rdelmonico

01-25-2014, 02:47 AM

arXiv:gr-qc/0609001v2 17 May 2007

Fractal Holography: a geometric

re-interpretation of cosmological large scale

structure

J. R. Mureika

Department of Physics, Loyola Marymount University, Los Angeles, CA 90045-8227

Email: jmureika@lmu.edu

Abstract The fractal dimension of large-scale galaxy clustering has been reported to be

roughly DF 2 from a wide range of redshift surveys. If correct, this statistic is of interest

for two main reasons: fractal scaling is an implicit representation of information content, and

also the value itself is a geometric signature of area. It is proposed that a fractal distribution

of galaxies may thus be interpreted as a signature of holography (“fractal holography”),

providing more support for current theories of holographic cosmologies. Implications for entropy

bounds are addressed. In particular, because of spatial scale invariance in the matter

distribution, it is shown that violations of the spherical entropy bound can be removed. This

holographic condition instead becomes a rigid constraint on the nature of the matter density

and distribution in the universe. Inclusion of a dark matter distribution is also discussed,

based on theoretical considerations of possible universal CDM density profiles.

PACS: 98.80.Jk, 98.62.Py

Keywords: large scale structure of the universe, galaxies, dark matter, holography

1 Introduction

The popular notions of fractals revolve around spatial power law scaling, physical selfsimilarity,

and structural recursiveness [1]. Mathematically, this relationship assumes the

general form

N(r) rDF . (1)

where DF is the fractal dimension and r is the scale measure. The quantity N(r) represents

the characteristic of the distribution that exhibits the fractal behavior. Measurement of

fractal statistics for a wide range of physical phenomena has been addressed over the years,

ranging from the shape of coastlines to the structure of clouds, and pertinent to this paper,

the large scale distribution of visible matter in the universe [2].

It should be emphasized that the meaning of the fractal dimension is not only statistical,

but it also has geometric significance. Topological considerations constrain the fractal dimension

of a distribution to be less than (or equal) to that of the space in which the structure is

embedded [3]. Furthermore, when a fractal dimension coincides with an integer dimension,

it is possible to make the association between the structure under consideration and the

geometry associated with the dimension. That is, a distribution with fractal dimension of

1

Survey DF Approx. Size

CfA1 1.7 (0.2) 1800

CfA2 2 11000

SSRS1 2.0 (0.1) 1700

SSRS2 2 3600

LEDA 2.1 (0.2) 75000

IRAS 1.2/2 Jy 2.2 (0.2) 5000

Perseus-Pisces 2.1 3300

ESP 1.8 (0.2) 3600

Las Campa˜nas (LCRS) 2.2 (0.2) 25000

SDSS (r1) 2 2 × 105 − 1.5 × 106

Table 1: Galaxy fractal dimension calculations for various redshift surveys (compiled from

[5],[6],[7],[10]).

DF = 0 is described as a point distribution, DF = 1 a linear distribution, DF = 2 a surface

distribution, and DF = 3 a volumetric or space-filling distribution.

A hierarchically-structured universe is a recurrent theme in our understanding of Nature.

Since at least the early sky maps of Charlier [4], it has been suggested that galaxies do not

cluster in a random fashion, but rather appear in clumps interspersed by voids. The advent

of deep sky redshift surveys brought with it a surge interest surrounding the exact nature

of large-scale galaxy distributions in the observable universe. An overwhelming number of

independent estimates of the galaxy clustering fractal dimension, obtained from a variety of

sources seem to unanimously suggest that this statistic has a value of or around DF = 2. Up

to the release of the SDSS data, the various redshift surveys had probed depths up to at least

10h−1 Mpc and confirmed the fractal scaling behavior. Extrapolating the analysis to include

superclustering structure suggested this behavior continued well up to 100 − 1000 h−1 Mpc

[5], with no apparent transition to homogeneity.

The newest SDSS redshift data confirms the DF 2 to a high precision up to 20h−1 Mpc,

but with the correlation weakening to homogeneity at distances of 70h−1 Mpc. Alternate

analyses suggest that the transition to homogeneity is not observed, but instead the fractal

scaling continues up to 200h−1 Mpc [6]. The authors of Reference [7] perform a two-dimension

multifractal analysis on SDSS projected data, deducing that the appropriate scaling is confirmed

to dimensions Dq 2 (1.7, 2.2) for all positive and negative q.

On the whole, this observational data suggests a (local) violation of the cosmological

principle. Geometrically speaking, a homogeneous visible universe should manifest itself as

an N(r) r3 distribution. In terms of the fractal dimension, this is a volumetric scaling with

a dimension DF = 3, synonymous with the lack of any preferential direction. The origins

of the observed large scale structure in the universe are unknown, although it is commonly

believed that it has arisen from anisotropically-distributed quantum fluctuations in the preinflation

epoch. The recent analysis of the SDSS data [8] coupled with emerging CMB data

from WMAP [9] data has helped to isolate probable cosmological parameters to determine

viability of formation models.

Although the origin of this structure is still a mystery, the most successful models are

2

CDM gravitational collapse scenarios in the early universe [11]. The basic framework of

such cluster-formation models relies on perturbations in the local background curvature due

to quantum fluctuations during the inflation epoch. Such models successfully reproduce a

wide range of observable clustering features, including power spectra and correlation functions,

as well in many cases the cited fractal dimensions for galaxy clustering [12, 13, 14, 15].

More recently, such models have been extended to galaxy and quasar clustering [16], with the

added advantage of introducing universal dark energy constraints. In this sense, the CDM

scenarios are the model logical and consistent explanation for the present inhomogeneous

matter distributions. In fact, very recent observational data [17] supports the filamentary

dark matter “scaffolding” that are a consequence of the models.

Despite the fact that the apparent success of the CDM models seems to put to rest any

mysteries surrounding the origin of large scale structure formation, this paper will focus on

a much different mechanism. which can be explained by adopting a new perspective on how

matter and gravitation are allowed to behave? Fractal behavior is generally not associated

with equilibrium growth, and thus most models of large scale structure evolution do not

predict its existence. As the aforementioned evidence suggests, however there appears to be

a uniformly-defined fractal distribution of matter in the universe, at the very least up to some

as-yet unkown scale length. The use of entropy to represent fractal structure stems from

the implicit relation between entropy and information (this is discussed in the concluding

section of this paper). Fractal – and moreover multifractal – statistics quantify the nature

in which information is encoded or distributed in a system. It is the intention of this paper

to highlight this connection between information, entropy, and fractality.

Before proceeding, it would be negligent to ignore the controversy surrounding the fractal

interpretation of large scale structure, which is by no means an established fact. Dubbed

the “fractal debate”, there has been an ongoing discussion of whether or not the analyses

indicating fractal clustering have been performed with the proper data treatment, most

notably a reliable estimate of the redshift distance from otherwise two-dimensional projections.

