pietronero, R. H. Sanders,

Astron. Astrophys. 200, L32 (1988). 13. N. Kaiser, Astrophys. J. 284, L9 (1984). 14. J. Neyman and E. L. Scott, ibid. 116, 144 (1952). 15. For a textbook review, see E. Kolb and M. S.

Turner, The Early Universe (Addison-Wesley, San Francisco, 1989).

16. C. Frenk, S. White, G. Efstathiou, M. Davis, Astro- phys. J. 351, 10 (1990).

17. T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981); T. A. Witten, in Chance and Matter: Proceedings of the Les Houches Summer School 1986, J. Soulettie, J. Vannimenus, R. Stora, Eds. (North-Holland, Amsterdam, 1987), p. 159, and the reference therein; T. A. Wiften and M. E. Cates, Science 232, 1607 (1986).

18. R. C. Ball and T. A. Witten, Phys. Rev. A 29, 2966

(1984).

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..................... ... ............................ .... . . ................ . .. . .. . ...... ... ........ . ... . ... . .

........................... ............... .............. R EPORTS ........................ ......... ......................

19. C. T. Hill, D. N. Schramm, J. Fry, Comments Nucl. Part. Phys. 19, 25 (1989).

20. E. Bertschinger, Astrophys. J. 316, 489 (1987). 21. J. P. Ostriker, C. Thompson, E. Witten, Phys. Lett.

180B, 231 (1986); J. P. Ostriker and L. L. Cowie, Astrophys. J. 243, L127 (1981).

22. Ya. B. Zeldovich, J. Einasto, S. F. Shandarin, Nature 300, 407 (1982).

23. N. Turok, Phys. Rev. Lett. 63, 2625 (1989).

24. M. S. Turner, R. Watkins, L. Widrow, Astrophys. J. 367, L43 (1991).

25. A. Stebbins and M. Turner, ibid. 339, L13 (1989). 26. This work was supported in part by NSF grant

90-22629 and NASA grant NAGW 1321 at the University of Chicago and by DOE and NASA grant NAGW 2381 at NASA/Fermilab.

27 December 1991; accepted 3 March 1992

Metallo-Carbohedrenes [M8C12+ (M = V, Zr, Hf, and Ti)]: A Class of Stable Molecular Cluster Ions

B. C. Guo, S. Wei, J. Purnell, S. Buzza, A. W. Castleman, Jr.*

Findings of magic peaks corresponding to M8C12+ (M = V, Zr, and Hf) formed from reactions of the respective metals with various small hydrocarbons, in conjunction with recent findings for the titanium system, establish metallo-carbohedrenes as a stable general class of molecular cluster ions. A dodecahedral structure of Th point symmetry accounts for the stability of these ionic clusters.

We report findings of magic peaks corre- sponding to M8C12+ (where M is V, Zr, and Hf ) which, along with prior observations for Ti8C12+, now establish metallo-carbohe- drenes as a class of stable molecular cluster ions. The question of whether a general class of such species exists was raised following recent reporting (1) of a prominent (magic) peak in the distributions of titanium-carbon clusters generated through reactions of the metal with hydrocarbons using a laser vapor- ization source. In view of the nature of the species, we raised the issue of whether Ti8C12+ might be the first observed member of a new class of molecular clusters com- prised of a cage-like network of carbon and metal atoms, possibly arranged in the form of a pentagonal dodecahedron. If metallo-car- bohedrenes do exist, it is expected that other early transition metals should be capable of forming molecules of a similar type which would also display an unusual stability. A short while after the observation of Ti8C12+, we extended our work to other transition metal systems, with particular attention to vanadium, and thereafter zirconium and hafnium. Like Ti8C12+, all M8C12+ (M is V, Zr, or Hf elements) also are found to display an enhanced stability.

The experiments were conducted with both a double mass spectrometer (MS/MS) system (2) and a time-of-flight (TOF) mass spectrometer (3) coupled with a laser vapor- ization source. Ionic species comprised of transition metal atoms and carbons are pro- duced with a versatile laser-induced plasma reaction concept (1). Employing a simple laser vaporization device (4, 5), the meth- odology enables the generation of pure met-

Department of Chemistry, The Pennsylvania State University, University Park, PA 16802.

*To whom correspondence should be addressed.

al-carbon and metal-nitrogen clusters in ei- ther neutral or ionized form. The details of the technique will be given elsewhere (6, 7). Briefly, a high power laser is used to irradiate the surface of the metal. In the presence of a plasma containing both neutral and ionic metal species, fast dehydrogena- tion reactions with hydrocarbons occur. As a result, in many cases the hydrocarbons lose all hydrogens and pure metal-carbon clusters are generated. The distribution of the ionic species are analyzed with a quadrupole or TOF mass spectrometer.

Figure 1 shows a typical mass spectrum of vanadium-carbon cationic clusters produced from reactions with CH4. Other small hy- drocarbons yield a similar cluster distribu- tion. The TOF spectrum was obtained with an electric pulser to attract the ionic clusters from the source and analyze them via TOF mass spectrometry. It is evident in this spec- trum that the peak at a mass of 552 atomic mass units (amu) (magic peak) displays en- hanced abundance compared to proximate clusters. Because the reactions involve three elements, the molecule corresponding to the magic peak could, in principle, have the molecular formula VaCbHc, where a, b, and c are the number of vanadium, carbon, and hydrogen atoms contained in the molecule, respectively. However, the isotope-labeling experiments made with hydrocarbons con- taining deuterium and 13C establish that the molecule has no hydrogen atoms at all and contains exactly 12 carbon atoms. Based on these facts and its mass position, the mole- cule is assigned as V8C12.

Figures 2 and 3 display the mass spectra of zirconium and hafnium-carbon cluster cat- ions, respectively. These spectra were ob- tained under the same experimental condi- tions used to obtain Fig. 1, except the use of the zirconium or hafnium rods instead of the

vanadium rod. Interestingly, the two spectra are seen to truncate at Zr8C12+ and Hf8C12+. It is well established that the intensity anomalies (magic numbers) ob- served in a mass spectrum of clusters reflect the stability of the corresponding cluster (8). Magic numbers do not always become man- ifested as prominent peaks, but more typi- cally as a discontinuity, namely truncation in the present case, in an otherwise smooth- ly varying distribution, indicating the forma- tion of geometric structures of special stabil- ity. Hence, the truncation seen in Figs. 2 and 3 indicates that Zr8C12+ and Hf8C12+ also display magic behavior.

Because Zr and Hf have a similar elec- tronic structure to that of Ti, it is expected that the dodecahedron model proposed for Ti8C12+ can rationalize the magic nature of the corresponding species, M8C12+. As for the ionic form of V8C12, although the vana- dium atom has one more electron than Ti, we believe that its geometric structure should also be dodecahedral, in which the vanadium atoms occupy eight unique posi- tions. In order to gain supporting evidence for the proposed structure, we conducted titration experiments with ND3 under ther- mal reaction conditions. In conducting

E-

z -t

E-

500 600 700 800 900 1000

MASS (amu)

Fig. 1. Time-of-flight mass spectrum of vanadi- um-carbon cluster cations. The labeled magic peak is V8C12+. Note that there are other prom- inent peaks proceeding the magic M8C12+ which are precursors involved in the mecha- nism of formation of the cage-like metallo-car- bohedrenes. Species with one- and two-carbon atoms attached to M8C12+ are also visible, where some carbons remain on the magic structure upon its closing (9). Other precursors to the magic peak are seen, such as (7, 12).

SCIENCE * VOL. 256 * 24 APRIL 1992 515

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