6 Yadav, Bagla and Khandai 5 SUMMARY ACKNOWLEDGMENTS REFERENCES summarised these findings in Figure 2. The top panel in the fig- ure shows the deviation and the dispersion as estimated from an N-Body simulation. In this case the estimated scale of homogene- ity 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 con- tain perturbations at very large scales. If we patch the power spec- trum 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 Δ D $_{q}$with 2 σ $^{Δ}$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). In this paper we have verified the relation between the two poi nt correlation function and fractal dimensions for a simulated dis- tribution 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 obser- vations (Einasto et al. 1997). 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 statis- tical 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 ex- plicitly that the uncertainty scales with bias in the same manner as the offset of fractal dimension, thus the true scale of homogene- ity is not sensitive to which objects are used as long as the survey volume is large enough to contain a sufficient number. The spi rit of this paper is to estimate the fractal dimensions for an ideal ob- servation, and not to worry about the limitations of current obser- vations. The connection between the fractal dimension and corre- lation 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 ob- servations, 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 es- timated scale of homogeneity is in some sense the upper limit of what can be determined as the scale of homogeneity from observa- tions. 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 homo - geneity is that it can be defined self consistently within thesetting 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. Th e scale of homogeneity is the same for the entire spectrum of fractal di- Computational work for this study was carried out at the clus- ter computing facility in the Harish-Chandra Research Institute (http://cluster.hri.res.in/). This research has made use of NASA’s Astrophysics Data System. The authors thank Prof. Jaan Einasto for useful suggestions. 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 ho- mogeneity 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 obser- vational dataset there are additional contributions to σ Δ $_{D$_{q}$}$. Any increase in the value of σ $^{Δ}$D$_{q}$ leads to a smaller scale of homo- geneity. In this sense our estimate is the ideal scale of homogeneity and may be treated as an upper limit. Note that the additional con- tributions to σ $^{Δ}$D$_{q}$ are required to increase its value by more than an order of magnitude above our determination for the scale of ho- mogeneity 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. 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