There are clearly two opposing opinions on the exact nature of any characteristic scale

lengths and clustering that might exist in clustering. The interested reader is directed to

some comprehensive and competing summaries of both sides of the debate in such references

as [5, 18, 19, 20, 21]. The availability of more redshift data will either confirm, deny, or

further muddle this issue. For the purposes of this discussion, however, the fractal model

will be assumed to be correct.

More pertinent to this paper, it should be noted that fractals themselves are members of

a fundamental class of statistical object whose basis lies in the heart of information theory. A

q-multifractal is defined by the measure partition Z(q, r) = Pi[pi(r)]q, and dimensions Dq =

1

q−1 d log[Z(q,r)]

dlog[r] . where q is any integer and pi is the local spatial density of the fractal object

within a sphere of radius r [3]. The traditional fractal dimension is obtained in the limit

q = 0, but when q ! 1 this quantity becomes I(r) = −Pi pi(r) log pi(r) , D1 = d log I(r)

d log r .

The function I(r) is the Shannon information entropy [22]. In the case of a monofractal

distribution, all dimensions Dq collapse to the single value D0, which is the fractal dimension.

In this respect, the fractal dimension may be seen as a representation of a system’s entropy

measure and content.

3

2 Information theory and the holographic principle

Information content and entropy have entered the area of the long-standing dichotomy between

classical and quantum gravity. In early works by Beckenstein [23] it was suggested

that the maximum entropy contained within a black hole was determined not by its volume,

but rather the horizon area AH. This limit, known as the Beckenstein bound, placed stringent

constraints on how information could distribute itself within a region of space. It was

further generalized as the spherical entropy bound,

S(V )

A

4

(2)

where S(V ) is the entropy contained in a volume of space V , and A is the area of the

(spacelike) boundary of V (in units of the Planck area).

Bousso has shown that each of these entropy bounds can be understood as classes of a

more general theory known as the holographic principle (HP) [24]. An unproven hypothesis,

the HP suggests that there exists a deeper geometric origin for the total number of possible

quantum states which can occupy a spatial region. In its most general formulation, the HP

states that S(B) @B/4, where B is some region, @B its boundary, and S(B) the entropy

contained in B. Although most instances are subject to specific failures, the most radical

formulation of the HP – Bousso’s Covariant Entropy Conjecture – proposes that the entropy

bounded by a region B defined by the light sheets of a backward-pointing null cone obeys

a holographic-type relationship. The various entropy bounds which form the holographic

principle place rigid constraints on the number of possible entropy states which can occupy

a region of space [24].

The HP is novel in its motivations: the physics of a spatial n-dimensional region are defined

by dynamical systems which exist exclusive on the region’s (n-1)-dimensional boundary.

The most promising support of this theory is the AdS/CFT correspondence [25], which explicitly

connects via a one-to-one correspondence the framework of a 5D string theory in

anti-deSitter space with a conformal quantum field theory on the 4D boundary.

3 Cosmology and holography

Fischler et al. [26, 27, 28, 29] have proposed an extensive cosmological version of the theory,

primarily based in part on the spherical entropy bound. Dubbed “Holographic Cosmology”,

the framework presents an alternate inflationary evolutionary model for large scale structure

in which structure evolves from evaporating primordial quantum black holes. This model

not only matches current observation but also explains the flatness and horizon problems.

A similar proposal based on the original work of Susskind and Fischler is discussed in [30],

which proposes a “cosmic holography” bound in FRWuniverses of positive, flat, and negative

curvature. Further comparisons and contrasts between holography and cosmology are offered

in [31], in which constraints from inflation are the focus. Similarly, the authors of [32] also

show that power spectrum correlations and suppression in the CMB may by holographic in

origin. References [33] discuss holographic implications for (2+1)-dimensional cosmological

models.

4

As derived in [26], assuming a homogeneous and isotropic universe with constant mean

density, it is possible to define a (co-moving) entropy density such that the total entropy

with a volume V is [24]

S = ZV

d3xph (3)

which can be written S = V , where is the (volumetric) entropy density. The entropy

condition is thus

V

A(V )

4

, (4)

where A(V ) = 4r2 is the bounding area of the volume V (in flat space). Including the

r-dependence, the entropy bound is

r3

3r2

4

, (5)

and so it can easily be shown [24] that the spacelike entropy bound is violated for sufficiently

large values of r > 3/4.

The aforementioned models rely on quantum mechanical entities (black holes) to describe

the basic units of the entropy bound, while proposals like the one presented herein assume

that galaxies are the key. The application of holographic bounds makes the assumption that

these objects are themselves in some way a fundamental “unit” of entropy. It is, however,

reasonable to make this a priori postulate.

It has long been theoreized that galaxies themselves are the result of gravitational clustering

around supermassive black holes that have either grown from accretion, or from combining

with smaller primordial black holes (see the various references [34, 35] and references

therein). Although supermassive black holes had been thought only to populate the bulges

of active galactic nuclei, growing evidence suggests that such objects may also be found in

type I Seyfert galaxies [36] and high-redshift blazars [37]. If one considers black holes to be

the “seeds” of every galaxy, then these objects may be understood to represent a universal

evolution toward a maximal spatial entropy state, independent of the type or size of galaxy.

4 A fractal connection to holography?

The work presented herein is similar in inspiration to that of Fischler et al., and like those

referenced works promotes the notion that holography should be a viable candidate for a

constraint of structure evolution. Holographic constraints will first be applied to visible

matter, but a discussion including dark matter distributions will follow in Section 5. Since it

is somewhat different from the previous cosmological holographies proposed in the literature,

it might be appropriate to label this version as “fractal holography”.

It is the different spatial dependences of area and volume that allows the inequality (4)

to be violated. With respect to Equation 1, however, one can reformulate the “problem” of

large scale fractal clustering by focusing not on the apparent break from homogeneity and

isotropic scaling, but rather by highlighting the specific geometry of the scaling. Since the

redshift surveys cited in Table 1 indicate that large-scale matter is distributed according to

a DF = 2 scaling, it seems more appropriate to describe the entropic content by the mean

5

“surface” entropy density . That is, each object contributes an average entropy S(V )/N

and there are N rDF objects. It should be noted that the fractal power laws are derived

from average density considerations, so such an argument is certainly well-founded. Fractal

large scale structure thus states that within a sphere of radius r, the number of cosmological

objects is a function of area (r2).

Thus, within a spherical volume of radius r, the number of galaxies N(r) must be proportional

to the surface area of the region’s boundary,

N(r) / @V (r) = A(r) , (6)

so that the entropy contained with a region V is

S(V ) = A , (7)

where > 0 is the proportionality constant. The above relation suggests that the distribution

of matter in the universe has perhaps a more fundamental and geometric origin.

