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Fractal Dimension as a measure of the scale of Homogeneity

In the multi-fractal analysis of large scale matter distribution, the scale of transition to homogeneity is defined as the scale above which the fractal dimension of underlying point distribution is equal to the ambient dimension of the space in which points are distributed. With finite sized weakly clustered distribution of tracers obtained from galaxy redshift surveys it is difficult to achieve this equality. Recently we have defined the scale of homogeneity to be the scale above which the deviation of fractal dimension from the ambient dimension becomes smaller than the statistical dispersion. In this paper we use the relation between the fractal dimensions and the correlation function to compute the dispersion for any given model in the limit of weak clustering amplitude. We compare the deviation and dispersion for the LCDM model and discuss the implication of this comparison for the expected scale of homogeneity in the concordant model of cosmology. We estimate the upper limit to the scale of homogeneity to be close to 260 Mpc/h for the LCDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only statistical contribution to the dispersion in fractal dimension and we have ignored cosmic variance and contributions due to survey geometry and the selection function. We find that as long as non linear correction are insignificant, scale of homogeneity as defined above does not change with epoch. The scale of homogeneity depends very weakly on the choice of tracer of the density field. Thus the suggested definition of the scale of homogeneity is fairly robust.


INTRODUCTION
One of the primary aims of galaxy redshift surveys is to determine the distribution of luminous matter in the Universe (Huchra et al. 1983;Lilly et al. 1995;Shectman et al. 1996;Huchra et al. 1999;York et al. 2000;Colless et al. 2001;Giavalisco et al. 2004;Le Fèvre et al. 2005;Scoville et al. 2007). These surveys have revealed a large variety of structures starting from groups and clusters of galaxies, extending to super-clusters and an interconnected network of filaments which appears to extend across very large scales (Bond, Kofman, & Pogosyan 1996;Springel et al. 2005;Faucher-Giguère, Lidz, & Hernquist 2008). We expect the galaxy distribution to be homogeneous on large scales. In fact, the assumption of large scale homogeneity and isotropy of the universe is the basis of most cosmological models (Einstein 1917). In addition to determining the large scale structures, the redshift surveys of galaxies can also be used to verify whether the galaxy distribution does indeed become homogeneous at some scale (Einasto & Gramann 1993;Martinez et al. 1998;Yadav et al. 2005;Sylos Labini et al. 2009a,b). Fractal dimensions of the galaxy matter distribution can be used to test the conjecture of homogeneity. One advantage of using fractal dimensions over other analyses is that in the former we don't require the assumption of an average density in the point set (Jones et al. 2004).
When doing our analysis we often work with volume limited sub-samples extracted from the the full magnitude samples of the galaxies. This is done in order to avoid an explicit use of the selection function. The volume limited sub-samples constructed in this manner from a flux limited parent sample naturally have a much smaller number of galaxies. This was found to be too restrictive for the earliest surveys and corrections for varying selection function were used explicitly in order to determine the scale of homogeneity (Bharadwaj et al. 1999;Amendola & Palladino 1999). But with modern galaxy redshift surveys, this limitation is less severe.
Making a volume limited sub-sample requires assumption of a cosmological model and this may be thought of as an undesirable feature of data analysis. However, if we directly use raw data and do not account for a redshift dependent selection function in any way, it is obvious that the selection function will dominate in any large scale description of the distribution of galaxies. This is only to be expected as we see only brighter galaxies at larger distances in a flux or magnitude limited sample.
For a given survey one computes the fractal dimension of the point set under consideration and the scale beyond which the fractal dimension is equal to the ambient dimension of the space in which the particles are distributed is referred to as the scale of homogeneity of that distribution. Mathematically the fractal dimension is defined for an infinite set of points. Given that the observational samples are finite there is a need to understand the deviations in the fractal dimensions arising due to the finite number of points. In practice we should identify a scale to be the scale of homogeneity where the rms error in the fractal dimension is comparable to or greater than the deviation of the fractal dimension from the ambient dimension (Bagla et al. 2008). It is not possible to distinguish between a given point set and a homogeneous point set with the same number of points in the same volume beyond the scale of homogeneity .
