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Register a new userRenyi entropy as a measure of cosmic homogeneity
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Biswajit Pandey
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We propose a method for testing homogeneity in three dimensional spatial distributions using Renyi entropy. We apply the proposed method to data from cosmological N-body simulations and Monte Carlo simulations of homogeneous Poisson point process. We show that the method can effectively characterize the inhomogeneities and identify any transition scale to homogeneity, if present in such distributions. The proposed method can be used to study the cosmic homogeneity in present and future generation galaxy redshift surveys.
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We analyze a set of volume limited sample of galaxies from the SDSS to study the issue of cosmic homogeneity. We use the Renyi entropy of different order to probe the inhomogeneties present in the galaxy distributions. We also calculate the Renyi diveregence to quantify the deviations of the galaxy distribution from a homogeneous Poisson distribution on different length scales. We separately carry out the analysis using the overlapping spheres and the independent voxels. Our analysis suggests that the scale of homogeneity is underestimated in the smaller galaxy samples due to the suppression of inhomogeneities by the overlapping of the measuring speheres. We find that an analysis with the independent voxels and/or use of a significantly larger galaxy sample can help to circumvent or mitigate this problem. Combining the results from these analyses, we find that the galaxy distribution in the SDSS becomes homogeneous on a length scale beyond $140 , h^{-1}, {rm Mpc}$.
Cosmology relies on the Cosmological Principle, i.e., the hypothesis that the Universe is homogeneous and isotropic on large scales. This implies in particular that the counts of galaxies should approach a homogeneous scaling with volume at sufficiently large scales. Testing homogeneity is crucial to obtain a correct interpretation of the physical assumptions underlying the current cosmic acceleration and structure formation of the Universe. In this Letter, we use the Baryon Oscillation Spectroscopic Survey to make the first spectroscopic and model-independent measurements of the angular homogeneity scale $theta_{rm h}$. Applying four statistical estimators, we show that the angular distribution of galaxies in the range 0.46 < z < 0.62 is consistent with homogeneity at large scales, and that $theta_{rm h}$ varies with redshift, indicating a smoother Universe in the past. These results are in agreement with the foundations of the standard cosmological paradigm.
According to the cosmological principle, galaxy cluster sizes and cluster densities, when averaged over sufficiently large volumes of space, are expected to be constant everywhere, except for a slow variation with look-back time (redshift). Thus, average cluster sizes or correlation lengths provide a means of testing for homogeneity that is almost free of selection biases. Using ~10^6 galaxies from the SDSS DR7 survey, I show that regions of space separated by ~2 Gpc/h have the same average cluster size and density to 5 - 10 percent. I show that the average cluster size, averaged over many galaxies, remains constant to less than 10 percent from small redshifts out to redshifts of 0.25. The evolution of the cluster sizes with increasing redshift gives fair agreement when the same analysis is applied to the Millennium Simulation. However, the MS does not replicate the increase in cluster amplitudes with redshift seen in the SDSS data. This increase is shown to be caused by the changing composition of the SDSS sample with increasing redshifts. There is no evidence to support a model that attributes the SN Ia dimming to our happening to live in a large, nearly spherical void.
Despite its fundamental importance in cosmology, there have been very few straight-forward tests of the cosmological principle. Such tests are especially timely because of the hemispherical asymmetry in the cosmic microwave background recently observed by the Planck collaboration. Most tests to date looked at the redshift dependence of cosmological parameters. These are subject to large systematic effects that require modeling and bias corrections. Unlike previous tests, the tests described here compare galaxy distributions in equal volumes at the same redshift z. This allows a straight-forward test and z-dependent biases are not a problem. Using ~10^6 galaxies from the SDSS DR7 survey, I show that re- gions of space separated by ~2 Gpc have the same average galaxy correlation radii, amplitudes, and number density to within approx. 5%, which is consistent with standard model expectations.
Nearly a century after the discovery that we live in an expanding Universe, and two decades after the discovery of accelerating cosmic expansion, there remains no direct detection of this acceleration via redshift drift - a change in the cosmological expansion velocity versus time. Because cosmological redshift drift directly determines the Hubble parameter H(z), it is arguably the cleanest possible measurement of the expansion history, and has the potential to constrain dark energy models (e.g. Kim et al. 2015). The challenge is that the signal is small - the best observational constraint presently has an uncertainty several orders of magnitude larger than the expected signal (Darling 2012). Nonetheless, direct detection of redshift drift is becoming feasible, with upcoming facilities such as the ESO-ELT and SKA projecting possible detection within two to three decades. This timescale is uncomfortably long given the potential of this cosmological test. With dedicated experiments it should be possible to rapidly accelerate progress and detect redshift drift with only a five-year observational baseline. Such a facility would also be ideal for precision radial velocity measurements of exoplanets, which could be obtained as a byproduct of the ongoing calibration measurements for the experiment.
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