In this case, a holographic-type space-like entropy bound is precisely given as

S(V ) = ¯A

A

4

, (8)

where for simplicity the proportionality constant has been absorbed into the surface density

term, ¯ = . Due to the A r2 dependence of each component of this inequality, the

spatial dependence vanishes and what is left is a truly scale-invariant bound. Specifically,

the violation of the space-like entropy bound is eliminated, and instead is replaced by rigid

constraints on the surface entropy density, and hence the geometry of the matter distribution:

¯

1

4

. (9)

What might be the value of ? The fractal distribution of galaxies extends to about

10 Mpc, or 1058 Planck length units. The area of the bounding sphere is thus on the order of

10116 area units. The entropy content of the entire visible universe is on the order of 1090 [31],

so even if a sizable fraction is represented in this fractal distribution, this implies the “surface”

density is no greater than 10−24 or so. The value of the proportionality constant thus

is the key to the inequality. Unless is of exceedingly high order of magnitude, though, it

is unlikely that this bound will ever be violated.

4.1 Entropy bounds for fractal dimensions near 2

Although the observational evidence points to a fractal scaling dimensions of DF = 2, this

exact geometric signature could be a coincidence. If such is the case, then the spherical

entropy bound (8) will not be scale invariant. However, implications of the bound become

even more interesting if one follows the prescription for non-integer scaling dimensions around

DF = 2.

Following this philosophy, (8) can be expressed as

rDF

A

4

r2−DF (10)

6

where is the “fractal number density” of the distribution. So, violations of the entropy

bound will occur whenever (up to geometric factors)

r > 1

1/(DF−2)

. (11)

If the fractal dimension is slightly higher than 2, the bound will ultimately be violated

for a large enough sphere. However, the relative radius of the sphere will be much greater

than in the case of homogeneity. In the case DF = 3, this reduces to the violation derived

in [26].

4.2 Transitions to homogeneity

Current observation suggests that the fractal distribution of matter may transition to homogeneity

at large distances. In this case, the entropy density (7) and scale-invariant entropy

bound (8) are no longer applicable, at least on a global scale.

In addition to the references discussed in Section 1, a recent analysis [38] has determined

that number of luminous red galaxies (LRGs) shows a well-defined DF = 2 fractal behavior

up to scale lengths of at least 20 h−1 Mpc, and a smooth transition to homogeneity (DF = 3)

beyond scales of 70 h−1 Mpc. Reference [39] confirms DF 2 fractal clustering behavior

to a scale of 40 Mpc, using independent analysis techniques such as the nearest neighbor

probability density, the conditional density, and the reduced two-point correlation function.

In terms of the holographic model, in the simplest of cases, consider a spherical region of

radius R (in flat space) in which the distribution of matter is fractal with DF = 2. Within

this region, the entropy constrain obeys that described in Equation 7, i.e. SF(r) ¯rD

F .

For separation scales r > R, the distribution resembles a homogeneous one, and the total

entropy is now described by a volumetric distribution with density . At the transition scale

r = R, these descriptions of entropy must agree. That is, the total entropy within a sphere

of radius r = R should be SF(R) = SH(R), where SF(R) = 4¯R2 and SH = 4/3R3. This

implies that R = 3¯

.

Potential violations of the holographic principle are now re-introduced at scales r > R, as

described by Equation 5. However, the requirement of statistical continuity in the description

of the entropy also requires that S′

F(R) = S′

H(R). This provides an additional constraint

equation on the value of R, in this case R = 2¯

, assuming the only radial dependence in the

holographic bound is in the geometric term (area and volume).

Hence, fractal holography implicitly supports the well-known observation of a slow transition

to homogeneity over distances of several megaparsecs. A similar holographic model

could represent the transition entropy in the most general form STrans(r) (r)rD(r), where

both the entropy “density” ((r)) and the fractal dimension itself (D(r)) become functions of

the scale length. This can be cast as a boundary-constrained problem, with the requirements

SF(R1) = STrans(R1), SH(R2) = STrans(R2), S′

F(R1) = S′

Trans(R1), S′

H(R2) = S′

Trans(R2). Appropriate

constraints on the values of D(r) and its derivative, for example, could be isolated

from correlation and fractal analyses.

It should also be noted that a critical discussion of the meaning of “transition to homogeneity”

found in [40] indicates that there could be two possible interpretations of what it

7

means to transition to homogeneity. This could be in terms of a trivial correlation function

at r = R (the usual transition boundary), as well as a more long-range scale that could

be greater than the horizon distance. Regions measured at scales R < r < are not strictly

homogeneous, but rather are analogous to a fluid at the critical point.

An oft-cited argument for the necessity of a transition to homogeneity stems from the

exceedingly homogeneous distribution of anisotropies in the cosmic microwave background

(CMB). It is important to note, however, that inferring homogeneity of a three-dimensional

distribution from its two-dimensional projection is not trivial. From the point of view of

fractal statistics, for example, a homogeneous surface distribution would coincide with a

fractal dimension of DS = 2. The fractal projection theorem [41] states that for any fractal

with dimension DF , its projection onto a sub-plane of dimension DP DF will itself have

a dimension of DP . That is, the dimensionality of the projective distribution “saturates”

the sub-plane. It is thus possible that the perceived homogeneity of the CMB may not truly

correlate to the homogeneity of volumetric spatial distributions at the time of last scattering.

In this case, the original distribution of anisotropies may well have obeyed a holographic-type

“area” constraint, like that discussed herein.

4.3 Scale evolution and fractal holography

There is ongoing debate as to whether or not the fractal distribution of visible matter extends

indefinitely to beyond 1000 h−1 Mpc. If in fact the entire universe is governed by the DF 2

fractal distribution, the fractal holographic condition can place a bound on its expansion rate.

For a sphere whose radius RH is the horizon distance, the usual holographic bound in d

spatial dimensions is [26]

RH(t)d < [a(t)RH(t)]d−1 . (12)

with a(t) tp the scale factor of the universe, p an expansion parameter, and RH(t) =

Rt

0 a(t′)dt t1−p. The left hand term in the inequality assumes that the entropy scales

volumetrically. Since is small, it has been demonstrated that the inequality is satisfied

throughout the history of the universe if p > 1/d.

Adopting the fractal interpretation and setting d = 3, this becomes

¯RH(t)2 < [a(t)RH(t)]2 . (13)

The new constraint on the evolution parameter is thus ¯ < t2p, which is almost certainly

always satisfied for an arbitrary choice of p.

5 Inclusion of dark matter

So far, the discussion of fractal holography has excluded dark matter. Clearly, any viable

model for large scale structure must replicate more than the visible matter distributions,

but also the “invisible” ones. Although the spatial structure of halo dark matter density

profiles can easily be inferred from galaxy rotation curves, it is uncertain exactly what the

large scale distribution looks like.

Since dark matter is believed to make up well over 90% of the material content of the

universe, no cosmological model would be complete without paying due attention to this

8

mystery. Unfortunately, not much is known about the actual form of the distribution of

dark matter in the universe. The best models we have for density distributions are those of

“small scale” dark matter halo structures derived from galaxy rotation curves, such as the

NFW profile [42], which suggest density profiles of the form DM r−2. While these reflect

the distribution of halo dark matter, they unfortunately offer no insight into the larger scale

structure.