In all practical cases of interest the fractal dimension is never equal to the ambient integer dimension of the space. Bagla et al. (2008) have shown that these deviations in fractal dimension occur mainly due to weak clustering present in the galaxy distribution, with a smaller contribution arising due to the finite number of galaxies in the distribution. This work assumed the standard cosmological model in order to derive a relation between the fractal dimension on one hand, and, a combination of number density of tracers of the density field and clustering on the other. It was assumed that the clustering is weak at scales of interest. In this paper, we revisit the expected deviation of fractal dimension for a finite distribution of weakly clustered points and verify the relations derived in Bagla et al. (2008). For this purpose we have used a particle distribution from a large volume N −Body simulations. Further, we generalise the relation between clustering and fractal dimension to compute the rms error in determination of fractal dimensions and clustering from data. We then discuss the implications of this for the expected scale of homogeneity in the standard cosmological model. We also comment on some recent determinations of the scale of homogeneity from observations of galaxy distribution in redshift surveys.
The plan of the paper is as follows. In section §2 we present a brief introduction to fractals and fractal dimensions. Subsection §2.1 describes the fractal dimension for a weakly clustered distribution of finite number of points. Section §3 discusses the calculation of variance in ξ and hence the variance in offset to fractal dimension. We present the results in section §4, and conclude with a summary of the paper in §5.

FRACTALS AND FRACTAL DIMENSIONS
The name fractal was introduced by Benoit B. Mandelbrot (Mandelbrot 1982) to characterise geometrical figures which may not be smooth or regular. One of the definitions of a fractal is that it is a shape made of parts similar to the whole in some way. It is useful to regard a fractal as a set of points that has properties such as those listed below, rather than to look for a more precise definition which will almost certainly exclude some interesting cases. A set F is a fractal if it satisfies most of the following (Falconer 2003): (i) F has a fine structure, i.e., detail on arbitrarily small scales.
(ii) F is too irregular to be described in traditional geometrical language, both locally and globally.
(iii) F has some form of self-similarity, perhaps approximate or statistical.
(iv) In most cases of interest F is defined in a very simple way, perhaps recursively.
Fractals are characterised using the so called fractal dimensions. These can be defined in different ways, which do not necessarily coincide with one another. Therefore, an important aspect of studying a fractal structure is the choice of a definition for fractal dimension that best applies to the case in study.
Fractal dimension provides a description of how much space a point set fills. It is also a measure of the prominence of the irregularities of a point set when viewed at a given scale. We may regard the dimension as an index of complexity. We can, in principle, use the concept of fractal dimensions to characterise any point set.
The simplest definition of the fractal dimension is the so called Box counting dimension. Here we place a number of mutually exclusive boxes that cover the region in space containing the point set and count the number of boxes that contain some of the points of the fractal. The fraction of non-empty boxes clearly depends on the size of boxes. Box counting dimension of a fractal distribution is defined in terms of non empty boxes N (r) of radius r required to cover the distribution. If we define D b to be the box counting dimension (Barabási & Stanley 1995). In general D b is a function of scale. One of the difficulties with such a definition is that it does not depend on the number of particles inside the boxes and rather depends only on the number of boxes. It provides very limited information about the degree of clumpiness of the distribution and is more of a geometrical measure. To get more detailed information on clustering of the distribution we use the concept of correlation dimension. We have chosen the correlation dimension, among various other definitions (see e.g. Borgani 1995;Martínez & Saar 2002), due to its mathematical simplicity and the ease with which it can be adapted to calculations for a given point set. The formal definition of correlation dimension demands that the number of particles in the distribution should be infinite (e.g. N → ∞ in equation 2). We use a working definition for a finite point set. Calculation of the correlation dimension requires the introduction of correlation integral given by Here we assume that we have N points in the distribution. M is the number of centres on which spheres of radius r have been placed with the requirement that the entire sphere lies inside the distribution of points. Spheres are centred on points within the point set and it is clear that M < N . Here ni(r) denotes the number of particles within a distance r from i th point: where Θ(x) is the Heaviside function. If we consider the distribution function for the number of points inside such spherical cells, we can rewrite C2 in terms of the probability of finding n particles in a sphere of radius r.
where P (n; r, N ) is the normalised probability of getting n out of N points as neighbours inside a radius r of any of the points. The correlation dimension D2 of the distribution of points can be defined via the power law scaling of correlation integral, i.e., Since the scaling behaviour of C2 can be different at different scales, we expect the correlation dimension to be a function of scale. For the special case of a homogeneous distribution we see that the correlation dimension equals the ambient dimension for an infinite set of points. C2(r) defined in the manner given by equation 4 provides the average of the distribution function of P (n; r, N ). In order to characterise the distribution of points, we need information about all the moments of the distribution function. This leads us to the generalised dimension Dq, also known as Minkowski-Bouligand dimension. This is a generalisation of the correlation dimension D2. The correlation integral can be generalised to define Cq(r) as which is used to define the Minkowski-Bouligand dimension We expect the generalised dimension to be scale dependent in general. If the value of Dq(r) is independent of both q as well as r then the point distribution is called a mono-fractal. If we are dealing with a finite number of points in a finite volume then we can always define an average density. This allows us to relate the generalised correlation integral with correlation functions. For q > 1, the generalised correlation integral and the Minkowski-Bouligand dimension can be related to a combination of n-point correlation functions with the highest n equal to q and the smallest n equal to 2. The contribution to Cq is dominated by regions of higher number density of points for q ≫ 1 whereas for q ≪ 0 the contribution is dominated by regions of very low number density. This clearly implies that the full spectrum of generalised dimension gives us information about the entire distribution: regions containing clusters of points as well as voids that have few points.
