Alexei A. Starobinsky — Page 6

arXiv1994

Equilibrium State of a Massless Self-Interacting Scalar Field in the De Sitter Background

Behaviour of a weekly self-interacting scalar field with a small mass in the de Sitter background is investigated using the stochastic approach (including the case of a double-well interaction potential). Existence of the de Sitter invariant equilibrium quantum state of the scalar field in the presence of interaction is shown for any sign of the mass term. The stochastic approach is further developed to produce a method of calculation of an arbitrary anomalously large correlation function of the scalar field in the de Sitter background, and expressions for the two-point correlation function in the equilibrium state, correlation time and spatial physical correlation radius are presented. The latter does not depend on time that implies that the characteristic size of domains with positive and negative values of the scalar field remains the same on average in the equilibrium state in spite of the expansion of the $t=const$ hypersurface of the de Sitter space-time.

astro-ph↗

arXiv1994

Non-Linear Approximations to Gravitational Instability: A Comparison in the Quasi-Linear Regime

We compare different nonlinear approximations to gravitational clustering in the weakly nonlinear regime, using as a comparative statistic the evolution of non-Gaussianity which can be characterised by a set of numbers $S_p$ describing connected moments of the density field at the lowest order in $<δ^2>$: $<δ^n>_c \simeq S_n<δ^2>^{n-1}$. Generalizing earlier work by Bernardeau (1992) we develop an ansatz to evaluate all $S_p$ in a given approximation by means of a generating function which can be shown to satisfy the equations of motion of a homogeneous spherical density enhancement in that approximation. On the basis of the values of we show that approximations formulated in Lagrangian space (such as the Zeldovich approximation and its extensions) are considerably more accurate than those formulated in Eulerian space such as the Frozen Flow and Linear Potential approximations. In particular we find that the $n$th order Lagrangian perturbation approximation correctly reproduces the first $n+1$ parameters $S_n$. We also evaluate the density probability distribution function for the different approximations in the quasi-linear regime and compare our results with an exact analytic treatment in the case of the Zeldovich approximation.

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