Abstract:
In the near future a new generation of CCD based galaxy surveys will enable high precision determination of the N-point correlation functions. The resulting information will help to resolve the ambiguities associated with two-point correlation functions thus constraining theories of structure formation, biasing, and Gaussianity of initial conditions independently of the value of $\Omega$. As one the most successful methods to extract the amplitude of higher order correlations is based on measuring the distribution of counts in cells, this work presents an advanced way of measuring it with unprecedented accuracy. Szapudi and Colombi (1996, hereafter \cite{sc96}) identified the main sources of theoretical errors in extracting counts in cells from galaxy catalogs. One of these sources, termed as measurement error, stems from the fact that conventional methods use a finite number of sampling cells to estimate counts in cells. This effect can be circumvented by using an infinite number of cells. This paper presents an algorithm, which, in practice achieves this goal, i.e. it is equivalent to throwing an infinite number of sampling cells in finite time. The errors associated with sampling cells are completely eliminated by this procedure which will be essential for the accurate analysis of future surveys.

Abstract:
The first non-trivial cumulant correlator of the galaxy density field $Q_{21}$ is examined from the point of view of biasing. It is shown that to leading order it depends on two biasing parameters $b$, and $b_2$, and on $q_{21}$, the underlying cumulant correlator of the mass. As the skewness $Q_3$ has analogous properties, the slope of the correlation function $-\gamma$, $Q_3$, and $Q_{21}$ uniquely determine the bias parameter on a particular scale to be $b = \gamma/6(Q_{21}-Q_3)$, when working in the context of gravitational instability with Gaussian initial conditions. Thus on large scales, easily accessible with the future Sloan Digital Sky Survey and the 2 Degree Field Survey, it will be possible to extract $b$, and $b_2$ from simple counts in cells measurements. Moreover, the higher order cumulants, $Q_N$, successively determine the higher order biasing parameters. From these it is possible to predict higher order cumulant correlators as well. Comparing the predictions with the measurements will provide internal consistency checks on the validity of the assumptions in the theory, most notably perturbation theory of the growth of fluctuations by gravity and Gaussian initial conditions. Since the method is insensitive to $\Omega$, it can be successfully combined with results from velocity fields, which determine $\Omega^{0.6}/b$, to measure the total density parameter in the universe.

Abstract:
We present a simple and accurate method to constrain galaxy bias based on the distribution of counts in cells. The most unique feature of our technique is that it is applicable to non-linear scales, where both dark matter statistics and the nature of galaxy bias are fairly complex. First, we estimate the underlying continuous distribution function from precise counts-in-cells measurements assuming local Poisson sampling. Then a robust, non-parametric inversion of the bias function is recovered from the comparison of the cumulative distributions in simulated dark matter and galaxy catalogs. Obtaining continuous statistics from the discrete counts is the most delicate novel part of our recipe. It corresponds to a deconvolution of a (Poisson) kernel. For this we present two alternatives: a model independent algorithm based on Richardson-Lucy iteration, and a solution using a parametric skewed lognormal model. We find that the latter is an excellent approximation for the dark matter distribution, but the model independent iterative procedure is more suitable for galaxies. Tests based on high resolution dark matter simulations and corresponding mock galaxy catalogs show that we can reconstruct the non-linear bias function down to highly non-linear scales with high precision in the range of $-1 \le \delta \le 5$. As far as the stochasticity of the bias, we have found a remarkably simple and accurate formula based on Poisson noise, which provides an excellent approximation for the scatter around the mean non-linear bias function. In addition we have found that redshift distortions have a negligible effect on our bias reconstruction, therefore our recipe can be safely applied to redshift surveys.

Abstract:
We measure the angular power spectrum of the WMAP first-year temperature anisotropy maps. We use SpICE (Spatially Inhomogeneous Correlation Estimator) to estimate Cl's for multipoles l=2-900 from all possible cross-correlation channels. Except for the map-making stage, our measurements provide an independent analysis of that by Hinshaw etal (2003). Despite the different methods used, there is virtually no difference between the two measurements for l < 700 ; the highest l's are still compatible within 1-sigma errors. We use a novel intra-bin variance method to constrain Cl errors in a model independent way. When applied to WMAP data, the intra-bin variance estimator yields diagonal errors 10% larger than those reported by the WMAP team for 100 < l < 450. This translates into a 2.4 sigma detection of systematics since no difference is expected between the SpICE and the WMAP team estimator window functions in this multipole range. With our measurement of the Cl's and errors, we get chi^2/d.o.f. = 1.042 for a best-fit LCDM model, which has a 14% probability, whereas the WMAP team obtained chi^2/d.o.f. = 1.066, which has a 5% probability. We assess the impact of our results on cosmological parameters using Markov Chain Monte Carlo simulations. From WMAP data alone, assuming spatially flat power law LCDM models, we obtain the reionization optical depth tau = 0.145 +/- 0.067, spectral index n_s = 0.99 +/- 0.04, Hubble constant h = 0.67 +/- 0.05, baryon density Omega_b h^2 = 0.0218 +/- 0.0014, cold dark matter density Omega_{cdm} h^2 = 0.122 +/- 0.018, and sigma_8 = 0.92 +/- 0.12, consistent with a reionization redshift z_{re} = 16 +/- 5 (68% CL).