This allows us to connect the concept of fractal dimensions with the statistical measures used to quantify the distribution of matter at large scales in the universe. In the situation where the galaxy distribution is homogeneous and isotropic on large scales, we intuitively expect Dq ≃ D = 3 independent of the value of q. Whereas at smaller scales we expect to see a spectrum of values for the fractal dimension, all different from 3. It is of considerable interest to find out the scale where we can consider the universe to be homogeneous. We address this issue using theoretical models in this paper.
In an earlier paper we have derived a leading order expression for generalised correlation integral for homogeneous as well as weakly clustered distributions of points (Bagla et al. 2008). We found that even for a homogeneous distribution of finite number of points the Minkowski-Bouligand dimension, Dq(r), is not exactly equal to the ambient dimension, D. The difference in these two quantities arises due to discreteness effects. Hence for a finite sample of points, the correct benchmark is not D but the estimated value of Dq for a homogeneous distribution of same number of points in the same volume. An interesting aside is that the correction due to a finite size sample always leads to a smaller value for Dq(r) than D. As expected, this correction is small if the average number of points in spheres is large, i.e.,N ≫ 1. Bagla et al. (2008) have demonstrated that the general expression for m th order moment of a weakly clustered distribution of points is given by: is the average number of particles in a randomly placed sphere and the volume averaged two point correlation function respectively. The generalised correlation integral (eq :6) for this distribution can now be written as The third term on the right hand side in fact encapsulates the contribution of clustering. This differs from the last term in the corresponding expression for a homogeneous distribution as in that case the "clustering" is only due to cells being centred at points whereas in this case the locations of every pair of points has a weak correlation. It is worth noting that the highest order term of order O ξ 2 has a factor O q 3 and hence can become important for sufficiently large q. This may be quantified by stating that qξ ≪ 1 is the more relevant small parameter for this expansion. The Minkowski-Bouligand dimension for this system can be expressed as: For a weakly clustered distribution we note that • For hierarchical clustering, both terms have the same sign and lead to a smaller value for Dq as compared to D.
• Unless the correlation function has a feature at some scale, smaller correlation corresponds to a smaller correction to the Minkowski-Bouligand dimension.
• If the correlation function has a feature then it is possible to have a small correction term (∆Dq) clus for a relatively large ξ. In such a situation, the relation between ξ and (∆Dq) clus is no longer one to one. In such a case (∆Dq) clus does not vary monotonically with scale. The model described here has been validated with the multinomial-multi-fractal model (see e.g. Bagla et al. 2008). In the following discussion, we validate our model with the help of a large volume N-Body simulation in order to check it in the setting where we wish to use it.

N -Body Simulations
We use a the TreePM code for cosmological simulations (Bagla 2002;Bagla & Ray 2003;Khandai & Bagla 2009) to simulate the distribution of particles. The simulations were run with the set of cosmological parameters favoured by WMAP5 (Komatsu et al. 2009) as the best fit for the ΛCDM class of models: Ωnr = 0.2565, ΩΛ = 0.7435, ns = 0.963, σ8 = 0.796, h = 0.719, and, Ω b h 2 = 0.02273. The simulations were done with 512 3 particles in a comoving cube of side 1024h −1 M pc.
We computed the two point correlation function ξ(r) directly from the N −Body simulation output by using a subsample of particles (Peebles 1980;Kaiser 1984). The volume averaged correlation functionξ(r) follows from ξ(r) using equation 9. Figure 1 presents the comparison between ∆Dq estimated directly using equation 7, from the N −Body simulation, and, the values computed using equation 11. For the last curve we use the values of ξ(r) andξ(r) computed from the simulation. We find that the two curves track each other and the differences are less than 10% at all scales larger than 60 h −1 Mpc. This is fairly impressive given that we have only taken the leading order contributions into account. This validates the relation between the correlation functions and the fractal dimensions in the limit of weak clustering. We find that as we go to larger scales, ∆Dq becomes smaller but does not vanish. One may then ask, is there no scale where the universe becomes homogeneous? The answer to this question lies in a comparison of the offset ∆Dq with the dispersion expected due to statistical errors, we discuss this in detail in the following section.