Based on a simple inverse-square density profile for dark matter, the authors of Reference

[43] have shown that a fractal distribution of galaxies is not inconsistent with a

homogeneous distribution of all matter, by virtue of the fact that the dark matter density

profile is the functional reciprocal of the galaxy number count. The authors further note that

this implies a different fractal correlations for luminous matter (DF = 2) and dark matter

(DF = 3).

Some numerical simulations of CDM cosmologies suggest that dark matter halo profiles

should roughly echo that of the matter distribution in the universe [44]. In particular, a

universal density profile derived from N-body simulations has shown that dark matter may

cluster in a hierarchical fashion [45].

It has more recently been suggested that all baryonic matter can be distributed in an

r2 fractal manner, by appealing to alternative theories such as Modified Field Theory [46].

Such a description is consistent with both the Cosmological Principle, as well as the Silk

Effect [47], and can produce a gravitationally-stable fractal clustering (the interested reader

is referred to [46] for further details).

For simplicity, assume the density profile of dark matter is hierarchical, according to the

power law DM r−

. This implies that the number of objects within a region of radius

r is NDM r3−

, which we may associated with a fractal dimension DDM = 3 −

. The

value of

ranges depending on the literature source, from

= 1.5 [48] to

= 2.1 − 2.5,

thus the corresponding fractal dimensions would range between DDM 1.5 − 2.5. In this

case, if DDM 2, the holographic constraint behaves as with luminous matter (and thus is

potentially not violated).

The most promising glimpse at possible larger-scale distributions of dark matter has

been reported by the COSMOS collaboration [17]. These results suggest a largely filamentary

clustering of dark matter in “rods”, with the more familiar halo clusters forming at

“vertices” of several rods. The extact spatial extent of the observed filaments is difficult to

extract from the present data, and specific clustering measurements are currently underway1.

Preliminary results suggest that the dark matter filaments can extend up to 30 Mpc or more

[17]. Nevertheless, the global geometry of such inhomogeneous filamentary structures would

be consistently with a linear dark matter scaling (similar to a fractal scaling of N(r) r),

which would not violate any holographic constraints.

5.1 General density distributions and holographic charge

As a final comment on the geometric re-interpretation of the holographic principle, consider

the general case of matter distributions with fractal dimensions DF = 0, 1, 2,and 3. The

holographic constraints are thus dimensionally written (omitting some constants)

1R. Massey, personal communication

9

S0 , S1 , S2 , S3

A

4

, r , r2 , r3 r2

r2 ,

r

, , r 1 (14)

The above expressions, cast in this manner, now become geometrically reminiscent of

field strengths for various charge distributions. The geometry in question corresponds to

the dimensionality of the fractal: point, linear, surface, and interior field of a continuous

distribution. The functional similarity to Gauss’ Law comes from the area term of the

inequality, and in this sense one might interpret Equation 14 as some variety of “holographic

field” equation (although the similarities are most likely only superficial).

6 Conclusions and future directions

The inspiration for cosmological holography rests in the notion that quantum scale entropic

physics can be arbitrarily expanded to any stable gravitational system. It is thus possible that

such holographic constraints in the early universe led to the formation and non-homogeneous

distribution of anisotropies. Furthermore, the previous discussion suggests that the link

between holographic area bounds and fractal DF = 2 scaling may be related. The galaxy

number counts can be directly related to the entropy content by cosmological considerations

like those discussed herein.

How is such a theory useful to further understanding cosmology and the origins of large

scale structure? At the very least, it helps to place somewhat rigid constraints on the nature

of the clustering, as discussed in Section 4.2 – that is, where the clustering can be represented

as “fractal”, where it might be homogeneous, and over how far a distance the transition can

occur. Hence, a cosmological model can be built using the holographic constraints placed on

the density distributions and their associated spatial extents. In this sense, such a theory is

not limited to only the “fractal” region of galaxy distributions, but can extend indefinitely

and still provide useful boundary constraints.

Nothing in this proposal changes the fundamental origins of clustering. Galaxies can still

form as standard theory dictates. Per the discussion in Section 3, galaxies do not replace

anything as the fundamental unit of structure, since they themselves are likely “perturbations”

to optimal clustering around black holes. Since holographic theories have evolved

from the study of such objects, it is not out of the question to expect galaxies to abide by

some similar form of such principles.

Granted, the success or failure of any holography-inspired theory hinges on the success of

the holographic principle itself. At present, this is still a largely-hypothetical and unproven

conjecture with limited (but growing) theoretical support. Since the holographic bounds are

derived as a fundamental “entropy saturation limit” of the universe, violations thereof would

likely imply that the base assumption is incorrect.

This paper has not considered the holographic bounds introduced in open and closed universes,

but there are several reasons for this omission. First, the wealth of observational data

suggests that the universe is most likely flat. Secondly, while it is still possible to measure

10

fractal dimensions in curved spaces using geodesic radii (via methods such as correlation

analyses), the elegant geometric interpretation of the fractal dimension (Equation 1) is not

as easily realized.

Thus the connection between information theory, gravitation, and geometry is a common

“theme” for fractal large scale structure. At the very least, the observed fractal distribution

behavior of galaxies could be understood to be a large scale bookend principle to holography.

Redshift survey results provide strong evidence that the number counts scale as an area, but

in order to verify a deeper connection future analyses should also focus on the pre-factor of

the fractal relationship. Fractal clustering of large-scale structure may well represent either a

manifestation of holographic entropy bounds, or the end result of a cosmological holography

model, and future studies should adopt such a re-interpretation to explore new implications

of the data.

Acknowledgments

Thanks to Richard Massey (Caltech) for some insight regarding the COSMOS collaboration

data. JRM is support by a Research Corporation Cottrell College Science Award Grant.

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[18] Scaramella, R. et al., Astron. Astrophys. 334, 404 (1998)

[19] Martinez, V. J., Science 284, 445 (1999)

[20] Wu, K. K. S., Lahav, O., and Rees, M. J., Nature 397, 225, (1999)

[21] Abdalla, E., Mohayaee, R., and Ribeiro, M. B., Fractals 9 (2001), 451-462

[22] van der Lubbe, J. C., Information Theory (1997), Cambridge University Press

[23] Beckenstein, J. D., Phys. Rev. D 23 (1981), 287; Phys. Rev. D 30, 1669 (1984)

[24] Bousso, R., Rev. Mod. Phys. 74 (2002), 825-874

[25] Maldacena, J., Adv. Theor. Math. Phys. 2 (1998) 231; Maldacena, J., Int. J. Theor.

Phys. 38 (1998) 1113; Petersen, J. L., Int. J. Mod. Phys. A 14 (1999) 3597; additional

introductions to and overviews of the AdS/CFT Correspondence may also be found in

the TASI lecture series at hep-th/0309246, by J. Maldacena and references therein.