Abstract:
Beyond the linear regime, the power spectrum and higher order moments of the matter field no longer capture all cosmological information encoded in density fluctuations. While non-linear transforms have been proposed to extract this information lost to traditional methods, up to now, the way to generalize these techniques to discrete processes was unclear; ad hoc extensions had some success. We pointed out in Carron and Szapudi (2013) that the logarithmic transform approximates extremely well the optimal "sufficient statistics", observables that extract all information from the (continuous) matter field. Building on these results, we generalize optimal transforms to discrete galaxy fields. We focus our calculations on the Poisson sampling of an underlying lognormal density field. We solve and test the one-point case in detail, and sketch out the sufficient observables for the multi-point case. Moreover, we present an accurate approximation to the sufficient observables in terms of the mean and spectrum of a non-linearly transformed field. We find that the corresponding optimal non-linear transformation is directly related to the maximum a posteriori Bayesian reconstruction of the underlying continuous field with a lognormal prior as put forward in Kitaura et al (2010). Thus simple recipes for realizing the sufficient observables can be built on previously proposed algorithms that have been successfully implemented and tested in simulations.

Abstract:
We introduce the theory of non-linear cosmological perturbations using the correspondence limit of the Schr\"odinger equation. The resulting formalism is equivalent to using the collisionless Boltzman (or Vlasov) equations which remain valid during the whole evolution, even after shell crossing. Other formulations of perturbation theory explicitly break down at shell crossing, e.g. Eulerean perturbation theory, which describes gravitational collapse in the fluid limit. This paper lays the groundwork by introducing the new formalism, calculating the perturbation theory kernels which form the basis of all subsequent calculations. We also establish the connection with conventional perturbation theories, by showing that third order tree level results, such as bispectrum, skewness, cumulant correlators, three-point function are exactly reproduced in the appropriate expansion of our results. We explicitly show that cumulants up to N=5 predicted by Eulerian perturbation theory for the dark matter field $\delta$ are exactly recovered in the corresponding limit. A logarithmic mapping of the field naturally arises in the Schr\"odinger context, which means that tree level perturbation theory translates into (possibly incomplete) loop corrections for the conventional perturbation theory. We show that the first loop correction for the variance is $\sigma^2 = \sigma_L^2+ (-1.14+n)\sigma_L^4$ for a field with spectral index $n$. This yields 1.86 and 0.86 for $n=-3,-2$ respectively, and to be compared with the exact loop order corrections 1.82, and 0.88. Thus our tree-level theory recovers the dominant part of first order loop corrections of the conventional theory, while including (partial) loop corrections to infinite order in terms of $\delta$.

Abstract:
Recently, several studies proposed non-linear transformations, such as a logarithmic or Gaussianization transformation, as efficient tools to recapture information about the (Gaussian) initial conditions. During non-linear evolution, part of the cosmologically relevant information leaks out from the second moment of the distribution. This information is accessible only through complex higher order moments or, in the worst case, becomes inaccessible to the hierarchy. The focus of this work is to investigate these transformations in the framework of Fisher information using cosmological perturbation theory of the matter field with Gaussian initial conditions. We show that at each order in perturbation theory, there is a polynomial of corresponding order exhausting the information on a given parameter. This polynomial can be interpreted as the Taylor expansion of the maximally efficient "sufficient" observable in the non-linear regime. We determine explicitly this maximally efficient observable for local transformations. Remarkably, this optimal transform is essentially the simple power transform with an exponent related to the slope of the power spectrum; when this is -1, it is indistinguishable from the logarithmic transform. This transform Gaussianizes the distribution, and recovers the linear density contrast. Thus a direct connection is revealed between undoing of the non-linear dynamics and the efficient capture of Fisher information. Our analytical results were compared with measurements from the Millennium Simulation density field. We found that our transforms remain very close to optimal even in the deeply non-linear regime with \sigma^2 \sim 10.

Abstract:
We define and study statistical ensembles of matter density profiles describing spherically symmetric, virialized dark matter haloes of finite extent with a given mass and total gravitational potential energy. Our ensembles include spatial degrees of freedom only, a microstate being a spherically symmetric matter density function. We provide an exact solution for the grand canonical partition functional, and show its equivalence to that of the microcanonical ensemble. We obtain analytically the mean profiles that correspond to an overwhelming majority of microstates. All such profiles have an infinitely deep potential well, with the singular isothermal sphere arising in the infinite temperature limit. Systems with virial radius larger than gravitational radius exhibit a localization of a finite fraction of the energy in the very center. The universal logarithmic inner slope of unity of the NFW haloes is predicted at any mass and energy if an upper bound is set to the maximal depth of the potential well. In this case, the statistically favored mean profiles compare well to the NFW profiles. For very massive haloes the agreement becomes exact.

Abstract:
We propose dual thermodynamics corresponding to black hole mechanics with the identifications E' -> A/4, S' -> M, and T' -> 1/T in Planck units. Here A, M and T are the horizon area, mass and Hawking temperature of a black hole and E', S' and T' are the energy, entropy and temperature of a corresponding dual quantum system. We show that, for a Schwarzschild black hole, the dual variables formally satisfy all three laws of thermodynamics, including the Planck-Nernst form of the third law requiring that the entropy tend to zero at low temperature. This is in contrast with traditional black hole thermodynamics, where the entropy is singular. Once the third law is satisfied, it is straightforward to construct simple (dual) quantum systems representing black hole mechanics. As an example, we construct toy models from one dimensional (Fermi or Bose) quantum gases with N ~ M in a Planck scale box. In addition to recovering black hole mechanics, we obtain quantum corrections to the entropy, including the logarithmic correction obtained by previous papers. The energy-entropy duality transforms a strongly interacting gravitational system (black hole) into a weakly interacting quantum system (quantum gas) and thus provides a natural framework for the quantum statistics underlying the holographic conjecture.

Abstract:
We describe a set of new estimators for the N-point correlation functions of point processes. The variance of these estimators is calculated for the Poisson and binomial cases. It is shown that the variance of the unbiased estimator converges to the continuum value much faster than with any previously used alternative, all terms with slower convergence exactly cancel. We compare our estimators with Ripley's $\tK_0$ and $\tK_2$.