VARIANCE IN FRACTAL DIMENSION
In this section we use the relation between the two point correlation function, number density of points and the fractal dimension to estimate the statistical error in determination of the fractal dimension. We have shown in Bagla et al. (2008) that for most tracers of the large scale density field in the universe, the contribution of the finite number density of points is much smaller than the contribution of clustering in terms of the deviation of fractal dimension from the ambient dimension. In the following discussion we assume that the contribution of a finite number density of points can be ignored. With this assumption, we elevate the relation between the fractal dimension and the two point correlation function to the dispersion of the two quantities. The statistical error in the Minkowski-Bouligand Dimension can then be estimated from the statistical error in the correlation function.
This implies that We can make use of the fact that |Cov(x, y)| ≤ σxσy where x and y are random variables. We get: If we find that one of the σξ(r) or σ ξ (r) is much larger than the other then we get: The problem is then reduced to the evaluation of statistical error in the correlation function. Starting point in the analytical estimate of the statistical error in ξ(r) is the assumption that the variance in the power spectrum is that expected for Gaussian fluctuations with a shot-noise component arising from the finite number of objects used to trace the density field (Feldman et al. 1994;Smith 2009): where V is the simulation volume, andn is the mean density of the objects considered (dark matter particles or halos). Angulo et al. (2008) found good agreement between this expression and the variance in P (k) measured from numerical simulations. In order to develop a consistent approach, we use 1/n = 0 in the following discussion.
The covariance of the two-point correlation function is defined by (Bernstein 1994;Cohn 2006;Smith et al. 2008): where the last term can be replaced by Eq. (16). The variance in the correlation function is simply σ 2 ξ (r) = C ξ (r, r). Kazin et al. (2009) have tested this formula against the variance derived from the mock catalogs for both dim and bright galaxy samples of SDSS, and found that at 50 < r < 100 h −1 Mpc the variance is consistent with equation 17. The direct application of Eq. (17) leads to a substantial over prediction of the variance, since it ignores the effect of binning in pair separation which reduces the covariance in the measurement (Cohn 2006;Smith et al. 2008).
An estimate of the correlation function in the i th pair separation binξi corresponds to the shell averaged correlation function where Vi is the volume of the shell. The covariance of this estimate is given by the following expression, e.g., see (Sánchez et al. 2008) wherē is the volume averaged Bessel function. So the variance inξ is σ 2 ξ (r) = Cξ(i, i). We find that at scales of interest σξ(r) ≪ σ ξ (r). From here, it is straightforward to compute the standard deviation in ∆Dq using equation 15.

SCALE OF HOMOGENEITY
We can describe a distribution of points to be homogeneous if the standard deviation of ∆Dq is greater than ∆Dq. Note that this formulation gives us a unique scale: above this scale it is not possible to distinguish between the given point distribution and a homogeneous distribution. In the cosmological context, we have bypassed the details of contribution to the variance arising from the survey geometry, survey size, etc. Clearly, if we were to take these contributions into account, the error in determination of ∆Dq will be larger and hence we will recover a smaller scale of homogeneity (Sarkar et al. 2009). This change results not from any change in the real scale of homogeneity but because of the limitations of observations. In our view, the real scale of homogeneity is one where these limitations do not matter. In the limit where we ignore these additional sources of errors, we have the following general conclusions: • As long as non-linear correction are not important, the scale of homogeneity does not change with epoch.
• In real space, the scale of homogeneity is independent of the tracer used, because the deviation ∆Dq as well as the dispersion in this quantity scale in the same manner with bias. This follows from the fact that bias in correlation function can be approximated by a constant number at a given epoch at sufficiently large scales (Bagla 1998;Seljak 2000).
• Redshift space distortions introduce some bias dependence in the scale of homogeneity.
• As long as our assumption of qξ ≪ 1 is valid, the scale of homogeneity is the same for all q.
These points highlight the robust and unique nature of the scale of homogeneity defined in the manner proposed in this work.