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[27] Banks, T. and Fischler, W., hep-th/0111142

[28] Banks, T. and Fischler, W., Phys. Scripta T117 (2005), 56-63

[29] Banks, T. and Fischler, W., hep-th/0405200

[30] Bak, D. and Rey, S.-J., Class. Quant. Grav. 17 (15) (2000), L83–L89

[31] Kaloper, N. and Linde, A, Phys. Rev. bf D60, 103509 (1999)

[32] Huang, Z.-Y., Wang, B., Abdalla, E., and Su, R.-K., to appear in J. Cosmol. Astropart.

Phys. (2006)

[33] Wang, B. and Abdalla, E., Phys. Lett. bf B466 (1999), 122-126; Phys. Lett. B471

(2000), 346–351

[34] Ghez, A. M.; Klein, B. L.; Morris, M.; Becklin, E. E., Astrophys. J. 509, 678 (1998)

[35] Melia, F. and Falcke, H., Ann. Rev. Astron. Astrophys˙39, 309–352 (2001)

[36] Filippenko, A. and Ho, L. C. Astrophys. J. 588, L13-L16 (2003)

[37] Romani, R. W. et al., Astrophys. J. 610 L9-L11 (2004)

12

[38] Hogg, D. et al., Astrophys. J. 624 (2005), 54–58

[39] Vasilyev, N, Baryshev, Yu. V., and Sylos-Labini, F. Astron. Astrophys. 447 (2006)

431-440

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[41] Vicsek, T., Fractal Growth Phenomena, World Scientific Press, Singapore (1989)

[42] Navarro, J. F., Frenk, C. S., White, S. D. M., Astrophys. J. 462 (1996), 563

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[48] Fukushige, T. and Makino, J., Astrophys. J. 557 (2001) 553-434

13

Fractal Holography: a geometric

re-interpretation of cosmological large scale

structure

J. R. Mureika

Department of Physics, Loyola Marymount University, Los Angeles, CA 90045-8227

Email: jmureika@lmu.edu

Abstract The fractal dimension of large-scale galaxy clustering has been reported to be

roughly DF 2 from a wide range of redshift surveys. If correct, this statistic is of interest

for two main reasons: fractal scaling is an implicit representation of information content, and

also the value itself is a geometric signature of area. It is proposed that a fractal distribution

of galaxies may thus be interpreted as a signature of holography (“fractal holography”),

providing more support for current theories of holographic cosmologies. Implications for entropy

bounds are addressed. In particular, because of spatial scale invariance in the matter

distribution, it is shown that violations of the spherical entropy bound can be removed. This

holographic condition instead becomes a rigid constraint on the nature of the matter density

and distribution in the universe. Inclusion of a dark matter distribution is also discussed,

based on theoretical considerations of possible universal CDM density profiles.

PACS: 98.80.Jk, 98.62.Py

Keywords: large scale structure of the universe, galaxies, dark matter, holography

1 Introduction

The popular notions of fractals revolve around spatial power law scaling, physical selfsimilarity,

and structural recursiveness [1]. Mathematically, this relationship assumes the

general form

N(r) rDF . (1)

where DF is the fractal dimension and r is the scale measure. The quantity N(r) represents

the characteristic of the distribution that exhibits the fractal behavior. Measurement of

fractal statistics for a wide range of physical phenomena has been addressed over the years,

ranging from the shape of coastlines to the structure of clouds, and pertinent to this paper,

the large scale distribution of visible matter in the universe [2].

It should be emphasized that the meaning of the fractal dimension is not only statistical,

but it also has geometric significance. Topological considerations constrain the fractal dimension

of a distribution to be less than (or equal) to that of the space in which the structure is

embedded [3]. Furthermore, when a fractal dimension coincides with an integer dimension,

it is possible to make the association between the structure under consideration and the

geometry associated with the dimension. That is, a distribution with fractal dimension of

1

Survey DF Approx. Size

CfA1 1.7 (0.2) 1800

CfA2 2 11000

SSRS1 2.0 (0.1) 1700

SSRS2 2 3600

LEDA 2.1 (0.2) 75000

IRAS 1.2/2 Jy 2.2 (0.2) 5000

Perseus-Pisces 2.1 3300

ESP 1.8 (0.2) 3600

Las Campa˜nas (LCRS) 2.2 (0.2) 25000

SDSS (r1) 2 2 × 105 − 1.5 × 106

Table 1: Galaxy fractal dimension calculations for various redshift surveys (compiled from

[5],[6],[7],[10]).

DF = 0 is described as a point distribution, DF = 1 a linear distribution, DF = 2 a surface

distribution, and DF = 3 a volumetric or space-filling distribution.

A hierarchically-structured universe is a recurrent theme in our understanding of Nature.

Since at least the early sky maps of Charlier [4], it has been suggested that galaxies do not

cluster in a random fashion, but rather appear in clumps interspersed by voids. The advent

of deep sky redshift surveys brought with it a surge interest surrounding the exact nature

of large-scale galaxy distributions in the observable universe. An overwhelming number of

independent estimates of the galaxy clustering fractal dimension, obtained from a variety of

sources seem to unanimously suggest that this statistic has a value of or around DF = 2. Up

to the release of the SDSS data, the various redshift surveys had probed depths up to at least

10h−1 Mpc and confirmed the fractal scaling behavior. Extrapolating the analysis to include

superclustering structure suggested this behavior continued well up to 100 − 1000 h−1 Mpc

[5], with no apparent transition to homogeneity.

The newest SDSS redshift data confirms the DF 2 to a high precision up to 20h−1 Mpc,

but with the correlation weakening to homogeneity at distances of 70h−1 Mpc. Alternate

analyses suggest that the transition to homogeneity is not observed, but instead the fractal

scaling continues up to 200h−1 Mpc [6]. The authors of Reference [7] perform a two-dimension

multifractal analysis on SDSS projected data, deducing that the appropriate scaling is confirmed

to dimensions Dq 2 (1.7, 2.2) for all positive and negative q.

On the whole, this observational data suggests a (local) violation of the cosmological

principle. Geometrically speaking, a homogeneous visible universe should manifest itself as

an N(r) r3 distribution. In terms of the fractal dimension, this is a volumetric scaling with

a dimension DF = 3, synonymous with the lack of any preferential direction. The origins

of the observed large scale structure in the universe are unknown, although it is commonly

believed that it has arisen from anisotropically-distributed quantum fluctuations in the preinflation

epoch. The recent analysis of the SDSS data [8] coupled with emerging CMB data

from WMAP [9] data has helped to isolate probable cosmological parameters to determine

viability of formation models.

Although the origin of this structure is still a mystery, the most successful models are

2

CDM gravitational collapse scenarios in the early universe [11]. The basic framework of

such cluster-formation models relies on perturbations in the local background curvature due

to quantum fluctuations during the inflation epoch. Such models successfully reproduce a

wide range of observable clustering features, including power spectra and correlation functions,

as well in many cases the cited fractal dimensions for galaxy clustering [12, 13, 14, 15].

More recently, such models have been extended to galaxy and quasar clustering [16], with the

added advantage of introducing universal dark energy constraints. In this sense, the CDM

scenarios are the model logical and consistent explanation for the present inhomogeneous

matter distributions. In fact, very recent observational data [17] supports the filamentary

dark matter “scaffolding” that are a consequence of the models.

