26 research outputs found
Statistical properties of the Burgers equation with Brownian initial velocity
We study the one-dimensional Burgers equation in the inviscid limit for
Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian
motion that starts from the origin x=0). We obtain the one-point distribution
of the velocity field in closed analytical form. In the limit where we are far
from the origin, we also obtain the two-point and higher-order distributions.
We show how they factorize and recover the statistical invariance through
translations for the distributions of velocity increments and Lagrangian
increments. We also derive the velocity structure functions and we recover the
bifractality of the inverse Lagrangian map. Then, for the case where the
initial density is uniform, we obtain the distribution of the density field and
its -point correlations. In the same limit, we derive the point
distributions of the Lagrangian displacement field and the properties of
shocks. We note that both the stable-clustering ansatz and the Press-Schechter
mass function, that are widely used in the cosmological context, happen to be
exact for this one-dimensional version of the adhesion model.Comment: 42 pages, published in J. Stat. Phy
Pdf's of Derivatives and Increments for Decaying Burgers Turbulence
A Lagrangian method is used to show that the power-law with a -7/2 exponent
in the negative tail of the pdf of the velocity gradient and of velocity
increments, predicted by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78,
1904) for forced Burgers turbulence, is also present in the unforced case. The
theory is extended to the second-order space derivative whose pdf has power-law
tails with exponent -2 at both large positive and negative values and to the
time derivatives. Pdf's of space and time derivatives have the same
(asymptotic) functional forms. This is interpreted in terms of a "random Taylor
hypothesis".Comment: LATEX 8 pages, 3 figures, to appear in Phys. Rev.
The multi-stream flows and the dynamics of the cosmic web
A new numerical technique to identify the cosmic web is proposed. It is based
on locating multi-stream flows, i.e. the places where the velocity field is
multi-valued. The method is local in Eulerian space, simple and computaionally
efficient. This technique uses the velocities of particles and thus takes into
account the dynamical information. This is in contrast with the majority of
standard methods that use the coordinates of particles only. Two quantities are
computed in every mesh cell: the mean and variance of the velocity field. In
the cells where the velocity is single-valued the variance must be equal to
zero exactly, therefore the cells with non-zero variance are identified as
multi-stream flows. The technique has been tested in a N-body simulation of the
\L CDM model. The preliminary analysis has shown that numerical noise does not
pose a significant problem. The web identified by the new method has been
compared with the web identified by the standard technique using only the
particle coordinates. The comparison has shown overall similarity of two webs
as expected, however they by no means are identical. For example, the
isocontours of the corresponding fields have significantly different shapes and
some density peaks of similar heights exhibit significant differences in the
velocity variance and vice versa. This suggest that the density and velocity
variance have a significant degree of independence. The shape of the
two-dimensional pdf of density and velocity variance confirms this proposition.
Thus, we conclude that the dynamical information probed by this technique
introduces an additional dimension into analysis of the web.Comment: 19 pages, 10 figure
Non--Linear Evolution of Cosmological Perturbations
In these lecture notes I review the theory of the non--linear evolution of
cosmological perturbations in a self--gravitating collisionless medium, with
vanishing vorticity. The problem is first analyzed in the context of the
Newtonian approximation, where the basic properties of the Zel'dovich,
frozen--flow and adhesion algorithms are introduced. An exact general
relativistic formalism is then presented and it is shown how the Newtonian
limit, both in Lagrangian and Eulerian coordinates, can be recovered. A general
discussion on the possible role of possible relativistic effects in the
cosmological structure formation context is finally given.Comment: 17 pages, tex using lecproc.cmm (enclosed), no figures. To appear in
Proc. Laredo Summer School "The universe at high-z, large scale structure and
the cosmic microwave background". Eds. E. Martinez-Gonzalez and J.L. Sanz.
Lecture Notes in Physics (Springer-Verlag
Stress effects in structure formation
Residual velocity dispersion in cold dark matter induces stresses which lead
to effects that are absent in the idealized dust model. A previous Newtonian
analysis showed how this approach can provide a theoretical foundation for the
phenomenological adhesion model. We develop a relativistic kinetic theory
generalization which also incorporates the anisotropic velocity dispersion that
will typically be present. In addition to density perturbations, we consider
the rotational and shape distortion properties of clustering. These quantities
together characterize the linear development of density inhomogeneity, and we
find exact solutions for their evolution. As expected, the corrections are
small and arise only in the decaying modes, but their effect is interesting.
One of the modes for density perturbations decays less rapidly than the
standard decaying mode. The new rotational mode generates precession of the
axis of rotation. The new shape modes produce additional distortion that
remains frozen in during the subsequent (linear) evolution, despite the rapid
decay of the terms that caused it.Comment: significantly improved discussion of kinetic theory of CDM velocity
dispersion; to appear Phys. Rev.
Statistics and geometry of cosmic voids
We introduce new statistical methods for the study of cosmic voids, focusing
on the statistics of largest size voids. We distinguish three different types
of distributions of voids, namely, Poisson-like, lognormal-like and Pareto-like
distributions. The last two distributions are connected with two types of
fractal geometry of the matter distribution. Scaling voids with Pareto
distribution appear in fractal distributions with box-counting dimension
smaller than three (its maximum value), whereas the lognormal void distribution
corresponds to multifractals with box-counting dimension equal to three.
Moreover, voids of the former type persist in the continuum limit, namely, as
the number density of observable objects grows, giving rise to lacunar
fractals, whereas voids of the latter type disappear in the continuum limit,
giving rise to non-lacunar (multi)fractals. We propose both lacunar and
non-lacunar multifractal models of the cosmic web structure of the Universe. A
non-lacunar multifractal model is supported by current galaxy surveys as well
as cosmological -body simulations. This model suggests, in particular, that
small dark matter halos and, arguably, faint galaxies are present in cosmic
voids.Comment: 39 pages, 8 EPS figures, supersedes arXiv:0802.038
Universality classes in Burgers turbulence
We establish necessary and sufficient conditions for the shock statistics to
approach self-similar form in Burgers turbulence with L\'{e}vy process initial
data. The proof relies upon an elegant closure theorem of Bertoin and Carraro
and Duchon that reduces the study of shock statistics to Smoluchowski's
coagulation equation with additive kernel, and upon our previous
characterization of the domains of attraction of self-similar solutions for
this equation
Dark Energy and Gravity
I review the problem of dark energy focusing on the cosmological constant as
the candidate and discuss its implications for the nature of gravity. Part 1
briefly overviews the currently popular `concordance cosmology' and summarises
the evidence for dark energy. It also provides the observational and
theoretical arguments in favour of the cosmological constant as the candidate
and emphasises why no other approach really solves the conceptual problems
usually attributed to the cosmological constant. Part 2 describes some of the
approaches to understand the nature of the cosmological constant and attempts
to extract the key ingredients which must be present in any viable solution. I
argue that (i)the cosmological constant problem cannot be satisfactorily solved
until gravitational action is made invariant under the shift of the matter
lagrangian by a constant and (ii) this cannot happen if the metric is the
dynamical variable. Hence the cosmological constant problem essentially has to
do with our (mis)understanding of the nature of gravity. Part 3 discusses an
alternative perspective on gravity in which the action is explicitly invariant
under the above transformation. Extremizing this action leads to an equation
determining the background geometry which gives Einstein's theory at the lowest
order with Lanczos-Lovelock type corrections. (Condensed abstract).Comment: Invited Review for a special Gen.Rel.Grav. issue on Dark Energy,
edited by G.F.R.Ellis, R.Maartens and H.Nicolai; revtex; 22 pages; 2 figure