We now turn to more specific conclusions in the cosmological setting below. We have calculated σ∆D q and ∆Dq using the power spectrum of a large N-body simulation as well as using the linear power spectrum obtained from WMAP5 parameters. We have Figure 2. Variation of ∆Dq and its predicted standard deviation with scale is shown in these plots. The top panel shows these for the ΛCDM simulation described in the text, the middle panel shows the same using the linearly evolved power spectrum and the lower panel again shows the same quantity with data from simulations patched with the linearly evolved power spectrum at large scales. ∆Dq/(0.5qD) is shown using a solid curve in each panel, the dashed line shows the dispersion σ ∆Dq /(0.5qD) and the dotdashed line shows 2σ ∆Dq /(0.5qD). The intersection of ∆Dq/(0.5qD) and σ ∆Dq /(0.5qD) is the scale beyond which we cannot distinguish between the ΛCDM model and a homogeneous distribution. summarised these findings in Figure 2. The top panel in the figure shows the deviation and the dispersion as estimated from an N-Body simulation. In this case the estimated scale of homogeneity is just above 320 h −1 Mpc. The corresponding calculation with the linear theory gives a scale of just over 280 h −1 Mpc. Part of the reason for this difference is that the simulation does not contain perturbations at very large scales. If we patch the power spectrum derived from simulations with the linearly extrapolated power spectrum of fluctuations at these scales then we get 260 h −1 Mpc as the scale of homogeneity, broadly consistent with the value derived from linear theory. Curves for this last estimate are shown in the lowest panel of Figure 2. If we compare ∆Dq with 2σ∆D q instead, then we get scales of 230, 215 and 220 h −1 Mpc, respectively, in the three cases. As expected, increasing the error or dispersion leads to a smaller scale of homogeneity. Once again, the answers derived using different approaches are broadly consistent with each other and even the small deviations can be understood in terms of generic aspects of non-linear evolution of perturbations (Bharadwaj 1996;Bagla & Padmanabhan 1997;Eisenstein, Seo, & White 2007).

SUMMARY
In this paper we have verified the relation between the two point correlation function and fractal dimensions for a simulated distribution of matter at very large scales. This relation indicates (Bagla et al. 2008) that if we use a strongly biased tracer, such as Luminous Red Galaxies or clusters of galaxies then the deviation of the fractal dimension from 3 is larger. This is consistent with observations . It is also worthwhile to mention that several observations have reported detection of excess clustering at a scale of 120 h −1 Mpc (Broadhurst et al. 1990;Einasto et al. 1997) and distances between the largest overdensities have been observed to exceed the scale of homogeneity derived from most observations (Einasto 2009).
In this paper, we have also used the relation to estimate statistical uncertainty in determination of fractal dimensions. The scale of homogeneity is then taken to be the scale where this uncertainty is the same as the offset of fractal dimension from 3. We show explicitly that the uncertainty scales with bias in the same manner as the offset of fractal dimension, thus the true scale of homogeneity is not sensitive to which objects are used as long as the survey volume is large enough to contain a sufficient number. The spirit of this paper is to estimate the fractal dimensions for an ideal observation, and not to worry about the limitations of current observations. The connection between the fractal dimension and correlation function allows us to make this leap in the limit of weak clustering that is clearly applicable at large scales. Applying this to a cosmological situation with the model favoured by WMAP-5 observations, we have estimated the scale of homogeneity to be close to 260 h −1 Mpc. As we have ignored several sources of uncertainty that are likely to be present in most observational data sets, this estimated scale of homogeneity is in some sense the upper limit of what can be determined as the scale of homogeneity from observations. It is comforting to note that the scale of homogeneity is much smaller than the Hubble scale.
An attractive feature of this way of defining the scale of homogeneity is that it can be defined self consistently within the setting of the cosmological model with density perturbations. Further, this scale is independent of epoch and largely independent of the choice of tracer for the large scale density field in the universe. The scale of homogeneity is the same for the entire spectrum of fractal di-mensions, within the constraints of the underlying assumption that q |ξ| ≪ 1, making this a unique scale in the problem.
A comparison with recent determination of the scale of homogeneity from observations is pertinent. Observational analyses have shown that the scale of homogeneity may be as small as 60 − 70 h −1 Mpc (Yadav et al. 2005;Hogg et al. 2005;Sarkar et al. 2009). This is much smaller than the scale of homogeneity we have found using our method. However, we have ignored the effect of survey geometry and size, and hence in the analysis of any observational dataset there are additional contributions to σ∆D q . Any increase in the value of σ∆D q leads to a smaller scale of homogeneity. In this sense our estimate is the ideal scale of homogeneity and may be treated as an upper limit. Note that the additional contributions to σ∆D q are required to increase its value by more than an order of magnitude above our determination for the scale of homogeneity to be as small as 70 h −1 Mpc. It should be possible to demonstrate consistency with our calculation through an explicit estimate of errors in observational survey. In the long term we expect that an increase in survey depth and size will gradually lead to lowering of errors from other sources and we should obtain a larger scale of homogeneity.