Despite the fact that the apparent success of the CDM models seems to put to rest any

mysteries surrounding the origin of large scale structure formation, this paper will focus on

a much different mechanism. which can be explained by adopting a new perspective on how

matter and gravitation are allowed to behave? Fractal behavior is generally not associated

with equilibrium growth, and thus most models of large scale structure evolution do not

predict its existence. As the aforementioned evidence suggests, however there appears to be

a uniformly-defined fractal distribution of matter in the universe, at the very least up to some

as-yet unkown scale length. The use of entropy to represent fractal structure stems from

the implicit relation between entropy and information (this is discussed in the concluding

section of this paper). Fractal – and moreover multifractal – statistics quantify the nature

in which information is encoded or distributed in a system. It is the intention of this paper

to highlight this connection between information, entropy, and fractality.

Before proceeding, it would be negligent to ignore the controversy surrounding the fractal

interpretation of large scale structure, which is by no means an established fact. Dubbed

the “fractal debate”, there has been an ongoing discussion of whether or not the analyses

indicating fractal clustering have been performed with the proper data treatment, most

notably a reliable estimate of the redshift distance from otherwise two-dimensional projections.

There are clearly two opposing opinions on the exact nature of any characteristic scale

lengths and clustering that might exist in clustering. The interested reader is directed to

some comprehensive and competing summaries of both sides of the debate in such references

as [5, 18, 19, 20, 21]. The availability of more redshift data will either confirm, deny, or

further muddle this issue. For the purposes of this discussion, however, the fractal model

will be assumed to be correct.

More pertinent to this paper, it should be noted that fractals themselves are members of

a fundamental class of statistical object whose basis lies in the heart of information theory. A

q-multifractal is defined by the measure partition Z(q, r) = Pi[pi(r)]q, and dimensions Dq =

1

q−1 d log[Z(q,r)]

dlog[r] . where q is any integer and pi is the local spatial density of the fractal object

within a sphere of radius r [3]. The traditional fractal dimension is obtained in the limit

q = 0, but when q ! 1 this quantity becomes I(r) = −Pi pi(r) log pi(r) , D1 = d log I(r)

d log r .

The function I(r) is the Shannon information entropy [22]. In the case of a monofractal

distribution, all dimensions Dq collapse to the single value D0, which is the fractal dimension.

In this respect, the fractal dimension may be seen as a representation of a system’s entropy

measure and content.

3

2 Information theory and the holographic principle

Information content and entropy have entered the area of the long-standing dichotomy between

classical and quantum gravity. In early works by Beckenstein [23] it was suggested

that the maximum entropy contained within a black hole was determined not by its volume,

but rather the horizon area AH. This limit, known as the Beckenstein bound, placed stringent

constraints on how information could distribute itself within a region of space. It was

further generalized as the spherical entropy bound,

S(V )

A

4

(2)

where S(V ) is the entropy contained in a volume of space V , and A is the area of the

(spacelike) boundary of V (in units of the Planck area).

Bousso has shown that each of these entropy bounds can be understood as classes of a

more general theory known as the holographic principle (HP) [24]. An unproven hypothesis,

the HP suggests that there exists a deeper geometric origin for the total number of possible

quantum states which can occupy a spatial region. In its most general formulation, the HP

states that S(B) @B/4, where B is some region, @B its boundary, and S(B) the entropy

contained in B. Although most instances are subject to specific failures, the most radical

formulation of the HP – Bousso’s Covariant Entropy Conjecture – proposes that the entropy

bounded by a region B defined by the light sheets of a backward-pointing null cone obeys

a holographic-type relationship. The various entropy bounds which form the holographic

principle place rigid constraints on the number of possible entropy states which can occupy

a region of space [24].

The HP is novel in its motivations: the physics of a spatial n-dimensional region are defined

by dynamical systems which exist exclusive on the region’s (n-1)-dimensional boundary.

The most promising support of this theory is the AdS/CFT correspondence [25], which explicitly

connects via a one-to-one correspondence the framework of a 5D string theory in

anti-deSitter space with a conformal quantum field theory on the 4D boundary.

3 Cosmology and holography

Fischler et al. [26, 27, 28, 29] have proposed an extensive cosmological version of the theory,

primarily based in part on the spherical entropy bound. Dubbed “Holographic Cosmology”,

the framework presents an alternate inflationary evolutionary model for large scale structure

in which structure evolves from evaporating primordial quantum black holes. This model

not only matches current observation but also explains the flatness and horizon problems.

A similar proposal based on the original work of Susskind and Fischler is discussed in [30],

which proposes a “cosmic holography” bound in FRWuniverses of positive, flat, and negative

curvature. Further comparisons and contrasts between holography and cosmology are offered

in [31], in which constraints from inflation are the focus. Similarly, the authors of [32] also

show that power spectrum correlations and suppression in the CMB may by holographic in

origin. References [33] discuss holographic implications for (2+1)-dimensional cosmological

models.

4

As derived in [26], assuming a homogeneous and isotropic universe with constant mean

density, it is possible to define a (co-moving) entropy density such that the total entropy

with a volume V is [24]

S = ZV

d3xph (3)

which can be written S = V , where is the (volumetric) entropy density. The entropy

condition is thus

V

A(V )

4

, (4)

where A(V ) = 4r2 is the bounding area of the volume V (in flat space). Including the

r-dependence, the entropy bound is

r3

3r2

4

, (5)

and so it can easily be shown [24] that the spacelike entropy bound is violated for sufficiently

large values of r > 3/4.

The aforementioned models rely on quantum mechanical entities (black holes) to describe

the basic units of the entropy bound, while proposals like the one presented herein assume

that galaxies are the key. The application of holographic bounds makes the assumption that

these objects are themselves in some way a fundamental “unit” of entropy. It is, however,

reasonable to make this a priori postulate.

It has long been theoreized that galaxies themselves are the result of gravitational clustering

around supermassive black holes that have either grown from accretion, or from combining

with smaller primordial black holes (see the various references [34, 35] and references

therein). Although supermassive black holes had been thought only to populate the bulges

of active galactic nuclei, growing evidence suggests that such objects may also be found in

type I Seyfert galaxies [36] and high-redshift blazars [37]. If one considers black holes to be

the “seeds” of every galaxy, then these objects may be understood to represent a universal

evolution toward a maximal spatial entropy state, independent of the type or size of galaxy.

4 A fractal connection to holography?

The work presented herein is similar in inspiration to that of Fischler et al., and like those

referenced works promotes the notion that holography should be a viable candidate for a

constraint of structure evolution. Holographic constraints will first be applied to visible

matter, but a discussion including dark matter distributions will follow in Section 5. Since it

is somewhat different from the previous cosmological holographies proposed in the literature,

it might be appropriate to label this version as “fractal holography”.

