Solving Stochastic Inflation for Arbitrary Potentials

A perturbative method for solving the Langevin equation of inflationary cosmology in presence of backreaction is presented. In the Gaussian approximation, the method permits an explicit calculation of the probability distribution of the inflaton field for an arbitrary potential, with or without the volume effects taken into account. The perturbative method is then applied to various concrete models namely large field, small field, hybrid and running mass inflation. New results on the stochastic behavior of the inflaton field in those models are obtained. In particular, it is confirmed that the stochastic effects can be important in new inflation while it is demonstrated they are negligible in (vacuum dominated) hybrid inflation. The case of stochastic running mass inflation is discussed in some details and it is argued that quantum effects blur the distinction between the four classical versions of this model. It is also shown that the self-reproducing regime is likely to be important in this case.Comment: 17 pages, 9 figure

Stochasticity in halo formation and the excursion set approach

The simplest stochastic halo formation models assume that the traceless part of the shear field acts to increase the initial overdensity (or decrease the underdensity) that a protohalo (or protovoid) must have if it is to form by the present time. Equivalently, it is the difference between the overdensity and (the square root of the) shear that must be larger than a threshold value. To estimate the effect this has on halo abundances using the excursion set approach, we must solve for the first crossing distribution of a barrier of constant height by the random walks associated with the difference, which is now (even for Gaussian initial conditions) a non-Gaussian variate. The correlation properties of such non-Gaussian walks are inherited from those of the density and the shear, and, since they are independent processes, the solution is in fact remarkably simple. We show that this provides an easy way to understand why earlier heuristic arguments about the nature of the solution worked so well. In addition to modelling halos and voids, this potentially simplifies models of the abundance and spatial distribution of filaments and sheets in the cosmic web.Comment: 5 pages, 1 figure. Matches published versio

The importance of stepping up in the excursion set approach

Recently, we provided a simple but accurate formula which closely approximates the first crossing distribution associated with random walks having correlated steps. The approximation is accurate for the wide range of barrier shapes of current interest and is based on the requirement that, in addition to having the right height, the walk must cross the barrier going upwards. Therefore, it only requires knowledge of the bivariate distribution of the walk height and slope, and is particularly useful for excursion set models of the massive end of the halo mass function. However, it diverges at lower masses. We show how to cure this divergence by using a formulation which requires knowledge of just one other variable. While our analysis is general, we use examples based on Gaussian initial conditions to illustrate our results. Our formulation, which is simple and fast, yields excellent agreement with the considerably more computationally expensive Monte-Carlo solution of the first crossing distribution, for a wide variety of moving barriers, even at very low masses.Comment: 10 pages, 5 figure

The excursion set approach in non-Gaussian random fields

Insight into a number of interesting questions in cosmology can be obtained from the first crossing distributions of physically motivated barriers by random walks with correlated steps. We write the first crossing distribution as a formal series, ordered by the number of times a walk upcrosses the barrier. Since the fraction of walks with many upcrossings is negligible if the walk has not taken many steps, the leading order term in this series is the most relevant for understanding the massive objects of most interest in cosmology. This first term only requires knowledge of the bivariate distribution of the walk height and slope, and provides an excellent approximation to the first crossing distribution for all barriers and smoothing filters of current interest. We show that this simplicity survives when extending the approach to the case of non-Gaussian random fields. For non-Gaussian fields which are obtained by deterministic transformations of a Gaussian, the first crossing distribution is simply related to that for Gaussian walks crossing a suitably rescaled barrier. Our analysis shows that this is a useful way to think of the generic case as well. Although our study is motivated by the possibility that the primordial fluctuation field was non-Gaussian, our results are general. In particular, they do not assume the non-Gaussianity is small, so they may be viewed as the solution to an excursion set analysis of the late-time, nonlinear fluctuation field rather than the initial one. They are also useful for models in which the barrier height is determined by quantities other than the initial density, since most other physically motivated variables (such as the shear) are usually stochastic and non-Gaussian. We use the Lognormal transformation to illustrate some of our arguments.Comment: 14 pages, new sections and figures describing new results, discussion and references adde

