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Register a new userDensity distribution of the cosmological matter field
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The one-point probability distribution function (PDF) of the matter density field in the universe is a fundamental property that plays an essential role in cosmology for estimates such as gravitational weak lensing, non-linear clustering, massive production of mock galaxy catalogs, and testing predictions of cosmological models. Here we make a comprehensive analysis of the dark matter PDF using a suite of 7000 N-body simulations that covers a wide range of numerical and cosmological parameters. We find that the PDF has a simple shape: it declines with density as a power-law P~rho**(-2), which is exponentially suppressed on both small and large densities. The proposed double-exponential approximation provides an accurate fit to all our N-body results for small filtering scales R< 5Mpc/h with rms density fluctuations sigma>1. In combination with the spherical infall model that works well for small fluctuations sigma<1, the PDF is now approximated with just few percent errors over the range of twelve orders of magnitude -- a remarkable example of precision cosmology. We find that at 5-10% level the PDF explicitly depends on redshift (at fixed sigma) and on cosmological density parameter Omega_m. We test different existing analytical approximations and find that the often used log-normal approximation is always 3-5 times less accurate than either the double-exponential approximation or the spherical infall model.
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Non-Gaussianity of the cosmological matter density field can be largely reduced by a local Gaussianization transformation (and its approximations such as the logrithmic transformation). Such behavior can be recasted as the Gaussian copula hypothesis, and has been verified to very high accuracy at two-point level. On the other hand, statistically significant non-Gaussianities in the Gaussianized field have been detected in simulations. We point out that, this apparent inconsistency is caused by the very limited degrees of freedom in the copula function, which make it misleading as a diagnosis of residual non-Gaussianity in the Gaussianized field. Using the copula density, we highlight the departure from Gaussianity. We further quantify its impact in the predicted n-point correlation functions. We explore a remedy of the Gaussian copula hypothesis, which alleviates but not completely solves the above problems.
The cosmological dark matter field is not completely described by its hierarchy of $N$-point functions, a non-perturbative effect with the consequence that only part of the theory can be probed with the hierarchy. We give here an exact characterization of the joint information of the full set of $N$-point correlators of the lognormal field. The lognormal field is the archetypal example of a field where this effect occurs, and, at the same time, one of the few tractable and insightful available models to specify fully the statistical properties of the evolved matter density field beyond the perturbative regime. Nonlinear growth in the Universe in that model is set letting the log-density field probability density functional evolve keeping its Gaussian shape, according to the diffusion equation in Euclidean space. We show that the hierarchy probes a different evolution equation, the diffusion equation defined not in Euclidean space but on the compact torus, with uniformity as the long-term solution. The extraction of the hierarchy of correlators can be recast in the form of a nonlinear transformation applied to the field, wrapping, undergoing a sharp transition towards complete disorder in the deeply nonlinear regime, where all memory of the initial conditions is lost.
We introduce the position-dependent probability distribution function (PDF) of the smoothed matter field as a cosmological observable. In comparison to the PDF itself, the spatial variation of the position-dependent PDF is simpler to model and has distinct dependence on cosmological parameters. We demonstrate that the position-dependent PDF is characterized by variations in the local mean density, and we compute the linear response of the PDF to the local density using separate universe N-body simulations. The linear response of the PDF to the local density field can be thought of as the linear bias of regions of the matter field selected based on density. We provide a model for the linear response, which accurately predicts our simulation measurements. We also validate our results and test the separate universe consistency relation for the local PDF using global universe simulations. We find excellent agreement between the two, and we demonstrate that the separate universe method gives a lower variance determination of the linear response.
The classical cosmological V/Vm-test is introduced. Use of the differential distribution p(V/Vm) of the V/Vm-variable rather than just the mean <V/Vm> leads directly to the cosmological number density without any need for assumptions about the cosmological evolution of the underlying (quasar) population. Calculation of this number density n(z) from p(V/Vm) is illustrated using the best sample that was available in 1981, when this method was developed. This sample of 76 quasars is clearly too small for any meaningful results. The method will be later applied to a much larger cosmological sample to infer the cosmological number density n(z) as a function of the depth z. Keywords: V/Vm . luminosity volume . cosmological number density . V/Vm distribution
Using distribution p(V/Vm) of V/Vm rather than just mean <V/Vm> in V/Vm-test leads directly to cosmological number density n(z). Calculation of n(z) from p(V/Vm) is illustrated using best sample (of 76 quasars) available in 1981, when method was developed. This is only illustrative, sample being too small for any meaningful results. Keywords: V/Vm . luminosity volume . cosmological number density . V/Vm distribution
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