It is the different spatial dependences of area and volume that allows the inequality (4)

to be violated. With respect to Equation 1, however, one can reformulate the “problem” of

large scale fractal clustering by focusing not on the apparent break from homogeneity and

isotropic scaling, but rather by highlighting the specific geometry of the scaling. Since the

redshift surveys cited in Table 1 indicate that large-scale matter is distributed according to

a DF = 2 scaling, it seems more appropriate to describe the entropic content by the mean

5

“surface” entropy density . That is, each object contributes an average entropy S(V )/N

and there are N rDF objects. It should be noted that the fractal power laws are derived

from average density considerations, so such an argument is certainly well-founded. Fractal

large scale structure thus states that within a sphere of radius r, the number of cosmological

objects is a function of area (r2).

Thus, within a spherical volume of radius r, the number of galaxies N(r) must be proportional

to the surface area of the region’s boundary,

N(r) / @V (r) = A(r) , (6)

so that the entropy contained with a region V is

S(V ) = A , (7)

where > 0 is the proportionality constant. The above relation suggests that the distribution

of matter in the universe has perhaps a more fundamental and geometric origin.

In this case, a holographic-type space-like entropy bound is precisely given as

S(V ) = ¯A

A

4

, (8)

where for simplicity the proportionality constant has been absorbed into the surface density

term, ¯ = . Due to the A r2 dependence of each component of this inequality, the

spatial dependence vanishes and what is left is a truly scale-invariant bound. Specifically,

the violation of the space-like entropy bound is eliminated, and instead is replaced by rigid

constraints on the surface entropy density, and hence the geometry of the matter distribution:

¯

1

4

. (9)

What might be the value of ? The fractal distribution of galaxies extends to about

10 Mpc, or 1058 Planck length units. The area of the bounding sphere is thus on the order of

10116 area units. The entropy content of the entire visible universe is on the order of 1090 [31],

so even if a sizable fraction is represented in this fractal distribution, this implies the “surface”

density is no greater than 10−24 or so. The value of the proportionality constant thus

is the key to the inequality. Unless is of exceedingly high order of magnitude, though, it

is unlikely that this bound will ever be violated.

4.1 Entropy bounds for fractal dimensions near 2

Although the observational evidence points to a fractal scaling dimensions of DF = 2, this

exact geometric signature could be a coincidence. If such is the case, then the spherical

entropy bound (8) will not be scale invariant. However, implications of the bound become

even more interesting if one follows the prescription for non-integer scaling dimensions around

DF = 2.

Following this philosophy, (8) can be expressed as

rDF

A

4

r2−DF (10)

6

where is the “fractal number density” of the distribution. So, violations of the entropy

bound will occur whenever (up to geometric factors)

r > 1

1/(DF−2)

. (11)

If the fractal dimension is slightly higher than 2, the bound will ultimately be violated

for a large enough sphere. However, the relative radius of the sphere will be much greater

than in the case of homogeneity. In the case DF = 3, this reduces to the violation derived

in [26].

4.2 Transitions to homogeneity

Current observation suggests that the fractal distribution of matter may transition to homogeneity

at large distances. In this case, the entropy density (7) and scale-invariant entropy

bound (8) are no longer applicable, at least on a global scale.

In addition to the references discussed in Section 1, a recent analysis [38] has determined

that number of luminous red galaxies (LRGs) shows a well-defined DF = 2 fractal behavior

up to scale lengths of at least 20 h−1 Mpc, and a smooth transition to homogeneity (DF = 3)

beyond scales of 70 h−1 Mpc. Reference [39] confirms DF 2 fractal clustering behavior

to a scale of 40 Mpc, using independent analysis techniques such as the nearest neighbor

probability density, the conditional density, and the reduced two-point correlation function.

In terms of the holographic model, in the simplest of cases, consider a spherical region of

radius R (in flat space) in which the distribution of matter is fractal with DF = 2. Within

this region, the entropy constrain obeys that described in Equation 7, i.e. SF(r) ¯rD

F .

For separation scales r > R, the distribution resembles a homogeneous one, and the total

entropy is now described by a volumetric distribution with density . At the transition scale

r = R, these descriptions of entropy must agree. That is, the total entropy within a sphere

of radius r = R should be SF(R) = SH(R), where SF(R) = 4¯R2 and SH = 4/3R3. This

implies that R = 3¯

.

Potential violations of the holographic principle are now re-introduced at scales r > R, as

described by Equation 5. However, the requirement of statistical continuity in the description

of the entropy also requires that S′

F(R) = S′

H(R). This provides an additional constraint

equation on the value of R, in this case R = 2¯

, assuming the only radial dependence in the

holographic bound is in the geometric term (area and volume).

Hence, fractal holography implicitly supports the well-known observation of a slow transition

to homogeneity over distances of several megaparsecs. A similar holographic model

could represent the transition entropy in the most general form STrans(r) (r)rD(r), where

both the entropy “density” ((r)) and the fractal dimension itself (D(r)) become functions of

the scale length. This can be cast as a boundary-constrained problem, with the requirements

SF(R1) = STrans(R1), SH(R2) = STrans(R2), S′

F(R1) = S′

Trans(R1), S′

H(R2) = S′

Trans(R2). Appropriate

constraints on the values of D(r) and its derivative, for example, could be isolated

from correlation and fractal analyses.

It should also be noted that a critical discussion of the meaning of “transition to homogeneity”

found in [40] indicates that there could be two possible interpretations of what it

7

means to transition to homogeneity. This could be in terms of a trivial correlation function

at r = R (the usual transition boundary), as well as a more long-range scale that could

be greater than the horizon distance. Regions measured at scales R < r < are not strictly

homogeneous, but rather are analogous to a fluid at the critical point.

An oft-cited argument for the necessity of a transition to homogeneity stems from the

exceedingly homogeneous distribution of anisotropies in the cosmic microwave background

(CMB). It is important to note, however, that inferring homogeneity of a three-dimensional

distribution from its two-dimensional projection is not trivial. From the point of view of

fractal statistics, for example, a homogeneous surface distribution would coincide with a

fractal dimension of DS = 2. The fractal projection theorem [41] states that for any fractal

with dimension DF , its projection onto a sub-plane of dimension DP DF will itself have

a dimension of DP . That is, the dimensionality of the projective distribution “saturates”

the sub-plane. It is thus possible that the perceived homogeneity of the CMB may not truly

correlate to the homogeneity of volumetric spatial distributions at the time of last scattering.

In this case, the original distribution of anisotropies may well have obeyed a holographic-type

“area” constraint, like that discussed herein.

4.3 Scale evolution and fractal holography

There is ongoing debate as to whether or not the fractal distribution of visible matter extends

indefinitely to beyond 1000 h−1 Mpc. If in fact the entire universe is governed by the DF 2

fractal distribution, the fractal holographic condition can place a bound on its expansion rate.

For a sphere whose radius RH is the horizon distance, the usual holographic bound in d

spatial dimensions is [26]

RH(t)d < [a(t)RH(t)]d−1 . (12)

with a(t) tp the scale factor of the universe, p an expansion parameter, and RH(t) =

Rt

0 a(t′)dt t1−p. The left hand term in the inequality assumes that the entropy scales

volumetrically. Since is small, it has been demonstrated that the inequality is satisfied

throughout the history of the universe if p > 1/d.