Large-scale assembly bias of dark matter halos

We present precise measurements of the assembly bias of dark matter halos, i.e. the dependence of halo bias on other properties than the mass, using curved "separate universe" N-body simulations which effectively incorporate an infinite-wavelength matter overdensity into the background density. This method measures the LIMD (local-in-matter-density) bias parameters $b_n$ in the large-scale limit. We focus on the dependence of the first two Eulerian biases $b^E_1$ and $b^E_2$ on four halo properties: the concentration, spin, mass accretion rate, and ellipticity. We quantitatively compare our results with previous works in which assembly bias was measured on fairly small scales. Despite this difference, our findings are in good agreement with previous results. We also look at the joint dependence of bias on two halo properties in addition to the mass. Finally, using the excursion set peaks model, we attempt to shed new insights on how assembly bias arises in this analytical model.Comment: 30 pages, 21 figures ; v2 : added references (sec. 1, 5), clarifications throughout ; v3 : clarifications throughout, version accepted by JCA

Influence of Super-Horizon Scales on Cosmological Observables Generated during Inflation

Using the techniques of out-of-equilibrium field theory, we study the influence on the properties of cosmological perturbations generated during inflation on observable scales coming from fluctuations corresponding today to scales much bigger than the present Hubble radius. We write the effective action for the coarse-grained inflaton perturbations integrating out the sub-horizon modes, which manifest themselves as a colored noise and lead to memory effects. Using the simple model of a scalar field with cubic self-interactions evolving in a fixed de Sitter background, we evaluate the two- and three-point correlation function on observable scales. Our basic procedure shows that perturbations do preserve some memory of the super-horizon-scale dynamics, in the form of scale-dependent imprints in the statistical moments. In particular, we find a blue tilt of the power-spectrum on large scales, in agreement with the recent results of the WMAP collaboration which show a suppression of the lower multipoles in the Cosmic Microwave Background anisotropies, and a substantial enhancement of the intrinsic non-Gaussianity on large scalesComment: 19 pages, 5 figures. One reference adde

On the Reliability of the Langevin Pertubative Solution in Stochastic Inflation

A method to estimate the reliability of a perturbative expansion of the stochastic inflationary Langevin equation is presented and discussed. The method is applied to various inflationary scenarios, as large field, small field and running mass models. It is demonstrated that the perturbative approach is more reliable than could be naively suspected and, in general, only breaks down at the very end of inflation.Comment: 7 pages, 3 figure

Getting in shape with minimal energy. A variational principle for protohaloes

In analytical models of structure formation, protohalos are routinely assumed to be peaks of the smoothed initial density field, with the smoothing filter being spherically symmetric. This works reasonably well for identifying a protohalo's center of mass, but not its shape. To provide a more realistic description of protohalo boundaries, one must go beyond the spherical picture. We suggest that this can be done by looking for regions of fixed volume, but arbitrary shape, that minimize the enclosed energy. Such regions are surrounded by surfaces over which (a slightly modified version of) the gravitational potential is constant. We show that these equipotential surfaces provide an excellent description of protohalo shapes, orientations and associated torques.Comment: 5 pages, 6 figure

Scale dependent halo bias in the excursion set approach

If one accounts for correlations between scales, then nonlocal, k-dependent halo bias is part and parcel of the excursion set approach, and hence of halo model predictions for galaxy bias. We present an analysis that distinguishes between a number of different effects, each one of which contributes to scale-dependent bias in real space. We show how to isolate these effects and remove the scale dependence, order by order, by cross-correlating the halo field with suitably transformed versions of the mass field. These transformations may be thought of as simple one-point, two-scale measurements that allow one to estimate quantities which are usually constrained using n-point statistics. As part of our analysis, we present a simple analytic approximation for the first crossing distribution of walks with correlated steps which are constrained to pass through a specified point, and demonstrate its accuracy. Although we concentrate on nonlinear, nonlocal bias with respect to a Gaussian random field, we show how to generalize our analysis to more general fields.Comment: 16 pages, 10 figures; v2 -- minor changes, typos fixed, references added, accepted in MNRA

One step beyond: The excursion set approach with correlated steps

We provide a simple formula that accurately approximates the first crossing distribution of barriers having a wide variety of shapes, by random walks with a wide range of correlations between steps. Special cases of it are useful for estimating halo abundances, evolution, and bias, as well as the nonlinear counts in cells distribution. We discuss how it can be extended to allow for the dependence of the barrier on quantities other than overdensity, to construct an excursion set model for peaks, and to show why assembly and scale dependent bias are generic even at the linear level.Comment: 5 pages, 1 figure. Uses mn2e class styl