Adopting the fractal interpretation and setting d = 3, this becomes

¯RH(t)2 < [a(t)RH(t)]2 . (13)

The new constraint on the evolution parameter is thus ¯ < t2p, which is almost certainly

always satisfied for an arbitrary choice of p.

5 Inclusion of dark matter

So far, the discussion of fractal holography has excluded dark matter. Clearly, any viable

model for large scale structure must replicate more than the visible matter distributions,

but also the “invisible” ones. Although the spatial structure of halo dark matter density

profiles can easily be inferred from galaxy rotation curves, it is uncertain exactly what the

large scale distribution looks like.

Since dark matter is believed to make up well over 90% of the material content of the

universe, no cosmological model would be complete without paying due attention to this

8

mystery. Unfortunately, not much is known about the actual form of the distribution of

dark matter in the universe. The best models we have for density distributions are those of

“small scale” dark matter halo structures derived from galaxy rotation curves, such as the

NFW profile [42], which suggest density profiles of the form DM r−2. While these reflect

the distribution of halo dark matter, they unfortunately offer no insight into the larger scale

structure.

Based on a simple inverse-square density profile for dark matter, the authors of Reference

[43] have shown that a fractal distribution of galaxies is not inconsistent with a

homogeneous distribution of all matter, by virtue of the fact that the dark matter density

profile is the functional reciprocal of the galaxy number count. The authors further note that

this implies a different fractal correlations for luminous matter (DF = 2) and dark matter

(DF = 3).

Some numerical simulations of CDM cosmologies suggest that dark matter halo profiles

should roughly echo that of the matter distribution in the universe [44]. In particular, a

universal density profile derived from N-body simulations has shown that dark matter may

cluster in a hierarchical fashion [45].

It has more recently been suggested that all baryonic matter can be distributed in an

r2 fractal manner, by appealing to alternative theories such as Modified Field Theory [46].

Such a description is consistent with both the Cosmological Principle, as well as the Silk

Effect [47], and can produce a gravitationally-stable fractal clustering (the interested reader

is referred to [46] for further details).

For simplicity, assume the density profile of dark matter is hierarchical, according to the

power law DM r−

. This implies that the number of objects within a region of radius

r is NDM r3−

, which we may associated with a fractal dimension DDM = 3 −

. The

value of

ranges depending on the literature source, from

= 1.5 [48] to

= 2.1 − 2.5,

thus the corresponding fractal dimensions would range between DDM 1.5 − 2.5. In this

case, if DDM 2, the holographic constraint behaves as with luminous matter (and thus is

potentially not violated).

The most promising glimpse at possible larger-scale distributions of dark matter has

been reported by the COSMOS collaboration [17]. These results suggest a largely filamentary

clustering of dark matter in “rods”, with the more familiar halo clusters forming at

“vertices” of several rods. The extact spatial extent of the observed filaments is difficult to

extract from the present data, and specific clustering measurements are currently underway1.

Preliminary results suggest that the dark matter filaments can extend up to 30 Mpc or more

[17]. Nevertheless, the global geometry of such inhomogeneous filamentary structures would

be consistently with a linear dark matter scaling (similar to a fractal scaling of N(r) r),

which would not violate any holographic constraints.

5.1 General density distributions and holographic charge

As a final comment on the geometric re-interpretation of the holographic principle, consider

the general case of matter distributions with fractal dimensions DF = 0, 1, 2,and 3. The

holographic constraints are thus dimensionally written (omitting some constants)

1R. Massey, personal communication

9

S0 , S1 , S2 , S3

A

4

, r , r2 , r3 r2

r2 ,

r

, , r 1 (14)

The above expressions, cast in this manner, now become geometrically reminiscent of

field strengths for various charge distributions. The geometry in question corresponds to

the dimensionality of the fractal: point, linear, surface, and interior field of a continuous

distribution. The functional similarity to Gauss’ Law comes from the area term of the

inequality, and in this sense one might interpret Equation 14 as some variety of “holographic

field” equation (although the similarities are most likely only superficial).

6 Conclusions and future directions

The inspiration for cosmological holography rests in the notion that quantum scale entropic

physics can be arbitrarily expanded to any stable gravitational system. It is thus possible that

such holographic constraints in the early universe led to the formation and non-homogeneous

distribution of anisotropies. Furthermore, the previous discussion suggests that the link

between holographic area bounds and fractal DF = 2 scaling may be related. The galaxy

number counts can be directly related to the entropy content by cosmological considerations

like those discussed herein.

How is such a theory useful to further understanding cosmology and the origins of large

scale structure? At the very least, it helps to place somewhat rigid constraints on the nature

of the clustering, as discussed in Section 4.2 – that is, where the clustering can be represented

as “fractal”, where it might be homogeneous, and over how far a distance the transition can

occur. Hence, a cosmological model can be built using the holographic constraints placed on

the density distributions and their associated spatial extents. In this sense, such a theory is

not limited to only the “fractal” region of galaxy distributions, but can extend indefinitely

and still provide useful boundary constraints.

Nothing in this proposal changes the fundamental origins of clustering. Galaxies can still

form as standard theory dictates. Per the discussion in Section 3, galaxies do not replace

anything as the fundamental unit of structure, since they themselves are likely “perturbations”

to optimal clustering around black holes. Since holographic theories have evolved

from the study of such objects, it is not out of the question to expect galaxies to abide by

some similar form of such principles.

Granted, the success or failure of any holography-inspired theory hinges on the success of

the holographic principle itself. At present, this is still a largely-hypothetical and unproven

conjecture with limited (but growing) theoretical support. Since the holographic bounds are

derived as a fundamental “entropy saturation limit” of the universe, violations thereof would

likely imply that the base assumption is incorrect.

This paper has not considered the holographic bounds introduced in open and closed universes,

but there are several reasons for this omission. First, the wealth of observational data

suggests that the universe is most likely flat. Secondly, while it is still possible to measure

10

fractal dimensions in curved spaces using geodesic radii (via methods such as correlation

analyses), the elegant geometric interpretation of the fractal dimension (Equation 1) is not

as easily realized.

Thus the connection between information theory, gravitation, and geometry is a common

“theme” for fractal large scale structure. At the very least, the observed fractal distribution

behavior of galaxies could be understood to be a large scale bookend principle to holography.

Redshift survey results provide strong evidence that the number counts scale as an area, but

in order to verify a deeper connection future analyses should also focus on the pre-factor of

the fractal relationship. Fractal clustering of large-scale structure may well represent either a

manifestation of holographic entropy bounds, or the end result of a cosmological holography

model, and future studies should adopt such a re-interpretation to explore new implications

of the data.

Acknowledgments

Thanks to Richard Massey (Caltech) for some insight regarding the COSMOS collaboration

data. JRM is support by a Research Corporation Cottrell College Science Award Grant.

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